Space, time, and space-time in craniofacial growth

American Journal of ORTHODONTICS Volume 77,

ORIGINAL

Number 6 June, 1900

ARTICLES

Space, time, and space-time in craniofacial growth Melvin L. Moss, Richard Skalak, Gautam Dasgupta, and Henning Vilmann New York, N. Y., and Copenhagen, Denmark Craniofacial growth is expressed as a patterned series of changes in size, shape, and location of cranial structure in space-time. The study of this growth is aided by the reduction of cranial structure to a series of points whose paths may be mathematical!\ modeled. The conventions of roentgenographic cephalometry are demonstrated to be capable only of demonstrating spatial changes occurring during jinite intervals of time and incapable of demonstrating point kinematics in space-time. The dtperences between anatomic and material points are emphasized. The location of any point is described by four coordinate values, three spatial and one temporal; hence, cranial growth occurs in a four-dimensional space-time. Models of space-time are depicted in which various projections of the point paths in space-time to space only are illustrated. Analyses of molluscan bivalve and of rat parietal bone growth suggest that the representation of time as an angular coordinate in a polar coordinate projection provides a growth model that closely simulates the biologic and physical realities of growth. It is noted that the point paths in this model are closely described b) logarithmic spirals, a mathematical description that has additional biologic connotations. Since the conventional methods of cranial growth analysis, explicit in roentgenographic cephalometry, do not directly represent a time dimension, an expunsion of these methods seems warranted. Key words: Growth, space-time, kinematics, logarithmic spiral, point paths

From the Departmentsof Anatomy, Civil Engineeringand EngineeringMechanics, and Bioengineer ing Institute, Columbia University, and Departmentof Anatomy, Royal Dental College, Copenhagen. This study was aided in part by Grant IROI DEO5145-01,National Institutes of Health. 0002-9416/80/060591+22$02.20/0

@ 1980 The C. V. Mosby

Co.

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592 Moxs et (11 The study of cranial growth consists of the identification, measurement, analysis, and interpretation of the changes in size, shape, and location of cranial skeletal structures occurring during finite intervals of time; the study of cephalic growth requires a similar. additional, study of the nonskeletal components of the head. Roentgenographic cephalometry (RC), is a generic term for a variety of conceptually and technically related methods frequently used to study mammalian cranial growth. ’ The data of RC are derived from the study of the changes in relative position, produced by growth, of identifiable cranial points. Recently the epistomologic bases of RC have been questioned with respect to the study of (1) change in cranial shape and (2) point transformation. It has been stated that RC does not provide a theoretical basis for study of either cranial structure shape or change in shape,2*3 and recently we suggested that RC is incapable of demonstrating the transformation of cranial points in space-time. 4 Since we believe that the latter topic is of significance to students of cranial growth, and since that paper” was concise, we here extend our description of craniofacial growth as a process occurring in space-time. Points Cranial structure, as visualized in a roentgenogram, can be reduced to a number of points whose relative positions are transformed during growth, a technique common in solid and fluid mechanics. There are two distinctly different types of point observed in cranial growth studies: 1. Material points. These are points made up of some physical material that can be identified and constantly tracked during growth. A material point may consist of a metallic implant. Alternatively, a small group of cells or a discrete region of bone whose location is made visible by dyes, chemicals, or radioactive isotopes may serve as a material point. The composition of the material point is constant with time, although its location may change. 2. Anaromic points. These are points capable of constant identification on the basis of position, boundaries, or functions. An anatomic point is identified as an invariant anatomic location. An anatomic point is not necessarily a material point, although it may be. The material composition of an anatomic point may change with time, although its anatomic location does not. Intravitally stained osseous material points,“-’ as well as metallic implants,‘-” are frequently used to track material point paths. Anatomic cranial points are of two varieties: at a specifically defined location OIZthe surface of a bone (gonion, menton, anterior nasal spine) and a specific location between two bones (nasion, bregma) or a point located within a synchondrosis. If growth takes place uniformly throughout a tissue, it may be expected that material and anatomic points will move together (as in interstitial growth). However, in appositional growth, by accretion on the outer edge of a tissue, anatomic and material points do not follow the same path. The usual RC graphic figures showing both material and anatomic point motions with time do not distinguish between the paths of these two types of point and hence lose some information. Roentgenographic cephalometric conventions Using either cross-sectional or longitudinal growth data, RC methods typically study temporally successive x-ray films by selecting one common fixed (registration) point and

