Some perspectives on modeling leukemia

Mathematical Biosciences 150 (1998) 113±130

Some perspectives on modeling leukemia Evans K. Afenya a

a,*

, Daniel E. Bentil

b,1

Department of Mathematics, Elmhurst College, 190 Prospect Avenue, Elmhurst, IL 60126, USA b Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue, Burlington, VT 05401, USA Received 15 July 1996; received in revised form 28 January 1998

Abstract A di€usion model of leukemia is presented. The space-occupying e€ects of leukemic cells during leukemic expansion is investigated. The analyses and simulations of the model suggest that acute leukemia is a state in which positions inhabited by colonies of normal cells are invaded by emerging colonies of abnormal cells. Normal cells are then driven to a state of extinction as leukemic cells evolve toward high and dominant steady state levels. Ó 1998 Elsevier Science Inc. All rights reserved.

1. Introduction Leukemia can be described as the disorganization of the hematopoietic system in which a malignant clone of cells acts to impede the growth of normal hemopoietic tissue. In leukemia the malignant clone displays certain characteristics. These include the ability to expand at the expense of normal myeloid or lymphoid lines, and the ability to suppress or impair normal myeloid or lymphoid cell growth [1]. Leukemias are named and grouped according to the kind of hematopoietic cell that is primarily involved. They can be acute or chronic and may take intermediate and variable forms. Without treatment, even the chronic disorders may be fatal [1,2].

* Corresponding author. Tel.: +1-630 617 3572; fax: +1-630 617 3739; e-mail: [email protected]. 1 Tel.: +1-802 656 3832; fax: +1-802 656 2552; e-mail: [email protected].

0025-5564/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 5 - 5 5 6 4 ( 9 8 ) 1 0 0 0 5 - 6

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Leukemia is one of the twelve leading cancers among men and women in the United States [3]. In 1996, 27 600 new cases of leukemia were diagnosed in the United States [3] and out of this number 21,000 deaths were expected. According to the American Cancer Society, an estimated 28 300 cases of leukemia were diagnosed in 1997 with 21,310 deaths expected [4]. Five-year survival rates for acute myelogenous leukemia (AML), which is common among adults, currently stand at 11.4% [3]. These ®gures suggest that the ®ght against leukemia should be continued and should be waged on all fronts. It is against this background that this investigative activity assumes signi®cance. The pathophysiology of the acute leukemias is directly related to the impact of the expanding cell number [1]. In acute myeloblastic leukemia, for example, the process of blast di€erentiation into mature forms of neutrophil precursors is curtailed. This leads to a proliferation and accumulation of abnormal cells in the bone marrow and the blood. These abnormal cells may cause bone marrow failure and in®ltrate other organs of the body. They may also occupy and proliferate in the peripheral blood and predispose the individual to serious fatalities that include infection, pancytopenia, sepsis, weakness, fatigue, hemorrhage, and ®nally death. It is known that approximately 1±2 kg of acute leukemic cells, comprising about 1 to 2 ´ 1012 cells, are sucient to cause death. This cell number occupies a volume of about 1.7 l, which is about the total marrow volume of an average adult [1]. To date, there have been advances in the detection, treatment, and overall management of the leukemias. Improvements in the detection of minimal residual disease, for example, are due to polymerase chain reaction (PCR) techniques that can detect one malignant cell in 100 000 cells [1]. Also, advances in cytogenetics and molecular biology are elucidating the etiology of the leukemias. These advances o€er the hope that management of some of the acute leukemias may lead to improved therapeutic strategies. Despite these advances, however, instances where malignancy escapes detection over short and long periods of time are still common, and this is one of the challenges facing biomedical science. As Schrier [1] points out, the design of management programs for leukemia patients should involve attempts to arrive at de®nitive diagnosis by using clinical information, classical morphology, cytochemistry, cytogenetics, molecular biology, and cell surface marker analysis. It should then be determined whether it is feasible to attempt to cure the disease given the nature of the illness, the therapy required, and the age and general condition of the patient. These assertions underscore the importance of mathematical modeling of the leukemic processes. In some recent articles [5±7], we proposed and investigated models that could be used to describe acute leukemia (AL). We considered models in which leukemic and normal cells evolve only with respect to time. As a follow-up to those articles, we shall focus on studying the spatial spread of normal and leukemic cells in AL. By so doing, we hope to extend our conclusions from

