Graph Theory and the EntropyConcept in Histochemistry

Graph Theory and the Entropy Concept in Histochemistry Theoretical Considerations, Application in Histopathology and the Combination with Receptor-specific Approaches

KLAUS KAYSER· HANS-JOACHIM GABIUS

With 37 Figures and 33 Tables

GUSTAV FISCHER

Stuttgart Jena Lubeck Ulm

KLAUS KAYSER, Prof. Dr. med. Dr. rer. nat. Direktor der Abteilung Pathologie, Thoraxklinik, AmalienstraBe 5, D-69126 Heidelberg, Germany HANS-JOACHIM GABIUS, Prof. Dr. rer. nat. Direktor des Instituts fur Physiologische Chemie, Tierarztliche Fakultat, Ludwig-Maximilians-V niversitat, VeterinarstraBe 13, D-80539 Munchen, Germany Preface and Acknowledgements Progress in science seems to behave similar to the theory of fractals: within a limited space of nature the increase of knowledge induces additional specialities, covering the lesser "space" the more advanced the research is. These areas do not necessarily stay separated but can invigorate a vivid exchange of ideas that may eventually lead to qualitative increases in the status of knowledge. Focusing on this theme, our review is designed to illustrate how two - at first glimpse separated fields can be integrated harmoniously, namely the theoretical concepts of graph theory and thermodynamics and quantitative morphometry, which in the next step is combined with receptordependent staining by immuno- or ligandohistochemical methods. The presented treatise contains fruits of the harvest of a long-lasting and enjoyable scientific collaboration, and would not have been brought into this form without the highly appreciated input of several friends: we are especially grateful to Gian Kayser and John Moyers who prepared the figures and the layout, to Prof. Dr. K. Sandau, Dr. S. Andre and Dr. S. Gabius for critical and insightful discussions and to C. Kayser and R. Ohl who assisted in various technical matters such as proof corrections and typing. The most grateful we are to our families for their patience, tolerance and support during the periods of time which have been and are spent to play with ideas and validate the results. May the book stimulate the reader to look beyond the "fences of his home" and to become encouraged to confidently embark on an interdisciplinary journey into uncharted territory. Dedicated to Prof. Dr. F. Cramer on the occasion of his 75 th birthday.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Kayser, Klaus: Graph theory and the entropy concept in histochemistry: theoretical considerations, application in histopathology and the combination with receptor-specific approaches; with 33 tables / Klaus Kayser; Hans-Joachim Gabius. - Stuttgart; Jena; Lubeck; Ulm : G. Fischer, 1997 Progress in histochemistry and cytochemistry; Vo!. 32, No.2) ISBN 3-437-21338-5 Library of Congress Card-No. 88-204 69

© Gustav Fischer Verlag· Stuttgart· Jena . Lubeck' Vim' 1997 Wollgrasweg 49, D-70599 Stuttgart Aile Rechte vorbehalten Gesamtherstellung: Laupp & Gobel, Nehren/Tubingen Printed in Germany

Contents 1 2 3 4 5 6 7 7.1 8 8.1 9 9.1 9.2 10 11 11.1 12 13 14 15 16 16.1 17 18

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . Biological structures - basic considerations . . . Biological structures - mathematical description Texture. . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure and energy . . . . . . . . . . . . . . . . . . . Current of entropy (Entropiefluss) . . . . . . . . . Orders of textures . . . . . . . . . . . . . . . . . . . . . Clinical applications Graph theory . . . . . . . . . . . . . . . . . . . . . . . . Clinical applications. . . . . . . . . . . . . . . . . . . . DNA content analysis . . . . . . . . . . . . . . . . . . Basic considerations on DNA cytometry . . . . . Clinical applications Calculation of structural entropy Calculation of current of entropy .. Clinical applications Cluster detection Basic stereological considerations Quantitative histochemistry. . . . . . . . . . . . . . Ligandohistochemical staining techniques A brief survey of animallectins . . . . . . . . . . . . Clinical applications Combined histochemical analysis . . . . . . . . . . Conclusion and perspectives. . . . . . . . . . . . . .

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Tables 1 to 33

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

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Subject index

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Abbreviations APA Ara Asn ConA DBA DSA E-PHA Fuc GAG Gal GalNAc GC GIcNAc Glc GSA II HPA Hyl Hyp IHCs LCA LFA L-PHA LTA Man MPA NANA OHCs PNA PSA PWM RCA-I SBA Ser SJA SuccWGA Thr TM UEA-I VVA WGA Xyl

Abrus precatorius agglutinin L-Arabinose

Asparagine Canavalia ensiformis agglutinin Dolichos biflorus agglutinin Datura stramonium agglutinin Phaseolus vulgaris agglutinin (erythroagglutinin) L-Fucose Glycosaminoglycan D-Galactose N -acetyl-D-galactosamine Glycoconjugate N -acetyl-D-glucosamine D-Glucose Griffonia simplicifolia agglutinin Helix pomatia agglutinin Hydroxylysine Hydroxyproline Inner hair cells Lens culinaris agglutinin Limax flavus agglutinin Phaseolus vulgaris agglutinin (leucoagglutinin) Lotus tetragonolobus agglutinin D-Mannose Madura pomifera agglutinin N-acetyl-neuraminic acid Outer hair cells Arachis hypogaea agglutinin Pisum sativum agglutinin Phytolacca americana agglutinin Ricinus communis agglutinin Glycine max agglutinin Serine Sophora japonica agglutinin Succinylated WGA Threonine Tectorial membrane Ulex europaeus agglutinin Vicia villosa agglutinin Triticum vulgare agglutinin D-Xylose

1 Introduction Histochemistry presents a static picture of a certain tissue to the observer. Refined techniques, as for example described in detail in other volumes of this series, allow to localize distinct epitopes in the tissue section. Beyond the visualization of selected determinants, which is the basis for morphological and diagnostic evaluations, the aspect of the structural organization of cell populations deserves to be addressed with elaborated algorithms and techniques. Therefore, it is our aim to engender interest in the application of the presented mathematical and physical concepts in histomorphological analysis. Although the tissue section certainly remains a snapshot of a dynamic development, syntactic structure analysis and morphometry encompassing - at first surprisingly - the estimation of thermodynamical parameters such as entropy and also the assessment of a DNA content-related parameter, i. e. integrated optical density, will gain evidence on the degree of order established in the cellular organization. Following the outline of the theoretical foundation and its application to histochemistry with focus on lung pathology, it is an intriguing possibility to correlate the expression of supposedly functionally relevant sites with the defined parameters. Initial studies in this area have centered around the detection of receptors for oligosaccarides (lectins), which are involved in such diverse processes as inter- and intracellular glycoprotein trafficking, regulation of proliferation or cell adhesion (for review, see GABIUS 1997; GABIUS and GABIUS 1993, 1997). It is our firm conviction that this interdisciplinary approach within histochemistry will contribute to pave the way for an improved understanding of the biochemical factors, which govern the generation of the presented tissue architecture. In this sense, we wish to provide a spark to ignite vigorous efforts in this area which holds the promise for far-reaching insights into biological structures.

2 Biological structures - basic considerations The physical laws which seem to «regulate» all phenomena in nature are from a basic point of view a correlation between the three-dimensional parameters of space and the time. Within this frame established by a set of four independent variables we can subdivide categories of correlations according to the interdependence of these coordinates, referring to the dimensions of space as one subgroup. They define the spatial positioning of any center of gravity. If the associations possess the same set of values independent from the time, we are dealing with reversible events. Irreversible incidents occur when spatial patterns have an inherent tendency to change with time without including the possibility of returning to any former configuration within their dynamic development. The basic laws in classical physics describe continuous reversible functions whose fundamentals are summarized in the field theory (LANDAU and LIFSCHITZ 1967) which includes the electromagnetic field theory and theory of gravity as well as the theory of

2 . K. Kayser and H.-J. Gabius

relativity (EINSTEIN 1955). They can be expressed by a set of mathematical functions which can be chosen to describe a homogeneous and isotropic system in both space and time (of a given body A). We can then «form» a new body C by bringing the body Bin contact with the body A, and we can define the space (and time) of body C. These systems are called inertial systems (LANDAU and LIFSCHITZ 1967). All physical laws and all properties of space and time are identical in all chosen inertial systems (Galilee's principle of relativity). In addition, they can be «attached» to each other, i. e., can be expanded (and shrink) continuously. To comprehend processes to the highest extent possible with lasting value, it is of great importance to define the «volume» of this fourdimensional space or, in other words, the characteristics of the physical laws especially close to or at the boundary of their range of validity. Commonly, we are not forced to consider additions to classical physics, unless we deal with (sub)atomic particles. The elaboration of a system for the mathematical description of e. g. the movement of bodies is of salient importance for technical development. It is an intriguing and at first glimpse trivial question whether and to which extent this set of laws will govern processes in life sciences. Indubitably, in biology physical laws are valid, and the biological functions are based upon the principles of thermodynamics and theoretical chemistry. However, it is not sufficient to use solely these laws in order to describe a biological system in its entirety. The limitations why physical laws are only of limited value to predict the behavior of living systems can be seen in the so-called boundary effects. Focusing e. g. on the cell surface with its numerous interaction and signalling systems which can even keep up an intensive cross-talk, it deserves emphasis that the interactions of the various participants of the individual molecular groups can change the parameters and consecutively the functionality of the whole system. The interactions, therefore, control phenotypic aspects of the whole system, which stays «alive» as long as the interactions assure a feedbackward mechanism, i. e. stabilize their surfaces. Such a system will «die» when surfaces or certain energy levels can no longer be established or kept constant within certain variations. Such a «breakdown» or «unlimited expansion» is associated with a continuous «vanishing» or «growing» of the space defined by the system, and commonly called a feed-forward mechanism. Feed-backward mechanisms are closely related with a controlled exchange of energy, and changes of boundary parameters influence the system considerably, not only in its function but also in geometrical parameters such as size, form, or distribution of structures inside the system. In addition, these interactions cause a phenomenon which ensures the continuity of the living system, namely the generation of new (nearly) identical systems. It will be one of the main aims of this treatise to describe the geometrical manifestation of these properties (structures), which is assessable by histochemistry, and then to try to delineate the influences of biochemical parameters on the geometrical structures by combined analysis. It is our firm conviction that such an approach to correlate parameter sets will supersede the traditional practice of separate investigations and will eventually mature to fruition by providing evidence for causative relationships between the histological pattern and distribution of biochemi-

Graph theory and the entropy concept in histochemistry . 3

cally defined epitopes. In addition, it might present a broader insight into energetic aspects of creation and stability of structures, and provide an analytical tool to foresee potential breakdowns or their «biological power» to destroy their environments. In biology, breakdown or uncontrolled growth of structures are the descriptives of inflammation and cancer, and a «covering» system C is forced to vanish to a volume equal or close to zero, if its embedded structures A, B, ... vanish. The meaning of space is, in these terms, not solely associated with the «ordinary» three-dimensional space and time. Other features can be chosen as embedding coordinates (for example energy, entropy, timedependent functions such as velocity of movements, currents, etc.). One possible «basic mathematical layout», as described in the next chapters, tries to handle these considerations in an analytical manner, and is connected with our principal persuasion that a precise understanding (and handling) of biological laws include(s) the interrelation of function and structure between embedded continuous units, and the discontinuities at their boundaries with their surrounding environment. We will see later that the accurate description of «shrinkage» of a space (time) volume to zero (as outlined in the theory of a chaotic system) will not necessarily form a chaos (that is an unpredictable system) but can generate certain new (and predictable) structures, if distinct constraints exist.

3 Biological structures - mathematical description Considering a small time-space volume which defines a specific time-space function, for example within an indefinitely small volume (point with a short time range), it may be useful to analyze the behavior of this function at its boundary of validity. The general laws of these boundaries have been described in various mathematical models, for example in the theory of numbers or of integration. The conditions for the existence of these boundaries (Cauchy's criteria) have to be obtained from the complete theory of real numbers. However, not even the existence of the boundary can be derived from the behavior of the internal laws. These internal laws are commonly considered to be independent from the boundary, and, vice versa, cannot predict its existence and properties. The general mathematical procedures define a set of (continuous) functions in a predefined space (space of definition). However, no mathematical theory exists to our knowledge which attempts to describe the relations between the «space of definition» and a set of parameters of functions defined in that space. Assuming that the small point follows a continuous law within the definite range, its behavior may change abruptly when arriving at the boundary. A continuous expansion of the laws into or beyond the boundary is not possible. We can, however, define a new range (area, volume) with new (or similar) laws at or beyond the boundary. The penetration of the point into the new range is not predictable and may follow non-defined or nonexisting laws; however, if it reaches the new range, it will behave according to the new laws. This principle is schematically explained in Fig. 1. The mathematical description can

4 . K. Kayser and H.-J. Gabius

Fig. 1. Two different finite spaces of reference are shown: arbitrarily distributed «dark balls» of different sizes which are not permitted to pass the boundary are defined in the inner space (disk). Outside the disk particles of different shape are computed which cannot pass' the boundary and move into the area of the disk (program «figure» written in DIAS, Towersoft, Berlin).

employ several possible procedures which may permit a construction of «passing rules». An example for a simple solution is a function which is characterized by «the average» of both functions, namely those defined in space 1 and those defined in space 2. We may, in addition, require that our point behaves according to the laws of range 1, if it is still close to the boundary in space two. Another attempt would be to associate the functions at the boundary to the «quantity» of points passing from space 1 into space 2 which might influence the <
Graph theory and the entropy concept in histochemistry . 5

or irregularities called bifurcation points the entropy of the system increases, and the time-dependence of such a system is irreversible. It is reassuring to mention in this context that in physics similar observations have been made: for example the difference of energy levels of electron shells decreases with increasing atomic number. All structures (and functions) within these systems are well defined within a given space-time volume (space of reference). It is now reasonable to proceed by asking how these properties behave at their boundary, for example explicitly: what happens to the bodies A and B close before they unite to form the body C? At first, we have to clarify the term «boundary». The following statements seem to permit a practical application: A boundary is a discontinuity which limits the description of certain features (general physical laws, mathematical functions, biological functions, etc.), or parameters (solid phase - fluid - gas phase, viscosity, velocity, etc.) and can, in principle, exist on the different space axes independently. As already recalled in the introductory statement, in nature we are (commonly) dealing with three indistinguishable space dimensions and one time dimension. These boundaries can be «stable» or «non-stable». «Stable» denotes the lack of influence by internal or external factors. In the following a stable boundary is called a surface. Definition:

(a) Surface. Given a continuous space-time function f(x, y, z, t) defined in a limited volume X-Y-Z-T, then sets of values X, Y, Z are called a surface S, if f(x, y, z, t) > 0 is continuous for x E X, Y E Y, z E Z, and f(x, y, z, t) = 0 for x $ X, y $ Y, z $ Z. X, Y, Zare called astable surface S of X, Y, Z, ifX(t) = X, yet) = Y, Z(t) Zfor all t < T.

=

(b) Limited (restricted) surface: Given two «different» points, for example, blue (b) and red (r) which are located in two different volumes displaying two different surfaces S(b) and S(r), then S(b) is called a limited surface, if S(b) < S(r) and feb) = fer) for all x, y, z $ S(b) and x, y, z < S(r). The behavior of the blue points changes to that of the red points, if they pass S(b) and enter the space of the red point. Of course, the blue points are no longer distinguishable from the red points within the space section of the red points. Derivative: A restricted surface Sr is the boundary of a volume Am if and only if

Sr(x, y, z, t)

=f(x, y, z, t) with x, y, z < Ao

Sr(x, y, z, t) = g(x, y, z, t) with x, y, z > A o , and g(x, y, z, t)"# f(x, y, z, t)

(1) (2)

These definitions have the following implications: (a) A hierarchy of structures can be defined independent of their actual size and properties (Ao E At E A 2 ••• E An)'

6 . K. Kayser and H.-J. Gabius

(b) Clearly separated ranges of validity of functions and structures can be derived. (c) Changes of surfaces dependent upon time or function can be computed. With the definition of a surface we have established the prerequisites to describe «more complex» situations, for example «textures» which play an important role in any biological system.

4 Texture A texture is a set of stable (restricted) surfaces which have certain features «in common», for example a conglomerate of cells with either some «identical physiological functions» or well-defined interactions in between. In addition, we have to demand a «stable geometrical relation» between the surfaces (which might not necessarily be in contact with each other). A geometrical stable relation implies a distance function Fd which has to exist «in and between» all defined surfaces for a time t ~ T. Remarks: If we freeze an image at time tl> the «recognition» or segmentation of features exists at time tl' A texture is then the geometrical association of segmented areas (their boundaries) in between or of segmented «lines». Thus, the definition of textures implies the definition of «association», i. e. the introduction of neighborhood (for example a distance function). The distance function can certainly be replaced by other relationships, for example «interaction functions», discontinuous relationships, or «similarity considerations». Appropriate techniques are based upon the graph theory, geometrical statistics, or cluster analysis (see chapter Graph theory), and can lead to the recognition of additional relationships (textures), if they define a new space with included surfaces (see chapter Order of textures). In order to create a texture we require a space V for suitable placement of the volume segments defined by surfaces S and we have to define a function Fi which denotes the volumes to be included (or excluded). Fi(xmj, ymi, zmi)

=Fd

(3)

for all surfaces Si E "y, xmi, ymi, zmi < V, and Fk(xmk' ymk, zmk) =0 for all surfaces Sk $ V, xmk, ymk, zmk > V.

Definition: A texture Tx is a function of different surfaces Si, Sj which have a defined distance function Fd: Tx

=Fd(Si)

(4)

For example, a distance function can be the «geometrical distance» G for d(x, y, z)

=d(x, y, z)

=min (Xii' Yij, Zij), and (Xij , Yij , Zjj) being points in a metric space.

Graph theory and the entropy concept in histochemistry . 7

The surface S(Tx) of a texture Tx can again be defined by a local maximum (or minimum) of its distance function, for example S(Tx) = f(A(Fd» with A(Fd) = ft(xm, ym, zm). Examples of the use of distance functions in the attempt to derive structures from geometrical clusters in a predefined space are discussed in the chapters on graph theory and cluster detection. One might try to define a surface function FS which depends upon the distribution of the red and blue balls (points), for example upon their density within a certain area «around» the surface. A characteristic situation is given, when the number of the «red points» exceeds an upper limit of the concentration and when consecutively the restricted (inner) surface will expand (or shrink). The expanding surface will lead to a stage at which the indistinguishable red and blue balls at time trb outside the former surface will become distinguishable, i. e. some red balls will change their color to blue. As a consequence, the number of red balls will decrease (or vice versa in the case of a shrinking surface), and the surface might shrink again, if the concentration of red balls cannot hold the size of the surface any longer. The situation (showing only a single type of particles) is demonstrated in Fig. 2. The question arises at which «concentration» the size alteration (expanding or shrinking) of the surface will stop. Likewise, it is pertinent to ask which constructions are adequate to describe the situation. In biology one is tempted to try a continuously increasing function (or family of functions) within the inner surface and a «nearly constant» function at the outer range. A heuristic function is a population function first described by Verhulst (1845, as cited by ELBERT et al. 1994). This model analyzes the number of individuals (for example cells) at a time t + 1 generated from individuals (e. g. cells) at time t. It is characterized by two constraints: the birth rate, i. e. the number of cells produced within a cell cycle, and a control factor which would restrict unlimited growth (for example conditions of supply of nutrition, steering functions of neighboring cells, cell death, etc.). Given the number of cells (x(t» at the generation t, the number of cells x(t + 1) in the time span of the next generation t + 1 can be calculated by the equation x(t+1)

=r' x(t){1-x(t)}

(5)

x(t, t + 1, etc.) is normalized, i. e. 0 ::::; x(t) ::::; 1. The schematic courses of these functions are illustrated in Fig. 3. The family of equations, given in (5), is known as a logistic or quadratic map, and corresponds to non-linear differential equation (5). The result of equation (5) can be drawn as a parabola {x(t) - (X(t»2}. The birth rate, i. e. parameter r, controls the model: if r < 1, then the subsequent generations will become smaller and smaller, and the population will succumb. If we choose 1 < r < 3, then the population will grow toward about 2/3 of

8 . K. Kayser and H.-J. Gabius

a

b

c Fig. 2. The disk (space 1) contains densely packed balls which can only pass the surface with a certain velocity and density at the boundary, and induce a pressure on the space of reference (2a). The inner pressure depends upon the volume-density of the particles and the «resistance» of the surface, and will expand the surface (2b). As a result more particles will leave the space of reference. The density of particles inside the space will decrease and that outside of this space will increase. As a result, the boundary will shrink again (2c).

the original population. For 3 < r < 4 the system becomes unstable, and two initially stable points of the period two are created, i. e. the number of cells oscillates between two different values. If we choose r :::: 4, the population will increase indefinitely. The reader who is familiar with the theory of dynamic systems (theory of deterministic chaos; see

Graph theory and the entropy concept in histochemistry . 9

a

C x(t+ 1)

x(t+ 1)

x(I)

x(I)

d x(!+ 1)

b x(t+ 1)

x(I)

x(!)

Fig. 3. Graph of the development of a population x(t) according to the differential equation x(t + 1) = r . x(t) - r' (x(t)jl. The development of successive time points (logistic map) follows a parabola with a smooth maximum around r/4 at x = 0.5. If we choose r < 1 (3a), the population will vanish. If we choose 1 < r ~ 3, the population will become stable at a point x(to)' %(3b). For r > 3 the formerly stable point I will become instable, and two new stable points I and II are created (3c). If we choose r > 4 (3d), the population will grow rapidly, and no stable state can be reached.

for example SAUNDERS 1980 for an excellent introduction and further details) will realize that we have introduced the definition of trajectories and attractors: if we choose 1 > r, the definite and stable value of the population will become x(fp) =0 independent from the starting point. This point x(fp) is called attractor, because the trajectories will end in this point. The same holds true for 1 < r < 3, since all trajectories will end in a resulting population measuring 2/3 of the original one.