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one common (orientation) line segment. All films, or their tracings, are then superimposed with the location of the common point (for example, sella point) and the direction of the line segment (for example, nasion-sella point) fixed in direction. These choices imply a choice of reference axes with origin at the fixed point, x axis in the chosen direction and y axis perpendicular to the x axis. The location of all other cranial points studied, at each “age” is displayed on a common two-dimensional projection plane, that is, relative to the chosen axes. The collection of cranial points so displayed is termed a growth cross section. The location of a point in space at a given time is determined by its coordinates relative to fixed axes (that is, to the frame of reference). A change in point location is a transformation of coordinates and may be produced by any combination of translation and rotation. The location of the same point at a later time can be expressed as a function of time and of its initial coordinates. A line, straight or curved, connecting successive locations of a given point is called a point path or trajectory. It is customary in RC to display point transformations due to growth and consequent changes in cranial size and shape on two-dimensional planes, although it is implicitly understood that growth occurs in a three-dimensional, Euclidean space. This space is defined by Cartesian coordinates, x, y, and z. The coordinate planes may be visualized as consisting of three mutually perpendicular planes, where, for example, x-y, y-z, and X-Z can represent sagittal, frontal (coronal), and horizontal planes, respectively. RC studies only the spatial changes produced by point transformations. These changes can be studied as alterations of distances or angles. Customarily, when the transformations of single points (for example, bregma, nasion, or gonion) are studied the lengths of the point paths between temporally successive locations are measured. When pairs of points (for example, nasion-sella point, anterior nasal spine-posterior nasal spine) are studied, it is possible to study the distances between successive pairs of point locations, as well as the changes in length (elongations) of the line segment. The origin (chosen fixed point) can be one of the pair. Finally, when the location of a collection of three points is considered, relative angular changes can be observed, in addition to the growth changes produced by the line segment elongation (for instance, sella point-nasion-anterior nasal spine). Again, the origin may be one of these points. At present, changes in cranial shape are studied by two principal methodsmathematical and graphic. The latter often involves some type of transformation of a two-dimensional grid.2, 15-21 In RC the location of any point is given by its spatial coordinates. The time intervals between films are recorded but are not usually intrinsically related to the point paths displayed on the common projection plane. Rather, time is considered as an independent variable and is used, with dimensional and angular growth data, to calculate and graphically illustrate curves of growth and rates and accelerations of growth, absolute and relative. The conventions of RC in using a two-dimensional common projection plane lead to other difficulties. It is recognized that conventional radiography uses a central projection and that all cranial points not projected by the central beam are inevitably distorted. A number of methods are available for correction. There is also the problem of bilateral point images projected to the sagittal plane (for example, gonion); here an “averaging” of location is customarily used. More serious is the impossibility, in any two-dimensional

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Fig. 1A. The growth behavior of an anatomic point (P,) in space-time and in space only during the period ~~-7, is illustrated. The growth cross sections (T, and Q) may be considered as representing sequential x-rays obtained by a convenUonal roentgenographic cephalometric technique in which the superimposition of the two films is equivalent to the view on the common (space) projection plane.

projection, of differentiating between and correcting for the differences between the projected and actual lengths of a line segment that is not parallel to the projection plane (for example, gonion-gnathion). All of these problems are related to the loss of coordinate value information for the third dimension. To recapture the three-dimensional image, it is customary to take a second series of films in a plane perpendicular to that of the first. Still, the mental synthesis of a three-dimensional object from the viewing of the images projected onto two perpendicular two-dimensional planes is not easily accomplished.22 The use of an isometric view of three-dimensional space on a two-dimensional plane may not help appreciably. These problems can be overcome by the use of computerized stereometry, where graphic display of the images of cranial points or of geometric constructions in three dimensions is possible, and these images can be presented as rotated in space. It is also possible to produce, by computers, pairs of stereo images for viewing with a stereoscope. However, these methods are equally incapable of demonstrating cranial point growth behavior in space-time since they are constrained by the conventions of RC to projections onto space.23-28

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Fig. 16. This figure indicates that in roentgenographic cephalometry only the spatial distance of point transformation and the time elapsed are considered, and that these methods are incapable of demonstrating the kinematics of the point in space-time.

Space, time, and space-time

The common-senseview of the nature of the real world is that spaceis naturally describedby the three-dimensionalstructureof Euclideangeometryand that, within this space,a materialparticle may be locatedby its Cartesiancoordinates.Time is commonly perceivedto havea different nature.Time is usuallyconsideredto consistof a sequenceof equalinstantsor intervalsof time, as measuredby somesuitableexternalevent suchas a clock, planetaryrevolution, stellar transit, or atomic disintegration,that form a temporal order. These intervals of time are usually regardedas existing independentlyof the existenceof any particlesin this space.Suchconceptsof spaceand of time are intuitively believedto be true.throughoutthe universe,and they form the basisfor classic, Newtonian mechanics.Theseconceptsappearto be naturalto the m ind, but this common-sense view doesnot necessarilycorrespondto the reality of the structureof spaceor of the natureof time. With respectto both time and space,alternativeviewpointsare possible,particularly from the standpointof relativistic physics. It is sufficient to recognizethat a numberof non-Euclideangeometriesprovidedifferent conceptsof the structureof space.8Similarly, the natureof time is not simple but is a problemof profoundimportancein philosophy, physics, cosmology, and biology.30-37The conventionsof RC are consonantwith the common-sense view of spaceand time. The view that space and time are independentattributes of nature is rejected by relativistic physics. Here it is consideredincorrect to separatespaceand time conceptually. Rather,it is believedthat spaceandtime aretwo aspectsof a single, unified manifold termed space-time.Neither spacenor time is viewed as independent;rather, they are combinedin a four-dimensionalcontinuumconsistingof one time and threespacedimensions34-36in which four coordinatesare required to locate an event in the physical world.3’, 38This combinationof spaceand time derivesfrom the fact that both are linked togetherin the study of the motion of physical particles (and of light).3sThe view that objective natureis naturally describedin space-timeis the currently acceptedviewpoint. In this view, spaceis a three-dimensionalcross sectionof space-time,and each space

596 Moss rt (il.

Fig. 2A. A typical shell (valve) of a bivalved mollusc is seen from above. A series of generating curves are seen, all of which are space curves except the plane curve indicated by the vertical line extending from the origin or umbo (0) and terminating at the location of that curve at the valve edge. This is the location of the anatomic point A and material point M, at time (TV).Two former valve edges are shown on the valve surface, and the former location of anatomic point A, is indicated by a dot.