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previous articles [5±7] to cover cases in which the spread of normal and leukemic cells a€ect the etiology and dynamics of the disease. 2. A di€usion-oriented model It has been postulated [8±11] that the origins of AL can be found in pluripotent stem cells. Therefore, it is suggestive that in the AL state a pluripotent stem cell in the bone marrow becomes malignant, proliferates and gives birth to leukemic blasts. These blasts also proliferate and, unlike in normal granulocytopoiesis, are unable to di€erentiate into mature forms of neutrophil precursors. These abnormal cells eventually enter the blood stream and the presence of large amounts of immature neutrophils in the blood exposes the human body to serious fatalities. This renders the immune system incapable of ®ghting disease or warding o€ di€erent types of bodily infections. Also, leukemic cells inhibit the colony-forming capabilities of the normal proliferative cells. Consequently, the body's hematopoietic system becomes disorganized. Proceeding from the postulates and medical information provided in Refs. [5±21], we make the following assumptions: 1. The populations of normal stem cells, myeloblasts, promyelocytes, myelocytes, metamyelocytes, band forms, and segmented neutrophils and normal neutrophils in the blood form a single normal cell population; and the leukemic stem cells and blasts form a single leukemic cell population. 2. Normal and leukemic cell populations follow a process of Gompertzian growth. 3. Leukemic cells exhibit a negative and inhibiting e€ect on the growth and development of the normal cells through interactive and other mechanisms. 4. Normal cells are destroyed due to inhibition from the leukemic cells, and also due to the natural causes of cell death (apoptosis). 5. Leukemic cells become inactive and die as a result of self-inhibition, and of natural causes. 6. The spread of normal and leukemic cells over a representative spatial section of the body, most importantly the marrow and the blood, follows a di€usive process. With particular regards to assumption 2, it is important to note that the body's homeostatic processes may be mediated and supported by normal cells. The physiological concept of homeostasis refers to the tendency to maintain a relatively constant internal milieu in the face of changing environmental conditions. Homeostasis may, therefore, involve the evolution towards a steady state [22] in which the production rate of normal cells may be high when the normal population is low and low when the population is large. Thus, an appropriate way of quantifying and describing normal cell behavior may be to assume that

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normal cells obey Gompertzian dynamics. Gompertzian growth of leukemic cells can be inferred from the work of Skipper and Perry [23] in their studies of a patient with relapsed leukemia. Assumption 6 follows from the notion that normal or leukemic cells display the tendencies of traveling to various sites of the human body. Based upon these assumptions, we let N(s, t) represent the population of normal cells and L(s, t) represent the population of leukemic cells at time t, and s (a vector) the body's space of distribution. The space of distribution may represent the geometrical sites within the body that may be occupied by normal as well as leukemic cells. The dynamics for the spread of acute leukemia may follow simple rates of change of cell populations, which are stated as follows: [Rate of change of the leukemic cell population] ˆ [Gompertzian growth of leukemic cells] ) [Leukemic cell loss] + [Leukemic cell di€usion]; [Rate of change of the normal cell population] ˆ [Gompertzian growth of normal cells] ) [Normal cell loss] ) [Normal cell interaction with leukemic cells] + [Normal cell di€usion]. In mathematical terms, we obtain the following system of equations:   A ÿ fL ‡ dl r2 L; g > 0; f > 0; …1† Lt ˆ gL loge L   An ÿ bN ÿ cNL ‡ dn r2 N ; a > 0; b > 0; c > 0 Nt ˆ aN loge …2† N with initial conditions N …s; 0† ˆ N0 ;

L…s; 0† ˆ L0 ;

…3†

where the parameters A and An are the asymptotic bounds on the leukemic and normal cell populations, respectively. Because leukemic activity is taking place in the body, which can be described as a closed environment [24], we impose zero ¯ux boundary conditions of the form …n
r†W ˆ 0;

…4†

where W ˆ (L, N) and n is the unit outward normal. The parameter a is the intrinsic growth rate or growth speed of the normal cells and b is their death rate. Here, c is a measure of the degree of inhibition of the normal cells with respect to the leukemic cells. The parameter g is the intrinsic growth rate or growth speed of the leukemic cells and f is their death rate. The di€usive coecients for the leukemic and normal cell populations are given by dl and dn , respectively. Eqs. (1)±(4) constitute the model system we shall use to describe the spatiotemporal and stationary dynamics of the spread of acute leukemia. We give an analytical appraisal of the model below.

