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Mathematical description and analysis of cell cycle kinetics and the application to Ehrlich ascites tumor
3. theor. Biol. (1975) 50,437-459
Mathematical Description and Analysis of Cell Cycle Kinetics and the ApplicMion to Ehrlich Ascites Tubor M. KIM SchooI of Electrical Engineering, Cornell University, Zthaca, New York 14850, U.S.A. K. BAHRAMI Jet Propulsion Laboratory, Pasadena, California,
U.S.A.
K. B. Woo Division of Cancer Treatment, National Cancer Institute, National Institutes of Health, Bethesda, Maryland 20014, U.S.A. (Received 11 March 1974, and in revisedform
16 August 1974)
The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processesof cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell sire, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined. 1. Introduction
The cell cycle kinetics of various cell populakions has been a subject of extensive experimental studies under various environmental conditions (see 437
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Baserga, 1965; Cleaver, 1967). These studies have played an important role in elucidating growth of human cancer. In modeling the behavior of cancer cell populations, knowledge of various parameters associated with growth of the population is required. The labeled mitosis method using tritiated thymidine as a radioactive label (Cleaver, 1967) has been widely employed in experimentally studying the cell cycle kinetics. Theoretical techniques for determining some of these parameters from experimental results have been developed (see Lebowitz & Rubinow, 1969; Fried, 1970; Steel, 1968; Takahaski, 1968). Also Shackney (1973) studied theoretical aspects of proliferation characteristics of tumor cell populations in terms of various cell cycle kinetics parameters. Hahn (1966, 1970) and Stewart & Hahn (1971) theoretically discussed the time course behavior of the cell age distribution of tumor cell population. With advances in experimental techniques, such as a cell separation techniques based on sucrose gradient sedimentation for determining cell size distribution (Omine & Perry, 1973) and a microfluorometer technique for cell DNA distributions (Tobey & Crissman, 1973), there is potential for fast sampling of tumor cells to determine the cell cycle kinetics parameters in terms of cell size and cell DNA distributions in conjunction with radioautograph. This paper discusses a mathematical model of the cell cycle kinetics of cell populations to assist interpreting experimental data obtained with these techniques. The cell cycle kinetics of cell populations will be represented by nonlinear difference equations, and its characteristics will be studied via application to Ehrlich ascites tumor (EAT). 2. Mathematical Description of Dynamics ‘of Cell Population Systems
Processes of cell division, cell accumulation, and various types of cell loss contribute to the growth of cell population systems. Determination of the extent and the relative importance of each of these processes is complicated by the fact that some or all of these processes may depend on other cell cycle parameters of the cell populations system. In certain cell population systems, such as Ehrlich ascites tumor, the growth is fast; initially due to rapid cellular division of a large number of cells and that this rapid phase is followed by a phase of progressive decrease in the rate of growth toward a constant population. Retardation of the rate of growth may be attributed to several factors. One factor is a progressive decline in the number of the dividing cells and consequently a decline in the growth fraction. Another possible factor is the gradual prolongation of different phases of the cell cycle with the aging of the system. These factors make the behavior of the system to be represented more appropriately by nonlinear equations,
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MODEL
In the model used for the description of cell cycle kinetics, the total population is considered to consist of two groupslof cells, namely proliferating cells and nondividing cells, for which the state vectors are defined. The proliferating cells pass through all phases of the cell cycle (G,, S, G,, and M phases) and are capable of dividing into two daughter cells. The nondividing cells include cells that either temporarily or permanently have lost their capability to proliferate, dead cells that are awaiting clearance, and cells that may be in a resting state. During any specified length of time, the proliferating cells may die or become nondividing. Similarly, during any specified length of time, the nondividing cells may die or regain their proliferating ability. The schematic diagram of the cell cycle kinetics is illustrated in Fig. 1. For the model the mean generation time of the cell, denoted by TO, is divided into n equal intervals AT,, called biological unit time (or unit time for short. Since To may in general be a function of time, i.e., To = T,(t), then AT, is also a function oft, i.e., AT, = T,(t). Biological age of the tumor is denoted by T and is defined as T = kAT,. It is then defined that each of these intervals constitute a cell age compartment. All cells within the intetval of one specific age compartment are considered as one subpopulation and Proliferoiing
g&p
I
Division
----+---S----t----------+tc---z+M--i ----+---S---t----------+tc---z+M--i
!+------I
--,I
2
3
.
Dead cells
.
.
.
1
Resting or arrested ceils
S
c
I I I-
-
Nonproliferating
group -I
FIG. 1. The schematic diagram of the cell cycle kinetics of cell populations.
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are characterized by the mean cell age of that compartment. Then the discrete time model for the kinetics of cellular proliferation, is represented as follows : x(k+l)
= $,(k+l, k)x(k)+@,,(k+l, k)w(k) (1) w(k+ 1) = @,,(k+l, k)x(k)+Q,,(k+ 1, k)w(k) (2) where x(k) is the n-dimensional cell age state vector at time k for the proliferating cells and w(k) is the m-dimensional state vector for the nonproliferating cells. In x(k) the jth element denotes the number of cells in the jth cell subpopulation at time k. The state transition matrixes QpP, ap,,,, a,.,,, and = (Yij(k + 1, k)) where y,(k+ 1, k) is the probability that during the time AT, from time k to k+ 1 cells in the nonproliferating population regain their proliferating ability and enter the proliferating group. Note that ai,,,, and @)mpare not identical. It is assumed that when cells in the nonproliferating population enter the proliferating group they have the same age as when they left the proliferating group. The matrix Q,,(k+ 1, k) is defined to be QP,,(k + 1) k) = (Oij(k + 1, k)) where 8,j(k+ 1, k) is the probability that during the time period AT, from time k to k+ 1 cells in the .ith state of nonproliferating group move to the ith state of the nonproliferating group, e.g., the dormant GO phase (the .$h state) to the dead compartment (the ith state). At time k the aggregate of
CELL
proliferating
CYCLE
cells, denoted by Z,(k)
KXNBTIC
xXk).
(5)
cells, denoted by Z,,,(k) is
Z,(k) = f,, w,(k) and the total population,
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is
Z,(k) = jl The aggregate of nonproliferating
MODEL
(6)
denoted by Z(k) is Z(k) = Z,(k)+Z,(k).
(7) The cell size distribution of a cell population is a cell cycle kinetic parameter of considerable interest. Experimental techniques (e.g., automatic counter, hemocytometer, and sucrose sedimentation technique for cell separation) exist for determination of cell size distribution. Given information on the size distribution of cells of any given age, the discrete time model described by equations (1) and (2) can be extended to account for the cell size distribution. In this case, the rate of increase in average cell size of a cell should be known as the cell advances in cycle. Scherbaum & Rasch (1957) have assumed hypothetical curves for the growth in mean size of a single cell as it completes its movement through the cell cycle. Cell size compartments are defined as follows. Let the mean initial size of the cell, i.e., the mean size of the cell just after mitosis, be a,, and let the mean final size of the cell, i.e., the mean size of the cell just before dividing, be C,,. The interval i$- COis divided into rnr equal intervals with each of length (Z,- &)/ml. Each of these intervals constitutes a size compartment. The mean size of the cells in size compartment i is I+ = CO+ i(Ca-5Jm,. Some of the young cells may have a smaller size than i+, and some of the cells close to division may have a larger size than &. To accommodate such cells the size compartments are extended to the left of 6, and to the right of 0”. The criteria for defining the size state of a cell is its position in the size compartments. The size state vector y(k) is defined as Y= = b,(k), y,W),~. -3y,(k),. - ->YmWl where y’ is the transpose of y and v*(k) is the total number of cells in the ith cell size compartment. The size vector y(k) indicates the distribution of cells based on the size at the time instant k&It. Knowing the cell age state vector x(r), one can obtain the cell size state vector y(k) with the following assumptions. (1) A curve for the growth in the mean size of a single cell between successive age compartments in the curve shown in Fig. 2 (Scherbaum & Rasch, 1957) used in the present study. The curve shows a monotonic increase in the mean size of the cell as it advances through the cell cycle. The final size
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4
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8
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12
Age compartment
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I8
20
22
24
(TO ~-24)
FIG. 2. A curve (solid line) for the growth in the mean size of s single cell between successive age compartments. The dotted line is the diagonal reference line. (From Scherbaum & Rasch, 1957.)