Fig. 26. The same shell as viewed from the side. The logarithmic spiral shape of the generating curves is seen. Here the location of the anatomic point A, at time (Q) is shown. The sites of location of that point at the earlier time (7, and T*) are shown, and are the sites of the material points M, and Ma respectively.

cross section is a collection of simultaneous events. 4o Biologists express similar views. “A stage is an arbitrarily cut section through the time-axis of an organism. A stage is thus really an abstraction of the four-dimensional space-time phenomenon which a living organism is. “41 An appreciation of space-time is most easily approached by those accustomed to RC by considering the following physical model of a hypothetic construction. Assume that a standardized sagittal roentgenogram is made of a patient every day for a year, and it is desired to study the growth behavior of a single point (say, anterior nasal spine). Let a threaded needle pierce this point on every film, and let these films be stacked parallel, and in temporal sequence, so that all film planes are perpendicular to the horizontal surface of a table. When the needle is drawn through each film successively, the thread marks the point path (trajectory) of the anterior nasal spine, not in space but in space-time.42 That this model can represent space-time may seem strange at first, since space-time is defined as a four-dimensional structure. However, since it is impossible to model a four-dimensional structure graphically, by convention a two- or three-dimensional model

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Fig. 3. Three hypothetical stages of growth of a bivalved molluscan shell are shown at times r,. r2, and Q. The valves are viewed laterally, as in Fig. 28, with the umbonal axis at 0. The anatomic location at the junction of the plane curve (as in Fig. 2A) with the aperture plane is the constant site of the anatomic point A,. Since this point of the shell is formed by a specific amount of calcium carbonate, it is SISO a material point; at time 7, the location of A, (7,) is also the location of M, (7,). With growth, the increased volume of enclosed molluscan viscera results in a rotation of the valve about an umbonal axis and the accretion of more shell material; here at time 7*, the former aperture plane is the line connecting M, (Q) and 0, while the new aperture plane connects 0 with A, (Q), which is also the site of M2 (TV),and similarly for time Q. More precisely, three cross sections of space-time are shown for fixed time with one space dimension also fixed. The figure shows three successive stages of lamellibranch shell growth, with the aperture plane (y-0) viewed on edge. (Figs. 3 and 5 to 16, inclusive, are either republished or adapted from Moss et al. (1980) by permission.)

in which time is substituted for one of the spatial axes is used. In the common view, used in RC, time lacks the dimensionality of space. In space-time, however, time also has a metric; that is, it has dimensionality. Hence, the substitution of one coordinate of space by a coordinate of time is consistent. In the hypothetic multiple-film model, while the x and y values on the films are spatial, the z value is temporal, and the horizontal length of the stacked films is a time axis. A graphic model of another hypothetic situation will further clarify the distinction between space and time and space-time. Consider two growth cross sections, representing x-ray films taken at times r1 and r2. On each cross section we find a single point (PI) and, with growth, this point is transformed (Fig. 1). This model, again, has two spatial axes, x and y, and a time axis, T. Here we separate the two growth cross sections by a distance along the time axis related to the time interval between the two films. In RC the films are superimposed, this being geometrically equivalent to projecting the two different locations of point P1 onto a common two-dimensional space plane. In RC we measure only the spatial distance of point transformation and record the interval of time elapsed. The line representing this spatial distance is commonly presumed to represent the “growth” of a point. However, these conceptually independent space and time data need to be represented in space-time to describe fully the growth behavior of the point, since growth is a pattern of point transformations in both space and time. Fig. 1 illustrates the transformation occurring along a line connecting the two point locations, Pi (rl)-P, (r2), in space-time. The spatial distance between these space-time point locations is actually a projection of the point path in space-time to the space plane

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Fig. 4. In the hypothetical study shown in Fig. 3, if the shell had been studied by a roentgenographic cephalometric technique, with the point 0 fixed and the plane O-A fixed in direction (that is, the shells SO placed that the aperture plane was parallel to the films), the growth behavior of the anatomic and material points would appear as shown here.

only. This figure shows that in RC only two sides of a triangle are considered, while in a space-time representation the third side of this triangle, the point path in space-time, becomes distinctly apparent for the first time. Having considered these illustrations of space-time, it is possible to describe its structure more fully. Space-time coordinates are ordered sets of spatial locations and instants of time. Together, these sets constitute the location of an event, defined here as something occurring at a definite moment in time and at a definite location in space. An event can be considered, for present purposes, as the location of a point in space-time that requires specification by four coordinates-three spatial and one temporal. It is proper to define space-time as the set (or collection) of all possible locations of events in the universe-past, present, and future. Given a material particle that persists (that is, has extension in time), it is evident that it is not sufficient to describe this particle as a single point. Rather, this particle must be described as a line, drawn in space-time, passing precisely through the proper location of these events. This is the “world line” of the particle, and this line completely describes all location information about that point. This line is represented by the third side of the triangle shown in Fig. 1. Mathematical modeling suggests several representations of space-time, and it is the task of the biologist to suggest which of them best describes ontogenetic reality. Fig. I shows a model of a Newtonian space-time, and it is appropriate for use in craniofacial growth studies since we deal here with material particles whose velocity is relatively low compared to that of light. In the real world the local conditions encountered during craniofacial growth are such that Euclidean geometry and Newtonian mechanics are sufficient for our purposes. While other space-time models exist,2g. ~9 38,3g we need not consider them further here. Molluscan shell growth. The differences between the display of growth in space only and in space-time are advantageously shown by analysis of shell growth in bivalved molluscs (Lamellibranchia). Two points require clarification first. PSEUDO-TIME. The use of a time dimension in modeling of growth requires a scale of