10 . K. Kayser and H.-J. Gabius

=

If we assume a «cellular birth rate» r 2, and we can determine the number of cells at time t, namely x(t), and in the next generation x(t + 1), we can derive the distance of the system from its final stage X(tf) by d{f(x(t), X(tf»}

=f(X(tf» - f(x(t + 1»/f(x(t»

(6)

If f(x(t + 1» is close to f(X(tf», the difference will become close to zero, if tf - (t + 1) ~ 0, the distance is close to the attractor 2/3, i. e. 0 < d(f(x(tf), x(t + 1») < %. We now assume that the volume Vp which harbors the population x(t) is a function g of x(t) Vp(t)

=g{x(t)}

(7)

g{x(t + 1)} can be chosen that Vp(t+1)

=V{p(t)} + a(t)· [d{f(x(t + 1), x(t»)}]

(8)

a(t) =f(x(t»/V; V> 0; i. e. Vp increases with increasing growth of the population x(t) and its population density. Surprisingly, the situation changes, if a(t) reaches the critical point r1. At this point a bifurcation is created, a next one at the distance do

=£2 1(%) - % 0

-

The distances do and the levels of entropy in relation to the function x(t) are plotted in Fig. 4. As x(t) is related to the fraction of cellular volume within an organism, one might ask about the conditions of a stable proliferation within the given organism, i. e. the given structure. If S is the space interval (volume) which is produced within the time interval T, then the new system C =S . T is the rectangle space-time which defines the «development function» f (STOYAN and STOYAN 1992). f has to be continuous and - after a certain time - discontinuous; otherwise the «old cell» will not be defined and the «new cell» not be created. We start to increase the volume of the «old cell» which can be computed by the «continuous part» of the function f (for example moving along the axis r in Fig. 4), and we will reach the point R] which is the discontinuous «part» of function f. Mathematically, this point can be defined by a space fraction DV which shrinks to zero at this point. This non-defined «zero»-space volume can be used as starting point for a new space-time relation C. However, as the space fraction shrinks to zero, the time fraction T becomes non-defined. We have an undefined «surface» or a new set of undefined parameters which will be responsible for certain stable components outside their area of definition. In the theory of deterministic chaos several «beautiful figures» can be derived from the characteristics of various applied functions f. Graphic examples are depicted in Fig. 5. Their obvious attractivity and principal importance notwithstanding, they will not be discussed in detail, because we are focusing on a different problem: We have to analyze a <
Graph theory and the entropy concept in histochemistry . 11

a x(t)

b E

Fig. 4. The relationship of x(t) and r are characterized by the accumulation points of a logistic map. For r < 3 the population will reach only one stable point (with an entropy E = L{p(tj) . In p(tj)} close to zero). For r > 3 we will obtain oscillations between two points (first bifurcation), and the entropy E achieves a higher level (4b). With increasing r additional oscillations will appear (period 4, 8,16, etc.), and the entropy attains elevated levels at the bifurcation points (adapted from SCHUSTER 1994).

state of a certain three-dimensional image which consists of a compact association of various discrete elements. These elements are referred to as cell compartments such as nuclei, cells, cell clusters such as vessels, nerves, etc. The chaos theory is one tool to attempt to describe the further development of structures from an unregulated chaotic system. From the viewpoint of cytology, histology and histopathology we have to answer two different basic questions, if we want to describe the potential development of the underlying biological structures, which - to our experience - can proceed from a

12 . K. Kayser and H.-J. Gabius

a

b

Fig. 5. Examples of computation of reiterative figures. Sa: Generation of «asymmetric» growing branches of so-called fractals of a bifurcation tree (according to PEITGEN and SAUPE, 1988). The algorithm is closed after the fifth and third generation, respectively. sb: Sierpinski fractal which displays the points of a self-correlation area defined by points m in a d-dimensional space. They are covered by d-dimensional balls of the diameter L. If L > 0, then the number of balls N(L) which are necessary to cover the points diverges proportional to L-D. Sierpinski fractals are computed with the dimension 0 = log3/iog2 = 1.5849. It should be noted that the fractals continue to grow indefinitely, although the covered space is limited.

non-structured system into a structured one, and decay from a structured one into a lessstructured or even chaotic system. (a) Development ofa non-structured into a structured system

What are the conditions for structures to originate from a non-structured system? The occurrence of complex structures, as presented routinely to the cyto- and histochemist, makes obvious that rules and forces govern their generation reliably. However, the nature of these factors is ill-defined compared to the maturity of the phenomenological system description. A reasonable approach to tackle the problem is to apply the analysis of physical processes such as noise, oscillations, and synergistic effects (resonance) to this problem. A non-structured system corresponds to noise which can oscillate under certain conditions (small changes of energy flow) and add up to a resonance phenomenon. Mathematically, the first prerequisite which has to be fulfilled by a chaotic or noisy system to show periodic structures is the non-linearity of the system. Only non-linear systems can be chaotic and may display a deterministic (regular) chaos. The reason for

Graph theory and the entropy concept in histochemistry . 13

the irregularity (non-continuous function) is the property of non-linear systems to separate neighboring trajectories exponentially within a short time, i. e. the space of the trajectories is «thinned» and «ruptures». This event induces a «barrier» or discontinuity. A simple and instructive example is the Rayleigh-Benard system (SCHUSTER 1994) in a cell which is based upon the Fouriertransformation of a signal x(t) x(t) =IimH~ I dt . x(t) . exp(iwt)

(9)

w = Fourier frequency of the transformed signal x(t)

If we have multiple periodic movements, then the frequency spectrum can be expressed by P(w)

= x(w) 2 1

1

(10)

P(w) being the «power» of the frequency w.

A completely chaotic system displays solely noise, whereas multiple periodic movements can be characterized by peaks within the frequency spectrum, for example of Brownian motion computed by iterative transformations, as illustrated in Fig. 6.

Brownian motion In one dimension

Fig. 6. Brownian motion in one dimension. A chaotic movement can also be created by a <
14 . K. Kayser and H.-J. Gabius

Another quite simple experiment has been described by Benard (LIBCHABER and MAURER 1980). The experiment is as follows: We increase the temperature of a certain volume of fluid (with a positive volume expansion coefficient) with a heater from the bottom. The warmer fluid has the tendency to move to the upper parts of the fluid, and the cold fluid due to its density will occupy the bottom layers, where it is exposed to the source of heat. The whole procedure is disturbed by the viscosity of the fluid. If the difference of temperature is small compared to the viscosity, the fluid will remain undisturbed. If one increases the temperature, a new stationary state is reached with «rolls of convection». The experiment can be described by the following movement equations (Lorenz equations): dYdt = - S • x + S • Y dY/dt = r . x - y - x . z dZ/dt=X·y-b·z

(11)

sand b are constants which describe the material properties of the system (viscosity etc.), and r represents an external control factor which is proportional to the temperature difference !1T. x is proportional to the circulation velocity, y to the difference in temperature between the ascending and descending fluids, and z to the aberration of the temperature profile from its equilibrium. The situation is graphically depicted in Fig. 7. It is clear that we have obtained stable spatial structures within a system which is primarily continuous in the involved parameters. They include space, gravity field, temperature, viscosity, volume expansion, and time. The behavior of the system is obviously controlled by an (external) parameter p which is related to the viscosity and the difference in temperature at the bottom and at the top. Introducing a phase space ph, we can summarize the above listed equations to dph/dt = F(ph,V, p)

(12)

Whether the system is stable, i. e. whether it creates a (new) structure or not, depends only upon the parameter p. A small variation in p may again induce chaos in an already structured system. With higher temperature the stability of the procured structures is increased, if a negative correlation exists between viscosity and heat. (b) Decay, ageing, or loss ofexternal energy can destroy already established structures

How can regulated (structured) systems decay into less ordered or non-regulated systems? Evidently, this question is of salient importance to the occurrence of nonordered or disoriented tissue growth, as seen in cancer. The concept of structural stability is related to the observation that we obtain similar results, if we repeat an experiment under at least closely related conditions. If we repeat an experiment several times under approximately the same conditions, and we obtain results with increasing divergence, we

Graph theory and the entropy concept in histochemistry . 15

h

e

a

t

Fig. 7. Rolls of convection as an example of structures which are created by a continuous flow of energy (heat) into a fluid with a certain viscosity and heat conductivity.

are probably dealing with a system which possesses no or only a weak structural stability, i. e. a stability which can be disturbed measurably by the experiment. Without any reference to biochemical properties a fundamental answer is already given in equation (12). This equation describes a dissipated stream. The structures are related to the parameter p and to the size of the volume element V. This element V has a surface S which encloses V and the phase space ph. As the volume diminishes, the stream of the three-dimensional arrangement through its surface S induces a point cluster of a dimension < 2 without a remaining volume (the volume of a space ph is equal to zero!). Therefore, the trajectories are forced into an «attractor» which still defines the surface of the phase space. The movement cannot yet be predicted, because its course is very sensitive to the starting conditions. For practical reasons we can conclude: (1) To create new structures in a continuous system we need a non-linear equation system or a family of functions which are continuous within the system and discontinuous at the boundary of the system. The continuous function will «break» at the boundary which is controlled by the function itself, and a new space-time volume is established. (In biology, this procedure can be applied to describe the creation of new cells). (2) The structures (boundaries) of a system will break down, if the space-time volume is reduced below a lower limit. This situation is, however, not avoidable in a thermody-

16 . K. Kayser and H.-J. Gabius

namically open system, which is located in a space with constant (or increasing) entropy. With increasing entropy of the inner system its time - space volume will decrease. These properties offer the potential to measure the «alteration» of structures within a system by the assessment of changes of structural entropy (see chapter Current ofentropy). (3) Structures and textures are related with «contents» of energy, which differ from their surrounding environment. In addition, a certain amount of energy is needed to create structures, and the decay of structures is associated with an increase of entropy. In chemistry, we can directly measure the energy which is necessary to form a new structure (for example of a molecule). In more complex and dynamic (living) systems of spatial organization we are frequently confronted with the situation that we are dealing with a given structure (for example at light microscopical level) and we have to interpret this texture. How can we determine the efficiency of the biological system to build the corresponding texture in terms of energy and entropy? This efficiency should imply biological importance as all textures in living organisms do exist in a very reproducible and reasonably stable manner as long as they possess the required functions, i. e., as long as they are considered to be «healthy». Is there a chance to delineate levels of energy and entropy from textures and apply the obtained quantitative values to biological functions?

5 Structure and energy As demonstrated in the Rayleigh-Benard experiment, a structure can be created within a system in which only continuous forces act. A continuous (nearly) constant energy supply is necessary to maintain the established structures, as the acting dispersal forces of the dynamic system will induce the decay into a chaotic system again, when the energy supply is altered or completely removed. How can we obtain the energy needed to maintain the initial degree of organization by an analysis of the existing structures? Obviously, there is a connection between the structures and the introduced energy. The smaller the «rolls» in the fluid are, the greater the extent of the introduced heat is, if the viscosity remains the same. If the viscosity decreases with increasing temperature of the fluid, again the number of «rolls» within a predefined fluid volume will increase. If we are dealing with large and complicated structures, it is probably impossible to calculate the amount of the absolute energy which is necessary to create them. To accurately know the absolute value of the transferred energy is, however, not essential, if only differences between comparable structures (for example cancer cells and cells of normal tissue) are to be analyzed. These are several approaches to describe the regularity or uniformity of a structured system. The simplest approach is the following procedure: We consider a line of a definite length and ask for a plane that could cover this line. One might think of an irregularly formed line similar to a coast line which should be covered by a (minimum) number of

Graph theory and the entropy concept in histochemistry· 17

boxes to be counted. The smaller the size e of the boxes, the more accurate the definition of the length of the curve (line) is. If we have the length L of the curve (L > e), and we need N . We) boxes, then L

=N . We)

(13)

holds. For example, a line measures 10 em, and the length of the boxes is 1 em, then we need 10 boxes to cover the line. The situation is demonstrated in Fig. 8. If we have an area covered by a volume,

(14) is the corresponding equation; and for a D-dimensional figure N(e)

=LD . We)D

(15)

This equation can be written into D

=log(N(e»)/log(L) + logWe)

(16)

It can further be transformed into D 1im > 0 = In{N(e)}llnWe)

(17)

Fig. 8. To measure the length L of a line a number of boxes N(b) of size b is needed to cover the line. The length L can be estimated by knowing both the number of boxes necessary to cover the line, and the size of the boxes. The smaller the size of the boxes, the more accurate the estimation of the length L will be.

18 . K. Kayser and H.-J. Gabius

which yields the capacity of a system. If the boxes are not equally filled and contain a mass Pi (Pi> 0), we then weight the original size e of the boxes with their mass Pi> i. e. N(e) = L(D) . L (Pi) . VI; (Pi' e). The corresponding equation will be D d =L Pi 'In(t;p)/ln(t;e)

(18)

which can also be written D 1im e->O = L (Pi ·In(pyln(e))

(19)

This equation is already evocative of that of the entropy which is basically

En =L Pi . In(pJ Concerning our intention to aim at the application of these concepts in histochemistry it is not essential to further elaborate the intricate theoretical foundation. The cardinal aspect is to keep in mind that the regularity of a structure is closely associated with an equation which expresses the entropy. What is the general importance of entropy? The term entropy is commonly used in two different respects: (a) from the viewpoint of the energy of a (closed or open) system: The entropy describes the total of free (potentially transferrable into any other kind) energy which has been transformed into a non-reversible form, i. e. heat. According to the principle of Claudius the state of maximum entropy within a closed system is the most probable one, and all energy-transforming processes will stop, when this point has been reached. Spoken in terms of «classical thermodynamics», the source of the increase of the entropy can be seen in the - continuous and unavoidable - dissipation of energy. If we apply this statement within the model of chaos, it depicts the shrinkage of the space-time volume (see chapter Texture). Whether this state still follows the principle of a chaotic system, has to be left open, i. e., whether there exist randomly distributed states with «entropy peaks» which can accumulate and then form new «energy structures». Remarks: In reality, a second source of increase of entropy exists, the ageing of the machine. A machine is a cyclic working mechanism which is associated with a loss of structures, i. e. a degradation of structured substance. If we are dealing with a biological or cybernetic system, we can, in addition, interpret the situation as loss ofmemory which is also associated with a loss of structured substance. Again, the «loss of energy» may follow some regulations for its chaotic decay (so-called chaotic regulations), and may create a memory. However, as we have seen above, even if we assume a continuous loss of degree of organization, i. e., a continuous increase of entropy, certain «stable» states within the system can be present which form «substructures» within the «currents of entropy» or «states of a local minimum». If we consider the entropy as a function which «follows» the rules of noise and oscillation (or might possess certain resonance states), we can describe the situation by an equation of differential calculus, namely dEn/dt

=a . En + b

(20),

Graph theory and the entropy concept in histochemistry . 19

and, again, we will find a number of discrete states which do not fit into the classic theory of a continuous increase of entropy within the system. The creation of certain «structures» within a chaotic or decaying system is, therefore, correlated with discrete levels of local energy (entropy) minima. These structures are not predefined, but the energy needed to generate them can be deviated, if we are confronted with two circumstances: a random process of small particles (for example noise), and a continuous field force (for example an electromagnetic field or a field of gravity). Each structure is associated with an abrupt increase of the level of entropy at its boundary, and a continuous removal of entropy through its surface (TRINTSCHER 1967). In addition, we are confronted with a second phenomenon: The number of these states will grow with increasing age of the system, i. e. the more distant it is from its original state, the greater the number of stable states (structures) is. These findings might explain the increasing complexity of biological systems within the development of life. It is also conducive to explain the increasing amount of «information exchange» during the development of biological systems: The greater the number of structures, the greater the number of surfaces, and, consequently, the number of «through-passing» energy streams, i. e. the amount of information. (b) from the viewpoint of information theory (SHANNON and WEAVER 1949): There is no principal mathematical difference between the entropy defined by the theory of thermodynamics and that defined in information theory. The basic principle is illustrated in Fig. 9. The equations are the same, only the meaning is different: In information theory the values Pi describe the «width» of the information channels or the probability to obtain a signal in the channel i, in the theory of thermodynamics they refer to the width of the channels whose work processes will generate heat. In reversible systems it is sufficient to describe the «content of entropy» within a system in order to measure its age or amount of «free energy». In thermodynamically open (irreversible) system, however, the cardinal question of its «stability» is connected with the measure of its own entropy and that of its environment. This consideration leads to the concept of current of entropy (Entropiefluss).

6 Current of entropy (Entropiefluss) The classical theory of thermodynamics describes a closed or «stable» system, i. e. a system which does not include the parameter time. The theory of thermodynamics of irreversible processes includes the parameter time describing the behavior of open systems. If we have an isolated system which cannot exchange any substances or energy with the surrounding medium, a constant stage which is independent from the parameter time can only be reached, if all energy has been transformed into heat. This statement is not at odds with the already mentioned fact that local and quasi-constant minima of entropy may exist in a closed system. There might be the situation that a

20 . K. Kayser and H.-]. Gabius

a

c

Fig. 9. Model of information. Thc capacity of information of thc box in Fig.9a has two stages Ci: One needs at least one question and one answer to localize the animal (right/left?). The box in Fig.9b has four stages Ci, and at least two questions have to be answered to find the animal (right/left?; up/down?). Fig.9c displays 16 stages Ci, and the minimum number of questions n(q) and answers which are necessary to find the animal is 4: (2 4), or n(q) = Id(YeJ The increase in information (negative entropy) is E(I) = - L (Yei) ·ld(VcJ

Graph theory and the entropy concept in histochemistry . 21

local minimum of energy can form a structure that «delays» the transformation or spatial flow of energy. This situation is equivalent with the establishment of a new «open system», its characteristics being dependent upon the created structure, its energy levels, and those of the surrounding environment. Vice versa, local energy or entropy minima can only exist, if there is a «delay» of energy transport. Thus, a thermodynamically open system can only exist, if and only if (a) it has a surface or structure, and (b) it is surrounded by or in connection with another (open or closed) system. These conditions imply that with increase in time a decrease of the volume of the open system ensues, i. e., its energy levels become «closer». In other words, structures of a thermodynamically open (biological) system become closer or more detailed the «younger» they are. It should be mentioned that in nature heat production is dependent upon the existence of certain energy levels either in form of molecules or electromagnetic fields, i. e. a spatially equal distribution of energy does not seem to appear to be present in nature. In our environment we have to deal with the following situation: we have numerous quasi-constant structures which are positioned in a quasi-constant environment. This space possesses numerous quasi-constant structures, and is, again, located within a quasi-constant system, and so on. The time periods Ti of these systems can be grouped according to the constancy of their structures, and commonly fulfill the following prerequisite in accordance with the above mentioned statement: (21)

If a system Sj is embedded in another system So (Sj E So), then To :::: Tj. In other words, the structures of the outer system So last longer (survive) than the structures of the inner system Sj. In addition, if the outer system So breaks down, the inner system Sj will either break down, or become a closed system with respect to the outer one. For example, a person will die, if either the environment «breaks» (can no longer provide the vital functions) or if the «interaction» with the environment (energy exchange) is disrupted (for example stop of gas exchange, fluid exchange, uptake of food, etc.). Between these systems streams of energy are exchanged. The flow of free energy F 12 (passing between V t and V 2) has to pass the surfaces of the involved systems. It usually flows from the outer to the inner one, or has the direction toward the embedded one. Thus, we can write E(1, T j)

=E(1, To) + dE(12, dT)

(22)

E(l, Ti) is the energy of system 1 at time Ti , E(l, To) is the energy of system 1 at time To dE(12, dT) is the difference of energy passing from system 2 to system 1 within the time dT. Since the total of free energy is related to the entropy of the system, the exchange of free energy is accompanied by exchange of entropy, and the total entropy of an embedded system is related to its stage at the beginning and the amount of entropy which has

22 . K. Kayser and H.-J. Gabius

been transferred between the two involved systems within the time dT. 1£ the surrounding system 2 is closed, the affinity between the two systems will decrease as the stream of entropy is reduced (En1 tends to reach En2, and the difference gets smaller as long as the exchange of entropy lasts). This is the theorem of PRIGOGINE (1947). The structures of system 1 and system 2 correspond to local minima of the respective levels of entropy, and structural difference between the system 1 and system 2 can be measured, if the current of entropy is known. How can we determine the structural entropy of a discontinuous quasi-constant system (for example the cellular system of an organ)? According to the theory of ONSAGER (1931) the current of entropy CEn represents the total heat of a thermodynamically open system which has been produced by its energy processes and which has to be removed through its surface S. As we have discussed above, the system tries to minimize its current of entropy during its lifetime. This means that the system is in an equilibrium stage with its environment, if the current of entropy is zero or at its minimum stage. However, there is one serious problem that we have to take into consideration, if we are dealing with a biological system. Each living organism does not only consist of one thermodynamically open system, but of numerous of such systems, which are (a) placed in a hierarchical order, and/or (b) spatially separated from each other with obvious connections in between. Examples of the first situation (a) are mitochondria in a cell which itself is part of a gland. The latter might be localized in a bronchus, which is a distinct part of the lung. The lungs are again a clearly separated and (from the viewpoint of energy exchange) independent constituent of the body. Examples of the second situation (b) are neighboring cells, neighboring bronchi, etc. 1£ we want to estimate the «stability» of a biological system, we have to define at which «level» the system is to be described, i. e. we have to introduce terms like neighborhood and «orders ofstructures (texture)>>.

7 Orders of textures This term was first introduced by KAYSER and HOFFGEN (1984), and has been proven to be a valuable concept not only to describe the algorithms of diagnostic procedures,

but also to estimate the <
Definition: A «still» image is an image which contains «separated items» or basic units within a two-dimensional space which can be segmented by an appropriate algorithm. Examples are nuclei, cells, vessels, nerves, etc. Segmentation is equivalent to the «extraction» of

Graph theory and the entropy concept in histochemistry . 23

certain structures which exist in an image. A segmentation without predefined structures is not reproducible and only possible at random. Principally, not only time-independent structures (of a frozen image) but also time-dependent structures can be segmented. Examples are movements of persons, scenes, or frequency analysis of acoustically transferred information. Let C h C2> .•• C n be a set of different basic units. We usually perform measurements to distinguish C j from C k in order to specify certain units, for example epithelial cells, lymphocytes, or endothelial cells. We can, however, also try to analyze their spatial relation, e. g. whether they are arranged in a distinct manner such as arrangements in chains, rings, or clusters. We can also analyze properties of subunits such as the color or intensity of staining after a histochemical reaction of the cytoplasm, or the size of the nucleus. We can, in addition, distinguish different «qualities», for example presence of tumor cells or other cell types, and state of viability, as inferred by degree of apoptosis or necrosis, etc. without quantitative measurements of their features. This procedure results in a «yes-no» differentiation (projection) of a predefined basic unit.

(a) First order structure Let C j be a basic unit of an image, and Psci properties inside C j , Puci properties of C j (shape, brightness, ...), and Pici properties between different C j (distance, neighborhood, etc.). C j is called afirst order structure, if an algorithm is present to measure Psci, Puci in a reproducible manner. This preposition usually holds true for any histological image (slide). Under certain circumstances it is not necessary to measure Psci or Puci in order to define C j • For example, the cells of a gland can be defined by their geometrical arrangement (ring-like texture), because only this type of cells is arranged in this manner. If we know that we are analyzing the scene of a «disco», and we are requested to determine the intervals with music and those of the breaks, it is not necessary to measure the size of persons, but only to look for the «distribution» of persons sitting at the tables or crowding on the dancing floor. (b) Second order structure

If all cells C j appear to be arranged in a regular manner, we can try to discern a pattern of the spatial distribution of these cells, and to look at the structure built only by these cells. Therefore, it is the aim to define any occurring regularity which is established by these cells only. A proper mathematical technique to recognize these regularities is the graph theory. A second order structure is defined by S(C j , Ps)

=L (C

j)

with Ps

=f(Puci)

(23)

Ps depends upon the neighborhood condition which is used. There are two neighborhood algorithms which seem to be appropriate at the light microscopical level: a) the tes-

24 . K. Kayser and H.-J. Gabius

selation analysis based upon «Voronoi's dot pattern process» (VORONOI 1902), and b) O'Callaghan's neighborhood condition (O'CALLAGHAN 1975). Both techniques require the measurement of the «geometric center» of all first order structures, i. e. the «basic» unit has to shrink to its center point CPo This center point must not necessarily be the «real geometric center», but might be weighted by additional factors such as location of the «darkest point within the cell», the «distribution of the chromatin pattern», etc. Voronoi's procedure tries to maximize the area covered by concentric circles around the CP j with the additional condition that the circles do not have to overlap. The procedure starts with growing circles around each point CP, and the growth of a circle CPj is stopped, if it touches another circle. Those points CPjk with connected circles are called neighboring points. Voronoi's procedure is useful for «solid structures» such as brain, skin, or liver (see Fig. 10, Fig. 11). O'Callaghan's neighborhood definition is useful for adenomatous textures, and is based upon two constraints:

Voronoi tessellation of a point process

Fig. 10. Voronoi tessellation of a point process. Neighboring points are defined by a common edge. The size of the points (nuclei) and the size of the Voronoi cells can be chosen within an upper and lower limit for practical clinical applications.