of the cell is twice that of its initial size. This curve is chosen because this mean growth rate gives a good fit in simulation of the cell size distribution of EAT under exponential growth. However, there is a need for determination of the mean growth rate for different environmental conditions or different types of cell populations. (2) It is assumed that the size distribution cells in each age compartment is Gaussian. The mean value of the distribution is obtained from assumption (I), and the coefficient of variation is assumed to be the same foi all size compat tments and assumed to be known for the particular cell under consideration. Thus, knowing the cell age state vectors x(k) and w(k), one can obtain the cell size vector y(k) by a linear transformation YW = W)W + pnlow~) (8) where P(k) and P,,,(k) are the transformation matrices, and may be constant or a function of the time, depending upon environmental conditions. Experimental techniques, such as microfluoremetry (Tobey & Crissman, 1972) are available for the determination of DNA contents of individual cells in a large population. Thus, it is of interest to determine theoretically the cell DNA distribution of the population in conjunction with the cell age
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and cell size distributions. In determining the cell DNA distribution for a cell population the DNA state of a cell is defined in the same manner as the cell size state was defined. The initial DNA content of a cell at the beginning of the S-phase is normalized to 1. The DNA content of the cell at the end of the S-phase is 2. The span of DNA content during the S-phase, from 1 to 2, will be divided into r equal intervals, each of length AI = I/r. The first interval (0, 1) defines compartment 1. It corresponds to all cells in the Gi phase. Interval (1 +(i-2)Al, 1 +(i- 1)AZ) for i = 2,3,. . .r defined compartments 2 to r. Intervals (1 + (r - l)AZ, 1 + rAZ) and (2, co) define compartments r+ 1 and r+2, respectively. The latter compartments corresponds to cells in the Gz phase. The DNA state of a cell is defined as the cell DNA compartment, to which the specific cells belong, The DNA vector at time k is defined as, ZT = [GW, Z,(k), . . . , Z,(k), . . . , UWl where Zdk) is the number of cells being at DNA state i at time k. Elements of z(k) form the cell distribution according to DNA content. At any discrete time k a relationship between the cell age vector and the cell DNA vector exists, Z(k) = QW + Q,wW (9) where Q and QM are linear transformation matrices. The entries of matrix Q are obtained from the knowledge of the DNA synthesis during the S-phase. Also the entries of QM for nonproliferating cells are determined from the DNA content compartment at which cells are transferred from the proliferating group to the nonproliferating group. In the specific case of exponentially growing cell populations without cell death and dormant (resting) phase, only proliferating cells exist. Then the model is simplified and all the elements in transition matrices cP,,,U&,and Q,,,,,,, in equations (1) and (2) become zero. Only $(k+ 1, k) of a,, in equation (3) have nonzero elements. This case has been studied by Hahn (1966) and Kim, Bahrami & Woo (1973). In the general case, transfer of cells between the proliferating group and nonproliferating group occurs. Then these transition matrices contain parameters corresponding to the probabilities of certain transfer rates, such as cell death, dormant cells in the resting stage regaining proliferating capabilities, and cells in the proliferating group losing dividing ability. These parameters may be constant or become a function of certain cell cycle parameters, such as the total cell population. Certain aspects of transitions from proliferating to nonproliferating states have been studied by Shackney (1973) and Stewart & Hahn (1971). The model will be analyzed and studied via application to the cell cycle kinetics of Ehrlich ascites tumor (EAT).
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3. Characteristics
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of Ehrlicb Ascites Tumor
Investigation of the pattern of cellular proliferation of the EAT cells in the peritoneal cavity has shown that the growth of EAT cells is initially very rapid due to rapid cellular proliferation and that this rapid phase is followed by a phase of progressive deceleration toward an asymptote (Andersson & Agrell, 1972; Baserga, 1963; Lala, 1971, 1972; Lala & Patt, 1968 ; Patt & Straube, 1956). Lala & Patt (1968) have indicated that the growth of an inoculum of lo6 cells is initially approximately exponential, but subsequently after 3 and 4 days a decline in the growth rate appears and continues until the terminal stage of the tumor. The number of tumor cells in the terminal stage is of the order of 10’ cells. A similar result has been observed by Andersson and Agrell (1972) for the growth of an inoculum of 4 x lo6 cells. The rate of growth of the tumor is rapid at first but it decreases with aging of the tumor. The total number of free tumor cells approaches asymptotically to about l-6 x 10’ in the terminal stage of the tumor. Retardation of the rate of growth of EAT cells with increasing size of the tumor is attributed to several factors. One possible factor is the gradual prolongation of different phases of the cell cycle with the aging of the tumor (Lala & Patf, 1968), although on one occasion it is reported that no appreciable change in the length of the DNA synthesis period was observed (Baserga, 1963). Another important factor is a progressive decline in the number of the dividing tumor cells and consequently a decline in the growth fraction (Andersson & Agrell, 1972; Baserga, 1963 ; Lala & Patt, 1968) as evidenced by a progressive decline in the number of the tumor cells that synthesize DNA (Baserga, 1963), and also by an accumulation and build up of cells in late stages of growth of the tumor. Experimental cell size distribution for tumor of various ages obtained by Andersson & Agrell (1972) clearly indicates that there is a definite shift in number of cells of smaller size to cells of larger size with aging of the tumor. This is furthermore evidenced by an increase in the average size and average DNA content of tumor cells with the aging of tumor. Factors affecting the rate of growth of EAT cells also include the rate of and the mechanism of cell death and the rate of dead cell clearance. There is evidence that there is preferentially larger cell death from the nondividing cell population compared to a smaller rate of cell death from the dividing cell population (Lala, 1972). Undoubtedly the simultaneous occurrence of several processes in the growth of EAT cell population makes it quite difficult to isolate the relative importance and extend to each of these processes. Meanwhile, the experimental studies of the growth pattern of EAT and the observations about the rate of growth of the cell population, variations in the mean generation time
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445
of the cells, changes in the mean size, mean DNA and RNA contents of the cells and finally changes in the cell size distribution have provided valuable data for the determination of the importance of various cell cycle kinetics parameters in growth of EAT. The elements of the transition matrices and the transformation matrices in equations (l)-(4), (8) and (9) are to be chosen for EAT. 4 in equation (3) is deli&, +(k+ 1, k) = 6A, where the matrix 6 is the dispersion operator defined in terms of CI the probability that cells in an age compartment advance two compartments after AT, and p the probability that cells in an age compartment do not advance to the next compartment after AT,. The matrices 8 and A are defined in Hahn (1966). The mean generation time, To, is divided into (n=) 24 equal parts, each part defines one cell age compartment. c1is chosen to be 0.3. The nonproliferating group is divided into three subgroups, and the state vector for the nondividing cells, w is defined: w=
Wl w, [Iw3
where the entries 1 and 2 of w correspond to cells in the resting states and entry 3 of w corresponds to the dead cells. Since there is a strong indication that the proliferating cells in the latter part of the cell cycle may transfer to the resting states (Andersson BEAgrell, 1972), entries 1 and 2 of w are selected to correspond to resting cells that have come from proliferating age compartments 23 and 24 as shown in Fig. 1. It is assumed that cell death in the proliferating population is uniform. The probability that a cell from the proliferating group dies in AT0 is dl. The probability that a cell from either wi or w2 dies in AT, is d2. The probability that a cell in age compartments 23 and 24 loses proliferating capacity and transfers to the nonproliferating compartment 1 or 2 is F. Then the following transition matrices are obtained. l-d2 0
(24 x 24) P(k+
1) =
d <‘Id, \ 0
T.B.