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Fig. 5. A roentgenographic cephalometric method of studying the growth behavior of the anatomic point A, and material points M, and M2 for interval rJ-r, is seen from a lateral view. The aperture plane O-A is held fixed in direction, and the location of the umbo (0) is fixed. The point paths in space of these points are shown; note that they are not identical. This figure represents the projection of material and anatomic point paths from space-time to space only (see Fig. 6). Superimposition of the valve shapes in Fig. 3 would yield this figure.

values. It is not always the most convenient or revealing to use the absolute time (T). Fortunately, it is permissible to define a useful pseudo-time (7). Pseudo-time, in general, is a monotonic function of absolute (real) time. Pseudo-time is defined here as such that equal percentages in one or more growth dimensions occur in the same pseudo-time increments. This defines pseudo-time in general: a unit of pseudo-time is such that a given dimension increases the same fixed percentage of its current length in a unit of pseudotime. PHYSICAL MODELS. Two models that include a pseudo-time dimension are found useful. In the first, pseudo-time is represented as a linear distance along an axis perpendicular to the planes of growth cross sections (Fig. 1). In the second, pseudo-time is represented as an angular coordinate in a polar coordinate system (Fig. 7). The curved valve (shell) surface displays a series of curvilinear markings, seen also as grooves and ridges, that radiate from a common origin, the umbo, and terminate at the valve margin (edge). The valve surface also displays a second series of markings that intersect the radiating series and mark the locations of previous valve edges (Fig. 2). The molluscan valve grows in length only at its edge, and these intersections are indications of cessation of growth, that is, resting lines. It is customary to say that the valve surface displays a family of generating lines, radiating from a common origin, and a family of intersecting directing curves. The generating lines are closely described by logarithmic spirals. 16, 43-47 As a response to the volumetric growth of the enclosed viscera, the mollusc grows by rotation of its two valves about a transverse umbonal axis and accretion of new valve material at the valve edge. At any moment of time, the two valve edges meet at an aperture plane that includes the umbonal axis (Fig. 3).

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\Material point path in space time projected to space only Fig. 6. When the three growth cross sections superimposed to produce Fig. 5 are displayed, with pseudotime represented along a linear axis (as in Fig. lA), this figure is produced. The distinction between anatomic and material point paths in space only, as in conventional roentgenographic cephalometry, and in space-time are shown. At time 7, the cross-section contains only point A, (T,), M, (T,), and the shell associated with it. The additional points and shell increments shown on that cross section result from their projection in space-time onto that space plane.

Consider the hypothetic example of a growing bivalved mollusc in which it is possible to place a metallic implant in the valve at desired time intervals. An identical anatomic location, precisely at the junction of the valve edge and a given generating line, is selected and is the position of the anatomic point A,. With growth, at three different ages, this location is the position of anatomic points AI (T,), A, (TV), A, (TJ, respectively. Since at each anatomic point there is a discrete bit of calcium carbonate in which an implant can be placed, every anatomic point is, simultaneously, a material point; for example, in Fig. 3,

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Fig. 7. Shell cross sections are shown in space-time where the angular coordinate represents time. On each plane (7) constant, the two coordinates r and z are used to plot the shell section.

A, (TV)and M, (TV)coincide. As the valve rotates during growth, the material point moves away from the aperture plane, and all points on the same aperture plane also move; that is, the material plane of these points rotates about the umbonal axis as a rigid body. With valve growth, new material is added at the valve edge and a new material point is formed at the same anatomic location; for example, M2 (TJ is coincident with A1 (7.J. Let us consider how the transformations of Al, M1, and M2 would be studied by RC. Assuming that the generating line studied was a plane spiral (it did not curve in space) and that the aperture plane was designated as the x-y plane, it is conformable to the conventions of RC to place the aperture (x-y) plane parallel to the film plane. The mollusc could also be studied if the film plane were placed perpendicular to the aperture plane and designated as the y-z plane. The tracings of films taken at times 71, TV,TV would be stacked and viewed on a common two-dimensional projection plane. In the x-y plane all positions of Al, M1, and M2 would appear on a single straight line (Fig. 4). Viewed on the y-z plane, where y-o is an edge view of the aperture plane, the difference between anatomic and material point paths in space becomes apparent (Fig. 5). Figs. 6 and 7 are graphic representations of this same valve growth in space-time. In Fig. 6 pseudo-time is represented as a linear distance perpendicular to the two-dimensional plane, while pseudo-time is represented as an angular coordinate (0) in a polar coordinate system (Fig. 7). In Fig. 6 the growth cross sections at TV, 72, r3 are shown and, in addition, the

Fig. 8. Projection of Fig. 7 onto a space-time plane, where 7 is an angular coordinate and r is the distance from the umbonal axis 0. Note that with proper scale for T, the anatomic point path A, (7) reproduces the shape of the shell in this projection. A25

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Flg. 9. Curvilinear anatomic point paths are observed when eight rat growth cross sections (7 to 150 days) are superimposed in norma lateralis (the x-y plane). The location of point 8 and the direction of line segment 8-9 are fixed. (For point descriptions see Vilmann. 59r81)In subsequent figures point A, is labeled A, for convenience.

cross-section at T, serves also as a common projection plane, as in Fig. 1. The paths of both material and anatomic points in space-time are shown, as in their projection to the space plane only. Note that a point path in space is properly defined as a projection ofa point path in space-time onto the space plane. In Fig. 7, the three growth cross sections on which are located the line segments of the aperture planes, O-A, (r,), O-A, (r2), and O-A, (~a), are rotated with respect to each other. The angles of rotation are directly related to the pseudo-time intervals (~~-7~)and (~a+-~). The anatomic point path in space-time is the curved line connecting A1 (T,), A,