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3. An analytical appraisal of the model To facilitate our analysis, we carry out an appropriate nondimensionalization of model system (1)±(4) with the view to obtaining various parameter relationships and reducing the number of parameters. We let L N a2 dn2 t g t ˆ ; s ˆ s; aˆ ; ; N ˆ ; A An dl dl f a b cA dn cˆ ; dˆ ; rˆ ; gˆ : bˆ ; dl dl dl dl dl The parameter g is crucial here; it is a measure of the di€usive ratio of the normal and leukemic cells. By dropping the asterisks for the sake of notational convenience, we obtain the following system of equations:   1 …5† ÿ bL ‡ r2 L; Lt ˆ aL loge L   1 Nt ˆ cN loge …6† ÿ dN ÿ rNL ‡ gr2 N N with L ˆ

L…s; 0† ˆ L0 ;

…7a†

N …s; 0† ˆ N0 ;

…7b†

…n
5†W ˆ 0;

…7c†

where W ˆ (L, N). When there is no spatial variation we obtain a system of ordinary di€erential equations of the form   dL 1 ˆ aL loge ÿ bL; …8† dt L   dN 1 ˆ cN loge ÿ dN ÿ rNL: …9† dt N By letting L…t† ˆ L and N …t† ˆ N in the uniform or homogeneous steady state situation and setting Eqs. (8) and (9) equal to zero, we obtain four equilibrium points, the only realistic ones of which are of the form
ÿ
ÿ L1 ; N1 ˆ eÿb=a ; 0 ; …10a†     ÿ
d ‡ reÿb=a : …10b† L2 ; N2 ˆ eÿb=a ; exp ÿ c Eqs. (8) and (9) can be transformed into a more tractable form by using the logarithmic transformations; x ˆ loge 1=L and y ˆ loge 1=N ; in which case we obtain

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dx ˆ ÿax ‡ b; dt

…11†

dy ˆ reÿx ÿ cy ‡ d; dt

…12†

with x…0† ˆ x0

and

y…0† ˆ y0 :

…13†

The general solution of the system (11)±(13) has the form   b b ÿat x …t † ˆ ‡ x 0 ÿ e ; a a 8 9 Zt Zt < = y …t† ˆ exp…ÿct† y0 ‡ d exp…cs† ds ‡ r exp ‰cs ÿ x…s†Š ds : : ; 0

…14†

…15†

0

From Eq. (14), we observe that as t ! 1; x…t† ! b=a: From Eq. (15), we note that for any 0
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et ˆ ÿae ‡ exx ; ÿ
jt ˆ ÿ d ‡ reÿb=a j ‡ gjxx ;

…16† …17†

ex …0† ˆ ex …s† ˆ jx …0† ˆ jx …s† ˆ 0:

…18†

This system produces solutions of the form e…x; t† ˆ Aeÿ…a‡k†t cos kx;   j…x; t† ˆ B exp ÿ …d ‡ reÿb=a ‡ gk†t cos kx;

…19† …20†

where A, B,  and k are constants. For steady state Eq. (10b), we now let  N …x; t† ˆ exp ÿ…d ‡ reÿb=a† =c ‡ j…x; t† with L(x, t) having the same form used for steady state Eq. (10a). Substituting these into Eqs. (5)±(7a), (7b) and (7c) and neglecting non-linear terms, we obtain et ˆ ÿae ‡ exx ;   d ‡ reÿb=a ‡ …d ÿ c†j ‡ gjxx ; jt ˆ ÿre exp ÿ c

…21† …22†

ex …0† ˆ ex …s† ˆ jx …0† ˆ jx …s† ˆ 0:

…23† ÿmt

This system admits solutions of the form e…x; t† ˆ Ce cos rx and j…x; t† ˆ Deÿmt cos rx where C, D, m, and r are unknowns to be determined. After substituting these into Eqs. (21) and (22), non-trivial solutions in which C ¹ 0 and D ¹ 0 will result so long as the following equation holds: m2 ÿ ‰a ‡ d ÿ c ÿ r2 …1 ÿ g†Šm ÿ …a ‡ r2 †…d ÿ c ÿ gr2 † ˆ 0:

…24†

Thus, the eigenvalues of system (21)±(23) are m1 ˆ ÿ…a ‡ r2 †;

…25a†

m2 ˆ d ÿ c ÿ gr2 :

…25b†

If the leukemic di€usive coecient is larger than that for the normal cell population and r2 is ®xed, then g is small. In this regard, the quantity gr2 in Eq. (25b) becomes negligible, and all the eigenvalues assume negative real parts if d < c. However, if d > c, then m2 > 0 and this eigenvalue will not have a negative real part. Now, by applying the boundary conditions in Eq. (23) to the solutions e…x; t† ˆ Ceÿmt cos rx and j…x; t† ˆ Deÿmt cos rx; we obtain r ˆ np/s, where n is a constant. If s is large and g has a ®xed value, then gr2 is negligible, and m1 < 0 and m2 < 0 if d < c. However, m1 < 0 and m2 > 0 if d > c. Thus, it can be observed from the solutions to system (21)±(23) that instabilities may arise when the eigenvalues, m1 and m2 , admit negative real parts. The analyses indicate that such phenomena would occur in association with the di€usive ratio of normal to leukemic cells or in association with the size of the space of distribution. Within this framework, it can be realized from Eqs. (19) and

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(20) that steady state Eq. (10a) would be stable to small space-time perturbations for all time. On another hand, analysis of Eqs. (25a) and (25b) suggest the possibility of having m1 < 0 and m2 < 0, notably over a large space of distribution. It is also possible to have m1 < 0 and m2 < 0 in the situation where the di€usive capabilities of the leukemic cells are far greater than those of the normal cells. This means that steady state Eq. (10b) would be unstable to small space-time perturbations. The situation in which steady state Eq. (10b) is stable in the homogeneous case, as is shown by the analysis of system (11)±(13), but unstable in the presence of spatial variations is an example of the phenomenon of di€usion driven instability, details of which can be found in Ref. [24]. This phenomenon occurs under the condition where the parameter g 6ˆ 1, which means dl ¹ dn from Eqs. (1) and (2). The stability analysis of the model system described by Eqs. (8) and (9) suggests that the normal and leukemic cells obey nonoscillatory dynamics. Therefore, to get a complete picture of the dynamics of the cell populations, we performed some numerical simulations. The parameter values which were used in this work were obtained from our recent articles [5±7] and inferred from other relevant papers [12±16,23]. These constituted a single set of constant values, and are given below: a ˆ 0:00333  0:05; b ˆ 0:0215  0:05; f ˆ 0:02  0:05; g ˆ 0:005  0:05; An ˆ 1:4  1012 ;

Al ˆ 3:0  1012 ;

c ˆ 5  10ÿ6 ;

dn ˆ 10ÿ6  …0:05  10ÿ3 †;

dl ˆ 10ÿ5  …0:05  10ÿ3 †: 4. Numerical simulations, results, and discussion A phase portrait of the normal and leukemic cells is depicted in Fig. 1. This is associated with model system Eqs. (5)±(7a), (7b) and (7c) without di€usion, which is described by ordinary di€erential Eqs. (8) and (9). Fig. 1 shows that as the leukemic population goes through a process of expansion, the normal population decreases and moves away from its carrying capacity. When leukemic cells are introduced into a population of hitherto normal cells, the normal cell population level decreases. This could be due to the inability of the immune system to deal with the abnormal situation because the leukemic cells may be actively participating in causing a malfunction of most of the organs engaged in normal granulopoiesis. We considered a situation in which ten leukemic cells were present when the normal cell population was close to its asymptotic bound, and carried out numerical simulations of model system (1)±(4) in one spatial dimension (with s ˆ x). Fig. 2(a) and (b) show the time evolution of the normal and leukemic cells in one spatial dimension. Here, the normal cell population decreases