0 l-d2
\ \
\
0 0
( \
dl+F ’ \ d,+F
1 (10)
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o...o 1 (3x24)
T(k+l)=
@,,(k+
F F+d,
0. ..O ; 0 ._------1 . . . II
(24 x 3)
K.
I
- 4 F+dl
B.
WOO
7 0 F F+d,
(12)
- 4 F+d,l
1, k) = 0
(13) where r is the clearance rate of dead cells in the nonproliferating compartment 3. dl and d2 are the probability of death for a proliferating cell and a dormant cell in the resting state, respectively, as defined previously. F is the probability that a proliferating cell becomes a dormant cell and transfer from the proliferating group into the nonproliferating group. 4. Dependence of Mean Generation Time of EAT Cells on the Age of the Tumor In this section attempts will be made to investigate the dependence of the mean generation time of EAT cells, T,, on the age of the tumor. It is of interest to know if it would be consistent with experimental data to assume that To remains invariant with aging of the tumor. On the other hand if To does not remain invariant, then what is the relationship between aging of the tumor and the change in To, and in what way is the change in To related to changes in cell production and cell death of EAT cells. Experimental EAT cell population growth curves and other data obtained by Andersson & Agrell (1972) will be used here for comparison with the results that will be obtained by adjusting the different parameters in the model. Experimental cell volume distributions obtained for EAT cells for tumors 6,9, 10, 11, 12 and 13 days old indicate that with aging of the tumor there is a progressive increase in the relative number of larger cells. The cell volume distribution for 6-day tumor has one peak whereas a second peak appears in the cell volume distribution for later days. Appearance of the second peak, which indicates the build-up of cells of larger size, could be interpreted as due to the resting cells. In other words some of the proliferating cells have left the proliferating group and have entered the nonproliferating group sometime after they have completed the S-phase (DNA synthesis phase) and prior to completion of the M-phase (mitosis). If, in fact, the second peak in the cell volume distribution of EAT cells is due to cells in the resting states, then the approximate growth fraction of the
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tumor cells may be calculated in the following fashion. A Gaussian distribution curve is fitted to the first peak and another to the second peak of EAT volume distribution. The area under the first Gaussian curve is assumed to represent the proliferating cells and the area under the second curve is assumed to represent the nondividing cells. The1 growth fraction denoted by f is defined as proliferating cells f= nonproliferating cells + proliferating cells’ Based on the above discussion, values off for the tumor at various ages are shown in Table 1. These values are based on experimental cell volume distribution of Andersson & Agrell (1972). TABLE
Growth
Age of tumor @wd : 10 11 12 13
Total
population (million cells) 600 1050 1150 1250 1350 1450
1
fraction of EAT cells Proliferating population (million cells) 486 819 828 712
Nonproliferating population (million cells) 114 231 322 538 702 841
Growth fraction Cfl O-81 0.78 0.72 o-57 O-48 0.42
In order to relate the initial population doubling time, Td, with the mean cell generation time, TO, certain assumptions are made. It is assumed that the initial size of the inoculum when the tumor is transplanted is 4 x lo6 cells and all these cells are proliferating cells. Age of the tumor denoted by t, is defined to be zero at the time of transplantation. It is assumed that the transplanted tumor is initially exponentially d@ibuted (for definition of exponentially distributed see one-day-old tumor in Fig. 4). It is furthermore assumed that the initial rate of cell death is negligible. With these assumptions, Td is related to TO in the following fashion. T,, log 2 Td =
n x log 2+log[l +a((2-‘/z”-21/z”)q ( ) k TO for II large, where acis the probability that cells in an age compartment compartments in AT,.
advance two
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Experimental data for growth of EAT with an initial size of 4 x IO6 cells indicate that the rate of growth of the population at the outset of tumor transplantation is very rapid, but subsequently it slows down (Andersson & Agrell, 1972). The doubling time of the population, Td, initially is about 8 hr. This indicates that the initial cell mean generation time of the proliferating cells is approximately 8 hr. If TO = 8 hr at I = 0 were to remain constant with aging of the tumor, that would be inconsistent with the much longer TO found by Baserga (1963) and Lala & Patt (1966) (see Table 2). TABLE
Duration
2
of different phases of EAT cells as reported in the literature
source Lala & Patt (1968) l-day Cday 7-day Baserga (1963)
To
TS
8 17 22 18
6 13 18 11
T G*+M
Td %I
Tca + M( %I
2 4 4 7
75 76.5 82 61
25 23.5 18 39
The growth fraction f for EAT population is close to 1 at the early age of the tumor, but with aging of the tumor it progressively declines (Table 1). For a 6-day old tumor f = 0.81 indicating that transition of cells from the proliferating group to the nonproliferating group up to day 6 has been small. Yet, the progressive decline in the rate of EAT population growth from day 0 to day 6 is quite pronounced (Andersson & Agrell, 1972). If this decline in the rate of growth is not due to transition of cells to the resting states, then it must be due to: (a) a prolongation of TO; and/or (b) occurrence of cell death. The procedure for determining the exact dependence of TO on age of the tumor or alternately on the size of the population, prior to day 6, is based on comparing the real time population growth with the biological time population growth. The following assumptions are made. (1) For t I 6 days transition of proliferating cells to resting states is small and can be neglected. (2) The probability of a proliferating cell dying in unit time AT, is d,(T). Probability d,(T) is the same for all proliferating cells. (3) The probability of a nonproliferating cell dying in unit time AT0 is d,(T). Probability dz(T) is the same for w1 and w2. (4) dl and d, are, in general, functions of 2, the total population. Motivation behind assumption 1 is that the growth fraction is close to 1 prior
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to day 6. Motivation behind assumptions 2 and 3 is that there is no indication of preferential age dependent cell death. Motivation behind assumption 4 is that the decrease in the growth fraction of EAT may be due to some inhibitor (Sama, 1973): Here it is assumed that the presence of the inhibitor is positively correlated with the total size of the population. In the model for EAT growth described by equations (1) and (2), quantities di and d2 are considered as parameters. Various equations for dl(Z) and d,(Z) are selected. Equations (1) and (2) are then solved for x(k) and w(k) by digital computer and Z(k) is obtained. Comparing Z(k) and Z(f) and utilizing the method described in Appendix A variation of the mean generation time TO with the total size of the tumor population is determined. A number of sets of the functions for d,, d,, and the rate of dead cell clearance were selected for simulation. In all cases the values of TO predicted increased monotonically with increasing size of the tumor, and were compared with the experimental values of TO obtained by L.ala & Patt (1968). The simulated results of the set of the functions which gave the best comparison with those reported by Lala & Patt (1968) are: l-day, the mean cell generation, TO, 8 hr (Lala & Patt, 1968) and 81.4hr (simulated); 4-day, T,, 17 hr (Lala & Patt, 1968) and 16 hr (simulated); 7-day, TO, 22 hr (Lala & Patt, 1968) and 24 hr (simulated). The relation between the mean generation time TO and the total population Z(h) is shown in Table 3. TABLE 3
Relation between the total population and mean generation time Total population
Z(h) (millions)
z-c4 4 kZ -c 32.4 32.4 IZ< 68.89 68.89 IZ< 117 11752-z looo
1oooIz
Mean
generation time TO(hr)
To = 8 TO = 0.52
log&Z)
-I- 7.68