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Fig. 10. The constancy of anatomic point location and the displacement of previously formed material points which occupied that location is shown in a rat by intravital injection of alizarin red S at birth (M,, Q), at 10 days (I%, Q), and at 40 days (IVJ, TV).The displacement of the parietal bone is a passive growth response to the expansion of the brain, while new bone formation is an active growth process. Here, as in the remaining figures of cranial growth, the calvarial bones are only schematically indicated, and the sites and effects of other osseous depositions and resorptions on the size and shape of the bone are not shown. The anatomic and material points are shown here in norma verticalis (y-z plane). (After Massler, M., and Schour, I.: Anat. Rec. 110: 83-101, 1951.)

(TV), and Ai (~a). In Fig. 8 we show an edge view of Fig. 7, produced by a projection of Fig. 8 onto a two-dimensional space-time plane in which T is the angular coordinate and T is the distance of A, from the umbonal axis. Because the aperture planes rotate in space-time in a way similar to the growth cross sections, the anatomic point path A (T) now reproduces the morphology of the valve shape. Evidently, the biologic reality of valve growth is better revealed only when the anatomic point path in space-time is

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Fig. 11. In this figure, the rat parietal bone growth pictured in Fig. 10 is studied in norma lateralis (the x-y plane). The displacements of earlier formed material points from the anatomic location are shown schematically, as is the anatomic point path, A, (T), in space only. Fig. 12. The curvilinear anatomic point path, A, (r), shown in Fig. 11 appears straight when the sagittal plane is viewed on edge and studied in norma frontalis (y-z plane) by the techniques of roentgenographic cephalometry.

considered and shown to be identical to the curved generating line on the valve surface. In a biologic and physical sense, the shape and size of the mollusc valve provide a calcified history of valve growth in space-time. This history cannot be recreated by RC but only by an appropriate space-time representation of valve growth. It can be shown that a similar statement is true also for mammalian cranial growth. Rut cranial growth. In a roentgenographic cephalometric study of mammalian cranial growth, it is frequently observed that the point paths in space of a number of cranial points are curvilinear. Such paths are common in RC growth studies of (1) human and nonhuman primates,48-55 (2) dog,56 and (3) rat.57-63

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Fig. 13. The growth behavior of material and anatomic points in rat parietal bone are depicted in norma frontalis (as they were shown in norma lateralis in Fig. 11). The successive increments of active bone growth in the intervals Q-Q and ~~-7,are seen. The line connecting M, at times 7,, Q, 73 is the material point path in space only, and the differing anatomic point path, A, (T), in space only, is shown also. In this figure only a portion of the parietal bone is depicted; the more lateral portions of the bone are omitted. This figure schematically represents the expansion of the calvaria during growth; the distances shown are arbitrary.

In our studies longitudinal data permitted construction of mean tracings of twentyseven male rats at 7, 14, 21, 30, 40, 60, 90, and 150 postnatal days. These eight growth cross sections were superimposed with point 8 fixed and line segment 8-9 fixed in direction. The curvilinear paths of the other cranial points were noted (Fig. 9). Curvilinear paths were observed also when other points and lines were fixed. The specific shapes of curves depend on the particular point and line segment chosen for common fixation. From the foregoing discussion, it is evident that these curvilinear paths represent the projection to the space plane only of the paths in space-time of the several cranial points studied. A demonstration of these point paths is facilitated by study of the parietal bone of the rat. Massler and Schour64 (Fig. 2) show the areas of this bone intravitally stained at three ages (Fig. 10). The anteromedial tip of this bone is a convenient anatomic location to study. At each age, the stained bone at this location, at a particular time, is a distinct anatomic point, Ai. This point was labeled A, in Fig. 9 for convenience because more points were shown there. Fig. 10 shows, in norma verticalis, the location of material points Mi, Ms, Ma, at the time (ta) at which the figure is made. At time TV, M3 and Al are at the same location in time and space, and M2 and M1 formerly occupied the same location at times 72 and TV, respectively. With the RC methods used in Fig. 9, the locations of the anatomic point A1 on the mid-sagittal plane (x-y) at times TV, TV, and r3 are shown. The fixed point and fixed line

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Fig. 14. When the three growth cross sections superimposed to produce Fig. 13 are displayed with

pseudo-time represented along a linear axis (as in Figs. 1A and 6) this figure is produced. The distinction between anatomic and material point paths in space only and in space-time are shown. At 7, the cross section contains only point A, (T,), M, (r,), and the none associated with it. The additional points and none increments shown on that cross section result from their projection in space-time onto that space plane.

sequents (As and Ate in Fig. 9) are presumed to lie on the x axis of the three-dimensional sketch in Fig. 11. The anatomic point path in space of A, is shown by the curved line connecting the location of AI at rl, r2, and 7%.Viewing Fig. 11 along the y axis produces Fig. 12. The y-z plane shows the frontal view. Since the x-y plane is now viewed on edge, the anatomic point path in space of AI is a straight line on the y axis of this projection. The transformations of anatomic and material points, and of the parietal bone itself,

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-A,(q),

M,(T,)

Fig. 15. Rat cranial cross sections (depicted in Fig. 14) are shown in space-time in a manner similar to that used in Fig. 7. Here the O-z axis represents a transverse basicranial plane intersecting line segment 8-9 in Fig. 9.