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Fig. 1. Phase portrait of a rapidly expanding leukemic population against a diminishing normal cell population. When the normal cell population is close to its carrying capacity and leukemic cells are introduced, a malfunction of the neutrophil production system occurs which leads to normal cell decline. The abscissa represents the normal cell population and the ordinate represents the leukemic population, all multiplied by 1010 , respectively. The arrow indicates the direction of time.

monotonically to a diminished level while the leukemic population increases towards a dominant steady state. These simulations demonstrate the evolution of the model system towards steady state Eq. (10a), in which there are no normal cells. From a biological viewpoint, it may be conjectured that the abrupt rise in leukemic cell numbers to dangerous levels could be due to a sudden and abrupt loss of regulatory control by the body's control mechanisms. The sharp and explosive increase in the leukemic population could also be due to the phenomenon of contact inhibition (this is re¯ected in a large value for the parameter r). In this case, a process of interaction takes place between leukemic and normal cells, which may lead to interference and disruption in the process of normal cell production and an increase in leukemic cell numbers. Fig. 3(a)±(d) show the behavior of the normal cells at t ˆ 40, 80, 200, 500 time units in a circular domain with radial symmetry. Associated with this is the behavior of the leukemic cells at those same time instances in the same circular domain (see Fig. 4(a)±(d)). In the simulations that yielded Figs. 3 and 4, we considered the situation in which the cells were con®gured in colonies. It can be observed in Figs. 3 and 4 that a representative colony of normal cells goes through a process of shrinkage when leukemic cells are introduced. This leukemic colony goes through a process of rapid expansion towards a level that dominates the normal neutrophil production system in space and time. The

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Fig. 2. Evolution of normal and leukemic cells in time and one-dimensional space. In non-dimensional space and time the normal cell population decreases monotonically and moves away from its carrying capacity in (a), upon the introduction of a rapidly proliferating leukemic cell population which assumes dominance, as shown in (b). The normal and leukemic populations represented on the vertical axes in (a) and (b) respectively, are multiplied by 1010 .

shrinkage of the normal cell colony as time progresses, ®nds its expression in the development of a ¯at circular portion of the apical structure and a thinning of the top portion of the structure as we move from Fig. 3(a)±(d). In the

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Fig. 3. Depletion of a representative colony of normal cells in a circular domain upon emergence of leukemic cells. This is measured at time units: (a) t ˆ 40; (b) t ˆ 80; (c) t ˆ 200; and (d) t ˆ 500. The depletion of the normal cell colony, as time progresses, can be observed from the formation of a ¯at portion of the apical structure and a thinning of this structure.

simulations yielding Fig. 4, a small colony of leukemic cells is represented by a bowl-like structure in a circular domain. The growth of this colony, as the normal cell colony shrinks at the same instances of time, is demonstrated through the formation of a 'more pronounced bowl' with a ¯at or plateau-like structure around its top edge. The continued expansion of this colony can be observed from the circularly pronounced nature of the plateau-like structure and the thinning of the base of this structure (bottom part of the bowl) as we move from Fig. 4(a)±(d). The two-dimensional simulations were carried out by solving the model system in a rectangular domain with zero ¯ux boundary conditions (with s ˆ (x, y)). We employed the Peaceman±Rachford Alternating Direction Implicit algorithm to simulate the full model system at various instances in time. Here, we also considered cell colony con®gurations. It can be observed from Fig. 5(a)± (d) that when leukemic cells emerge in their colonies, they invade the existing normal cell colonies and cause a shrinkage of these colonies. From a biological perspective, it may be conjectured that through the space-occupying e€ects of the leukemic cells, the normal cells get displaced from subendosteal sites where hematopoiesis is preferentially resident [7]. These space-occupying e€ects may