To = 3 logm(Z) + 3.95 To = 11.13 log,,(Z) - 10.92 To = 17.71 log,,(Z) - 24.65 To =28-S
5. Resting Cells
In the previous section it was noted that prior to day 6 the retardation in the growth of EAT cell population is basically due to the prolongation of the ccl1 cycle duration of the proliferating cells and cell death. In this section transition of proliferating cells to the resting states is included and its effects on the growth of the population is studied. Appearance of the second peak in the experimental cell volume distribution with the aging of the tumor
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(Andersson & Agrell, 1972) indicates that with the aging of the tumor there is a transition of the proliferating cells to the nonproliferating compartment and an accumulation of cells in the resting states. In the NCTC2472 ascites tumor, an accumulation of cells with a G2 DNA content also seemed to occur with increasing time after transplantation (Frindel, Valleron, Vassort & Tubiana, 1969). In the JB-I ascites tumor, it was shown that with increasing tumor age, there are first an accumulation of non-cycling cells with G, DNA content and in later stages also an accumulation of noncycling cells with Gz DNA content (Dombernowsky, Bichel & Hartman, 1974). The existence of noncycling cells with G, DNA content have been demonstrated in the sarcoma 180 ascites tumor and the Ehrlich ELD ascites tumor (DeCosse 8c Grelfant, 1968). In order to incorporate the effect of the resting cells in the model the following assumptions are made. (1) Transition of the proliferating cells to the resting states (denoted by G,, M + wl, w2) occur in an age specific manner, namely from age compartments 23 and 24. The probability of transition of a proliferating cell in compartment 23 to the resting state wi (also compartment 24 + w2) in AT, is denoted by F. (2) F is a function of the total cell population, i.e., F = F(Z). (3) The mean generation time of the proliferating cells increases according to the relation in Table 3. Motivation behind assumption 1 comes from consideration of the second peak in the experimental volume distribution of EAT (Andersson & Agrell, 1972). Assumption 2 is based on the experimental data indicating that the decrease in growth fraction of EAT cells is associated with crowding mediated by some type of inhibitor (Sarna, 1973). The increase in To in assumption 3 is consistent with the experimental data (Lala & Patt, 1968). The procedure for simulation of EAT cell growth is as follows. Initially certain functions for d,(Z), d,(Z), F(Z) and r(Z) are assumed and are used in computer simulation of the model. From this simulation the growth of the total population Z(T), the proliferating population Z,(T), and the nonproliferating population Z,(T) as a function of the biological time are obtained. Since the relationship between T and t is determined from assumption 3, the real time growth of the total population Z(t), the proliferating population Z,(t), and the nonproliferating population Z,(t) are calculated. If, in fact, functions d,(Z), d*(Z), F(Z) and r(Z) are properly selected then Z(t) will agree with the experimental data of Andersson & Agrell (1972), and the growthf(t) will agree with the values of experimental growth fraction shown in Table 1, otherwise those functions will be changed in a reasonable way until results consistent with experimental data are achieved. Again a number
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of different sets of the functions for d,, d,, Fi and y are used for simulation. The set of the functions which gave a good comparison is: dl = 0+034(10g z)1*25 d 2 = 0.01 F = O-07~~)@s+“.03(&-)
The simulation
r= 1.0 results of the total population
0
2
4
6 Days
after
Z(t), the proliferating
8
IO
12
popula-
14
inoculohion
Fro. 3. Comparison of experimental data with the simulated growth of case 3. Z = total population, Zp = population in proliferating group, Z, ,= populationin nonprolifexating group, and the subscriptE indicatesexperimentalresuk (The experimentaldata from Andersson& Agrell, 1972.)
tion Z,(t), and the nonproliierating population Z,(t) are compared with the experimental results (Andersson & Agrell, 1972) in Fig. 3. Up to now parameters d,(Z), d,(Z) and F(Z) have been expressed in units of cells per ATO, but in experimental cases rate of cell death is expressed in cells per unit time (e.g., per hour). In order to obtain the rate of proliferating cell death d:(a, the rate of G2, M-arrested cell death d;(Z), and the rate of
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transition of proliferating cells to the resting states F*(Z), are all expressed in real time, the following relationships are used. d:(Z) = d,(Z) n cells/hr T,(Z) df(Z) = d,(Z) &)
cells/h1
0
F*(Z) = F(Z) n cells/hr (16) To(Z) where T,(Z) is the mean generation time of the proliferating cells expressed in hours at the time that the total population is Z. Using equations (14), (15) and (16) the function for T,(Z), d:(Z) and F*(Z) is calculated. From this result, it was seen that the rate of proliferating cell death for a tumor of 150 million cells is O-0165 cells@, and the rate of nonproliferating cell death is 0.0175 cells/hr. The overall rate of cell death is O-0165 cells/hr. Lala & Patt (1968) interpreting their experimental findings, have arrived at an overall cell death rate of 0.013 cells/hr for a similar tumor. For a tumor of 600 million cells, values for dy and df are 0.014 and 0.01 cells/hr respectively, and the rate of overall cell death is 0.014 cells/hr overall cell death rate reported for a similar tumor (Lala & Patt, 1968). It should be noted here that if function d, was selected such that d;(Z) had a much lower value for low values of Z, the growth of the population would have changed very slightly. This is due to the fact that for low Z the number G,, M-arrested cells is only a small percentage of the total population. 6. Cell Age Distribution
of EAT Population
As it was established earlier, the growth of EAT population is not exponential. It is expected then, that the cell age distribution of cells in this population change with aging of the tumor. In this section the simulated cell age distribution is presented. It should be noted that the age refers to the biological age. It was assumed that at the time of tumor transplantation (day 0) the cell age distribution was that of an exponentially growing culture. The results of simulation shown in Fig. 4 shows that with aging of the tumor, the cell age distribution of proliferating cells starts from exponential distribution and gradually approaches uniform distribution. The overall cell distribution of the population can be obtained by including the age distribution of G,, M-arrested cells. The latter appears as a peak in the latter part of cell age distribution.
CELL
Day
I
CYCLE
KINETIC
4
6
i
T.
0 Cell
9
3.2
0
age distribution
FIG. 4. Time course behavior of cell age distribution. M-arrested cells.
7. Cell Size Dlstrlbution
453
MODEL
-,
proliferating
cells; ---,
GP,
of EAT Cell Popdation
In this section time course behavior of cell size distribution of EAT cells is determined. Also the change in the mean size and mean volume of EAT cell population is studied. Cell size distribution is obtained by the method described in section 2. In this method, which is based on performing a linear transformation on the age vector x(k) and w(k), it is assumed that the mean cell size of a newborn cell is normalized to 1 and the mean cell size just before division is 2 and that the size distribution of cells of any particular age is Gaussian with a coefficient of variation of 11.5 %. The curve for size growth versus age compartment in Fig. 2 (Scherbaum & Rasch, 1957) is used which gives good comparison with the experimental data (Andersson & Agrell, 1972). The simulated cell size distributions obtained for cells in 1, 4, 6, 10, 11, 12 and 13-day old tumors are shown in Fig. 5. The experimental results (solid circle) are shown in the figure. As it can be seen from this figure, there is an increase in the relative number of larger cells with aging of the tumor as manifested by the appearance of the second peak in the size distribution with
454
M.