shown in Fig. 11 result from the dynamics of neurocranial growth. It is understood that the calvarial bones exist within a neurocranial capsule whose inner layer is the dura mater and whose outer layer is the scalp. W ithin this capsule, the volume of the neural mass (brain and cerebrospinal fluid) increases with growth. This expansion causes the capsule to expand, passively moving the enclosed bones upward and outward. Concurrently, as the calvarial bones separate, active deposition occurs at the bone’s margins. In the interval T,-T~, these growth processesmove material point M, (TJ from the anatomic location A, (z-~)to position M1 (TV)and form a new material point M2 (7.J at the location of anatomic point A1 (TV). Continued growth, both passive and active, transforms the point M1 (TV) to position M, (TV), and the second material point Mz (TV) is transformed to position Mz (TV). In Fig. 11 the projection of the anterior portion of the parietal bone is shown extending to the left of A1 (TV) and the increments of bone added in the intervals ~~-7~ and ~~-7~ are indicated as the interrupted lines-for example, at the top of the figure extending between Ml (TV) and M2 (TV) and between M, (TV) and A, (TV), respectively. The addition of this graphic display of cranial growth dynamics to Fig. 12 produces Fig. 13. Here the passive and active growth of right parietal bone are perceived, as in an anteroposterior projection, on a common two-dimensional frontal (y-z) plane. The line joining M, (TV), M, (TV), and M1 (G-J is a material point path in space, as is the line connecting Mz (7.J and M2 (TV). The straight line joining A, (rl), A, (T*), and A, (TV) (Figs. 12 and 13) and the curved line connecting these same points in Fig. 11 are the same

(T,) , A,(r,), M&$1 ,c--M&Q j----M 13(T 1

--..-

/

Fig. 16. Projections of Fig. 15 onto a space plane, as in Fig. 8, produces this figure. Note that the anatomic point paths A, (t) resemble that observed in Fig. 8. This shows that the growth follows (approximately) a log spiral in space-time, as intuition suggests.

anatomic point path in space. Significantly, while the shape of the point path is related directly to the type of projection, the informational content of both lines is identical. The two space-time models used above are informative. With pseudo-time as a linear distance on an axis perpendicular to the y-z (frontal) plane, Fig. 14 is produced. Here the T, cross section also serves as a common projection plane, as in Fig. 6. The point paths of both material and anatomic points in space-time are shown, as is their projection to the space plane only. Note that the projection of cross sections ru and TVonto 7, effectively is equivalent to the conventional RC stacking of successive tracings of growth cross sections. It is now clear that the informational content of a point path in space-time is identical to that of the same point path in space only. However, it is impossible to recover all of this information with the use of only the point path in space; the kinematic behavior of the point in space-time is not directly revealed in RC projections. The representation of time as an angular coordinate in a polar coordinate projection produces Fig. 15. In this figure the r3 cross section is held vertical, and the locations of A, (TJ, A, (T*), and A1 (TJ are on the edge of each cross section, corresponding to both the y axis of Fig. 15 and to the similar line intersecting the three cross sections of Fig. 14. The dynamics underlying the kinematics of both the anatomic and material points shown here is identical to that discussed above. The anatomic point path in space-time A, (TV), Al (Td, A, (TV) and M2 (T-J, and M2 (73) are shown as curvilinear. Fig. 16 is an edge view of Fig. 14, and the anatomic point path in Fig. 15 is shown in a two-dimensional space-time. Consideration of Fig. 10 shows that this path, when considered in three-dimensional space, is a space curve and does not lie entirely in one line of space. This threedimensional space-time curve is shown in the polar coordinate representation of Fig. 16.

Spuce time and space-time in craniofacial growth

609

As noted above, the anatomic point path of A, (r) in Figs. 14 and 15 contain identical information; they are transformations of each other. Inspection of Figs. 8 and 16 shows that both anatomic point paths, in a polar coordinate representation, follow a logarithmic spiral in space-time, as intuition suggests. This is not to suggest that the rat neurocranium grows, as does the molluscan valve, by simple rotation about some basicranial axis analogous to the umbonal axis. Rather, these figures show that a fundamental attribute of rat anatomic point (and cranial) growth behavior, which is capable of description by a logarithmic spiral, can be demonstrated only by analysis of such growth in spacetime. This statement, capable of extension to biologic growth in general, has other significant implications. In particular, the properties of any biologic system whose growth is capable of display as a logarithmic spiral are capable of alternative descriptions by a series of similarity transformations, that is, other, interchangeable ways of expressing the informational content of a logarithmic spiral. Among these is that cranial growth is closely described by the allometric (relative) growth relationship.65a 66 This topic will be extensively reported elsewhere. We note here that a logarithmic spiral has been demonstrated to be a valid expression of human and nonhuman primate mandibular growth.67-70 The concept that biologic growth occurs in the four dimensions of space-time also permits a reconsideration of the nature of craniofacial growth patterns. Cranial points have intrinsic spatial and temporal attributes. Since time is a dimension providing coordinate values for point location, patterned cranial growth is best studied comprehensively with consideration of its intrinsic time dimension. Point paths in space-time offer a natural and comprehensive way to describe the change in skeletal size, shape, angulation, and position in any growth pattern. Both normal and abnormal patterns of cephalogenesis also reflect biologic and physical differences in point motions in space-time. While a considerable literature, in a variety of ways and in many disciplines, considers the “timing” of growth as significant,71 it has been difficult previously to exhibit time graphically.‘*, 73 The techniques demonstrated here may be useful in this regard. This approach is currently implemented by us in a number of mathematical models. Summary