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Fig. 4. Expansion of the emergent colony of leukemic cells. Associated with the diminishing behavior of the normal cells in Fig. 3 is the expansion of the emergent colony of leukemic cells in the same circular domain at the same instances of time: (a) t ˆ 40; (b) t ˆ 80; (c) t ˆ 200; and (d) t ˆ 500. The leukemic cells occupy and grow in the position of the normal cell colony thus causing a depletion of this colony of cells. Expansion of the leukemic colony can be observed from the formation of a ¯at circular edge at the top of the bowl-like structure. As time grows, the ¯at edge of the bowl becomes circularly pronounced and there is a thinning of the bottom portion of the bowl.

partly be a direct result of inhibition stemming from the expanding leukemic cell numbers. The rapid expansion of the leukemic population may also be due to nutrients available to the leukemic cells from the sites which were originally occupied by normal cells and from where they were displaced. Fig. 5(a) and (b) represent snapshots of respective contour and rectangular (surface) plots of the shrinking normal cell colonies as time becomes relatively large (a steady state situation). Fig. 5(c) and (d) represent snapshots of respective contour and rectangular plots of the expanding leukemic colonies, as time grows relatively large. These snapshots indicate that the normal and leukemic cells may be approaching steady state Eq. (10a), in which there are only leukemic cells and no normal cells. It is important to mention that the simulations were carried out by considering representative geometrical sites of the body that could be occupied by normal and leukemic cells. The bone marrow, for example, is said to be

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Fig. 5. Contour and rectangular plots of shrinking normal cell and expanding leukemic cell colonies. Upon emergence of leukemic cells, a process of invasion and occupation of positions inhabited by normal cells is initiated. As a result, the normal cells are displaced from regions of nutrient availability and this causes a shrinkage and extinction of normal cell colonies. (a) and (b) show the shrinking normal cell colonies and (c) and (d) show the leukemic colonies growing at the same sites that were originally inhabited by the normal cells. The vertical axes in (b) and (d) represent respective normal and leukemic populations multiplied by 1010 . The x- and y-axes are measured in nondimensional units.

surprisingly uniform [2] and the bone which encompasses it is tubular. From its organizational layout, this bone could be approximated by a cylindrical structure. Consequently, a cross-section of the bone marrow could be endowed with the properties of geometrical uniformity and could be approximated by a circular structure. Also, other areas of the body occupied by normal and leukemic cells could have nonuniform geometries and these could be approximated by two-dimensional geometrical shapes, the simplest of which could be rectangles. In Fig. 6 we demonstrate visualization plots for Fig. 5. Fig. 6(a) represents contour lines for leukemic cells that are 'projected' onto the surface plot for normal cells at the steady state. Similarly, contour lines representing normal

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Fig. 6. Visualization plots with axes not matched to scales. Contour lines representing leukemic cells are drawn on the surface plot for normal cells and vice versa. The colors represent di€erent contour levels. In (a), the tips of the inverted cone-like structures are inhabited by shrinking normal cell colonies and their middle and top portions are covered by emergent colonies of leukemic cells. In (b), the middle and top parts of the various hills are occupied by colonies of leukemic cells and their bottom portions are occupied by displaced and shrinking normal cell colonies. The vertical axes in (a) and (b) represent respective normal and leukemic populations multiplied by 1010 . The x- and y-axes are measured in non-dimensional units.

cells are projected onto the surface plot of leukemic cells in Fig. 6(b). In Fig. 6(a) the apexes of the inverted cone-like structures are inhabited by shrinking normal cell colonies and their middle and top portions are inhabited by expanding colonies of leukemic cells. The middle and top portions of the