Graph theory and the entropy concept in histochemistry . 25

Neighborhood condition according to O'Caliaghan /

/

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C: Central cell N: Neighboring cells 0: Cells not fulfulling the neighborhood condition: Distance d > d max (distance constraint), or angle a < amin (direction constraint) Fig. 11. Neighborhood condition as defined by O'CALLAGHAN (1971). The cells marked with N are neighbors of the center cell C, those marked with 0 are not (they are either hidden behind a neighbor or located at a distance beyond the neighboring range).

(1) the distance constraint which excludes all points from a neighborhood which are

positioned beyond an upper limit Lu (and, in a variation, closer than a lower limit Ll) from the point CPj ; (2) the direction constraint which excludes all points from a neighborhood which are «hidden» behind a (neighboring) cell, i. e. the angle a has to fulfill the constraint a(CP j), D(CPn' CPk) > LDa, if CPk is a neighbor of CP j

(24)

All points CPk which obey the described constraints are neighbors of the point CP j •

26 . K. Kayser and H.-J. Gabius

Having defined the set of neighbors S(C;), we can again perform the same procedure as described above and obtain S(C j )

=Si ={Pssi, Pusi, Pisi}

(25)

with Pssi = properties inside Si (substructures) Pusi = properties of Si (shape, size, minimum diameter, etc.) Pisi = properties between different S; (distance, neighborhood, etc.).

(c) Third order structure

Commonly, different C j build disparate Si. If Si created by different C j also have a regular geometrical relationship in common, these regularities can be summarized again to build a new (higher) structure using the same algorithm. Examples are circles of cells fully embedded one in another (epithelial cells surrounded by basal cells, or endothelial cells surrounded by smooth muscle cells, etc.) or «solid» structures surrounded by rings such as nerve fibers placed in a nerve sheet (myelin), as schematically shown in Fig. 12.lt should be noted that the concept is not restricted to «healthy structures». Epitheloid granulomas are an example of a non-healthy third order structure (KAYSER and HOFFGEN 1984). (d) Fourth and higher order structures

The described technique can be applied again to a set of third order structures, and a new level of structures is created. This level already includes visible and distinct parts of an organ, for example liver lobuli or glomerula. Moving on with application of the theorem, another order of structures can be described. In nature, the concept is limited by a «time constraint», placing emphasis on the fact that the regularities used for the construction of hierarchic structures should be time-independent, i. e. constant within the length of the period considered. The size and shape of any organ and of the whole body usually does not change during adulthood; however, the geometrical relation of a «person» to his neighboring «persons» changes considerably within the period under consideration. Thus, it does not make sense to extend the scope of the concept described beyond the «structures of a person», unless another time constraint is introduced. Remarks: The same concept can be applied to «time-dependent structures». Given a scene with movements mt(s, v, r, t), m2(s, v, r, t), m3(s, v, r, t), ..., mn(s, v, r, t), we can, for example, «clustep> all movements into the same direction r, or those starting at the same to> or those with the same velocity v, or those with the same geometrical properties s, and will obtain a new set of movements (higher order time-dependent structures). This concept is valuable to analyze the size of currents, and can also be used for artificial creation of scenes (virtual imaging, animation). Histochemically> especially with relevance for clinical purposes, the concept of orders of textures has the following advantages:

Graph theory and the entropy concept in histochemistry . 27

(a) Measurements performed on different orders can easily be combined. Usually, the features of a higher texture can be measured, and the existence of other (lower) textures within this texture can be analyzed and grouped according to various aims (diagnostic importance, prognosis, therapy, etc.). (b) Neighborhood conditions between basic units of various orders of textures can be exactly defined. (c) It can be used as a theoretical concept to enhance understanding of the relationship between structure and function within one order of texture. (d) The set of calculations can be applied to the interaction of textures at various orders, and - with similar considerations - a hierarchic concept of biochemical and physiological functions can be derived.

7.1 Clinical applications The concept of orders of (light microscopical) structures has been applied for diagnostic purposes to various kinds of human organs (KAYSER and HOFFGEN 1984; DusSERT et al. 1989; KAYSER et al. 1990). The main purpose can be ascribed to serve as a diagnostic support in difficult situations. In addition, the concept can be used for construction of neuronal networks to apply «mixed» digital and analogous (hybrid) algorithms to reach a fast and accurate diagnosis (KOLLES and DAUMAST-DuPORT 1995). KAYSER and HOFFGEN (1984) reported that the separate measurements of second and third order structures are an appropriate technique to distinguish the following classes of functions (diseases) on endometrial mucosa: normal endometrium (8d, 16d, 21d, 28d), glandular hyperplasia, glandular-cystic hyperplasia, adenomatous hyperplasia, atypical hyperplasia, cystic atrophy, carcinoma in situ, adenocarcinoma (Gl, G2, G3 according to the common grading of the WHO applied in HOFFGEN et al. 1983). The second order structures created by adenocarcinomas are apparently very similar to those of healthy tissue; the determination of first order structures is needed to separate these entities in a reproducible manner. However, most of the diagnoses depend on the description of third order structures. The measurements of normal, adenomatous and carcinomatous colon mucosa revealed analogous results (KAYSER et al. 1986a). Distinguishing these diagnostic groups was facilitated by a representation of the arrangement of glandular structures in the two-dimensional space (plane) (KAYSER et al. 1985, 1989a). The quantitative evaluation of the glandular structures revealed that adenomatous glands display a larger minimum diameter and a broader variance than those of normal tissue and adenocarcinomas (KAYSER and SCHLEGEL 1982). Another illuminating analysis within this respect has been performed on malignant diseases of pleura (KAYSER et al. 1987a). Separate measurements of first and third order structures of malignant cells within the pleura could distinguish epithelial mesothelioma from metastatic adenocarcinoma into the pleura with an accuracy which was at least comparable

28 . K. Kayser and H.-J. Gabius

b

c

d

Fi . 12. rder of tru cur . 12a di play a cr ti n fa p ripheral n rv. ach mented fib ria unit of a fir t rd r tru cure. The relation b ten n i hboring fiber i hownin 12b andi u edtoc mputeth nd rder tru tur (12 }.Thenerve i elf i a unit of a third ord r tru cur (a ording to Y ER and H" I'r E 19 4) (12d). or mpari n the graph of a me theli rna computed n a fir t (cum r celt) and third rder ( land-like texcure) eru cure are given in 12 and 12f ( ee n epa e).

to that of common immunohistochemistry and ligandohistochemistry in a learning set and in prospective data analysis (KAYSER et al. 1986a, b, 1987a, 1989a, b). In addition, it could be demonstrated that those cases which are difficult to separate by texture analysis displayed disparate features in the histochemical analysis (KAYSER et al. 1987a; see Fig. 13).

Graph theory and the entropy concept in histochemistry . 29

e

Fig. 12e+f.

In conclusion, the concept of hierarchic structures in histopathology permits separate and, from the statistical point of view, independent measurements of segmented features at different levels of magnification. Table 1 compiles the basic relationships between structure levels and diseases (of the endometrium) which hold also true for diseases of the colon and lung (KAYSER and HOFFGEN 1984). The main contributors for diagnostic classification can be commonly found at the level 2 and level 3. The term surface includes a distinct mark (boundary) between two (one embedded and one surrounding) spaces; the term texture a relationship between various «embedded" spaces (structures). Until now we have focused on the conceptual framework behind the relationships or procedures of mapping, and only marginally touched the actual handling of these relationships. The most common mathematical technique to describe connections or relationships between a set of various elements with close association to our common spatial experience is the so-called graph theory, and it is necessary to introduce the basic principles of this procedure to understand its clinically important implications.

30 . K. Kayser and H.-J. Gabius

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<> CEA- + TPAFig. 13. The separate measurements of different order structures offer a close association between the obtained data and the diagnostic classification. P(X/G) is the computed distance from the centroid, and peG/X) the probability of diagnosis, i. e., to classify a case as metastatic adenocarcinoma (pleuritis carcinomatosa) or mesothelioma. The more probable the diagnosis of a mesothelioma (metastatic adenocarcinoma), the more distinguished are the expressions of certain diagnostic markers (carcinoembryonic antigen CEA, and the tissue polypeptide antigen TPA, according to KAYSER et al. 1987a). Metastatic adenocarcinomas commonly express CEA, whereas mesotheliomas rather lack CEA. TPA is expressed by both tumor types in a comparable frequency.

Graph theory and the entropy concept in histochemistry . 31

8 Graph theory The graph theory is a mathematical procedure which analyzes relationships between certain elements with a network of «connections» (GIBLIN 1977). The advantage of the procedure is that it introduces a measure which is based upon the distance between certain elements, and that it deals with the real space concept, i. e. it is applicable in the two- or three-dimensional space. A graph consists of a finite set of basic elements which are called vertices (nodes) Vi> and a set of relations (arcs) ei between the nodes which can be described by an incidence function and are referred to as edges. This function groups the vertices to pairs (vj, eJ It is excluded by definition that the arcs join each other or cross in between except in a vertex. If the incidence function between two vertices exists, these vertices are connected by a line (edge). A graph is called complete, if every pair of vertices is joined by an edge. The incidence function can have a direction, i. e. indicate a flow or not. Then the graph is called either a directed or a non-directed graph. In histopathology, normally a non-directed graph is given, time functions being commonly neglected.

Definition: A path on a graph G from any vertex VI to a vertex Vn is a sequence of vertices Vi and edges ei {Vlel> V2 ell V3 e3,··· , vne n} with el

(26)

=(VI> V2); e2 =(Vb V3), ••• en-l =(Vn_1> Vn)

A simple path is a path with distinct edges el> ell ... en _I> and distinct vertices VI> V2, ... Vn _I> with the only exception that the following situation is given: Vn =Vn_I' A connected graph is a graph which fulfills the condition that a path from Vi to Vk for any two vertices Vi, vk is present; i. e. each vertex can be reached from another vertex by at least one sequence of {Vi, eJ A loop is a simple graph G {VI' Vb V3, ..., v n}with VI =vn.1t is clear that n > 2 follows for a loop. Examples of the discussed representations of graphs and trees are shown in Fig. 14. The basic principle to apply the graph theory in histochemistry follows the outlined scheme: The segmented objects of a histological slide (cells or their nuclei) are considered as vertices. If the objects have something in common, they will be connected by edges. For example, if we are dealing only with tumor cells Ctj, the individual elements of this cell population will be connected by edges, deliberately excluding other cell types in this section such as lymphocytes Clio If we want to measure organizational parameters of all cells C i irrespective of their histogenetic origin, the lymphocytes Cli will be included into the graph.

ser and 32 . K. Kay

s H.-]. Gabiu

a

b

2

2

3

4

4

6

7

7

c

2

d

2

4

4

5

5

II

7

6

7

Graph theory and the entropy concept in histochemistry . 33

e

f

2

3

3

4

4

5

5

6

7

6

7

Fig. 14. Various realizations of a graph. a: complete graph; b: connected graph; c: the graph contains two loops I and II; d: simple path; e: tree; f: minimum spanning tree according to the length of the edges.

The situation can be described mathematically by the following equation: L Cti = {(Vi, ei)}

(27)

with ej =1, if Vi =tumor cell, and ei =0, if Vi =lymphocyte. The simplest way to describe the population of tumor cells is by their position in a two-dimensional coordinate system x,y. Thus, we can write

L Cti ={(Xi, Yi, ei)}

(28)

The parameters to describe the (non-directed, non-weighted) graph are: 1. number of vertices Nv 2. number of edges Ne 3. cyclomatic number Nc which in a connected graph is defined as follows: Nc=Ne-Nv+ 1

(29)

The cyclomatic number measures the «connectivity» of a graph, i. e. how many (spatial) associations exist between the segmented objects.

34 . K. Kayser and H.-J. Gabius

a

b

y

y

• •

x

c

d

y

y

x

•

Graph theory and the entropy concept in histochemistry . 35

e v

Fig. 15. Construction of a graph and derived trees. lSa displays the segmented nuclei. In Isb the center of the nuclei are marked. The complete graph is demonstrated in ISc, that of a weighted tree (according to the size of the nuclei) in lsd. The minimum spanning tree of this configuration is shown in lSe.

x

4. Number of subgraphs Ngs A subgraph is a graph which is part of the complete graph (complete image), and which is more restricted than the complete set. For example, a simple subgraph is the graph consisting of two elements only with an edge (Vih eil) and (Vi2' eil) which leads to the equation:

Gs(2)

={(ViI> 1), (Vi2' 1)}

(30)

Another subgraph is, for example, the set of vertices having three neighbors. If we measure the length L of the edges ei> we can introduce the average length Lm of the edges. The measure ej . nv(Lm) (nv = number of vertices) is then a statistically «stable» limit to divide the graphs into separated subgraphs, and to search for spatial «clusters» (ZAHN 1971; KAYSER et al. 1992a, 1994a; MARCEPOIL and USSON 1992). Examples of the procedure are given in Fig. 15 and Fig. 16. 5. The construction of a tree A tree is a graph which consists only of connected vertices, and fulfills the additional condition that only one path exists to reach each vertex. These graphs do not contain closed rings (loops); i. e. a tree is a non-empty (at least two vertices) connected graph without any loops. Trees are characterized by the following theorem:

36 . K. Kayser and H.-J. Gabius

a

b

•

•

•

•

•

c

d

Fig. 16. Clusters of subgraphs obtained by decomposition of the minimum spanning tree (MST) using the length of edges for decomposition measure. 16a displays the segmented nuclei with marked center of gravity. The corresponding minimum spanning tree is shown in 16b, and the decomposition according to the length of the edges in 16c. Four separated clusters are obtained (16d).

Graph theory and the entropy concept in histochemistry . 37

Let G(v, e) be a graph. Then G is a tree, if and only if (a) G is connected and Ne Nv-1 (b) G does not contain cycles (loops), i. e. is acyclic, and Ne Nv -1; (c) a connection (path) between every pair of vertices in G is required.

=

=

6. The construction of the minimum spanning tree (MST): A minimum spanning tree is a subtree and contains all vertices of the given graph with the following constraints: (a) The distance between all vertices is minimal; (b) It is a tree (according to the definition in 5).

A minimum spanning tree of an adenocarcinoma of the lung computed according to the minimum distance of the centers of Feulgen-stained nuclei is drawn in Fig.!7. In contrast to other approaches in describing spatial relations (for example O'Callaghan's neighborhood definition) the characteristic (pseudoglandular) textures of this cell type cannot be recognized immediately. A minimum spanning tree can be constructed as follows: A starting vertex is selected, and the position of the nearest vertex is calculated. These two vertices are connected by an edge. In a second step, the nearest vertex to both connected vertices is calculated, and

Fig. 17. Minimum spanning tree of an adenocarcinoma of the lung.

38 . K. Kayser and H.-J. Gabius

this vertex is connected to its neighboring point of the two selected ones by an edge. In a third step, the vertex nearest to the three selected vertices is calculated, and the procedure is continued until all vertices are included into the minimum spanning tree. Another often applied algorithm is the adjustment of sequences of edges in an ascending order according to their vertices, and to select those edges with the nearest distance. 7. Distribution of the number of edges per vertex (stars)

8. Number of loops (closed paths) within a graph For practical reasons, the construction of a MST is often performed. Compared to the analysis of a complete graph, that of a MST has several advantages: (a) it is easy to handle; (b) it reflects properties of the complete graph and (c) it is stable despite spatial transformations. A non-directed non-weighted graph reflects a simple situation of a histological section: it represents the spatial situation of one (or several) sets of cells distributed in the tissue section. Numerous simple questions can be answered by the use of this technique: For example, the simple pattern of distribution of two different cell types with respect to the cell type of the neighboring cell (select those tumor cells which have a neighboring lymphocyte and calculate their number in comparison to that of tumor cells having a neighboring tumor cell) can be easily assessed. In addition, a further simple question, namely how «far» away from each other are the neighboring cells at average, can be answered by the construction of a non-weighted graph. However, other simple questions (for example, what is the amount of DNA in tumor cells with neighboring lymphocytes, etc.) require a different approach in calculation, namely the use of weighted graphs. Definition: A weighted graph is a graph which contains a «weight function» for each vertex. The weight function can be a simple number, for example the color or size of a nucleus, the distance to a neighboring vertex, or a set of weights (distinct associated parameters).

Gw =((Vj(Wk), ei»), i

=1,2, ... N; K =1,2, ... m

(31)

A weighted graph can in principle be handled similar to a non-weighted graph. It includes the computation of spatial relationships between different cell types (see Fig. 18). In histochemistry, it is useful to include certain spatial parameters (area, diameter, integrated optical density, integrated nuclear fluorescence, etc.) into the weights. The application of weighted graphs is the basis of the measurements described in the following chapters. It permits a detailed insight into structural and cellular (nuclear) properties of any tissue and its abnormalities such as malignant growth. However, to take advantage of the described mathematical tools for analysis of cell proliferation in relation to the expression of distinct biochemical properties such as histochemically visible binding capacities, the technique of static analysis of DNA content by measurements of the integrated optical density (IOD) has to be introduced.

Graph theory and the entropy concept in histochemistry . 39

8.1 Clinical applications The application of graph theory in texture analysis is also called syntactic structure analysis (Lu and Fu 1978; SERRA 1982). As already mentioned with respect to the concept of hierarchy in structures (and used for support in diagnostic classification), the technique of syntactic structure analysis was first applied as a helpful tool for discrimination of various types of diseases. SANFELIU et al. (1981) demonstrated that this method was appropriate to distinguish primary myogenic disorders from neurogenous muscle diseases. They analyzed the spatial pattern distribution of histochemically labeled muscle fibers in a set of patients with neurogenous and myogenic muscle dystrophy. Independent from this group a similar strategy was developed by KAYSER and SCHLEGEL (1982) and KAYSER et al. (1985). Concerning diseases of the lung the cyclomatic number, the distance to the nearest neighboring or second nearest neighboring tumor cell, the number of branching points, average number of vertices, of subgraphs, and that of nonbranching points were found to be of high discriminatory power in diagnosis (KAYSER et al. 1985, 1986a, b, 1987a, b, c). The predictive values of the cyclomatic number and the distance of nearest neighboring tumor cell could be underscored in a learning and a prospective data set (KAYSER et al. 1985). Results of reclassification by parameters of texture analysis obtained from 75 lung cancer cases are exemplarily compiled in Table 2, and those from 75 cases of diseases of the colon mucosa in Table 3. Three sets of diagnostic groups can be discerned: (I) the healthy tissue; (II) small cell anaplastic carcinomas and epidermoid carcinomas; (III) adenocarcinomas and large cell anaplastic carcinomas. The reader who is not familiar with diagnostic difficulties is referred to the problem to distinguish correctly in small biopsy specimens two different tumor entities with similar structural behavior, namely small cell anaplastic carcinomas versus poorly differentiated epidermoid carcinomas, and poorly differentiated adenocarcinoma versus large cell anaplastic carcinomas. The completely different structural properties of healthy lung tissue in comparison to those of primary lung carcinomas are exemplarily demonstrated in Fig. 19. The distance of the nearest neighboring cell in relation to the cyclomatic number differs remarkably in healthy lung tissue from that of lung carcinomas (Fig. 19). These findings are not surprising, because the syntactic structure analysis in principle mirrors the diagnostic algorithms used by pathologists, and the texture of healthy lung tissue differs from those of the carcinomas to a high extent (KAYSER 1987). A more detailed analysis of primary carcinoma and metastatic carcinoma into the lung pinpointed different features of the minimum spanning tree with respect to number of neighboring cells and their mean distance (see Table 4, KAYSER and STUTE 1989a). In addition, the main cell types of bronchial carcinomas can display diverse structural features, as exemplarily demonstrated in Table 5 and Table 6 according to KAYSER and STUTE (1989a). These results are of pivotal importance for understanding the growth behavior of malignancies in the lung, and support the notion that malignant tumors originating from different host organ sites will probably continue to exhibit certain structural features which might be

40 . K. Kayser and H.-J. Gabius

a

b

•

• •

•

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• •

••

Ly

•

•

••

Ly

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Graph theory and the entropy concept in histochemistry . 41

e

Fig. 18. Scheme of a weighted graph for analysis of spatial features between two different cell types. The segmented lymphocytes (small black disks, Ly) and segmented tumor cells (larger gray disks, Tu) are shown in 18a. The constructed minimum spanning tree of all cells is depicted in 18b. The minimum spanning tree of lymphocytes (18c) and that of tumor cells (18d) are confined to solely one cell type in the section. Edges to compute the minimum distances between the two cell types are illustrated in 18e.

instrumental to delineate the primary site of distant lesions or to separate primary from metastatic carcinomas of the lung (KAYSER and STUTE 1989b). The concept of syntactic structure analysis requires a segmentation based upon structural units of an image (usually cells). The segmentation procedure is most efficient, if only one main compartment of the cells is visible, for example the nucleus. In histology, certain histochemical procedures have been developed which stain specifically and quantitatively DNA (Feulgen stain, DAPI stain). These stains are the basis for quantitative measurements of DNA content of (tumor) cells (DNA cytometry). The application of syntactic structure analysis to images which only display DNA-dependent dye deposition can be instrumental for diagnostic purposes. Before this aspect is explained, it is instructive to outline the methodology of DNA analysis.