Craniofacial growth is expressed as a patterned series of changes in size, shape, and location of cranial structure in space-time. The study of this growth is aided by the reduction of cranial structure to a series of points whose paths may be mathematically modeled. The conventions of roentgenographic cephalometry are demonstrated to be capable only of demonstration of spatial changes occurring during finite intervals of time and incapable of demonstration of point kinematics in space-time. The differences between anatomic and material points is emphasized. The location of any point is described by four coordinate values, three spatial and one temporal; hence, cranial growth occurs in a four-dimensional space-time. Models of space-time are depicted in which various projections of the point paths in space-time to space only are illustrated. Analysis of molluscan bivalve growth and of rat parietal bone growth suggests that the representation of time as an angular coordinate in a polar coordinate projection provides a growth model that closely simulates the biologic and physical realities of growth. It is noted that the point paths in this model are closely described by logarithmic spirals, a mathematical description that has additional biologic connotations. Since the conventional

610

Moss ct rrl

methods directly

of cranial represent

growth

analysis,

a time dimension,

explicit

in roentgenographic

an expansion

of these methods

cephaiometry,

do not

seems warranted.

REFERENCES 1. Krogman, W. M., and Sassouni, V.: A syllabus of roentgenographic cephalometry, Philadelphia, 1957, Philadelphia Center for Research in Child Growth. 2. Bookstein, F. L.: The measurement of biological shape and shape change. Lecture notes in Biomathematics No. 24, Berlin, 1978, Springer Verlag. 3. Moyers, R. E., and Bookstein, F. L.: The inappropriateness of conventional cephalometrics. AM. J. ORTHOD.75: 599-617. 1979. 4. MOSS,M. L., Vilmann, H., Dasgupta, G., and Skalak, R.: Craniofacial growth in space-time, Monograph 10, Craniofacial Growth Series, Ann Arbor, 1980, Center for Human Growth and Development, University of Michigan. 5. Cleall, J. F.: Growth of the palate and maxillary dental arch, J. Dent. Res. 53: 226.1234, 1974. 6. Cleall, J. F., Jacobson, S. H., Chebib, F. S., and Berker, S.: Growth of the craniofacial complex in the rat, AM. J. ORTHOD.60: 368-381, 1971. 7. Cleall, J. F., Wilson, G. W., and Gamett, D. S.: Normal craniofacial skeletal growth of the rat, Am. J. Anthropol. 29: 225-242, 1968. 8. BjGrk, A.: Facial growth in man. studies with the aid of metallic implants, ActaOdontol. Stand. 13: 9-34, 1955. 9. Bjiirk, A.: The use of metallic implants in the study of facial growth in children: Method and application, Am. J. Phys. Anthropol. 29: 243.254, 1968. 10. Bjark, A., and Skieller, V.: Postnatal growth and development of the maxillary complex, Monograph 6, Craniofacial Growth Series, Ann Arbor, 1976, Center for Human Growth and Development University of Michigan, pp. 61-99. 11. Skieller, V.: Growth of stimulation of the upper face in the case of mesial occlusion analyzed by the implant method, Tandlaegebladet 75: 1296- 1306, 197 1. 12. Melsen, B.: The postnatal growth of the cranial base in Mucucu rhesus analyzed by the implant method, Tandlaegebladet 75: 1320- 1329, 1971. 13. Samat, B. G.: Growth of bones as revealed by implant markers in animals. Am. J. Phys. Anthropol. 2% 255-286, 1968. 14. McNamara, J. A., Riolo, M. L., and Enlow, D. H.: Growth of the maxillary complex in the rhesus monkey (Mac~acamului~u), Am. J. Phys. Anthropol. 44: 15-26, 1976. 15. Huxley, J.: Problems of relative growth, London, 1932, Methuen & Co. 16. Thompson, D. W.: On growth and form, ed. 2, Cambridge, 1952, The University Press. 17. Moorrees, C. F. A., and Lebret, L.: The mesh diagram and cephalometrics, Angle Orthod. 32: 214-231, 1962. 18. Howells, W. W.: Craniometry and multivariate analysis, Papers Peabody MUS. Harvard Univ. 57: l-43, 1966. 19. Oxnard, C.: Form and pattern in human evolution, Chicago, 1973, University of Chicago Press. 20. Lestel, P. E.: Some problems in the assessmentof morphological size and shape differences. In Year Book of Physical Anthropology, Chicago, 1974, Year Book Medical Publishers, Inc., vol. 18, pp. 140-162. 21. Oyen, 0. J., and Walker, A. C.: Stereometric craniometry, Am. J. Phys. Anthropol. 46: 177-182, 1977. 22. Marr, D., and Nishihara, H. K.: Representation and recognition of the spatial organization of threedimensional shapes, Proc. R. Sot. Lond. [Biol.] 200: 269-294, 1978. 23. Walker, G. F., and Kowalski, C. J.: A two-dimensional coordinate model for the quantification, description, analysis, prediction and simulation of craniofacial growth, Growth 35: 191-211, 1971. 24. Selvik, G.: A roentgen stereophotogrammetric method for the study of the kinematics of the skeletal system, M.D. thesis, University of Lund, 1974. 25. Rune, B., Jacobsson, S., Samas, K.-V., and Selvik, G.: Roentgen stereophotogrammetry applied to the cleft maxilla of infants. I, Implant technique, Stand. J. Plast. Reconstr. Surg. 11: 131-137, 1977. 26. Rune, B., Jacobsson, S., Nilsson, M., Samas, K.-V., and Selvik, G.: Roentgen stereophotogrammetry applied to the cleft maxilla of infants. 2. Motion between segments, Stand. J. Plast. Reconstr. Surg. 11: 139-146, 1977.