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various hills in Fig. 6(b) are occupied by colonies of leukemic cells and the very bottom portions are occupied by the displaced and shrinking normal cell colonies. These superimposed surface plots and contours suggest the persistence of steady state Eq. (10a), which shows that the total space of distribution is ultimately dominated by the leukemic cells. They also reveal the spatial heterogeneities that persist when leukemic cells are present, with aggregates of these abnormal cells being the main occupants of the space of distribution. The behavior observed in the ®gures and model analysis and predictions indicate that the leukemic state could generate an aberrant process of disorganization, which aids in altering and negatively a€ecting the normal neutrophil production process. We note here that sensitivity analyses of the model parameters did not a€ect the predictions of the model. 5. Concluding remarks Biologically, the perturbations of the steady states described by Eqs. (10a) and (10b) with the trigonometric spatial dependence (this is re¯ected in the solutions to Eqs. (16)±(18) and Eqs. (21)±(23)) suggest that there may be signi®cant accumulations of cells at certain sites and depletions at other sites. Through such processes, the leukemic cells may accumulate at sites most suitable for normal cells. The normal cell population may therefore get depleted. Consequently, it is indicative that spatial heterogeneities may persist when leukemic cells, which may aggregate very rapidly via di€usive mechanisms, are introduced into the body. Ultimately, the possible breakdown of steady state Eq. (10b) and the persistence of steady state Eq. (10a) may be a situation in which aggregates of leukemic cells dominate the whole space of distribution and drive the normal cells to complete extinction. In summary, the analyses and simulations of the model imply the following: 1. The model predicts the existence of two realistic sets of steady states; one in which there is a coexistence of normal and leukemic cells and the other in which there are leukemic cells and no normal cells. 2. The steady state that represents a situation of coexistence between normal and leukemic cells breaks down in the face of small space-time perturbations while the steady state with only leukemic cells and no normal cells persists. The persistence of this particular steady state indicates that there is an ultimate situation in which normal cells are replaced by leukemic cells. 3. The breakdown in the steady state of coexistence may occur when there is a large leukemic di€usive coecient compared to a small normal di€usive coecient. It may also occur when the size of the space of distribution is relatively large. This may mean that in the leukemic state normal cell survival cannot occur unless this state is itself disrupted by an external agent (in this case, a drug) [25].

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4. There may be accumulations of cells at certain sites and depletions at other sites and through such processes leukemic cells may occupy sites of the normal cells as the propagation of spatial heterogeneities occur. As a result, it may be suggested that the positions occupied by the leukemic cells as they expand, may be very fertile areas that are rich in nutrients needed for hematopoiesis. This is because those positions used to be occupied by the displaced normal cells. Thus, leukemic cell numbers may increase very rapidly. 5. The rapid increase in leukemic cell numbers may also be tied to the phenomenon of contact inhibition through which normal cell production is disrupted and leukemic cell growth is stimulated possibly over a wide region of space. 6. Upon introduction of leukemic cells, existing normal cell colonies go through a process of shrinkage as their positions are invaded by emerging colonies of abnormal cells. The resulting leukemic dominance may cause damage to and disturb the colony-forming capabilities of the normal cells. Proceeding from the points enumerated above, it is appropriate to suggest that through certain di€usive processes and mechanisms, the normal cells are displaced from their positions by colonies of leukemic cells, over a wide region of space, and are driven to extinction. Essentially, the leukemic colonies display a tendency to invade the spaces 'designated' for normal cell growth. Also, over a region of space, the rapid increase in the leukemic population over a period of time may result in a high leukemic cell density. This could lead to a migration of the leukemic cells, possibly through a di€usive process (as we have described in this article), to regions of low cell density and nutrient availability. This could account for the reasons why other organs of the body become clogged with masses of abnormal cells, as is noted in Ref. [26]. It is important to mention that the predictions of the model may hold for some acute leukemias but not for the chronic leukemias in which there is a much gradual progression towards leukemic dominance. The model further suggests that cell behavior in the acute leukemias follows nonoscillatory dynamics but this may not be the case in chronic leukemias. A case in point is the oscillations observed in chronic myelogenous leukemia [22]. Thus, the model could be useful in studies involving chemotherapy and radiotherapy of acute leukemia, particularly when it comes to issues of optimal drug delivery during treatment. Acknowledgements This work was partially supported by a University of Vermont Grant (1996) to Daniel Bentil through which a travel grant was made available to Evans Afenya. Daniel Bentil was partially supported by National Science Foundation (NSF) grant number DMS 9696130. The authors are grateful to Professors F.K.A. Allotey, of the Ghana Institute of Mathematics, and C.P. Calder on,

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of the University of Illinois at Chicago, for many helpful discussions. The authors are also grateful to Dr W. Hartz, attending physician and consulting hematologist at Sherman Hospital in Elgin, Illinois, for providing useful information on leukemia. We would also like to express our gratitude to the anonymous referees whose comments and suggestions aided in the revision of this article.

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