9 DNA content analysis In a normal cell, the content of DNA of the nucleus is defined by the constant set of chromosomes, and thus stable during the whole life of the cell. As long as the cell stays in the Go/G t phase, the double-stranded structure of the DNA helix accounts for a sharp peak (approximately Gaussian distribution) of the DNA, if determined by optical tech-

42 . K. Kayser and H.-J. Gabius I/J

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cyclomatic number

-e-s- healthy . .x. . e~idermoid

-~

.. ....::- aaeno

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-:-:- small cell

-~

large cell

nc nc nc nc nc

15 15 15 15 15

r·0.65 r c 0.81 r c 0.61 r c 0.72 r c 0.63

Fig. 19. Linear regression curves of the cyclomatic number and of the mean distance between nearest neighboring cells in healthy lung tissue and primary lung carcinomas (N = 75). Note that the graph of healthy lung tissue displays a different behavior, although the relation is linear in all analyzed cases (according to KAYSER et al. 1985).

niques. This peak is called 2C peak (diploid peak, 2N chromosomes). The cells which actively replicate intranuclear DNA have entered a so-called S-phase, and then a postsynthetic Grphase, where their nuclei have acquired 4N chromosomes, doubling the normal DNA content in preparation for mitosis. The DNA content as well as the number and appearance of chromosomes can be noticeably altered in tumor cells, and the occurrence of tumor cells displaying 12N DNA content is not a rare event. The Fig.20 and Fig.21 demonstrate the distribution of signals in static DNA measurements on material from pleural effusions. A benign pleural effusion with numerous inflammatory cells and desquamated mesothelial cells harbors nearly all mesothelial cells in the 2C peak of the nuclear DNA distribution as measured by static DNA cytometry (Fig. 20). In contrast, the DNA histogram of a malignant pleural effusion associated with a metastatic adenocarcinoma of the lungs reveals numerous proliferating cells and those with an abnormal DNA content> SC (Fig. 21).

9.1 Basic considerations on DNA cytometry In the following chapter it is intended to give a mathematical description for measurements of the DNA content in cell nuclei by combining a routine staining protocol such

Graph theory and the entropy concept in histochemistry . 43

Cytometric DNA Analysis

Cell Count 2.83c

Fig. 20. DNA histogram of a pleural effusion with a non-specific inflammation. Nearly all cells display a 2C peak, and 96.6% of cells are found to be in the G1-phase.

as the Feulgen protocol or exposure to a fluorescent dye and a suitable imagemonitoring system (static DNA cytometry). The following prerequisites have to be met, if the obtained DNA distribution profiles are considered to possess a biologically meaningful information: (a) The nuclei of «normal» human cells contain the same range of amount of DNA which depends only upon the position of the individual cell within the cell cycle at the time of the measurement (fixation), the differences of the sex chromosome being to small to be detected by the recording systems. (b) The cells repeat the individual stages of the cell cycle which can be roughly divided into three clearly separated parts: 1. resting cell (Go) with constant DNA content (Cl); 2. replication cycle of DNA (constant increase of DNA content until it is doubled, S-phase) (C2); 3. reorganization ofthe cell and cell division (doubled DNA content) (C3). (c) No other cells are present or their occurrence is negligible (for example necrotic or apoptotic cells undergoing DNA degradation).

44 . K. Kayser and H.-J. Gabius

Ce 11

Count

D:58.4X

3.82c

4c 6c 8e !~=----~'===;r:----"

__-J4-.Jl..1 _ _ _ 6. 1.ee

C2

Fig. 21. DNA histogram of a malignant pleural effusion. About 5% of tumor cells are in the Sphase, and the proliferation index is calculated to be 19.6% (S-phase and G 2/M phase).

Definitions The following symbols are used: DS Cjt hrj Vrj htj Vtj As Nr Nt

= amount of DNA in a single nucleus (for example in picograms) = stage of cell cycle i (1,2,3) of a cell at time t

= 1/Vrj =duration of cell cycle i (i = 1,2,3) of the reference cells = velocity of the cell cycle i (i = 1,2,3); hrj =constant for each i = 1/Vtj =duration of cell cycle i (i = 1,2,3) of the tumor cells = velocity of cell cycle i; htj = constant for each i

= absorption of single nuclei stained by Feulgen's procedure

=absolute number of measured reference cells within a given area (volume) of tissue

= absolute number of measured (tumor) cells within a given area (volume) of tissue.

Derivations

=

(1) Nr L [Nr(C it )]; The total number of all measured reference cells is the sum of the cells which are in the various cycle stages.

Graph theory and the entropy concept in histochemistry . 45

(2) N =L [Nt(C it)]; (tumor cells). (3) Nr(C it ) =kr . hri; the number of reference cells at the cell cycle stage i is proportional to the duration of the cell cycle. (4) Nt(C it ) =kt· hti' (5) Cjt =P . As; the stage of the cell cycle is proportional the amount of DNA in the nuclei of the reference cells. (6) Nr(Cit)/Nr pr' Cit; the relative number of reference nuclei at the stage i is proportional to the duration of the stage i. (7) Cit(s) =As . N(Cit)/Nr . p; the cell cycle stage of the reference cell can be derived from the absolute DNA content and the frequency distribution of the DNA content of the reference cells. These equations permit the following deductions: (7a) As =Ci/p =Pr . Nr(Cit)/p . Nr; the relative number of reference cell nuclei with a DNA content As is proportional to the stage within the cell cycle. Remarks: The most important constraints are the constant cell cycle velocity for each cell, the same size of the nuclei during the cell cycles, and the absence of apoptotic cell nuclei.

=

Derivations for tumor cells

(8) Ats(cj) =AS(Ci) + DA(Ci); the amount of DNA in a tumor cell nucleus is a function of the stage in the cell cycle (time t) and the amount of abnormal DNA (aberrant number of chromosome sets or chromosomes) which can also depend upon the stage in the cell cycle. (9) AtS(Ci) =Cit' P + DA(cj); (10) At(Ci)/Ar(cj) =1 + DA(cj)/Ar(cj); the aberrant amount of DNA deviating from the normal content in tumor cells can be derived from the amount of DNA in reference cells, if the stage of the cell cycle in tumor cells and in reference cells is known. (11) AtS(Ci) =pt· Nt(cj)/Nt (12) DA(ci) =pt . Nt(Ci)/Nt - Cjt . p; the aberrant content of DNA in tumor cell nuclei can be derived from the DNA distribution of the reference cells and that of the measured tumor cell nuclei. This conclusion is only true, if the velocity of cell cycles of tumor cells is independent from the size of DNA content in cell nuclei. The registration of the DNA content is based upon the principle that certain substances can bind to DNA in a stoichiometric manner. The classical reagent is the Feulgen stain, whose application requires to separate the double-stranded DNA into two single strands. Technically, the separation of the two DNA strands is routinely performed by hydrolysis with 5N HCI at room temperature for 30 min. The acid hydrolysis of the nuclear DNA breaks the N-glycosidic base-deoxyribose bond, whereafter a reactive aldehyde group is exposed. This part of the pentose then reacts with the Schiff reagent, usually pararosaniline. The determination of the integrated optical density (absorption

46 . K. Kayser and H.-J. Gabius

of light at the isosbestic point, 555 nm) is representative for the total DNA content of the corresponding nucleus. The measurements should be performed with complete cell nuclei, i. e. with cytological specimens, if it is desired to compute the total DNA content. Interpretation of the results of the processing of histological slides has to take into consideration that they present nuclei, which are cut at various nuclear diameters, therefore not allowing to monitor the complete nucleus. These unavoidable impediments can only be addressed to a certain degree. Therefore, the respective analysis of histological slides is not as accurate as that performed on cytological specimens. The DNA data obtained from histological slides are called integrated optical density (IOD) in order to distinguish them from biochemical DNA measurements, which are able to yield the total DNA content of a tissue sample following its homogenization. This procedure, however, inevitably causes the loss of spatial patterns. An alternative to the Feulgen stain to visualize nucleic acid is the use of fluorescent dyes which bind stoichiometrically to DNA. One commonly used fluorochrome is DAPI (4',6-diamidino-2-phenylindole) (DANN et al. 1971; COLEMAN 1979; COLEMAN et al. 1981; HAMADA and FUJITA 1988). A comparison of measurements performed on conventional Feulgen - stained cytological specimens (tumor imprints of bronchial carcinomas) with those of DAPI - stained smears is given in Fig. 22. No statistically significant differences could be obtained, ascertaining the assumption that the two procedures will end up in equivalent sets of data (KAYSER et al. 1996a). The advantage of the DAPI stain is the fast processing which lasts only a few minutes. Thus, the DNA measurements based on the determination of the integrated nuclear fluorescence (INF) can be performed intraoperatively, and prognostic indicators for the survival of the patients can be evaluated during surgery guiding the surgeon (KAYSER et al. 1996a). The measurements can be carried out by flow cytometry or by (static) image analysis. Both techniques yield a graphical representation of the DNA contents whose actual meaning is then delineated. Usually, the DNA distribution displays certain clearly separated peaks which are called stem lines as long as they are not repetitive by a factor 2, i. e., a DNA distribution with two peaks at 2C and 4C is considered to consist of only one stem line (at 2C). The second parameter to be located is the appearance of the main peak. If it is not at the correct 2C location, the non-proliferating cells have an abnormal DNA content, and the whole population is called aneuploid. A third visible parameter is the appearance of additional peaks (outside the 4C peak) which indicate presence of an aberrant DNA content in the measured cells. The area of the curve between the 2C peak and the 4C peak indicates the amount of «cycling cells», and is called the S-phase-related fraction. The limits are commonly set to 2.75 < IOD < 3.25. Another parameter is the DNA index DI which is defined as follows: DI

=L PVPn with Pi = measured DNA peak, Pn = DNA peak at 2C

(32)

A DI = 1 indicates a normal diploid cell population, DI = 2 a cell population with two peaks (at 2C and 4C), i. e. a tetraploid population. Data obtained from an unusual malig-

Graph theory and the entropy concept in histochemistry . 47

I="eulgen and formalin-fixed dapi s-phases in lung carcinomas {N=20) 20

1&

.

12

c

0)

.• ::>

8

o

o

4

8

12

1&

20

dapl

Fig. 22. Comparison of S-phase-related tumor cell fraction between measurements of integrated optical density (IOD, Feulgen-stained slides), and integrated nuclear fluorescence intensity (INF, DAPI-stained slides).

nant tumor of the chest wall (Askin tumor) are given in Fig. 23, and reveal that numerous tumor cells possess a DNA content equivalent to only one half of the normal chromosome set (peak at 1C). Using this approach a rare malignant teratoma of the lung could be classified. This mixed mesenchymal/epithelial malignancy displayed lymph node metastases solely from the (aneuploid) ectodermal component (KAYSER et al. 1993d). Other descriptive parameters are the «width» of the 2C peak as indicated by the coefficient of variation lCV, or the attempt to represent the curve by a set of sinus functions related to the width of the 2C peak and the intensity (height) and position of additional peaks (SCHENCK et al. 1997). The procedure is equivalent to a «simple Fourier analysis»; its clinical significance has still to be validated. A very useful parameter has been described by STENKVIST and STRANDE (1990), the DNA entropy. This parameter is derived from the entropy defined in information the-

48 . K. Kayser and H.-J. Gabius ASKIN tumor 100 histogram

18

16

14

12

I

to

IOD

Fig. 23. IOD histogram of an Askin tumor (Note that the first peak of the IOD histogram is around IC).

ory. It denotes the statistical information given by the «number of events» in each «channel» of the DNA distribution, and is defined as follows: E(IOD) =L Pi . In (Pi); Pi =probability of events in channel i, i

=1, 2, 3, .•., N

(33)

The entropy E(IOO) depends upon the number of channels N (usually 100) or the «fine tuning» of the distribution, and upon the upper limit of the ONA content in the overall distribution, the maximum DNA content (ONA(max» usually set to ONA(max) 12C. E(IOO) can be divided by N . In(N) for practical reasons, i. e., to disminish the influence of the «fine tuning» (KAYSER et al. 1993a, b).

=

9.2 Clinical applications The clinical applications of DNA cytometry have two main goals: (a) to increase the reliability to separate benign from malignant tumors, and (b) to support the estimation of prognosis of patients with malignant tumors. At present, most of the experts working

Graph theory and the entropy concept in histochemistry . 49 III

the field of DNA cytometry agree to the following statements 1995):

(SCHULTE

and

BaCKING

1. The majority of benign tumors display an euploid DNA distribution. 2. Most of the malignant tumors have an aneuploid DNA distribution. 3. The level or reproducibility of the measurements is more than satisfactory for detailed clinical analysis, as exemplarily shown in Table 7 (KAYSER et al. 1994a, b). These statements imply that an aneuploid DNA distribution is highly suggestive for presentation of a malignant disease, whereas an euploid DNA distribution does not exclude a malignancy. The detailed interpretation of a DNA measurement has to take several clinical features into consideration such as the patients' sex and age, clinical history, and the origin of the measured cells (body organ). For example, aneuploid DNA configurations are nearly exclusively associated with a carcinoma in specimens of the bladder, breast, or lung; however, they do not exclude a benign pleural effusion (KAYSER et al. 1995). About 10% of pleural effusions with aneuploid cells reveal no indications of a malignant pleural tumor, and the patients generally have a history of tuberculosis or repetitive therapies such as pleurodesis. As already explained, measurements of integrated optical density performed on histological slides are not as accurate as those performed on cytological specimens. Nonethe-

2E -1

o

o

2c

4c

6c

8c

IOc

Fig. 24. IOD histogram of a large cell anaplastic carcinoma of the lung prior to induction chemotherapy.

50 . K. Kayser and H.-J. Gabius

less, presence of different stem lines and the percentage of tumor cells within the Sphase-related tumor cell fraction, and that with an IOD > 3C or > 5C can reliably be computed, as shown in Fig. 24. Moreover, the preferred growth directions and the grading of the tumor cells can be estimated (RODENACKER et al. 1992). The IOD assessment of tumor cells can also be exploited with respect to the nature of the nearest neighboring cells, for example lymphocytes or tumor cells, as demonstrated in Fig. 25. These measurements have proven to permit the evaluation of prognostic estimators even in small histological specimens (KAYSER et al. 1994a, b). Of prognostic value for lung carcinoma patients are the number of stem lines, and the percentage of tumor cells to be grouped into the S-phase (S-phase-related tumor cell fraction (SPRF)), and the percentage of tumor cells with an IOD > 5C (see Figs. 26-28). The measurements of DNA distribution by integrated optical density (IOD) can, in addition, be used to gain a further insight into the behavior of malignant and benign tumors, for example of lung carcinomas. KAy-

1 neighbor

%0

%0

400

400

200

200

o

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o

2000

4000

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ADENOCA~CINOMA 3 neighbors

%0

200

200

2000

4000

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o

2000

4000

extinction

DIST~IBUTION all neighbors

%0

400

o

o

- DNA

400

o

2 neighbors

o

2000

4000

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Fig. 25. Calculated IOD distributions of an adenocarcinoma in relation to the number of neighboring cells. Adenocarcinoma cells are seen more frequently in the S-phase-related tumor cell fraction, if they have two neighboring tumor cells compared to tumor cells with only one neighboring tumor cell.

Graph theory and the entropy concept in histochemistry' 51

%

89 79 69 59

0

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+

!/r-oup 2

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!/r-oup 3

49 39 29 19

8

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,",onths

Fig. 26. Survival rates of potentially curatively operated bronchus carcinoma patients grouped according to the number of stem lines (group 1: 1 stem line; group 2: 2 stem lines; group 3: > 2 stem lines; KAYSER et al. 1994a).

et al. (1989a, 1994a, b) analyzed the tumor volume and the IOD of bronchial carcinomas in association to the inflammatory response of the host tissue. No influence of the apparently possible immune reaction determined by the density of immune-reactive cells within the bronchial carcinomas could be detected. However, with increasing stroma reaction the percentage of carcinoma cells with an IOD > 5C decreased remarkably (Table 8). The potency of bronchial carcinomas to destroy the original healthy interstitial lung tissue is also associated with IOD features (Table 8). The more complete the host tissue is altered by the tumor growth, the higher the percentage of tumor cells with an abnormal DNA content is (KAYSER et al. 1989a). In principle, bronchial carcinomas have about 10-16% of tumor cells in the S-phase-related tumor cell fraction, and about the same percentage of tumor cells with an IOD > 5C, changes with the cell type, pT and pN stages being apparent (Table 9 and 10). Within lymph node metastases these SER

52 . K. Kayser and H.-J. Gabius

%

11<1

8

16

23

31

39

47

55

62

o

group 1

+

group 2

x

group 3

70

78

Months

Fig. 27. Survival rates of potentially curatively operated bronchus carcinoma patients grouped according to the S-phase-related tumor cell fraction (group 1: < 15% S-phase; group 2: 15% < Sphase < 20%; group 3: S-phase > 20%; KAYSER et al. 1994a).

figures differ: the percentage of tumor cells with an laD> 5C is increased, and that of the cells within the S-phase-related tumor cell fraction decreased. An additional effect of induction therapy (preoperative cytostatic therapy which is applied to «stage down» the carcinomas, or to offer initially not surgically treatable patients the chance of a potentially curative surgery) on the laD distribution has been reported by KAYSER et al. (1994b). Induction therapy decreases the percentage of the S-phase-related tumor cell fraction from 12 to 7%, and increases the percentage of tumor cells with an laD> 5C from 10 to 16% at average (Table 11). No association with tumor volume and tumor stage or cell type has been noted. The results are in agreement with the general theory that cytostatic drug regimes mainly affect the pool of proliferating tumor cells, and might enhance the heterogeneity of malignant tumors (KAYSER et al. 1994b, c). It has already been emphasized that cytometric measurements can be performed intraopera-

Graph theory and the entropy concept in histochemistry . 53

%

J.9

8

16

23

31

39

407

55

62

o

group 1.

+

group 2

)(

group 3

79

78

...on ths

Fig. 28. Survival rates of potentially curatively operated bronchus carcinoma patients grouped according to the percentage of tumor cells with an IOD > 5C (group 1: < 5%; group 2: 5% < 5C < 20%; group 3: 5C > 20%; KAYSER et al. 1994a).

tively by use of fluorescent stains, for example DAPI (DANN et al. 1971; COLEMAN et al. 1981). The results obtained in a pioneering investigation in the field of lung cancer are comparable with the conventional Feulgen stains, as demonstrated in Table 12. There are no major differences when the staining time is reduced to 5 minutes, as shown in Table 13 and Table 14 (KAYSER et al. 1996a). The detailed results of a first trial are compiled in Table 15. The combined measurements of textural and nuclear features, i. e., the amount of DNA, already highlight the inherent clinical significance. The application of a (continuous) energy field (for example related to an electromagnetic/gravity-associated spatial relation) can then be used to calculate thermodynamically important properties of the biological system, such as structural entropy and current of entropy. The mathematical procedure is described in the next chapter.

54 . K. Kayser and H.-J. Gabius

10 Calculation of structural entropy The calculation of structural entropy is performed based upon the following assumptions: 1. The entropy of a regular or symmetric structure is constant, and can be set to zero. 2. It can be calculated for one order of structure only, for example epithelial cells. 3. It is regulated by the «steering mechanisms» of the cells, i. e. the nuclear content of cells such as by proliferating cells. 4. It depends upon the amount of DNA and upon the distances between the centers of nuclei. The total energy of such a system, referred to as E(MST), can be computed by the equation (34), if we assume that the laws of thermodynamics are obeyed (see Fig. 29): E(MST)

=k· I; ([D (IOD)/IOD]2 + [D (r)/r]2) j

j

(34)

with Dj(IOD) = difference of IOD between the nearest nuclei of the corresponding cells = average IOD of all nuclei of the corresponding cells IOD Dj(r) = difference of distance between the centers of nearest nuclei of the corresponding cells = average nearest distance between the centers of nuclei of the corresponding cells r k =constant, set to 1. E(MST) becomes zero, if all distances Dj(r) between nearest nuclei amount to the average distance between the nuclei, and the IOD of two neighboring cells is equal for the cells under consideration. The distinct parameters of the equation are dependent upon the organ under consideration, i. e. the average distance r between the centers of nuclei has to be measured independently for each organ. However, if r is known, the calculation of E(MST) leads to values, which can be compared independent of the specific organ structure in between (KAYSER et al. 1993a, b). The reader should be aware that it is surmised that only the physical energy which changes in relation to the square of the distance is taken into account.

11 Calculation of current of entropy According to the theory of ONSAGER (ONSAGER 1931; DE GROOT 1960; DE GROOT and MAZUR 1969) the total amount of irreversibly created heat has to be transferred through the surface of such a system. The distance of such a system from its equilibrium stage is correlated to the size of the current of entropy, i. e. the greater the distance, the greater the current of entropy is. The currents of entropy EF(IOD), EF(MST) for both E(IOD) and E(MST) can be described by the equations

Graph theory and the entropy concept in histochemistry . 55

1

d(l ;2)

2

Fig. 29. Calculation of structural entropy based upon the minimum spanning tree. The equation assumes a llr2 dependence of DNA content differences (IOD 1-IOD 2) and aberrations in the distances of gravity centers of nearest neighboring (tumor) cell nuclei (d(1;2)-d(2;3».

56 . K. Kayser and H.-J. Gabius

EF(IOD)

=E(IOO) . [~(IOD)/dt]/Su

= K· E(IOD) . SPRF . (R/r )3/R2

(35)

and

EF(MST) =E(MST)· SPRF· (R/r)3/R2 with R

(36)

=average

macroscopic diameter of the biological system under consideration (tumor) SPRF = S-phase-related fraction (percentage of cells in S-phase) = surface of the system under consideration Su = average distance of the centers of nuclei (calculated by use of MST) r !l [IOD/dt] = changes of IOD within the time interval dt

The equations (35) and (36) describe the transfer of heat through the surface (boundary) of the system, and do not take into account that all biological systems possess a vascular system, which is also able to transfer heat. If we are able to measure, in addition, the surface fraction of the vascular system, EF can be described more accurately:

EF(IOO) =E(IOO) . SPRF . (R/r )3/(SV . R 3 + R 2)

(37)

EF(MST) =E(MST)· SPRF· (R/r )3/(Sv· R3 + R2)

(38)

with Sv = surface fraction of the vessels within the system.

For a large size of R and for values of Sv obtained in reality (approximately 0.2-0.3) the second part of the divisor becomes small compared to the term which represents the vascularization. As we have already discussed, the current of entropy reflects the «distance» of the system from its equilibrium state irrespective of the status of its environment that can be a thermodynamically open or closed system. To illustrate the clinical relevance of these considerations based upon physical principles, some characteristics of these parameters are shown in Fig. 30. The entropy distribution of lung tumors closely follows a Gaussian shape (Fig.30a), that of the current of entropy has an exponential shoulder (Fig. 30b). The same holds true for the 100 entropy and current of 100 entropy (entropiefluss) (Fig.30c, 30d). The average distances of the edges which are used for the calculation of the structural entropy and currents of entropy are not independent of the number of neighboring cells, as demonstrated in Fig. 31. Thus, the structural entropy is closely associated with those structures commonly described in histopathology, namely glandular, papillary, or solid structures.

Graph theory and the entropy concept in histochemistry . S7 2E-1.

a

2E-1.

o 69 2E-1.

1.38

207

o o

276

c

4E-1.

1.29

258

387

51.6

88

1.76

265

353

1.9 442

d

50

1.9 1.00 1.50 200 252

Fig. 30. Graphs showing the distribution of IOD entropy, MST entropy, and currents of entropy (IOD) and (MST) in bronchial carcinomas (N=195). Fig.30a: IOD entropy; Fig.30b: MST entropy; Fig. 30c: current of IOD entropy; Fig. 30d: current of MST entropy (KAYSER et al. 1993b).