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27. Rune, B., Selvik, Cl., Kreiborg, S., Samas, K.-V., and Kagstrom, E.: Motion of bones and volume changes in the neurocranium after craniectomy in Crizon’s disease, J. Neurosurg. 50: 494-98, 1979. 28. Macagno, E. R., Levinthal, C., and Sobel, I.: Three-dimensional computer reconstruction of neurons and neuronal assemblies, Annu. Rev. Biophys. Bioeng. 8:323-351, 1979. 29. Sklar, L.: Space, time and spacetime, Berkeley, 1977, University of California Press. 30. Bergson, H.: Creative evolution, New York, 1911, Henry Holt & Company. 31. Reichenbach, H.: The philosophy of space and time, New York, 1958, Dorr Publishers. 32. Fischer, R.: Interdisciplinary perspectives of time, Ann. N. Y. Acad. Sci. 138: 371-373, 1967. 3:3. Grunbaum, A.: Philosophical problems of space and time, ed. 2, Boston Studies in the Philosophy of Science, Dordrecht, 1973, D. Reidel Publishing Company, vol. 12, pp. 1-884. 34. Whitrow, G. J.: The natural philosophy of time, London, 1961, Thomas Nelson & Sons. 35. Whitrow, G. J.: The nature of time, Baltimore, 1975, Penguin Books, Inc. 36. Geroch, R.: General relativity from A to B, Chicago, 1978, University of Chicago Press. 37. Richardson, I. W., and Rosen, A.: Aging and the metrics of time, J. Theor. Biol. 79: 415423, 1979. 38. Zukav, G.: The dancing of the Wu Li masters, New York, 1979, William Morrow. 39. Davies, P. C. W.: Space and time in the modem universe, Cambridge, 1977, Cambridge University Press. 40. Rucker, R. V. B.: Geometry, relativity and the fourth dimension, New York, 1977, Dover Publications. 41. deBeer, G.: Embryos and ancestors, ed. 3, Cambridge, 1958, Cambridge University Press. 42. Gamow, G.: One two three. infinity, New York, 1961, Bantam Books, The Viking Press. 43. Mosely, H.: On the geometrical forms of turbinated and discord shells, Philos. Trans. R. Sot. Lond. 138: 351-370, 1838. 44. Lison, L.: Recherches sur la forme at la mechanique de developpement des coquilles des Lamellibranches, Konink Belg. Inst. Naturwetensch. Verhandl. Ser. 2134: l-87, 1949. 45. Owen, G.: The shell in Lamellibranchia, Q. J. Micr. Sci. 94: 57-70, 1953. 46. Rudwick, M. J. S.: The growth and form of brachiopod shells, Geol. Mag. 96: l-24, 1959. 47. Stasek, C. R.: Geometrical form and gnomonic growth in the bivalved mollusks, J. Morphol. 112: 215231, 1963. 48. Popovich, F., and Thompson, G. W.: Craniofacial templates for orthodontic case analysis, AM. J. ORTHOD. 71: 406-420. 1977. 49. Delattre, A., and Fenart, R.: L’hominisation du crane, Ed. Centre National de la Tech. Scient., 1960. 50. Delattre, A.. and Fenart, R.: Formule generale de contruction de la courbe endocranienne sagittale chez les mammiferes ses applications, Arch. d’Anat. Pathol. 8: 1 l-26, 1960. 5 1. Delattre, A.. Fenart, R., Darras, P., Becart, P., and Pelerin, P.: Position vestibulaire du point 2 de Klaatsch au tours de quelques ontogeneses craniennes comparees, Bull. Sot. Anthropol. 6: 392-442, 1964. 52. Cousin, R. P.: Etude en projections sagittale de cranes d’enfants orientes dans les axes vestibulares, Doctoral thesis, University of Paris, 1969. 53. Cousin, R. I’., Dardenne, J., and Fenart, R.: Analyse de l’angle sphenoidal humain relativement aux axes vestibulaires, Bull. Acad. Sot. Lorrain. Sci. 9: 334-342, 1970. 54. Chalanset, C.: Utilization de la methode des ellipses equiprobables dans l’etude de la variabilite de points crania-faciaux, chez l’enfant de race blanche, au tours de la croissance. Orientation vestibulaire sur des projections sagittales, doctoral thesis, University of Paris VI, 1973. 55. Fenart, R., and Deblock, R.: Ontogenetic course of the teeth and the tooth germs of anthropods, studied in the vestibular axis, J. Hum. Evol. 1: 73-77, 1971. 56. McKeown, M.: Craniofacial variability and its relationship to disharmony of the jaws and teeth, J. Anat. 119: 579-588, 1975. 5’7. Destombes, P.: Documents sur l’hydrocephalie provoquee chez le rat, Annex de la These, University of Lille, 1972. 58. Terk, B.: Modifications apportees chez le rat, a l’evolution des dents, par l’hydrocephalie experimentale: Etude en orientation vestibulaire, doctoral thesis, University of Paris VI, 1973. 59. Vilmann, H.: The growth of the parietal bone in the albino rat studied by roentgencephalometry and vital staining, Arch. Oral Biol. 13: 887-901, 1968. 60. Vilmann, H.: The growth of the cranial base in the albino rat revealed by roentgencephalometry, J. Zool. Lond. 159: 283-291, 1969. 61. Vilmann, H.: The growth of the cranial vault in the albino rat, Arch. Oral Biol. 17: 399.414, 1972.

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