KIM,
K.
BAHRAMI
AND
K.
0I 2I 2 Ceil
B.
WOO
2
I
2
size (normalized)
FIG. 5. Time course behavior of simulated (solid line) and experimental (solid circle) cell size distribution of EAT population obtained by computer simulation. The experimental points (solid circle) are from Andersson & Agrell (1972).
ii 0’12 i 008 B cr’ 004 0
I
I
I
I
I
I
2
4
6
8
IO
I2
Days after
inoculation
FIG. 6. Simulated (solid line) and experimental (solid circle) graph of mean cell volume versus days after inoculation. The experimental points from Andersson & Agrell(l972).
CELL
CYCLE
KINETIC
455
MODEL
the aging of the tumor. The changes in the average volume of cells in EAT population with aging of the tumor are shown in Fig. 6. The simulated cell size distribution, shown in Fig. 5 compares favorably with the experimental results, However, the experimenta values of days 12 and 13 at the larger cell size part are higher than the simulated values. This may be due to the fact that dormant cells in the resting state’may continue to grow in size while the model assumes that the size of dormant cells remain constant. 8. Ceii DNA Distribution
of EAT
In this section the time course behavior of cell DNA distribution and also the change in DNA content per cell of EAT population are presented. The method for obtaining cell DNA distribution is based on performing a linear transformation on age vectors x(k) and w(k) discussed in section 2. Three different curves for DNA content of a cell according to its age are assumed (Fig. 7). Duration of S-phase is considered to last 314 of the generation time.
2.0 a ‘Z::
I.8 -
I
k-ssG~,+M+ Age comportment FIG.
7. Increase in average per cell DNA content for cells in EAT population.
For each of these assumptions simulated (predicted) cell DNA distributions are obtained. Cell DNA distributions for the rate of DNA synthesis that is large close to the end of the S-phase (assumption 3) are shown in Fig. 9. For each of these three assumptions, the variation of mean DNA content/cell with the aging of the tumor is calculated (Fig. 8). In all cases, the mean DNA content/cell increases monotonically with aging of the tumor. Figure 8 indicates that the results obtained with assumption 3 are close to the experimental observations.
456
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
2.0 .
Assumptions
I ond 2
8 !i 4 IO
0
I 2
I
I
I
I
I
4
6
8
IO
12
Days
after
inoculation
FIG. 8. Increase in averageper cell DNA content for cells in EAT population. The simulated curves are solid line. The experimental points are solid circles from Andersson & Agrell (1972).
Assun,pt
ion
3
2
I DNA
FIG.
2 content
I
21
21
21
2
(normalized)
9. Cell DNA distribution for Assumption 3.
9. Discussion
From the computer simulation of the model it may be concluded that the retardation in the rate of growth of EAT population is due to a combination of several factors. In the early stages of tumor growth, when the cell population has not reached 600 million cells, the primary reason for retardation of the rate of growth is the prolongation of the mean generation time of the proliferating cells. The prolongation of the mean generation time of the
CELL
CYCLE
KINETIC
MODEL
457
proliferating cells with increasing population is slow at the beginning of the tumor transplantation. This explains the exponential behavior of the population growth- at the time of tumor transplantation. The mitotic cycle of 8.4 hr is predicted for this stage of growth. Subsequently, at a tumor population of about 200 million cells the increase In the mean generation time of the proliferating cells becomes rapid. This corresponds to a pronounced decrease in the rate of tumor growth that occurs in experimental tumors at this stage of growth. On the whole the mean generation time of the proliferating cells does not remain constant as reported by Baserga (1963), but it increases (as reported by Lala & Patt, 1968) from 8.4 hr in the early stages of the tumor to 28 hr in the final stages of the tumor. At the time of the tumor transplantation the transition of cells to the resting states is very slow, and the growth fraution is very close to 1. With increasing of the size of the tumor, the transition of proliferating cells to the resting states increases, but even at a 600 million cell tumor the effect of this transition in retardation of the rate of growth of EAT is slight compared to the effect of the prolongation of the mean generation time of the proliferating cells. When the total population reaches 900-1000 million cells then the transition of proliferating cells to the resting states increases very sharply. This has the effect of reducing the growth fraction rapidly, and also altering the cell age and size and DNA distributions significantly. The rate of growth of population at any time, then, is a function of the total cell population. The rate of cell death from the proliferatingipopulation is small for early stages of the tumor, but it is higher and approaches a constant rate for older tumor. The rate of cell death in the nonprolifbrating group prior to day 6 does not effect the growth of the total population appreciably, since prior to day 6 the nonproliferating population is a small part of the total population. The rate of nonproliferating cell death from population of 600 million cells on does not change appreciably. The rate of the, overall cell death determined from the model are generally in good agreement with the experimental data. The effect of cells in Gz, M-arrest on the cell $ize distribution is the appearance of a second peak in the size distribution with the aging of the tumor, and also, increasing mean cell volume of cells in the tumor. It may be concluded from this study that the process of transition of proliferating cells to resting states depends on the, sire of the population of the tumor. The latter in turn depends on the change in growth media and the increase in some type of growth inhibitor brought about by aging of the tumor. If, in fact, the growth media is to be altered by addition or removal of certain chemicals, without changing the cell population, then the rate of transition of cells to resting states must accordingly be modified. This modification could easily be introduced in the model.
458
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BAHRAMI
AND
K.
B.
WOO
In the model used, it was assumed that the transition of proliferating cells to resting states occurs from proliferating cells in age compartments 23 and 24. The assumption was motivated by experimental data as well as the experimental cell volume distributions. If, in fact, the transition of proliferating cells to the resting states occurred somewhere else in the cell cycle, again the discrete-time model can easily be modified to accommodate these changes. As it is already evidenced, in the model used here for simulation of kinetics of EAT population, which is nonlinear, there are several parameters that are selected simultaneously. Since these parameters are interacting and the model is nonlinear, it may be very difficult, if not impossible, to find meaningful analytical relationships between them. In this study a general quantitative model for the dynamics of the cell cycle kinetics and proliferation is developed and it is used to analyze and simulate the cell cycle kinetics of Ehrlich ascites tumor and to predict various parameters of the cell cycle kinetics. The authors wish to express their thanks to Dr Seymour Perry, Deputy Director, Division of Cancer Treatment, National Cancer Institute, for support and interest in this work. REFERENCES ANDERSWN, G. K. A. & AGRELL, I. P. S. (1972). Virchows Arch. A&. BASERGA, R. (1963). Arch. Path. 75, 156. BASEKGA, R. (1965). Cancer Res. 25,581. BELL, G. I. &ANDERSON, E. C. (1969). Biophys. J. 4, 329. CLEAVER, J. E. (1967). Thymidine Metabolism and Cell Kinetics.
CeilPurh.
11, 1.