11.1 Clinical applications

To figure out whether these computations are of clinical relevance, we have initiated their correlation to the survival time of patients (KAYSER et a1. 1993a, b). Notably, the current of entropy is of prognostic value for lung cancer patients, whereas the degree of entropy appears to have no apparent predictive value (Fig. 32). The structural entropy

58 . K. Kayser and H.-J. Gabius

%

length of edges Fig. 31. Distribution of mean length of edges in relation to the number of neighboring cells in epidermoid bronchus carcinomas (N = 20).

reaches the lowest level in small cell anaplastic carcinomas, and the highest in intrapulmonary metastases (Table 16). It is also increased in lymph node metastases (Table 17), and does not depend on the cell type (Table 18), the tumor volume or the postsurgical tumor stages (KAYSER et al. 1993a, b, c). It is intriguing for readers interested in developmental biology that the monitoring of the given set of parameters is definitely not restricted to cancer analysis. Indeed, it can be proposed to be similarly rewarding in the area of analysis of organ development and restructuring. In an initial study measurements of human fetal lungs revealed that the structural entropy changes abruptly at a certain developmental stage (KAYSER et al. 1997). It occurs around the 18th week of gestation, and is an indicator for pronounced morphological changes in fetal lungs. At this age the lungs are subject to fundamental structural alterations, namely the step from the pseudoglandular period to the canalicular stage. The pseudoglandular stage is characterized by completing the conducting airways, and the branching airways have a glandular appearance, whereas the canalicular phase has already empty air spaces with an internal lining of the primitive acini. These structural changes of the lungs take place during the 17th-25th week of gestation, and the calculation of structural entropy reflects these changes in showing a sharp increase followed by a reduction of structural entropy (Fig. 33). As it will be discussed later (see chapter 16.1 Clinical applications), receptor-targeted histochemical approaches lead to congruous data with respect to molecular characteristics and expression of binding

Graph theory and the entropy concept in histochemistry . 59

o

yr-oup 1.

+

!1r-oup 2

X

yr-oup 3

30

20

10

""on t h!

Fig. 32. Survival rates of potentially curatively operated bronchus carcinoma patients grouped according to the amount of current of MST entropy (EF) (group 1: < 10; group 2: 10 < EF < 30; group 3: > 30; KAYSER et al. 1994a).

capacities for various carbohydrates, underscoring the value of this approach for studies in developmental biology and histopathology. Combined cytometric and histometric measurements permit a more detailed analysis of malignant tumors, for example lung carcinomas. Calculation of the area of the Voronoi cells can be combined with the measured nuclear area (Table 19), and the obtained correlations are valuable in difficult diagnostic cases (KAYSER et al. 1993d, 1996d). These measurements have also been helpful for the diagnosis of a rare Askin tumor, reaching a similar degree of specificity as the currently performed immunostains (Table 20). With respect to prognosis the evaluation of cytometric parameters in specimens from potentially curatively operated lung carcinoma patients even appears to deserve priority, followed by consideration of histometric parameters and the expression of distinct biochemical features (Table 21).

60 . K. Kayser and H.-J. Gabius

gestation age and structural entropy in fetal lungs (N=135l

700r------------=-----------,

580· . 0:: IZ W I I-

4&0

c.'

(f)

:::;;

340·

220·

100L.5

~

15

~

~

25

35

__'_

--'

45

55

weeks Fig. 33. Changes of structural entropy in huma fetal lungs according to gestation age (N = 135). The steep increase of structural entropy at the beginning of the canalicular phase of lung development, and its decline at the end of this period are especially noteworthy (KAYSER et al. 1997).

The cytometric approaches and that of syntactic structure analysis have thus proven to be important tools to answer various clinical questions. At this level of refinement the analysis treats the lesion as a unit, disregarding intralesional focal aberrations from the average structure. These discontinuities of features are, however, prominent in numerous diseases, for example lung carcinomas or soft tissue tumors. Since they might contain important biological information, the concept needs to be extended to address this issue. Indeed, it is not to difficult to perform adequate analyses of data sets obtained from syntactic structure analysis in search for spatial clusters, as outlined in the next chapter.

12 Cluster detection

=

As already described, a graph G (V, E) is represented by a set V of points (vertices) and E connections called edges. A subgraph H (\v, F) of G is a graph with W f V and F f E. An attribute of a graph is given by a set of attributed values Ai which can be metrical, ordinal or nominal. These values can be attributed to a vertex (A h A 2, ••• An) and to an edge (B h B2, ••• Bm ). If we compute a minimum spanning tree (MST), it can easily be demonstrated that its construction is strongly dependent upon the weight function given by the attributes. Given a distinct MST(Ai, Bk), it can be divided into subtrees by

=

Graph theory and the entropy concept in histochemistry . 61

removing the edges with the greatest weights. This procedure entails to obtain a number of subtrees or tree clusters TC1, ... TCr. At this stage we can again compare the subtrees according to simple descriptors of the graph G, for example number of vertices, number of edges, cyclomatic number, etc. Hereby, a recognition of subsets of the MST and a description about their regularities is facilitated. However, the problem remains how to decompose the complete MST. For this purpose we need a dissection function d. This function d can be defined according to the concept of ZAHN (1971) who used the function d(eo)

=m(eo) + 2s(eo)

(39)

m(eo) is the mean, and s(eo) is the standard deviation of the weights of the edges in a neighborhood N(eo). N(eo) is the set of all neighbors of eo and of all second neighbors of eo. The edge eo is removed, if its weight w is larger than its decomposition value d(eo). An illustration of applying this strategy for decomposition of a tumor is shown in Fig. 34.

Fig. 34. Computation of artificial cytoplasm boundaries and a quantitative measurement of cytoplasmic staining intensity yielding construction of a minimum spanning tree. The staining intensity is grouped into four semiquantitative levels (none, weak, moderate, and strong) according to the average staining intensity within the cellular area (white = no, pale = weak, dark = moderate, black = strong intensity). At least three clusters of white cells can be noted. Compare also to Fig. 16.

62 . K. Kayser and H.-J. Gabius

The inherent potential of this concept will be further discussed, when the biochemical background for probe selection has been provided. It should be noted at this point that we can, in a second step, define a center of such a tree cluster, for example by associating the positions of the corresponding vertices with certain attributes. We can then once more start to dissect the obtained cluster tree. The utility and practicability of this concept - at this stage centered entirely on spatial parameters - has been underscored by the description of soft tissue tumors (KAYSER et al. 1991 b, 1992c). We have discussed the fundamental tools for tissue analysis in order to detect cytometrically, structurally, and thermodynamically important features, including the search for spatial clusters. These algorithms distinguish between structural units of the same textural order, and are, in addition, conducive to define structural units of a higher order. However, in light microscopy we are often confronted with the problem to estimate «densities» and «relative sizes» of certain repetitive structures at various levels of structural order. For example, the estimation of the «relative volume» of vessels, necrotic foci, and connective tissue can be used to calculate the «real» cellular volume of a tumor. If we know the gross tumor volume and the «volume fractions» of necrotic parts, vessels, and connective tissue, we can easily compute the cellular tumor volume by multiplying the corresponding volume fractions with the gross tumor volume, and subtracting the result from the total volume. These estimations belong to the field of stereology, have been well documented and applied in numerous scientific disciplines including histology, biology, crystallography, and mineralogy, and are essential to derive three-dimensional data from a two-dimensional section, the study object of the histochemist.

13 Basic stereological considerations Stereology is considered to be the «oldest» attempt to collect data on structural components of a two-dimensional histological slide with the aim to translate this information into a three-dimensional picture. Point-counting and counting the number of intersections or predefined lines (test system) with the edges of the object (profiles) are instrumental technically to complete this task (WEIBEL 1980). If we want to measure (estimate) the size of a profile (area), we simply have to count the number of points at a given constant distance present within this area. If we have a total of P t points in a predefined area A, and we count Pi points within the object 0, then the area fraction Aa can be calculated according to

Aa = PYi't

(40)

which is equivalent to the volume fraction Vv, if the objects are distributed isotropically. Similarly, the surface area Sv can be estimated by the equation

(41)

Graph theory and the entropy concept in histochemistry . 63

if the surface under study is oriented randomly, uniformly, and isotropically (VIR). Ba is the boundary length of the object area O. It should be noted that surface areas are «scale-dependent», i. e. the higher the magnification, the larger the surface area. This statement is equivalent to the considerations given above (chapter Structure and energy) on measurements of the length of a structure by fractals (i. e., the smaller the boxes covering a structure, the more precise the estimation of the length of the structure is). This concept can also be applied to infer preferred growth directions in various systems such as tumor cells or trees (SANDAU 1989; RODENACKER et al. 1992). If we want to count particles within a given volume V, we have to consider that larger particles have a higher probability than smaller particles to be hit by a histological section. This problem is solved by the construction of a dissector consisting of two planes at a fixed distance h, which run through the interesting areas with a volume Va . h in parallel. Only those objects are counted which are «cut by the lower section» and they represent the numerical density of particles Nv. The mean corpuscular volume vm is calculated by vm = vV/Nv

(42)

i. e. by dividing the volume density Vv by the numerical density Nv. These techniques are now commonly implemented in computerized image-analyzing systems. Estimating the area of an object by segmentation of objects in a histological slide is equivalent to point-counting: each pixel inside the object represents Pi, and the total number of pixels within the image is equivalent to Pt. Problems of calibration and resolution occur as the pixel size determines the accuracy of the measurements. Thus, the magnification has to be adequately adapted to the size of the objects to be measured. The modern development of stereology offers simple and reproducible measurements for various tissues including gross handling of the specimen, sampling, and measurements (GUNDERSEN et al. 1988a, b). The tools provided by stereology are most frequently applied for the estimation of volume and numerical densities of different segmented objects, for example tumor cells, lymphocytes, or tumor cells with expression of certain functional properties such as the presence of the cell cycle-associated Ki-67 antigen. They complement the structural analysis of any sample under scrutiny and are likewise predicted to be valuable in combination with immuno- and ligandohistochemistry.

14 Quantitative histochemistry Target-specific tools (antibodies, non-immunoglobulin epitope-specific probes such as lectins, carrier-immobilized ligands, substrates, etc.) in combination with a convenient dye-generating system are commonly used in histochemistry. To distinguish the application of antibodies from that of biochemically active ligands, which will visualize specific

64 . K. Kayser and H.-J. Gabius

binding sites in the tissue when properly controlled and connected with an appropriate signal-generating system, we refer to this technique as ligandohistochemistry. The problem of quantitation in immuno- and ligandohistochemistry as well as enzyme histochemistry can be separated into two parts which call for different techniques for practical reasons: (a) histological slides with known size of the stained or stain-free objects; (b) histological slides with unknown size of the objects. The problem mentioned under (a) is equivalent to immunohistochemistry with particular staining of nuclei. This technique is most frequently applied in slides immunohistochemically stained with antibodies specific for steroid hormone receptors or proliferation markers such as Ki-67, and permits analysis of all nuclei which are visualized by a light hematoxylin counterstaining. In this instance, the size of the nuclei can be measured reliably. The determination of the staining intensity and number of stained nuclei can be performed according to the equations of stereology: Np --

VvP/

/Vmp

(43)

The numerical density of stained nuclei Np can be calculated by the computation of the volume fraction of stained nuclei Vvp divided through their corpuscular volume vrnp. This equation can be expanded to those nuclei which are colored with a certain intensity Ipi (for example, G; = 1,2,3,4: 1 = no staining, 2 = weak, 3 = moderate, 4 = strong), and an antibody-related score Sc can be introduced by grading Np and the different intensities Ipi, i. e. by classifying the intensity of the stained nuclei: (44)

Such a score has been reported to be very informative in grading the amount of expression of hormone receptors in breast carcinomas, even surpassing the accuracy of biochemical assays, after its introduction by KOHLER and BASSLER 1986 as well as REMMELE and STEGER 1987. The problem (b), i. e. the analysis of the stained object with unknown size, is more difficult to solve. The objects under scrutiny (commonly cytoplasm) can be completely negative, focally positive, or only include membrane compartments stained with different intensities. Rarely, the complete object (cytoplasm and nucleus) is specifically reactive with the probe. Even if these elements can be segmented according to their intensity, it is usually not known, whether they are of the same size as those objects which are negative or only partly positive. Since we cannot apply a technique to measure the size of the objects in detail (i. e. the cytoplasm of the interesting cells), the unknown actual size of the cells has to be inferred. The following model can be used for estimation of Vp (KAYSER et al. 1995a), if the following constraints are fulfilled: (1) Each cell located in the area to be measured (area of interest) can be detected by the image-analyzing system (i. e. a nuclear stain will only react with the nuclei).

Graph theory and the entropy concept in histochemistry . 65

(2) The average size of the cells is known, for example from additional measurements performed on hematoxylin-stained slides. (3) In each slide or smear the nearest neighboring cell of each cell can be defined, and the distance between the centers of gravity of the neighboring nuclei (in addition to their size) can be measured. We can now calculate a quantitative cytoplasmic immuno/ligandohistochemical staining intensity using the equation:

Ie =vrnp . L

[(~(Iei,Ib»]

(45)

Ie

= staining intensity of cell Ck /:;. (ICi' Ib) = L (Ie; - Ib) Ie; = intensity (gray value) of pixels within the area of the object Ib = intensity (gray value) outside the object (background)

vrnp =2Jt· r 2

(46)

r = radius, r min [6(c;, Ck), rav] rav =average cellular radius measured in hematoxylin-stained cells 6(cj, Ck) = weighted distance/2 between the nearest neighboring cells.

The weighting d(ci) of the cellular radius ri is performed as follows: d(ci) . rnj + d(ck) . rnk =rj + rk . (rn; + rnk)

(47)

m;, mk = nuclear radius of cell i, k (nearest neighboring cells) ri, rk = cellular radius of cell i, k.

Taking lung carcinomas as example, quantitative assessment of the detectable binding sites ofhistoblood group trisaccharides A, B, and H, respectively, has previously been reported (KAYSER et al. 1994d, e). Pertinent results of this study are listed in Tables 22-24. In other words, the weighting of the cellular size is performed according to the size of the nuclei of nearest neighboring cells and the constraint that the cellular size is not greater than the average size of all cells present in the area of interest. This procedure leads to a straightforward assessment of signal intensities in histochemistry. Moreover, the distances between cells with similar staining intensities can be evaluated, cellular clusters of strongly stained cells can be detected, and the volume densities or numerical densities can be determined using the techniques described in the previous chapters on graph theory, stereology. To further illustrate the actual application of these principles presented so far with emphasis on their theoretical background, examples will be given in the paragraph on ligandohistochemistry, even taking the next step by their combination with localization of any selected determinant. In this branch of histochemistry epitope-specific probes from distinct receptor-ligand recognition systems are purposefully employed to visualize the presence of the complementary binding partner in the tissue section. Either labeled probes whose functionality is not impaired by the chemical modification (lectins, peptides or nucleic acids with target selectivity) or low molecular

66 . K. Kayser and H.-J. Gabius

weight compounds such as carbohydrates or steroid hormones which are covalently linked to a histochemically inert and labeled carrier are employed in ligandohistochemistry. An outline of the routine application of this technique which is conceptually similar to immunohistochemistry is presented in the next paragraph.

15 Ligandohistochemical staining techniques Since the general concept does not at all limit scope of the production of targetspecific probes either by chemical synthesis or biochemical purification, initial screening is primarily necessary to determine whether the processing of frozen sections is required or whether the distinct molecular interaction is still operative in fixed and paraffinembedded tissue sections. In this case, it is advisable to compare the suitability of different fixatives, Bouin's fluid or buffered formalin (4-8%, pH 6.9-7.4) being documented to provide reliable results (DANGUY and GABIUS 1993; DANGUY et al. 1995, 1997). The technical procedure includes the following steps: 1. Deparaffinize and rehydrate the 4-6 [.lm thick slides by successive exposure to graded alcohol

(95, 70, 50%). 2. Wash the slides with phosphate-buffered saline (PBS) buffer for 5 min. 3. Block the endogenous peroxidase activity by incubation with 0.1 % methanolic hydrogen peroxide (15 min). 4. Wash the slides twice with PBS (5 min). 5. Treat the slides with 1% bovine serum albumin to minimize background staining due to nonspecific protein-protein interactions. 6. Wash the slides twice with PBS (5 min). 7. Incubate the sections with solution containing the labeled probe (at a suitable concentration) for an optimal period. Both factors need to be determined in preliminary studies. If the presence of Ca2 + -ions is indispensable, buffer substances such as Tris or Hepes substitute phosphate salts. 8. Wash the slides twice with PBS (5 min) to remove any trace of the labeled probe. 9. Apply a label-specific kit reagent system such as avidin-biotin complex for 30 min. If the section has a strong abundance of receptors which bind kit reagents such as the oligomannoside sugar chains of avidin or horseradish peroxidase adequate precautions are required (coincubation with 0.2 M D-mannose). 10. Wash the slides twice with PBS (5 min) to remove any trace of kit reagents. 11. Apply the solution of color-generating reagent (15 min) for development of the chromogenic product. 12. Wash the slides twice with PBS (5 min). 13. Perform a light counterstain with Meyer's hematoxylin. 14. Mount the slides. All steps are performed at room temperature.

The classification of the result into negative/positive is based on a semiquantitative judgement. Sections should be considered to be positive, i. e. binding capacities to the applied ligand are present, if all or clusters of the analyzed cells exhibit an intense color.

Graph theory and the entropy concept in histochemistry . 67

The concentration of the ligand can vary with the source of the tissue section in the range of 5-100 /!g/ml, and should be determined in an experimental series together with the adequate control reactions. Competitive inhibition assures the target specificity and application of e. g.ligand-free but labeled carrier excludes the binding to sites which recognize the label (e. g. biotin) or the polymeric backbone, yielding false-positive results. Having developed the theoretical background of the assessment of diverse parameters derived from syntactic structure analysis and monitoring of DNA-related features, it is an intriguing possibility to combine these approaches with ligand-dependent histochemistry. Only very few studies have so far addressed this issue, which affords the potential to establish a new branch of histochemical investigation. To place emphasis on this promising concept leading to a riveting endeavor, examples of such combined investigations are outlined. Since primarily the localization of binding sites for carbohydrate ligands was the biochemical factor, the next paragraph will contain essential background information on endogenous lectins.

16 A brief survey of animallectins In line with the general concept of this article a theoretical consideration will be the basis for the part that describes the applied work. It is self-evident that physiological processes depend upon biochemical information stored and transmitted by several classes of biopolymers. Nucleic acids and proteins figure prominently as hardware for this task. When the capacity for information storage is assessed for biological polymers, these biomolecules, however, are surpassed in this respect by oligosaccharides. Calculations of the number of isomers that can theoretically be generated have yielded a clear-cut answer concerning the coding potential of carbohydrate chains in comparison to amino acids (LAINE 1997). Whereas the hexamer synthesis with six amino acids will produce 46,656 different structures, the same starting condition for saccharides will ensure the staggering number of ::::: 1.05 X 10 12 isomers. It is therefore pertinent to carefully study the way information is stored and transmitted by oligosaccharides. Although some biochemically important substances differ only in - on a first glance - minor details from related but inert compounds, these small variations seem to control their basic biological importance. An example are the ABH histoblood group antigens. Their small structural difference, shown in Fig. 35, will affect physicochemical features rather marginally. Oligosaccharides such as these antigens form a structural element of cellular glycoconjugates which not only can modify properties such as solubility, but also serves as docking point for receptor proteins such as lectins. As compiled in Table 25, several techniques have been elaborated to detect sugar-binding activities. Lectins are commonly denoted as carbohydrate-binding proteins without enzymatic activity acting on their ligand. They must also clearly be separated from immunoglobulins. Combined chemical and histochemical techniques, providing access to carrier-immobilized polymers (neo-

68 . K. Kayser and H.-J. Gabius

A

GalNAc

~Gal~

t

GlcNAc-R

~Gal~

GlcNAc-R

Ul,2

Fuc

B

Gal

t

Ul,2

Fuc

H (0)

Gal~

t

GlcNAc-R

Ul,2

Fuc Fig. 35. Schematic representation of the structure of the ABH blood group antigens.

glycoconjugates) and applying them on cell and tissue specimens, ensure a wide range of studies encompassing histopathological monitoring (SCHREVEL et al. 1981; SINOWATZ et al. 1989; GABIUS and BARDOSI 1991; GABIUS et al. 1993, 1994; LEE and LEE 1994; BOVIN and GABIUS 1995; DAN GUY et al. 1995, 1997). Due to the theoretically derived enormous coding potential it is not surprising that research efforts over the last decade have revealed the salient contribution of recognitive protein-carbohydrate interactions to multifarious physiological processes such as intracellular trafficking of glycoproteins or mediation of cellular interactions (for review, see GABIUS and GABIUS 1993, 1997). Notably, this basic research offers the potential to turn these advances advisedly into progress with clinical relevance (GABIUS et al. 1995, 1996). In this context it is noteworthy that research on animallectins has already matured to a stage, where several families are defined in detailed molecular terms (GABIUS 1997). As shown in Table 26, five categories for animallectins are currently e~tablished. Their classification is often based upon structural characteristics of the carbohydrate recognition domain (CRD), even allowing to draw dendrograms of the relationship of the individual members of a family, as shown for C-type lectins (DRICKAMER 1993). Among the five groups, the C-type and I-type lectins as well as the galectins have members which are present in diverse tissue types and in tumors (Tables 27-29). In addition to the accepted categories, the ongoing analysis is sure to define further groups which will expand the present status, e. g. the molecular chaperones calnexin and calreticulin or other lectins involved in intracellular shuttling of glycoproteins (for review, see GABIUS 1997). As

Graph theory and the entropy concept in histochemistry . 69

already referred to, the custom-made design of neoglycoconjugates facilitates to acquire a visualization of binding activities in a section, unless they are blocked by high-affinity endogenous ligands or harmed by any step of the sample processing (GABIUS and BARDOSI 1991; DANGUY and GABIUS 1993; DANGUY et al. 1995, 1997). The distribution of staining, reflecting the localization of the receptor type(s), is then the basis to correlate a deliberately chosen feature, i. e. expression of a supposed functional determinant, with the parameter set, which has been described in the preceding paragraphs. Having introduced the versatile tools to visualize lectin presence, we can now proceed to present initial results obtained in this recently opened field.