Amsterdam: North
Holland Publishing Co. DE-E, J. J. & GELFANT, S. (1968). Science, N. Y. 162, 698. DOMBERNOWSKY, P., BICHEL, P. & HARTMAN, N. R. (1974). Cell Tissue Kinetics 7,47. FRIED, J. (1970). Muth. Biosci. 8,379. FRINDEL, E., VALLERON, A. J., VASWRT, F. & TUBIANA, M. (1969). CeN Tissue Kinetics 2,51. HAHN, G. (1966). Biophys. J. 6, 275. HAHN, G. (1970). Math. Biosci. 6, 295. KIM, M., BAHRAMI, K. & Woo, K. B. (1973). Proc. 26th A. Co& Engng Med. Biol. 15,72. LALA, P. K. (1971). In Methods in Cuncer Research (H. Busch, ed.). New York: Academic LALA, P. K. (1972). Cancer 29,261. LALA, P. K. & PATT, H. (1968). Cell Tissue Kinetics 1, 137. LE~O~ITZ, J. L. & RLJBINOW. S. I. (1969). J. theor. Biof. 23,99. OMINE, M. & PERRY, S. (1972). J. natn. Cancer Inst. 48, 697. PAN, H. M. & STRAUBE, R. L. (1956). Ann. N. Y. Acad. Sci. 63,728. SARNA, G. P. (1973). In press. SCHERBAUM, 0. & RANCH, G. (1957). Actu path. microbiof. scand. 41, 161. SHACKNEY, S. E. (1973). J. theor. Biol. 38,305. STEEL, G. G. (1968). Cell Tissue Kinetics 1, 193. STEWART, P. G. & HAHN, G. (1971). Cell Tissue Kinetics 4,279. TAKAHASHI,M. (1968). J. theor. Biol. 18,202. TOBEY, R. A. L CRISSMAN, H. A. (1972). Cuncer Res. 32,2726.
CELL
CYCLE
KINETIC
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MODEL
APPENDIX
Relationship between Proliferating Cells Mean Generation tie Age of lhmor
and
In this appendix, comparing Z(T) with the experimental Z(t), a relationship between proliferating cell mean generation time To and age of tumor is developed. It is required that the simulated growth Z(T) = Z(kAT,) = Z(k) coincide with the experimental growth Z(r). For any k = k,, the quantities Z(k,) and Z(k, + 1) are known from simulation of the model. Since Z(t) is monotonically increasing in t then there exists t, and ts, both unique and t, > t, such that z(h) = Z&) and Z(t,) = Z(k, + 1) Denote the rate of increase in Z(t) at t = t1 by r,(t,). For t, close to t,, using a lirst order approximation, the rate of growth is
On the other hand the rate of growth of Z(T) for T = k, AT, is
r,tWTJ
=
Ztki +l)--Z(h)
AT, Coincidence of Z(t) and Z(T) requires that
*
(442)
r,(h) = r,Uw%). (A3) Equations (Al), (AZ) and (A3) imply s AT, = (tz - tJ. (A4) From equation (A4) and the definition of AT, it follows that T,(t,) = nAT,(tr) = n(ts-ti). (A% Equation (A5) expresses the proliferating cells mean generation time at time tr in terms of f, and t2 where t, and tz correspond to discrete times kl and kl + 1, respectively.
Mathematical Description and Analysis of Cell Cycle Kinetics and the ApplicMion to Ehrlich Ascites Tubor M. KIM SchooI of Electrical Engineering, Cornell University, Zthaca, New York 14850, U.S.A. K. BAHRAMI Jet Propulsion Laboratory, Pasadena, California,
U.S.A.
K. B. Woo Division of Cancer Treatment, National Cancer Institute, National Institutes of Health, Bethesda, Maryland 20014, U.S.A. (Received 11 March 1974, and in revisedform
16 August 1974)
The linear and nonlinear aspects of the dynamics of the cell cycle kinetics of cell populations are studied. The dynamics are represented by difference equations. The characteristics of cell population systems are analyzed by applying the model to Ehrlich ascites tumor. The model applied for the simulations of the growth of Ehrlich ascites tumor cells incorporates processesof cell division, cell death, transition of cells to resting states and clearance of dead cells. Comparison of the results obtained with the model and the experimental data suggests that the duration of the mean generation time of the proliferating EAT cells increases with aging of the tumor. An attempt is made to relate the prolongation of cell mean generation time with processes of cell death and dead cell clearance. Studying the transition of cells to the resting states, it becomes apparent that in fact transition of proliferating cells to the resting states occurs somewhere close to the end of the cell cycle and with a rate that varies with the age of the tumor. Time course behavior of the cell age, cell sire, and cell DNA distribution with aging of the tumor are obtained. Variations in average size and average DNA contents are determined. 1. Introduction
The cell cycle kinetics of various cell populakions has been a subject of extensive experimental studies under various environmental conditions (see 437
438
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
Baserga, 1965; Cleaver, 1967). These studies have played an important role in elucidating growth of human cancer. In modeling the behavior of cancer cell populations, knowledge of various parameters associated with growth of the population is required. The labeled mitosis method using tritiated thymidine as a radioactive label (Cleaver, 1967) has been widely employed in experimentally studying the cell cycle kinetics. Theoretical techniques for determining some of these parameters from experimental results have been developed (see Lebowitz & Rubinow, 1969; Fried, 1970; Steel, 1968; Takahaski, 1968). Also Shackney (1973) studied theoretical aspects of proliferation characteristics of tumor cell populations in terms of various cell cycle kinetics parameters. Hahn (1966, 1970) and Stewart & Hahn (1971) theoretically discussed the time course behavior of the cell age distribution of tumor cell population. With advances in experimental techniques, such as a cell separation techniques based on sucrose gradient sedimentation for determining cell size distribution (Omine & Perry, 1973) and a microfluorometer technique for cell DNA distributions (Tobey & Crissman, 1973), there is potential for fast sampling of tumor cells to determine the cell cycle kinetics parameters in terms of cell size and cell DNA distributions in conjunction with radioautograph. This paper discusses a mathematical model of the cell cycle kinetics of cell populations to assist interpreting experimental data obtained with these techniques. The cell cycle kinetics of cell populations will be represented by nonlinear difference equations, and its characteristics will be studied via application to Ehrlich ascites tumor (EAT). 2. Mathematical Description of Dynamics ‘of Cell Population Systems
Processes of cell division, cell accumulation, and various types of cell loss contribute to the growth of cell population systems. Determination of the extent and the relative importance of each of these processes is complicated by the fact that some or all of these processes may depend on other cell cycle parameters of the cell populations system. In certain cell population systems, such as Ehrlich ascites tumor, the growth is fast; initially due to rapid cellular division of a large number of cells and that this rapid phase is followed by a phase of progressive decrease in the rate of growth toward a constant population. Retardation of the rate of growth may be attributed to several factors. One factor is a progressive decline in the number of the dividing cells and consequently a decline in the growth fraction. Another possible factor is the gradual prolongation of different phases of the cell cycle with the aging of the system. These factors make the behavior of the system to be represented more appropriately by nonlinear equations,
CELL
CYCLE
KINETIC
439
MODEL
In the model used for the description of cell cycle kinetics, the total population is considered to consist of two groupslof cells, namely proliferating cells and nondividing cells, for which the state vectors are defined. The proliferating cells pass through all phases of the cell cycle (G,, S, G,, and M phases) and are capable of dividing into two daughter cells. The nondividing cells include cells that either temporarily or permanently have lost their capability to proliferate, dead cells that are awaiting clearance, and cells that may be in a resting state. During any specified length of time, the proliferating cells may die or become nondividing. Similarly, during any specified length of time, the nondividing cells may die or regain their proliferating ability. The schematic diagram of the cell cycle kinetics is illustrated in Fig. 1. For the model the mean generation time of the cell, denoted by TO, is divided into n equal intervals AT,, called biological unit time (or unit time for short. Since To may in general be a function of time, i.e., To = T,(t), then AT, is also a function oft, i.e., AT, = T,(t). Biological age of the tumor is denoted by T and is defined as T = kAT,. It is then defined that each of these intervals constitute a cell age compartment. All cells within the intetval of one specific age compartment are considered as one subpopulation and Proliferoiing
g&p
I
Division
----+---S----t----------+tc---z+M--i ----+---S---t----------+tc---z+M--i
!+------I
--,I
2
3
.