16.1 Clinical applications The analysis of binding capacities histochemically detectable in human lung carcinomas and peripheral lung parenchyma has been proven to be a reliable and reproducible technique (KAYSER 1992; KAYSER et al. 1989b, 1992a, b, 1994d, e). The clinical importance can at present be underlined by the following results: (a) Expression of binding capacities is related to the tumor cell type and can therefore be used for diagnostic purposes. For example, small cell anaplastic lung carcinomas differ at least quantitatively in the extent of expression of binding capacities for various carbohydrate components from non-small cell lung carcinomas (KAYSER et al. 1989b, 1992a, b). N-Acetyl-D-glucosamine- and lysoganglioside GMt-bearing neoglycoproteins have been proven to distinguish mesothelioma from metastatic adenocarcinoma into the lung (KAYSER et al. 1992b). Binding of GMt-oligosaccharide can be seen in about 75-85% of the mesotheliomas, and only in about 10-15% of metastatic adenocarcinomas in contrast to binding of ~-N-acetyl-D-glucosamine,which reaches a level of 75% in metastatic adenocarcinomas and about 10% in mesotheliomas under identical conditions. The specificity and sensitivity of these glycohistochemical markers are at least comparable to the commonly used antibodies against CEA, Leu-1, or HEA. (b) Healthy lung tissue not affected by infectious diseases displays only weak binding capacities to the majority of simple sugar units such as mannose, fucose, maltose, or lactose. In the course of bacterial and viral infections these binding capacities are expressed in about 80-90% of the cases, and, interestingly, remain detectable for a long period of time even after complete remission of the infection. These findings can be used to distinguish inflammatory, non-infectious diseases such as autoimmune disorders from bacterial infections under the same experimental conditions (KAYSER et al. 1991a, 1994b, c). Moreover, it has been demonstrated that alterations of the lung parenchyma induced by cytostatic drug regimes can be reproducibly recognized by application of a set of markers including carrier-immobilized mannose or maltose, and labeled heparin (KAYSER et al. 1991 a, 1994c). Characteristic alterations in the profile of reactivity are likewise seen in samples with granulomatous disorders such as sarcoidosis and tuberculosis (KAYSER et al. 1991a).

70 . K. Kayser and H.-J. Gabius

(c) Certain stages in fetal lung development are closely associated with the expression of specific sugar-binding capacities (KAYSER et aI. 1995b). This result holds also true for other organs and species (DONALDO-JACINTO et aI. 1995). The major changes take place at the transitional stage from the pseudoglandular stage to the canalicular stage, i. e., between the 16th-20th week of gestation, as already commented upon (KAYSER et aI. 1997). These findings suggest that changes in defined binding capacities are closely related with structural reorganizations as defined in terms of structural entropy (see also chapter 11.1 Clinical applications). It is tempting to envision a functional correlation.

17 Combined histochemical analysis A large body of evidence intimates that the expression of blood group-related oligosaccharide structures is linked to the degree of invasive growth capacity and to prognosis (COON and WEINSTEIN 1986; DUBE 1987; LLOYD 1987; SELL 1990; MURAMATSU 1993; KING 1994; GARATT 1995; ORNTOFT and BECR 1995; DABELSTEEN 1996). Using the complementary approach, denoted as reverse glycohistochemistry (GABIUS et aI. 1993), we have shown that the expression of binding sites for blood group-related trisaccharides A and H, too, has prognostic relevance for patients with non-small cell bronchial carcinoma, inviting the assumption of a potentially productive protein-carbohydrate interplay in situ (KAYSER et aI. 1994d). Having detected this correlation between the presence of binding site(s) for a carbohydrate ligand and the survival of the patients with lung cancer, it goes without saying that parameters from IOD-measurements and syntactic structure analysis were set into a relation to this glycohistochemical feature. The presence of binding sites for the blood group H-trisaccharide, its computation to survival of patients being shown in Fig. 36, is significantly correlated with a lower percentage of tumor cells in the S-phase, a lower degree of aneuploidy and a lower current of MST-entropy than measured in receptor-negative tumor cells (KAYSER et aI., 1994e). Remarkably, no significant differences were seen in the tumor cell populations, which are categorized on the basis of their binding capacity to the blood group B-trisaccharide. Addition of a a1,3-linked galactose moiety to the core (H) structure, which is drawn in Fig. 35, thus translates into obvious differences, underscoring the inherent specificity of the reaction (Tables 30-32). An conceptually similar study has already been performed for carcinoids (KAYSER et aI. 1996b). It is interesting to note that typical and atypical carcinoids can be distinguished by the IOD entropy current and the distance between proliferating tumor cells (KAYSER et aI. 1996b). When the survival of patients was set into relation to these parameters, several factors reveal a positive or negative correlation. Similar to the size of the S-phase-related fraction the extent of binding sites for B- N -acetylgalactosamine is attributable to an unfavorable prognosis (Fig. 37). In tumor models in vitro and in comparison between primary and secondary lesions the specific binding capacity for this sugar unit was in line with metastasis formation (GABIUS and KAYSER 1989; GABIUS et aI.

Graph theory and the entropy concept in histochemistry . 71

%

19

99

°

group 1

+

group 2

29

19

8

16

23

31

39

47

55

62

79

78

Months

Fig. 36. Survival rates of potentially curatively operated bronchus carcinoma patients grouped according to the presence of binding sites of histoblood group H antigen (N = 195, according to KAYSER et al. 1994d).

1990). Entropy currents and distances between neighboring tumor cells are significantly correlated with favorable prognosis, as is the case for the expression of the heparinbinding lectin and binding sites for the lymphokine macrophage migration inhibitory factor (KAYSER et al. 1996b). These two proteins had been chosen, because the lectin, which binds heparan sulfate proteoglycans, can interfere with growth factor-dependent signal elicitation, and the presence of lymphokine has been unraveled as prognostic factor in non-small cell lung carcinomas (GABIUS et al. 1991; KOHNKE-GODT and GABIUS 1991; KAYSER et al. 1994d). Interestingly, galectin binding, too, has prognostic relevance for non-small cell lung carcinomas, emphasizing the potential role of lectin-glycoligand interplay for this tumor group (KAYSER et al. 1996c). Our initial study on bronchial carcinoma has revealed that survival, certain IOD- and MST-features and - among the receptors for ABH-histoblood group epitopes - the presence of H-trisaccharide-specific

72 . K. Kayser and H.-J. Gabius 1.2500

p-GALW\C 1=negattve 2=positlve

(/)

1.0000

,.....-,1'----.

Q.

r:.

0.7500

...

(/)

o

> >

0.5000

:J

0.2500

......------2

(/)

0.0000

L..-

O. 0

--'

20.0

40.0

60.0

80.0

100.0

Survlvaltlrne(rnonths)

120.0

Fig. 37. Survival rates of potentially curatively operated patients with carcinoids grouped according to the expression of binding sites for carrier-immobilized B-N-acetyl-D-galactosamine (N = 85, according to KAYSER et al. 1996b)

binding sites are statistically connected. To address the question whether the biochemical and morphometric features show correlations also in other types of carcinoma, we have performed this analysis for prostate cancer (KAYSER et al. 1995c). As shown in Table 33, the SC-exceeding rate shows the same behavior, as seen for bronchial carcinoma. In addition, the 2CV-index and the distance between aneuploidic tumor cells are significantly different, when tumor cell populations were grouped according to the glycohistochemical feature (Tables 30-33). Since we thus can answer the given question positively, encouraging further studies along this route, it is then tempting to propose that this reasoning can be similarly valid to processes in tissue development and restructuring. Using samples of human lung at various developmental stages, it has been documented that this set of methods is likewise readily applicable (KAYSER et al. 1997). Based on this initial experience, it is reasonable to expect a burgeoning growth in this area, considering different types of sample and histochemical tool.

18 Conclusion and perspectives The technical refinements of acquisition and handling of quantitative data in histochemistry have invited to further extend the scope of the exploitation of the available result sets. Beyond mere statements about the status of staining intensity of distinct cells

Graph theory and the entropy concept in histochemistry . 73

in a uniform or mixed population their spatial organization can be treated with groups of algorithms which are exquisitely elaborated within the framework of graph theory or thermodynamics. Although the complete theoretical foundation surely extends the range of back-of-the-envelope calculations, the given explanations have been designed to rebuff the notion that a perception gap between mathematical aficionados and histochemists must inevitably persist. Having documented the feasibility to meaningfully apply terms such as degree or current of entropy in histochemistry and having emphasized the potential relevance of introduction of these parameters for histopathological considerations aiming at prognostic evaluations, it is our firm conviction that the deliberate combination of these approaches with the assessment of expression of distinct epitopes not only adds a vitalizing factor to this branch of histochemistry but also holds enviable promise to unravel functional correlations, as illustrated by initial results in our treatise. Besides antibodies carrier-immobilized ligands such as neoglycoconjugates generated by tailor-made chemical synthesis account for the intriguingly wide variety of probes to visualize any receptor of choice. By mastering the chemistry of conjugate formation and successfully running the essential specificity control reactions, ligandohistochemistry equals immunohistochemistry in utility. Armed with the validated approaches to dissect receptor presence and the various aspects of structural organization quantitatively, we are therefore in a position to cheerfully address the question, which biochemical factors are relevant for the formation and/or stability of the presented tissue structure-derived parameters, e. g. in a section (or a series of slides) at a crucial step of developmental reorganization or in malignant transformation and the course of the disease. It is self-evident that any detectable correlation, which at that stage of investigation should be cautiously considered phenomenologic, will serve as a track to be pursued diligently in adequate cell biological models such as histotypic cultures to lead to tangible progress. Clues for target selection can be derived from various sources including molecular biology such as the application of the differential display reverse transcriptase polymerase chain reaction. With respect to tumor biology, the possibility should not be neglected that the definition of biochemical determinants with assumed impact on already pinpointed morphometric parameters as indicators of an unfavorable clinical course may be of therapeutic value. The study of lectin/glycoligand expression is just one example of the array of studies which appear to be indubitably plausible. Being confronted with the redundancy on the level of individual effectors for various functions and the inherent static nature of each section, the interpretation of the results of such studies is to be given with a grain of salt. However, the placement of these pieces of knowledge together with e. g. the evidence of cell biological work is proposed to contribute to solve the intricate puzzle on structurelfunction relationships extending to the level of tissues and their capacities for dynamic pattern formation including appearance of clinically important aberrations. In sum, the judgement appears to be reasonable that the appreciation of the outlined interdisciplinary methodology deserves to grow over the next decade.

74 . K. Kayser and H.-J. Gabius

Table 1. Main contribution of structure levels for diagnostic classification of diseases of the endometrium':'. Structure level

Diseases 1 functional disorder

2

3

+

+

infection, nonspecific

+

+

infection, specific

+

+ +

benign tumors (hyperplasia) . .. carCInoma In SItu

+

KAYSER

+

+ +

malignant tumors ':- according to

4

and

HOFFGEN

+

1984.

Table 2. Results of reclassification in bronchial carcinomas using the technique of syntactic structure analysis':-. Learning set (N = 75) Actual cases

Predicted cases

Cell type

healthy tissue

epidermoid carCInoma

adenocarcinoma

healthy tissue

15

0

0

epidermoid carCInoma

0

12

0

adenocarcinoma

0

0

13

0

2

small cell anaplastic carCInoma

0

2

0

13

0

large cell anaplastic carCInoma

0

0

0

0

15

small cell anaplastic carCInoma 0

large cell anaplastic carCInoma 0 2

Graph theory and the entropy concept in histochemistry . 75

Table 2. Continued. Prospective set (N = 25) Predicted cases

Actual cases healthy tissue

Cell type

epidermoid carcinoma

adenocarcinoma

small cell anaplastic carcinoma

large cell anaplastic carcinoma

healthy tissue

5

0

0

epidermoid carcinoma

0

4

0

0

adenocarcinoma

0

0

4

0

small cell anaplastic carcinoma

0

0

0

5

0

large cell anaplastic carcinoma

0

0

0

0

5

':. according to

KAYSER

0

0

et al. 1985.

Table 3. Actual and predicted cases of healthy, adenomatous and carcinomatous colon mucosa based solely upon syntactic structure analysis (graph theory; N =75Y. Predicted group healthy

adenoma

19 1

17

adenocarcinoma

learning set - healthy - adenoma - adenocarcinoma

o

1

o

1

2 19

1

o

prospective set

4

- healthy - adenoma - adenocarcinoma ':- according to

KAYSER

o o

et al. 1986a.

4 2

1 3

76 . K. Kayser and H.-J. Gabius

Table 4. Structural parameters obtained by application of the minimum spanning tree (MST) technique in primary and metastatic adenocarcinomas of the lung (N = 22, mean and standard deviation, distance in microns':'). Feature

Primary tumor

metastatic carcinoma

mean distance cells without neighbors (%) cells having one neighbor cells having two neighbors cells having three neighbors

11 ± 1 5±3 27 ± 17

12 ± 1 10 ± 6 36 ± 12 37±26 17 ± 7

" according to

KAYSER

48 ±23 20±9

and STUTE 1989b.

Table 5. Parameters of syntactic structure analysis grouped according to the cell type in primary bronchus carcinomas (N = 75, mean; confidence limits, p > 0.95; distances in microns). Cell type

healthy tissue

cyclomatic number

25 ± 2.5

distance of nearest neighbor

distance of 2nd nearest neighbor

distance of 3rd nearest neighbor

13

±0.8

17.9 ± 1.2

26.4 ± 1.8

epidermoid carCllloma

265 ± 25

8.4 ± 0.4

10.7 ± 0.6

12.7 ± 0.7

adenocarcinoma

114 ± 15

9.7 ± 0.6

13.1 ± 0.7

16.5 ± 0.8

small cell carCllloma

281 ± 13

7.1 ± 0.4

9.2 ± 0.6

11.0 ± 0.6

large cell anaplastic carCllloma

137 ± 16

11.3 ± 0.5

15.0 ± 0.7

18.1 ± 0.9

Graph theory and the entropy concept in histochemistry . 77

Table 6. Average number of vertices, subgraphs, and branching points in primary lung carcinomas grouped according to cell type (N = 120, mean; confidence limits p > 0.95; k = number of neighboring tumor cells)"". Cell type Feature

epidermoid carcinoma (N = 20)

vertices

192

± 19

adenocarcinoma (N = 20)

large cell anaplastic carcinoma (N =20)

small cell anaplastic carcinoma (N = 60)

197

155

377

± 18

± 15

± 18

number of subgraphs

4.7 ± 0.4

4.7 ± 0.4

5.1 ± 0.4

1.4 ± 0.1

number of elementary collapses (k = 1)

47.1 ± 5.9

49.7 ± 5.5

41.7 ± 5.8

92.5 ± 6.9

number of nonbranching points (k =2)

97.7 ± 16.9

98.2 ± 18.7

71.9 ± 12.2

20.2 ± 14.4

number of branching points (N = 2)

37.5 ± 6.7

39.8 ± 8.2

30.2 ± 6.3

79.3 ± 5.7

number of branching points (N =3)

1.6 ± 0.6

1.9 ± 0.5

1.7 ± 0.6

4.6 ± 0.6

':. according to

KAYSER

et al. 1987c.

Table 7. Reproducibility of laD and MST measurements (N distances in micronsY-o

= 25; mean and standard deviation;

Feature

previous measurements

12 months later

S-phase-related fraction

13

12

%>5C 2CY-STND laD entropy

±6

7 ±6 11

±5

2.2 ± 0.5

±6 8 ±7 12 ± 7 2.1 ± 0.5

laD entropiefluss (current)

13.2 ± 7

13.7 ± 6.5

distance tumor - tumor cell

12

±2

12

±2

distance tumor -lymphocyte

11

±2

11

±3

MST entropy

10.4±7.1

10.5 ± 8.1

MST entropiefluss (current)

23 ±13

25

± 14

". according to (KAYSER et al. 1994a). 2CY-STND : 2CY standard deviation. Remarks: To demonstrate the reproducibility and stability of the applied algorithms and used equipment, 25 cases out of the complete data set were randomly selected, and measured again after a time period of one year by a different person.

78 . K. Kayser and H.-J. Gabius Table 8. Features of integrated optical density (100) in relation to growth pattern and stroma reaction in operated bronchial carcinomas (N = 75; confidence limits p > 0.95 y-. N

% of 100 > 3C

% of 100 > 5C

DNA index

preserved

12

35.2-52.1

11.4-23.1

1.7-2.6

partly destroyed

29

42.7-55.5

15.7-27.3

1.8-2.5

completeley destroyed

31

48.9-78.3

17.1-46.9

2.1-2.7

none-slight

28

46.2-63.8

18.6-35.0

2.2-2.8

moderate

30

38.0-54.8

13.4-26.2

1.7-2.5

severe

17

33.1-47.9

7.9-17.3

1.8-2.4

Feature interstitial tissue

stroma reaction

':- according to

KAYSER

et al. 1989c.

Table 9. Features of integrated optical density in relation to cell type of operated bronchus carcinoma patients (confidence limits, p > 0.95; area in square microns). Cell type Feature

epidermoid carCInoma (N = 59)

adenocarCInoma (N = 56)

large cell anaplastic carCInoma (N = 32)

small cell anaplastic carCInoma (N = 48)

nuclear area

55 -60

73 -80

66 -75

33 -38

61 -78

% S-phases (SPRF)

10 -14

10 -14

10 -14

12 -16

8 -16

%>5C

5 -11

4 -9

8 -13

8 -13

11 -17

2CV-STND

7.5-13.5

6.1-12.7

7.6-14.3

3.6-11.6

7.3-9.6

100 entropy

1.9-2.4

2.0-2.5

2.0-2.5

2.1-2.5

2.0-2.7

100 entropiefluss

9.9-14.3

9.7-13.7

10.3-14.4

10.5-15.3

12.1-15.6

metastases

(N = 46)

Graph theory and the entropy concept in histochemistry . 79 Table 10. laD features and postsurgical tumor stages in potentially curatively operated lung carcinoma patients (confidence limits y-. Feature

S-phaserelated fraction

N

%>5C

2CV-STND entropy

entropiefluss (current)

pTstage -pT 1

43

9-11

12-16

7.5-9.1

2.2-2.6

7.6- 8.4

-pT2

99

10-13

12-16

8.2-9.4

2.1-2.5

12.5-13.5

-pT3

33

11-16

14-19

8.1-9.5

2.3-2.7

15.1-17.2

-pT 4

7

11-18

13-20

8.3-9.0

2.2-2.9

13.0-16.0

-pNO

67

9-12

13-17

8.1-9.1

2.0-2.4

9.1-11.8

-pN 1

48

10-16

11-16

7.6-8.7

2.1-2.5

15.2-17.5

-pN2

49

10-17

12-18

9.1-9.9

2.3-2.7

12.1-14.2

-pN3

17

11-18

13-19

8.4-9.6

2.4-2.9

8.0- 9.2

pN stage

':. according to

KAYSER

et al. 1994a.

Table 11. Features of integrated optical density (laD) in relation to cytostatic drug regimes (induction chemotherapy; bronchial carcinomas: N = 48, lymph node metastases: N = 28; confidence limits, p > 0.95; nuclear area in square micronsY-o laD-Feature

prior to chemotherapy

after chemotherapy

lymph node metastasis

% tumor cells

12.2-13.8

7.8- 9.6

8.7-11.3

>5C

6.2- 7.8

12.8-17.2

13.4-20.6

2CY-STND

8.6-10.7

5.9- 8.9

4.7- 9.5

entropy

2.2- 2.3

2.2- 2.5

2.2- 2.9

S-phase-related fraction

nuclear area

55 -60

,:. according to KAYSER et al. 1994b. 2CY-STND : 2CY standard deviation.

48 -57

46 -59

80 . K. Kayser and H.-J. Gabius Table 12. Comparison of cytometric data obtained from tumor imprints of primary bronchial carcinomas (N =20, mean and standard deviation; nuclear area in square micronsY-o Feature

Feulgen

DAPI, fixed

DAPI, unfixed

100

107 ±68

218

177 ± 112

±99

nuclear area

67.4 ± 14.2

66.7 ± 10.8

69

S-phase-related fraction

11.1 ± 2.5

12.2 ± 3.4

12.3 ± 3.5

% tumor cells> 5C

28.7 ± 15.0

29.8 ± 22.6

29.6 ±26

IOD entropy

16.9 ± 0.8

16.8± 1.4

15.3 ± 4.4

".- according to

KAYSER

± 14.6

et al. 1996a.

Table 13. Nuclear features taken from tumor-free lymph nodes in relation to staining time and staining procedures (N = 10; mean and standard deviation; nuclear area in sqare micronsy-"'-. Feature

Feulgen*

DAPI5*

DAP15fix*

DAPI30*

DAPI30fix*

stem lines S-phase-related fraction

0 ±1.3

0 ±0.7

±2.3

0 ± 1.6

±1.7

%>3C

3.5 ± 4.8

4.5 ±4.8

2.6 ± 5.3

2.7 ± 3.2

4.7 ± 4.0

%>5C

0

0

0

0

0

±4.8

14 ±5.2

entropy

1.5 ± 0.5

1.5 ± 0.5

2CV-STND

3.5 ± 2.5

3.5 ± 2.7

nuclear area

15

15

±5.3

± 5.1

16 ±5.4

1.4 ± 0.6

1.5 ± 0.5

1.5±0.5

3.3 ± 3.0

3.4 ± 2.1

3.4±2.1

". Feulgen, Feulgen stain (according to MIKEL et al. 1985) DAPI 5, DAPI stain, 5 min, air-dried DAPI 5 fix, DAPI stain, 5 min, formalin-fixed DAPI30, DAPI stain, 30 min, air-dried DAPI 30 fix, DAPI stain, 30 min, formalin-fixed 2CV STND, 2CV standard deviation. ,,-':- according to KAYSER et al. 1996b.

14

Graph theory and the entropy concept in histochemistry . 81

Table 14. Nuclear features of various intra-operatively analyzed malignant lung tumors in relation to the staining procedure (N = 40; mean and standard deviation; area in square microns; time in minutes). Feature

Feulgen'~

2.4 ± 0.7

Stem lines

DAPI5" 2.2 ± 0.5

DAPI5 fix" 2.0± 0.5

S-phase-related fraction

11

± 2.3

12

± 4.7

11

± 4.3

%>3C

64

± 16

61

±21

62

±20

%>5C

25

± 14

29

±23

27

±22

nuclear area

57 ±24

55

±25

56 ±23

IOD entropy

1.7± 0.8

1.7± 0.9

1.7± 0.9

2CV-STND

4.8± 3.0

5.8± 4.5

5.8± 4.1

16 ± 4

time of measurements ':. Feulgen, DAPI5, DAPI 5 fix,

15

± 4

16

± 4

Feulgen stain according to MIKEL et al. 1985 DAPI stain,S minutes, air-dried DAPI stain,S minutes, formalin-fixed time of measurements in minutes according to KAYSER et al. 1996a.

Table 15. Nuclear features of imprints taken from various intra-operatively analyzed adeno- and epidermoid carcinomas of the lung, and carcinomas of the thymus in relation to the staining procedure (DAPI stain for 5 minutes, air-dried; mean and standard deviation; area in square microns; time in minutes)". Feature

stem lines

adenocarcinoma (N = 19) 1.5± 0.6

epidermoid carcinoma (N=8)

thymus carcinoma (N =5)

2.3 ±0.9

2.7 ± 0.5

S-phase-related fraction

11

± 4.3

12

± 4.5

12

± 5.0

%>3C

63

± 16

64

±23

63

±38

%>5C

27

±20

28

±26

26

±13

nuclear area

65

±29

58

±27

53

±26

entropy

1.7± 0.8

1.7± 0.9

1.7± 0.9

2CV-STND

5.0± 8.0

5.5 ± 0.8

4.4 ± 0.9

time of measurements

16 ± 5

" according to KAYSER et al. 1996a. 2CV-STND: 2 CV standard deviation.