Dead cells
.
.
.
1
Resting or arrested ceils
S
c
I I I-
-
Nonproliferating
group -I
FIG. 1. The schematic diagram of the cell cycle kinetics of cell populations.
440
hi.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
are characterized by the mean cell age of that compartment. Then the discrete time model for the kinetics of cellular proliferation, is represented as follows : x(k+l)
= $,(k+l, k)x(k)+@,,(k+l, k)w(k) (1) w(k+ 1) = @,,(k+l, k)x(k)+Q,,(k+ 1, k)w(k) (2) where x(k) is the n-dimensional cell age state vector at time k for the proliferating cells and w(k) is the m-dimensional state vector for the nonproliferating cells. In x(k) the jth element denotes the number of cells in the jth cell subpopulation at time k. The state transition matrixes QpP, ap,,,, a,.,,, and
CELL
proliferating
CYCLE
cells, denoted by Z,(k)
KXNBTIC
xXk).
(5)
cells, denoted by Z,,,(k) is
Z,(k) = f,, w,(k) and the total population,
441
is
Z,(k) = jl The aggregate of nonproliferating
MODEL
(6)
denoted by Z(k) is Z(k) = Z,(k)+Z,(k).
(7) The cell size distribution of a cell population is a cell cycle kinetic parameter of considerable interest. Experimental techniques (e.g., automatic counter, hemocytometer, and sucrose sedimentation technique for cell separation) exist for determination of cell size distribution. Given information on the size distribution of cells of any given age, the discrete time model described by equations (1) and (2) can be extended to account for the cell size distribution. In this case, the rate of increase in average cell size of a cell should be known as the cell advances in cycle. Scherbaum & Rasch (1957) have assumed hypothetical curves for the growth in mean size of a single cell as it completes its movement through the cell cycle. Cell size compartments are defined as follows. Let the mean initial size of the cell, i.e., the mean size of the cell just after mitosis, be a,, and let the mean final size of the cell, i.e., the mean size of the cell just before dividing, be C,,. The interval i$- COis divided into rnr equal intervals with each of length (Z,- &)/ml. Each of these intervals constitutes a size compartment. The mean size of the cells in size compartment i is I+ = CO+ i(Ca-5Jm,. Some of the young cells may have a smaller size than i+, and some of the cells close to division may have a larger size than &. To accommodate such cells the size compartments are extended to the left of 6, and to the right of 0”. The criteria for defining the size state of a cell is its position in the size compartments. The size state vector y(k) is defined as Y= = b,(k), y,W),~. -3y,(k),. - ->YmWl where y’ is the transpose of y and v*(k) is the total number of cells in the ith cell size compartment. The size vector y(k) indicates the distribution of cells based on the size at the time instant k&It. Knowing the cell age state vector x(r), one can obtain the cell size state vector y(k) with the following assumptions. (1) A curve for the growth in the mean size of a single cell between successive age compartments in the curve shown in Fig. 2 (Scherbaum & Rasch, 1957) used in the present study. The curve shows a monotonic increase in the mean size of the cell as it advances through the cell cycle. The final size
442
M.
KIM,
2
4
K.
6
BAHRAMI
8
IO
AND
12
Age compartment
14
K.
16
B.
WOO
I8
20
22
24
(TO ~-24)
FIG. 2. A curve (solid line) for the growth in the mean size of s single cell between successive age compartments. The dotted line is the diagonal reference line. (From Scherbaum & Rasch, 1957.)
of the cell is twice that of its initial size. This curve is chosen because this mean growth rate gives a good fit in simulation of the cell size distribution of EAT under exponential growth. However, there is a need for determination of the mean growth rate for different environmental conditions or different types of cell populations. (2) It is assumed that the size distribution cells in each age compartment is Gaussian. The mean value of the distribution is obtained from assumption (I), and the coefficient of variation is assumed to be the same foi all size compat tments and assumed to be known for the particular cell under consideration. Thus, knowing the cell age state vectors x(k) and w(k), one can obtain the cell size vector y(k) by a linear transformation YW = W)W + pnlow~) (8) where P(k) and P,,,(k) are the transformation matrices, and may be constant or a function of the time, depending upon environmental conditions. Experimental techniques, such as microfluoremetry (Tobey & Crissman, 1972) are available for the determination of DNA contents of individual cells in a large population. Thus, it is of interest to determine theoretically the cell DNA distribution of the population in conjunction with the cell age
CELL
CYCLE
KINETIC
MODEL
443
and cell size distributions. In determining the cell DNA distribution for a cell population the DNA state of a cell is defined in the same manner as the cell size state was defined. The initial DNA content of a cell at the beginning of the S-phase is normalized to 1. The DNA content of the cell at the end of the S-phase is 2. The span of DNA content during the S-phase, from 1 to 2, will be divided into r equal intervals, each of length AI = I/r. The first interval (0, 1) defines compartment 1. It corresponds to all cells in the Gi phase. Interval (1 +(i-2)Al, 1 +(i- 1)AZ) for i = 2,3,. . .r defined compartments 2 to r. Intervals (1 + (r - l)AZ, 1 + rAZ) and (2, co) define compartments r+ 1 and r+2, respectively. The latter compartments corresponds to cells in the Gz phase. The DNA state of a cell is defined as the cell DNA compartment, to which the specific cells belong, The DNA vector at time k is defined as, ZT = [GW, Z,(k), . . . , Z,(k), . . . , UWl where Zdk) is the number of cells being at DNA state i at time k. Elements of z(k) form the cell distribution according to DNA content. At any discrete time k a relationship between the cell age vector and the cell DNA vector exists, Z(k) = QW + Q,wW (9) where Q and QM are linear transformation matrices. The entries of matrix Q are obtained from the knowledge of the DNA synthesis during the S-phase. Also the entries of QM for nonproliferating cells are determined from the DNA content compartment at which cells are transferred from the proliferating group to the nonproliferating group. In the specific case of exponentially growing cell populations without cell death and dormant (resting) phase, only proliferating cells exist. Then the model is simplified and all the elements in transition matrices cP,,,U&,and Q,,,,,,, in equations (1) and (2) become zero. Only $(k+ 1, k) of a,, in equation (3) have nonzero elements. This case has been studied by Hahn (1966) and Kim, Bahrami & Woo (1973). In the general case, transfer of cells between the proliferating group and nonproliferating group occurs. Then these transition matrices contain parameters corresponding to the probabilities of certain transfer rates, such as cell death, dormant cells in the resting stage regaining proliferating capabilities, and cells in the proliferating group losing dividing ability. These parameters may be constant or become a function of certain cell cycle parameters, such as the total cell population. Certain aspects of transitions from proliferating to nonproliferating states have been studied by Shackney (1973) and Stewart & Hahn (1971). The model will be analyzed and studied via application to the cell cycle kinetics of Ehrlich ascites tumor (EAT).
444
M.
KIM,
K.
BAHRAMI
3. Characteristics
AND
K.
B.