13

± 2

15

± 5

82 . K. Kayser and H.-J. Gabius

Table 16. Structural entropy and current of entropy (entropiefluss) grouped according to cell type and postsurgical tumor stages in bronchial carcinomas (N = 251, confidence limits, p > 0.95)';. Feature Feature

N

entropy

entropiefluss

59 56 48 32 46

8.4-11.7 9.0-12.3 9.2-12.6 5.2- 7.0 10.3-14.6

32-50 21-34 21-36 24-36 30-39

44 99 35 17

7.0-11.6 6.8-10.6 9.4-13.7 9.1-11.9

12-21 26-42 29-46 32-65

67 50 52 26

6.5-9.9 8.5-12.4 8.3-12.4 9.4-14.1

22-38 27-47 23-43 22-30

cell type (carcinoma)

- epidermoid -adeno -large cell - small cell - metastases pTstage

-pT 1 -pT2 -pT3 -pT4 pN stage

-pNO -pNl -pN2 -pN3 '; according to

KAYSER

et al. 1993b.

Graph theory and the entropy concept in histochemistry . 83

Table 17. Features of IOD and MST in potentially curatively operated lung carcinomas measured in tumor cells within the primary tumor and lymph node metastases (confidence limits, p > 0.95, distances in microns)". Feature

primary carcinoma (N = 195)

lymph node metastasis (N = 115)

IOD features 12 -14

8 -10

%>5C

6 - 8

17 -23

2CV-STND

8.5-10.7

3.1- 5.3

IOD entropy

2.2- 2.3

2.6- 2.8

S-phase-related fraction

MST features distance tumor - tumor cell

11 -12

14 -16

distance tumor cell- lymphocyte

10 -11

12 -14

distance proliferating tumor cells

37 -41

47 -59

distance tumor cells with an IOD > 5C

25 -28

37 -49

8.9-10.5

MST entropy

12.5-18.8

distance tumor - tumor cell: average length of edges between neighboring tumor cells, distance tumor cell-lymphocyte: average length of edges between tumor cell and nearest lymphocyte distance proliferating tumor cells: average length of edges between nearest proliferating tumor cells, distance tumor cells with an IOD > 5C: average length of edges between nearest tumor cells with anIOD>5C ;:. according to KAYSER et al. 1996a.

Table 18. MST features in relation to cell type in potentially curatively operated lung carcinoma (confidence limits, p > 0.95; distances in micronsY:-o Feature

epidermoid carcmoma (N = 59)

adenocarcinoma (N =56)

large cell anaplastic carcinoma (N =48)

distance tumor - tumor cell

11 -13

11 -13

12 -14

9 -11

distance tumor cell-lymphocyte

10 -12

10 -13

11 -13

10 -12

small cell anaplastic carcinoma (N = 32)

MSTentropy

7.9-12.6

7.8-12.7

8.4-13.6

5.9- 7.7

MST entropiefluss (current)

2.9- 4.3

2.7-4.1

3.1- 4.4

3.1- 4.4

;, according to

KAYSER

et al. 1994a.

84 . K. Kayser and H.-J. Gabius

Table 19. Histometric features in potentially curatively operated lung carcinomas grouped according to tumor cell type and non-involved lymph nodes (confidence limits, p > 0.95; length in microns; area in square microns)'f. lymph node (N = 115)

epidermoid carcinoma (N = 59)

adenocar- large cell small cell carcmoma anaplastic anaplastic (N =56) carcinoma carcinoma (N = 48) (N = 32)

cell area

40-41

157-161

243-253

249-260

85-88

ratio nucleus/cell

2.7-2.9

3.0-3.2

3.5-3.7

4.0-4.2

2.9-3.2

total length

307-323

27-129

7-45

16-60

111-321

length of edges

4.7-5.5

9.1-11.2

10.5-12.3

10.5-12.8

6.5-8.5

Feature

Voronoi

minimum spanning tree

':. according to

KAYSER

et al. 1993c.

Graph theory and the entropy concept in histochemistry . 85

Table 20. Cytometric and histometric findings in Askin tumors, small cell anaplastic lung carcinomas, and uninvolved lymph nodes (mean and standard deviation, distances in microns; area in square micronsY. Feature

caseO.A.

Askin (N =3)

small cell carcinoma (N =20)

lymph nodes (N =10)

nuclear area stem lines

18 ± 0.5 3 18

21 ±0.9 3-4

29 ±0.8 2-5

14 ±Oo4 1

18 ±5 2.8 ±Oo4

14 ±6 2.3 ± 0.5

0.5 ± 1.0

0.28 ± 0.13

0.37 ± 0.04

004 ± 0.02

0.24 ± 0.12 7.1 ±0.3 9.7 ±1.2

0046 ± 0.07 7.3 ±Oo4

0.5 ± 0.04 4.9 ± 0.1

6.1 ± 1.8

3.8 ± 0.9

S-phase-related fraction lOD entropy nucleus/cytoplasm ratio - Voronoi - Johnson Mehl distance tumor - tumor cell MST entropy

3.0

0.23 0.27

6.9±Oo4 10.6

1.5±Oo4

" according to KAYSER et al. 1996d. Voronoi: ratio of nuclear/cytoplasm area as calculated by the Voronoi cell. Johnson-Mehl: ratio of nuclear/cytoplasm area as calculated by the Johnson-Mehl cell. Note. The case O.A. fits best into the data obtained from rare Askin tumors (malignant neuroendocrine tumor of the chest wall in childhood). The clinical data (14 year old boy with chest pain, pleural effusion, and a solid mass in the left upper pleura and pulmonary apex) are in full aggreement with this diagnosis.

Table 21. Sequence of parameters associated with survival of potentially curatively operated lung carcinoma patients (in descending order of predictive powerY. 1. Cytometric parameters (S-phase-related tumor cell fraction, percentage of tumor cells with an lOD > 5C, number of stem lines; F = 15.2, P < 0.0001). 2. Histometric parameters (MST current of entropy, MST entropy, distance between proliferating tumor cells; F = 14.2, P < 0.0001). 3. Binding and presence of macrophage migration inhibitory factor (MlF) (ligandohistochemically analyzed by biotinylated MlF and sarcolectin; F = 13.5, P < 0.001). 4. Trisaccharide-binding capacity (ligand histochemically analyzed by carrier-immobilized blood group antigens A and H; F = 12.8, P < 0.01). 5. Clinical parameters (pN stage, pT stage, cell type; F = 1.9, P < 0.1). p: probability. F: statistical measure (censored non-hierarchic multivariant analysis). '; according to KAYSER et al. 1995a.

86 . K. Kayser and H.-J. Gabius

Table 22a. Expression of binding sites for histoblood group A trisaccharide (staining intensity with biotinylated, blood group A trisaccharide-exposing neoglycoconjugate) in lung cancer (N = 147, in percent of cellsr·. Cell type

Tumor cell area fraction (Aa)

Number of tumor cells displaying

(Aa, in percent)

no

weak

moderate

strong intensity

epidermoid (N = 30)

41

61

16

20

4

= 32) small cell (N = 27) large cell (N = 28) metastasis (N = 30)

40

69

12

18

o

41

81

5

13

42

62

14

23

41

63

11

21

adeno (N

4

Table 22b. Expression of binding sites for histoblood group H trisaccharide (staining intensity with biotinylated, blood group H trisaccharide-exposing neoglycoconjugate) in lung cancer (N = 147, in percent of cellsY. Cell type

Tumor cell area fraction (Aa)

Number of tumor cells displaying

(Aa, in percent)

no

weak

moderate strong intensity

epidermoid (N = 30)

41

63

18

15

= 32) small cell (N = 27) large cell (N = 28) metastasis (N = 30)

38

75

12

12

41

88

5

5

42

52

14

33

40

80

5

8

adeno (N

3 2 7

Graph theory and the entropy concept in histochemistry . 87

Table 22c. Expression of binding sites for macrophage migration inhibitory factor (MIF, staining intensity with biotinylated MIF, N = 147, in percent of cases)". Number of tumor cells displaying

Tumor cell area fraction (Aa)

Cell type

(Aa, in percent)

no

weak

moderate strong intensity

epidermoid (N = 30)

42

4

33

58

4

adeno (N = 32)

43

20

29

46

4

small cell (N = 27)

39

53

23

23

large cell (N = 28)

40

17

35

47

metastasis (N = 30)

43

27

30

41

2

Table 22d. Intracellular presence of macrophage migration inhibitory factor (staining intensity with biotinylated sarcolectin, N = 147, in percent of cases)". Tumor cell area fraction (Aa)

Cell type

Number of tumor cells displaying

(Aa, in percent)

no

weak

moderate strong intensity

epidermoid (N = 30)

43

14

32

46

7

adena (N = 32)

40

23

25

45

6

small cell (N = 27)

40

63

17

19

0

large cell (N = 28)

42

42

22

31

5

metastasis (N = 30)

38

17

26

49

7

':. according to

KAYSER

et al. 1995a.

88 ' K. Kayser and H.-]. Gabius

Table 23. Distances between tumor cells (centers of gravity) with various staining intensities in lung cancer, obtained with carrier-immobilized histoblood group A/H trisaccharides, labeled macrophage migration inhibitory factor, and sarcolectin (in microns, mean and standard deviation, N = 147)"'-, Staining intensity

Ligand none

weak

moderate

strong

HB-A-NEG -POS

38 ± 17 23 ± 11

37 ± 15

36 ± 10

15 ± 3

HB-H-NEG -POS

36 ± 14 26 ± 11

37± 15

36 ± 10

16 ± 3

MIF-NEG -POS

39 ± 12 25 ± 12

36 ± 15

36 ± 13

15 ± 3

SAR-NEG -POS

32 ± 14 22 ± 11

33 ± 13

35 ± 12

15 ± 5

':- according to

KAYSER

et al. 1995a.

Table 24. Expression of binding sites for histoblood group A (HB-A) and H (HB-H) trisaccharides, macrophage migration inhibitory factor (MIF), and sarcolectin (SAR) in lung cancer (in percentY:-. HB-A

Cell type

N

Epidermoid

30

Adeno Small Cell

32 27 28

44 47

30

30

Large Cell Metastasis ':- according to

KAYSER

HB-H

MIF

SAR

39

58

83

81

29

27

75

76

42 60 26

81 98

75 81

87

77

et al. 1995a.

Table 25. Methods used in the search for lectins':-. Tools

Parameter

multivalent glycans and (neo)glycoconjugates or defined cell populations

carbohydrate-dependent inhibition of lectinmediated glycan precipitation or cell agglutination

labelled (neo)glycoconjugates and - matrix-immobilized extract fractions or purified proteins

signal intensity

Graph theory and the entropy concept in histochemistry . 89 Table 25. Continued. Parameter

Tools

- cell populations - tissue sections - animal

labeling intensity staining intensity biodistribution of signal intensity

(neo)glycoconjugate-drug chimera and cell populations

cellular responses (cell viability etc.)

matrix-immobilized (neo)glycoconjugates and - cell populations - cell extracts

carbohydrate-inhibitable cell adhesion carbohydrate-elutable proteins

homology searches with - computer programs and knowledge of structural aspects of carbohydrate recognition domains - lectin motif-reactive antibody ':. according to

GAB IUS

homology score in sequence alignment or knowledge-based modeling extent of cross-reactivity

1997.

Table 26. Current categories for classification of various animallectins':·. Family

Structural Motif

Carbohydrate Ligand

Modular Arrangement

C-type

conserved CRD

variable (mannose, galactose, fucose, heparin tetrasaccharide)

yes

I-type

immunoglobulin-like CRD

variable (man6glcNAcz, HNK-1 epitope, hyaluronic acid, a2,3/a2,6-sialyllactose)

yes

galectins (S-type)

conserved CRD

~-galactosides

variable

pentraxins

pentameric subunit arrangement

4,6-cyclic acetal of ~-galactose, galactose, sulfated and phosphorylated monosaccharides

yes

P-type

homologous, not yet strictly defined CRD

mannose-6-phosphate-containing glycoproteins

yes

CRD: carbohydrate recognition domain. ':. according to GABIUS 1997.

90 . K. Kayser and H.-J. Gabius

Table 27. Members of the C-type family of mammalian lectins"". Name

Occurrence

Carbohydrate ligand

Structural peculiarity

asialoglycoprotein receptor

hepatocytes, testis and spermatozo:l

gal (avian homologue: glcNAc; potential alligator homologue: manlfuc)

type II transmembrane protein

asialoglycoprotein receptor

macrophages

gallgalNAc

type II transmembrane protein

Kupffer cell receptor

liver marcophages

galNAc, ~-fuc-containing neoglycoproteins

type II transmembrane protein

gp 120 receptor

placenta

man

type II transmembrane protem

low-affinity IgE receptor (CD23)

activated B cells and macrophages, eosinophils, platelets

gal

type II transmembrane protein

CD69

activated T and B cells, neutrophils, platelets, epidermal Langerhans cells

unknown

type II transmembrane protein

CD72

pan B cell

unknown (CDS glycopart?)

type II transmembrane protem

mast cell function-associated antigen

mast cells

unknown

type II transmembrane protein

NK cell lectin group (NKRPl,NKG2, Ly49, CD94)

NK cell, T cell subsets

mostly unknown (heparin tetrasaccharide; sulfated, fucosylated epitope)

type II transmembrane protein

proteoglycan core proteins or hyalectans (aggrecan, verSlcan, neurocan, brevican)

cartilage, fibroblasts, brain

gal, fuc, glcNAc (tenascin-R glycopart)

at C-terminal side variable number of EGF-like domains, the CRD, and complement-binding consensus repeats; central region with attachment sites for glycosaminoglycan chains, N-terminal hyaluronic acidbinding domain

Graph theory and the entropy concept in histochemistry . 91

Table 27. Continued. Name

Occurrence

Carbohydrate ligand

Structural peculiarity

selectins (L, P, E)

leukocytes (L), platelets (P), endothelial cells (E, P)

fucosylated, sialylated/sulfated epitopes

type I transmembrane proteins, two to nine complement-binding consensus repeats, an EGF-like domain and the distal CRD

mannose and phospholipaseA2 receptors

macrophages, retinal pigment epithelium, lymphoid dendritic cells, airway smooth muscle cells, kidney, corpora lutea; thymic epithelial cells (DEC-20S)

man

type I transmembrane proteins, fibronectin type II domain, eight or ten repeats of CRD

collectins (mannan-binding lectin; conglutinin, CL-43; surfactant proteins A and D)

collagen-like domain plasm~.,

liver

plasma alveolar surfactant and gastrointestinal mucosa

man, fuc, glcNAc gicNAc, man, fuc manNAc, fuc, mal mal, man, gal

tetranectin

plasma

unknown

plasminogen/kringle 4-binding domain

pancreatitisassociated protein

pancreas, ileum, jejunum, duodenum

unknown

signal peptide joined to CRD

pancreatic stone pancreas and thread (reg) proteins

unknown

signal peptide joined to CRD

HIP

lac

signal peptide joined to CRD

pancreas, small intestine, primary liver cancer

CRD: carbohydrate recognition domain. * according to GABIUS 1997.

92 . K. Kayser and H.-J. Gabius

Table 28. Members of the I-type subgroup of animallectins in the immunoglobulin superfamily"--. Name

Occurrence

Carbohydrate Ligand

Domain Structure

ICAM-1 (CD54)

endothelials cells, many activated cell types

hyaluronic acid

(C2h

PECAM-1 (CD31)

platelets, endothelial cells, myeloid and B lymphoid lineage cells

heparin

(C2)6

N-CAM

central and peripheral nervous system

oligomannosidic (and complex?) glycans, heparin

(C2)s

Po glycoprotein

peripheral nervous system

HNK-1 epitope

(V)\

myelin-associated glycoprotein

peripheral nervous system

Neu5Aca2-3gal

(C2)4 (V)\

sialoadhesin

macrophages in hemopoietic and secondary lymphoid tissues

Neu5Aca2-3gal

(C2)16 (V)I

CD22

mature B cells

Neu5Aca2-6gal

(C2)6 (V)l

CD33

myeloid progenitor cells, monocytes

Neu5Aca2-3gal

(C2)1 (V)l

'; according to

GABIUS

1997.

Graph theory and the entropy concept in histochemistry . 93

Table 29. Members of the galectin (S-type) family of mammalian lectins"'. Name

Occurrence

Structural Features

galectin-1 (galaptin, L-14)

many cell types

homodimer; one CRD per subunit (12-16 kDa): proto type

galectin-2

lower small intestine; clone from human hepatoma

homodimer; one CRD per subunit (43% sequence identity to galectin-1; 14kDa): proto type

galectin-3 (CBP35, Mac-2, IgE-binding protein, L-29, L-34)

many cell types

monomer with one CRD (oligomer formation in solution and on surfaces); pro-, tyr-, and gly-rich repeats in N-terminal section (29-37 kDa): chimera type

galectin-4

colon, small intestine, stomach, oral epithelium, esophagus

monomer with two partially homologous but distinct CRDs, connected by a linker region (36 kDa); proteolysis generates truncated proto type-like products; tandem-repeat type

galectin-5

blood cells

monomer with one CRD (17 kDa): proto type

galectin-6

small intestine, colon

tandem-repeat arrangement of two CRDs (33 kDa)

galectin-7

keratinocytes

one CRD (12.7 kDa): proto type

galectin-8

several tissues

homologous to galectin-4 and -6 (tandem-repeat arrangement of two CRDs with unique link peptide; 34 kDa)

Charcot-Leyden crystal protein

major autocrystallizing constituent of eosinophils and basophils

one CRD and lysophospholipase activity (16.5 kDa)

CRD: carbohydrate recognition domain. "-, according to GABIUS 1997.

94 . K. Kayser and H.-J. Gabius

Table 30. Cytometric features and binding of histoblood group trisaccharides in human lung carcinomas (N = 195; meany':-.

100 features

A+

A-

B+

B-

H+

H-

S-phase-related fraction

11.2

13.6"

12.3

13.1

11.1

13.7"

%>5C

7.4

8.5':'

7.2

8.1

6.9

8.4':'

2CY-STND

8.7

9.8"'-

9.5

9.3

8.6

9.8':-

entropy

2.2

2.3

2.2

2.2

2.1

2.3

14.4

12.9

13.5

13.4

13.6

13.4

entropiefluss (current of entropy)

A+, B+, H+: cases with expression of binding capacities for histoblood group A-, B-, H-trisaccharides. A-, B-, H-: cases without expression of binding capacities for histoblood group A-, B-, H-trisaccharides. 2CY-STND: 2CY standard deviation. * statistically significant (p < 0.05). ':'".- entropy according to STENKVIST and STRANDE 1990; KAYSER et al. 1994e.

Table 31. Histometric features obtained by syntactic structure analysis (minimum spanning tree) and binding of histoblood group trisaccharides in human lung carcinomas (N = 195; mean; distances in microns)"*. MST features

A+

A-

B+

B-

H+

H-

distance tumor - tumor

12

12

12

12

12

12

distance tumor -lymphocyte

12

11

11

11

12

11

distance of proliferating tumor cells

38

40"

39

39

38

40"

distance of tumor cells with an 100 >5C

28

25':-

26

26

28

26

entropy entropiefluss (current of entropy)

1.2 34

1.4 33

1.3 32

1.3 34

1.3 34

1.3 33

A+, B+, H+: cases with expression of binding capacities for histoblood group A-, B-, H-trisaccharides. A-, B-, H-: cases without expression of binding capacities for histoblood group A-, B-, H-trisaccharides. " statistically significant (p < 0.05). ':-" according to KAYSER et al. 1994e.

Graph theory and the entropy concept in histochemistry . 95

Table 32. Binding of synthetic hiostoblood group A-, B-, and H-trisaccharides in human lung carcinomas (histochemically positive cases, in percentY-o Feature

N

A(%)

B(%)

H(%)

patients

195

37

47

40

- men - women

151

37 39

49

39

43

46

44

cell type -

epidermoid carcinoma adenocarcinoma large cell anaplastic carcinoma small cell anaplastic carcinoma

51

37

58

51

56

29 47 44

42 47

27 60

38

31

35

46

40

39 38

54

37 41

38 32

blood group status

-A

79 34

-B

-H Y,-

according to

82 KAYSER

et al. 1994d.

43

96 . K. Kayser and H.-J. Gabius

Table 33. Features of integrated optical density (lOD) and syntactic structure analysis (MST) in relation to expression of binding sites for interleukin 2 (lL-2), Urtica dioica agglutinin (UDA), and histoblood group A- and H-trisaccharides (A, H) for prostatic adenocarcinomas (N = 20; mean, distances in micronsY-::·.

Probe

Feature lL-2-

lL-2+

S-phase-related fraction

8.8

7.8

8.7

7.4

8.2

8.3

8.0

8.4

lOD entropy

2.6

2.8

2.7

2.8

2.9

2.7

2.6

2.8

DNA-index

2.8

2.5

2.7

2.5

2.4

2.7

3.6

2.1 "-

% >5C

11.8

7.6':-

10.5

7.6

8.8

9.7

12.6

7.8':'

distance tumor - tumor

10.4

8.8':-

10.0

8.6':'

9.5

9.5

8.8

9.9

distance tumor -lymphocyte

10.0

8.6':-

10.0

7.9':'

9.1

9.2

8.3

9.8

distance of proliferating tumor cells

42.9

34.9':'

39.8

37.2

34.8

40.1

40.0

38.3

distance of tumor cells >5C

33.1

25.7""

28.2

30.4

24.0

30.7

25.1

31.F

MST entropy

64.6

33.8"

57.5

29.7':'

46.7

48.1

43.7

50.0

UDA- UDA+ A-

A+

H-

H+

lL-2+, UDA+, A+, H+: cases with histochemically detectable expression of corresponding binding sites. lL-2-, UDA-, A-, H-: cases without histochemically detectable expression of binding sites. ':- statistically significant (p < 0.05). ':-':- according to KAYSER et at. 1995c.

Graph theory and the entropy concept in histochemistry . 97

References BOVIN, N. V, GABIUS, H.-J.: Polymer-immobilized carbohydrate ligands: chemical tools for biochemistry and medical sciences. - Chern. Soc. Rev. 24, 413-421 (1995). COLEMAN, A. W.: Use of the fluochrome 4',6-diamidino-2-phenylindole in genetic and developmental studies of chloroplast DNA. - J. Cell BioI. 82, 299 (1979). COLEMAN, A. W., MAGUIRE, M.l, COLEMAN, J. R.: Mithramycin- and 4',6-diamidino-2-phenylindole (DAPl)-DNA staining for fluorescence microspectrophotometric measurements of DNA nuclei, plastids, and virus particles. - J. Histochem. Cytochem. 29, 959-968 (1981). COON, J. 5., WEINSTEIN, R. 5.: Blood group-related antigens as markers of malignant potential and heterogeneity in human carcinomas. - Hum. Pathol.17, 1089-1106 (1986). DABELSTEEN, E.: Cell surface carbohydrates as prognostic markers in human carcinomas. - J. Pathol. 179, 358-369 (1996). DANGUY, A., CAMBY, 1., SALMON, 1., KISS, R.: Modern glycohistochemistry: a major contribution to morphological investigations. - In: Glycosciences: Status and Perspectives (GABIUS, H.-J., GABIUS, 5., eds), pp. 547-562. - Chapman & Hall, London - Weinheim 1997. DANGUY, A., GABIUS, H.-J.: Lectins and neoglycoproteins - attractive molecules to study glycoconjugates and accessible sugar-binding sites of lower vertebrates histochemically. - In: Lectins and Glycobiology (GABIUS, H.-J., GABIUS, 5., eds), pp. 241-250. - Springer Verlag, Heidelberg - New York 1993. DANGUY, A., KAYSER, K., BOVIN, N. V, GABIUS, H.-J.: The relevance of neoglycoconjugates in histology and pathology. - Trends Glycosci. Glycotechnol. 7, 261-275 (1995). DANN, 0., BERGEN, G., DEMANT, E., VOLZ, G.: Trypanocide Diamidine des 2-Phenylbenzofurans, 2-Phenylindens und 2-Phenylindols. - Liebigs Ann. Chern. 61, 68-89 (1971). DE GROOT, S. R.: Thermodynamik irreversibler Prozesse. - BI Hochschultaschenbucher, Mannheim 1960. DE GROOT, S. R., MAZUR, P.: Grundlagen der Thermodynamik irreversibler Prozesse. - BI Hochschultaschenbucher, Mannheim 1969. DONALDO-JACINTO, 5., GABIUS, H.-J., KAYSER, K.: Do neoglycoprotein-binding capacities exhibit alterations during embryonic development of duck neurons, hepatocytes, myocardial cells and pneumocytes? - Electron. J. Pathol. Histol. 951-06 (1995). DRICKAMER, K.: Evolution of Ca 2+-dependent animallectins. - Progr. Nucl. Acid Res. Mol. BioI. 45,207-233 (1993). DUBE, V E.: The structural relationship of blood group-related oligosaccharides in human carcinoma to biological function: a perspective. - Cancer Metastasis Rev. 6, 541-557 (1987). DUSSERT, C., Kopp, E, GANDILHON, P., POUREAU-SCHNEIDER, N., RAsIGNI, M., PALMARI, J., RAsIGNI, G., MARTIN, P. M., LLEBARIA, A.: Toward a new approach in tumor cell heterogeneity studies using the concept of order. - Anal. Cell. Pathol. 1, 123-132 (1989). EHEMANN, V, MARKLE, C, KAYSER, K.: Diagnostic use of flow cytometry in pleural effusions. Proc. 9th Intern. Conf. Diag. Quant. Pathol., pp. 87 (1995). EINSTEIN, A.: The Meaning of Relativity. 5th Edition including the Relativistic Theory of Nonsymmetric Field. - Princeton University Press, Princeton 1955. ELBERT, T., RAy, W., KOWALIK, Z.J., SKINNER,J. E., GRAF, K. E., BIRBAUMER, N.: Chaos and physiology: deterministic chaos in excitable cell assemblies. - Physio!. Rev. 74, 1-45 (1994). GABIUS, H.-J.: Animallectins. - Eur. J. Biochem. 243, 543-576 (1997). GABIUS, H.-J., ANDRE,S., DANGUY, A., KAYSER, K., GABIUS, 5.: Detection and quantitation of carbohydrate-binding sites on cell surfaces and in tissue sections by neoglycoproteins. - Meth. Enzymol. 242,37-46 (1994).

98 . K. Kayser and H.-J. Gabius

GABIUS, H.-J., BARDOSI, A.: Neoglycoproteins as tools in glycohistochemistry. - Progr. Histochern. Cytochem. 22(3), 1-64 (1991). GABIUS, H.-J., GABIUS, S. (eds): Lectins and Glycobiology. - Springer Verlag, Heidelberg - New York 1993. - (eds): Glycosciences: Status and Perspectives. - Chapman & Hall, London - Weinheim 1997. GABIUS, H.-J., KAYSER, K.: Elucidation of similarities of sugar receptor (lectin) expression of human lung metastases from histogenetically different types of primary tumors. - Anticancer Res. 9, 1599-1604 (1989). GABIUS, S., KAYSER, K., BOVIN, N. YAMAZAKI, N., KOJIMA, S., KALTNER, H., GABIUS, H.-J.: Endogenous lectins and neoglycoconjugates: a sweet approach to tumor diagnosis and targeted drug delivery. - Eur. J. Pharm. Biopharm. 42, 250-261 (1996). GABIUS, H.-J., KAYSER, K., GAB IUS, S.: Protein - Zucker - Erkennung. - Naturwissenschaften 82, 533-543 (1995). GABIUS, H.-J., KOHNKE-GODT, B., LEICHSENRING, M., BARDOSI, A.: Heparin-binding lectin from human placenta as a tool for histochemical ligand localization and isolation. - J. Histochem. Cytochem.39, 1249-1256 (1991). GABlUS, S., SCHIRRMACHER, FRANZ, H., JOSHI, S. S., GABlUS, H.-J.: Analysis of cell surface sugar receptor expression by neoglycoenzyme binding and adhesion to plastic-immobilized neoglycoproteins for related weakly and strongly metastatic cell lines of murine tumor model systems. - Int. J. Cancer 46, 500-507 (1990). GARATT, G.: Blood group antigens as tumor markers, parasitic/viral/bacterial receptors, and their association with immunologically important proteins. - Immunol. Invest. 24, 213-232 (1995). GIBLIN, P.J.: Graphs, Surfaces and Homology. - Chapman & Hall, London 1977. GUNDERSEN, H.J. G., BAGGER, P., BENDTSEN, T. F., EVANS, S. M., KORBO, 1., MARCUSSEN, N., MOLLER, A., NIELSEN, K., NYENGAARD, J. R., PAKKENBERG, B., SORENSEN, F. B., VESTERBY, A., WEST, M.J.: The new stereological tools: dissector, fractionator, nucleator and point sampling intercepts and their use in pathological research and diagnosis. - APMIS 96, 857-881 (1988b). GUNDERSEN, H.J. G., BENDTSEN, T. F., KORBO, 1., MARCUSSEN, N., MOLLER, A., NIELSEN, K., NYENGAARD, J. R., PAKKENBERG, B., SORENSEN, F. B., VESTERBY, A., WEST, M. J.: Some new, simple and efficient stereological methods and their use in pathological research and diagnosis. - APMIS 96, 379-394 (1988a). HAMADA, S., FUJITA, S.: DAPI staining improved for quantitative cytofluometry. - Histochemistry 61, 1555-1562 (1988). HOFFGEN, H., KAYSER, K., SCHLEGEL, W., RUMMEL, H. H.: Low resolution structure analysis of corpus endometrium. - Path. Res. Pract.178, 131-134 (1983). KAYSER, K.: Neighborhood condition and application of syntactic structure analysis in histopathology. - Acta Stereol. 6, 373-384 (1987). - Analytical Lung Pathology. - Springer Verlag, Heidelberg - New York 1992. KAYSER, K., ANDRE, S., BOHM, G., DONALDO-JACINTO, S., FRITZ, P., KALTNER, H., KAYSER, G., KUNZE, W. P., NEHRLICH, A., ZENG, F.-Y., GABIUS, H.-J.: Developmental regulation of presence of binding sites for neoglycoproteins and endogenous lectins in various embryonic stages of human lung, liver, and heart. - Roux's Arch. Dev. BioI. 204,344-349 (1995b). KAYSER, K., BAUMGARTNER, K., GABIUS, H.-J.: Cytometry with DAPI-stained tumor imprints: a reliable tool for improved intraoperative analysis of lung neoplasms. - Anal. Quant. Cytol. Histol. 18, 115-120 (1996a). KAYSER, K., BERTHOLD, S., EICHHORN, S., KAYSER, c., ZIEHMS, S., GABIUS, H.-J.: Application of attributed graphs in diagnostic pathology. - Anal. Quant. Cytol. Histol. 18, 286-292 (1996d).

v.,

v.,

Graph theory and the entropy concept in histochemistry . 99

KAYSER, K., BOSSLET, K., EBERT, W., KIEFER, B.: Differential diagnosis of mesothelioma and metastatic adenocarcinoma of pleura by means of indirect immunoperoxidase technique combined with syntactic structure analysis. - Tumor Diagn. Ther. 8, 28-35 (1987a). KAYSER, K., BOVIN, N. KORCHAGINA, E. Y., ZEILINGER, c., ZENG, E- Y., GABIUS, H.-J.: Correlation of expression of binding sites for synthetic blood group A-, B-, and H-trisaccharides and for sarcolectin with survival of patients with bronchial carcinoma. - Eur. J. Cancer 30A, 653657 (1994d). KAYSER, K., BOVIN, N. ZENG, E- Y., ZEILINGER, c., GABIUS, H.-J.: Binding capacities to bloodgroup antigens A, Band H, DNA- and MST-measurements, and survival in bronchial carcinoma. - Radiol. Oncol. 28, 282-286 (1994e). KAYSER, K., BUBENZER, J., KAYSER, G., EICHHORN,S., ZEMLYANUKHINA, T. BOVIN, N. ANDRE,S., KOOPMANN, J., GABIUS, H.-J.: Expression of lectin-, interleukin-2- and histoblood group-binding sites in prostate cancer and its correlation with integrated optical density and syntactic structure analysis. - Anal. Quant. Cytol. Histol.17, 135-142 (1995c). KAYSER, K., CARL,S., GABIUS, H.-J., BAUER, H. G., SCHNEIDER, P.: Wie verandert adjuvante zytostatische Chemotherapie intrapulmonale Metastasen und tumorfreies Lungenparenchym? - Z. Herz-, Thorax-, Gefiilkhir. 8,79-84 (1994c). KAYSER, K., FITZER, M., BLZERBRUCK, H., BOSSLET, K., DRINGS, P.: TNM stage, immunohistology, syntactic structure analysis, and survival in patients with small cell anaplastic carcinoma of the lung. - J. Cancer Res. Clin. Oncol. 113, 473-480 (1987b). KAYSER, K., GABIUS, H.-J., CARL,S., BUBENZER, J.: Alterations in human lung after cytostatic therapy. - APMIS 99, 121-128 (1991a). KAYSER, K., GABIUS, H.-J., CIESIOLKA, T., EBERT, W., BACH,S.: Histopathologic evaluation of application of labeled neoglycoproteins in primary bronchus carcinoma. - Hum. Pathol. 20, 352-360 (1989b). KAYSER, K., GABIUS, H.-J., GABIUS, 5., HAGEMEYER, 0.: Analysis of tumor necrosis factor aspecific, lactose-specific, and mistletoe lectin-specific binding sites in human lung carcinomas by labelled ligands. - Virch. Arch. [PathoI. Anat.] 421,345-349 (1992a). KAYSER, K., GABIUS, H.-J., HAGEMEYER, 0.: Malignant teratoma of the lung with lymph node metastasis of the ectodermal compartment: a case report. - Anal. Cell. Pathol. 5, 31-37 (1993d). KAYSER, K., GABIUS, H.-J., RAHN, W., MARTIN, H., HAGEMEYER, 0.: Variations of binding of labelled tumor necrosis factor a, epidermal growth factor, ganglioside GMt> and Nacetylglucosamine, galactoside-specific mistletoe lectin and lectin-specific antibodies in mesothelioma and metastatic adenocarcinoma of the pleura. - Lung Cancer 8, 185-192 (1992b). KAYSER, K., HOFFGEN, H.: Pattern recognition in histopathology by orders of textures. - Med. Inform. 9, 55-59 (1984). KAYSER, K., JACINTO, S. D., BOHM, G., FRITZ, P., KUNZE, W. P., NEHRLICH, A., GABIUS, H.-J.: Application of computer-assisted morphometry to the analysis of prenatal development of human lung. - Anat. Histol. Embryol. 26, 135-139 (1997). KAYSER, K., KALTNER, H., DONG, X., KNAPP, M., SCHMETTOW, H.-K., VLASOVA, E. BOVIN, N. GABIUS, H.-J.: Prognostic relevance of detection of ligands for vertebrate galectins and a LewisY-specific monoclonal antibody: a prospective study of bronchial carcinoma patients treated surgically. - Int. J. Oncol. 9, 893-900 (1996c). KAYSER, K., KAYSER, G., BOVIN, N. GABIUS, H.-J.: Quantitative evaluation of ligandohistochemistry with cytoplasmic markers: program structure and application to lung carcinomas. Electron. J. Pathol. Histol. 954-02 (1995a). KAYSER, K., KAYSER, c., RAHN, W., BOVIN, N. GABIUS, H.-J.: Carcinoid tumors of the lung: immuno- and ligandohistochemistry, analysis of integrated optical density, syntactic structure

v.,

v.,

v.,

v.,

v.,

v.,

v.,

v.,

100 . K. Kayser and H.-J. Gabius

analysis, clinical data, and prognosis of patients treated surgically. - ]. Surg. Oncol. 63, 99-106 (1996b). KAYSER, K., KIEFER, B., BURKHARDT, H. U., SHAVER, M.: Syntactic structure analysis of bronchus carcinoma - first results. - Acta Stereol. 4, 249-253 (1985). KAYSER, K., KIEFER, B., MERKLE, N. M., VOLLHABER, H. H.: Strukturelle Bildanalyse als diagnostisches Hilfsmittel in der Histopathologie. - Tumor. Diagn. Ther. 7, 21-27 (1986b). KAYSER, K., KREMER, K., TACKE, M.: DNA- und MST-Entropie und EntropiefluB: neue Parameter zur biologischen Tumorcharakterisierung. - Zentralbl. Pathol. 139, 427-431 (1993a). - Integrated optical density and entropiefluss (current of entropy) in bronchial carcinoma. - In Vivo 7,387-392 (1993b). KAYSER, K., LIEWALD, F., KREMER, K., TACKE, M.: Integrated optical density (IOD), syntactic structure analysis, and survival in operated lung carcinoma patients. - Path. Res. Pract. 190, 1031-1038 (1994a). KAYSER, K., LIEWALD, F., KREMER, K., TACKE, M., STORCK, M., FABER, P., BONOMI, P.: Alteration of integrated optical density and intercellular structure after induction chemotherapy and survival in lung carcinoma patients treated surgically. - Anal. Quant. Cytol. Histol. 16, 18-24 (1994b). KAYSER, K., SANDAU, K., BOHM, G., KUNZE, K., PAUL,].: Analysis of soft tissue tumors by attributed minimum spanning tree. - Anal. Quant. Cytol. Histol. 13, 329-334 (1991b). KAYSER, K., SANDAU, K., PAUL, J., WEISSE, G.: An approach based on two-dimensional graph theory for structural cluster detection and its histopathological application. - ]. Microsc. 165, 281288 (1992c). KAYSER, K., SCHLEGEL, W.: Pattern recognition in histopathology: basic considerations. - Meth. Inform. Med. 21, 15-22 (1982). KAYSER, K., SCHMIDT, H., STUTE, H., BACH, S.: DNA-content, inflammatory tissue response and tumor size in human lung carcinoma. - Path. Res. Pract.185, 584-588 (1989c). KAYSER, K., SHAVER, M., MODLINGER, F., POSTL, K., MOYERS,].].: Neighborhood analysis of low magnification structures (glands) in healthy, adenomatous, and carcinomatous colon mucosa.Path. Res. Pract.181, 153-158 (1986a). KAYSER, K., STUTE, H.: Minimum spanning tree, Voronoi's tesselation and johnson-Mehl diagrams in human lung carcinoma. - Path. Res. Pract. 185, 729-734 (1989a). - DNA content and minimum spanning tree in primary and metastatic adenocarcinoma of the lung. - Acta Stereol. 8, 117-121 (1989b). KAYSER, K., STUTE, H., BUBENZER,j., PAUL,].: Combined morpho metrical and syntactic structure analysis as tools for histomorphological insight into human lung carcinoma growth. - Anal. Cell. Pathol. 2,167-178 (1990). KAYSER, K., STUTE, H., EBERT, W.: Analysis of branching points and minimum distance between neighboring cells in primary bronchus carcinoma. - Acta StereoI. 6, 207-212 (1987c). KAYSER, K., STUTE, H., NOETEL, M.: Simulation der histopathologischen Diagnostik mit Hilfe eines automatischen Bildanalysesystems (VISIAC). - Gegenbaurs morphol. jahrb.135, 19-24 (1989a). KAYSER, K., STUTE, H., TACKE, M.: Minimum spanning tree, integrated optical density and lymph node metastasis in bronchial carcinoma. - Anal. Cell. PathoI. 5, 225-234 (1993c). KING, M.-].: Blood group antigens as markers for normal differentiation and malignant change in human tissues. - Am.]. Clin. Pathol. 87, 129-139 (1987). KOHLER, G., BASSLER, R.: Ostrogcn-Rczeptor-Status in Mammakarzinomen. Ergebnisse vergleichender immunhistochemischer und biochemischer Untersuchungen. - Dtsch. Med. Wschr. 111,1954-1960 (1986). KOHNKE-GODT, B., GABIUS, H.-].: Heparin-binding lectin from human placenta: further characterization of ligand binding and structural properties and its relationship to histones and heparin-binding growth factors. - Biochemistry 30, 55-65 (1991).

Graph theory and the entropy concept in histochemistry' 101

KOLLES, H., DAuMAS-DuPORT, c.: A morphometric model for structure analysis of oligodendrogliomas. A study of 49 cases. - Electron. J. Pathol. Histol. 953-10 (1995). LAINE, R. A.: The information-storing potential of the sugar code. - In: Glycosciences: Status and Perspectives (GABIUS, H.-]., GABIUS, S., eds), pp. 1-14. - Chapman & Hall, London - Weinheim 1997. LANDAU, L. D., LIFSCHITZ, E. M.: Lehrbuch der theoretischen Physik. Vol. VIII. Elektrodynamik der Kontinua. - Akademie Verlag, Berlin 1967. LEE, Y. c., LEE, R. T. (eds): Neoglycoconjugates: Preparation and Applications. - Academic Press, San Diego 1994. LIBcHABER, A., MAURER,].: Une experience de Raleigh-Benard de geometrie reduite: multiplication, accrochage et demultipliation de frequences. - ]. Phys. (Paris) ColI. 41, C 3-51 (1980). LLOYD, K. 0.: Blood group antigens as markers for normal differentiation and malignant change in human tissues. - Am. ]. Clin. Pathol. 87, 129-139 (1987). Lu, S. Y., Fu, K. S.: A syntactic approach to texture analysis. - Compo Graphics Image Proc. 7, 303-330 (1978). MARCEPOIL, R., USSON, Y.: Methods for the study of cellular sociology: Voronoi diagrams and parametrization of the spatial relationships. - ]. Theor. BioI. 154, 359-369 (1992). MIKEL, U., FISHBEIN, W. N., BARH, G.: Some practical considerations in quantitative absorbance microspectrophotometry (Preparation techniques in DNA cytometry). - Anal. Quant. Cytol. Histol. 7,107-118 (1985). MURAMATSU, T.: Carbohydrate signals in metastasis and prognosis of human carcinomas. - Glycobiology 3,291-296 (1993). O'CALLAGHAN, ]. E: An alternative definition for neighborhood of a point. - IEEE Trans. Comput. 24,1121-1125 (1975). ONSAGER, L.: Reciprocal relations in irreversible processes. - Phys. Rev. 37,405-426 (1931). ORNTOFT, T. E, BEcH, E.: Circulating blood group-related carbohydrate antigens as tumor markers. - Glycoconjugate]. 12, 200-205 (1995). PEITGEN, H. D., SAUPE, D.: The Sciences of Fractal Images. - Springer Verlag, Heidelberg - New York 1988. PRIGOGINE, 1.: Etude Thermodynamique des Phenomenes Irn~versibles. - Dunod, Paris - Liege 1947. REMMELE, W., STEGNER, H. E.: Vorschlag zur einheitlichen Definition eines immunreaktiven Score (IRS) fur den immunhistochemischen bstrogenrezeptor-Nachweis (ER-ICA) im Mammakarzinomgewebe. - Pathologe 8, 138-140 (1987). RODENACKER, K., AUBELE, M., JUTTING, B., GAlS, P., BURGER, G., GOSSNER, W, OBERHOLZER, M.: Image cytometry in histological sections. - Acta Stereol. 11, 249-254 (1992). SANDAU, K.: Estimation of length and surface density using information about directions. - Acta Stereol. 8,95-100 (1989). SANFELIU, A., Fu, K. S., PREWITT,]. M. S.: An application of distance measure between graphs to the analysis of muscle tissue pattern recognition. - Saratoga Springs Meeting Proc. 86-89 (1981). SAUNDERS, P. T.: An Introduction to Catastrophe Theory. - Cambridge University Press, New York 1980. SCHENCK, u., JUTTING, u., RODENACKER, K.: Modelling, definition and applications of histogram features based on DNA values weighted by sine functions. - Anal. Quant. Cytol. Histol. 19, 443-452 (1997). SCHREVEL,]., GROSS, D., MONSIGNY, M.: Cytochemistry of cell glycoconjugates. - Progr. Histochern. Cytochem.14 (2), 1-269 (1981). SCHULTE, E., BaCKING, A.: Standardization of the Feulgen reaction including a quality assurance protocol for DNA image cytometry. - Electron.]. Pathol. Histol. 954-04 (1995).

102 . K. Kayser and H.-J. Gabius

SCHUSTER, H. G.: Deterministisches Chaos, eine Einfiihrung. - VCH Verlagsgesellschaft, Weinheim - New York 1994. SELL, S.: Cancer-associated carbohydrates identified by monoclonal antibodies. - Hum. Pathol. 21, 1003-1009 (1990). SERRA, J.: Image Analysis and Mathematical Morphology. - Academic Press, London 1982. SHANNON, C. T., WEAVER, W.: The Mathematical Theory of Communication. - University of Illinois Press, Urbana 1949. SINOWATZ, E, VOGLMAYR,J. K., GABIUS, H.-J., FRIESS, A. E.: Cytochemical analysis of mammalian sperm membranes. - Progr. Histochem. Cytochem. 19 (4), 1-74 (1989). STENKVIST, B., STRANDE, G.: Entropy as an algorithm for the statistical description of DNA cytometry data obtained from image analysis microscopy. - Anal. Cell. Pathol. 2, 159-166 (1990). STOYAN, D., STOYAN, H.: Fraktale - Formen - Punktfelder. - Akademie Verlag, Berlin 1992. TRINTSCHER, K. S. (ed.): Biologie und Information. - Teubner Verlagsgesellschaft, Leipzig 1967. VORONOI, G.: Nouvelles applications des parametres continus a la theorie des formes quadratiques, deuxiemes memoire: recherches sur les paralleIoedres primitifs. - J. Reine Angew. Math. 134, 188-287 (1902). WEIBEL, E. R.: Stereological Methods: Practical Methods for Biological Morphometry, Vol. 1. Academic Press, New York 1980. ZAHN, C. T.: Graph-theoretical methods for detecting and describing gestalt clusters. - IEEE Trans. Comput. C-20, 68-86 (1971).