WOO
of Ehrlicb Ascites Tumor
Investigation of the pattern of cellular proliferation of the EAT cells in the peritoneal cavity has shown that the growth of EAT cells is initially very rapid due to rapid cellular proliferation and that this rapid phase is followed by a phase of progressive deceleration toward an asymptote (Andersson & Agrell, 1972; Baserga, 1963; Lala, 1971, 1972; Lala & Patt, 1968 ; Patt & Straube, 1956). Lala & Patt (1968) have indicated that the growth of an inoculum of lo6 cells is initially approximately exponential, but subsequently after 3 and 4 days a decline in the growth rate appears and continues until the terminal stage of the tumor. The number of tumor cells in the terminal stage is of the order of 10’ cells. A similar result has been observed by Andersson and Agrell (1972) for the growth of an inoculum of 4 x lo6 cells. The rate of growth of the tumor is rapid at first but it decreases with aging of the tumor. The total number of free tumor cells approaches asymptotically to about l-6 x 10’ in the terminal stage of the tumor. Retardation of the rate of growth of EAT cells with increasing size of the tumor is attributed to several factors. One possible factor is the gradual prolongation of different phases of the cell cycle with the aging of the tumor (Lala & Patf, 1968), although on one occasion it is reported that no appreciable change in the length of the DNA synthesis period was observed (Baserga, 1963). Another important factor is a progressive decline in the number of the dividing tumor cells and consequently a decline in the growth fraction (Andersson & Agrell, 1972; Baserga, 1963 ; Lala & Patt, 1968) as evidenced by a progressive decline in the number of the tumor cells that synthesize DNA (Baserga, 1963), and also by an accumulation and build up of cells in late stages of growth of the tumor. Experimental cell size distribution for tumor of various ages obtained by Andersson & Agrell (1972) clearly indicates that there is a definite shift in number of cells of smaller size to cells of larger size with aging of the tumor. This is furthermore evidenced by an increase in the average size and average DNA content of tumor cells with the aging of tumor. Factors affecting the rate of growth of EAT cells also include the rate of and the mechanism of cell death and the rate of dead cell clearance. There is evidence that there is preferentially larger cell death from the nondividing cell population compared to a smaller rate of cell death from the dividing cell population (Lala, 1972). Undoubtedly the simultaneous occurrence of several processes in the growth of EAT cell population makes it quite difficult to isolate the relative importance and extend to each of these processes. Meanwhile, the experimental studies of the growth pattern of EAT and the observations about the rate of growth of the cell population, variations in the mean generation time
CELL
CYCLE
KINETIC
MODEL
445
of the cells, changes in the mean size, mean DNA and RNA contents of the cells and finally changes in the cell size distribution have provided valuable data for the determination of the importance of various cell cycle kinetics parameters in growth of EAT. The elements of the transition matrices and the transformation matrices in equations (l)-(4), (8) and (9) are to be chosen for EAT. 4 in equation (3) is deli&, +(k+ 1, k) = 6A, where the matrix 6 is the dispersion operator defined in terms of CI the probability that cells in an age compartment advance two compartments after AT, and p the probability that cells in an age compartment do not advance to the next compartment after AT,. The matrices 8 and A are defined in Hahn (1966). The mean generation time, To, is divided into (n=) 24 equal parts, each part defines one cell age compartment. c1is chosen to be 0.3. The nonproliferating group is divided into three subgroups, and the state vector for the nondividing cells, w is defined: w=
Wl w, [Iw3
where the entries 1 and 2 of w correspond to cells in the resting states and entry 3 of w corresponds to the dead cells. Since there is a strong indication that the proliferating cells in the latter part of the cell cycle may transfer to the resting states (Andersson BEAgrell, 1972), entries 1 and 2 of w are selected to correspond to resting cells that have come from proliferating age compartments 23 and 24 as shown in Fig. 1. It is assumed that cell death in the proliferating population is uniform. The probability that a cell from the proliferating group dies in AT0 is dl. The probability that a cell from either wi or w2 dies in AT, is d2. The probability that a cell in age compartments 23 and 24 loses proliferating capacity and transfers to the nonproliferating compartment 1 or 2 is F. Then the following transition matrices are obtained. l-d2 0
(24 x 24) P(k+
1) =
d <‘Id, \ 0
T.B.
0 l-d2
\ \
\
0 0
( \
dl+F ’ \ d,+F
1 (10)
446
M.
KIM,
K.
BAHRAMI
AND
o...o 1 (3x24)
T(k+l)=
@,,(k+
F F+d,
0. ..O ; 0 ._------1 . . . II
(24 x 3)
K.
I
- 4 F+dl
B.
WOO
7 0 F F+d,
(12)
- 4 F+d,l
1, k) = 0
(13) where r is the clearance rate of dead cells in the nonproliferating compartment 3. dl and d2 are the probability of death for a proliferating cell and a dormant cell in the resting state, respectively, as defined previously. F is the probability that a proliferating cell becomes a dormant cell and transfer from the proliferating group into the nonproliferating group. 4. Dependence of Mean Generation Time of EAT Cells on the Age of the Tumor In this section attempts will be made to investigate the dependence of the mean generation time of EAT cells, T,, on the age of the tumor. It is of interest to know if it would be consistent with experimental data to assume that To remains invariant with aging of the tumor. On the other hand if To does not remain invariant, then what is the relationship between aging of the tumor and the change in To, and in what way is the change in To related to changes in cell production and cell death of EAT cells. Experimental EAT cell population growth curves and other data obtained by Andersson & Agrell (1972) will be used here for comparison with the results that will be obtained by adjusting the different parameters in the model. Experimental cell volume distributions obtained for EAT cells for tumors 6,9, 10, 11, 12 and 13 days old indicate that with aging of the tumor there is a progressive increase in the relative number of larger cells. The cell volume distribution for 6-day tumor has one peak whereas a second peak appears in the cell volume distribution for later days. Appearance of the second peak, which indicates the build-up of cells of larger size, could be interpreted as due to the resting cells. In other words some of the proliferating cells have left the proliferating group and have entered the nonproliferating group sometime after they have completed the S-phase (DNA synthesis phase) and prior to completion of the M-phase (mitosis). If, in fact, the second peak in the cell volume distribution of EAT cells is due to cells in the resting states, then the approximate growth fraction of the
CELL
CYCLE
KINETIC
447
MODEL
tumor cells may be calculated in the following fashion. A Gaussian distribution curve is fitted to the first peak and another to the second peak of EAT volume distribution. The area under the first Gaussian curve is assumed to represent the proliferating cells and the area under the second curve is assumed to represent the nondividing cells. The1 growth fraction denoted by f is defined as proliferating cells f= nonproliferating cells + proliferating cells’ Based on the above discussion, values off for the tumor at various ages are shown in Table 1. These values are based on experimental cell volume distribution of Andersson & Agrell (1972). TABLE
Growth
Age of tumor @wd : 10 11 12 13
Total
population (million cells) 600 1050 1150 1250 1350 1450
1
fraction of EAT cells Proliferating population (million cells) 486 819 828 712
Nonproliferating population (million cells) 114 231 322 538 702 841
Growth fraction Cfl O-81 0.78 0.72 o-57 O-48 0.42
In order to relate the initial population doubling time, Td, with the mean cell generation time, TO, certain assumptions are made. It is assumed that the initial size of the inoculum when the tumor is transplanted is 4 x lo6 cells and all these cells are proliferating cells. Age of the tumor denoted by t, is defined to be zero at the time of transplantation. It is assumed that the transplanted tumor is initially exponentially d@ibuted (for definition of exponentially distributed see one-day-old tumor in Fig. 4). It is furthermore assumed that the initial rate of cell death is negligible. With these assumptions, Td is related to TO in the following fashion. T,, log 2 Td =
n x log 2+log[l +a((2-‘/z”-21/z”)q ( ) k TO for II large, where acis the probability that cells in an age compartment compartments in AT,.
advance two
448
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
Experimental data for growth of EAT with an initial size of 4 x IO6 cells indicate that the rate of growth of the population at the outset of tumor transplantation is very rapid, but subsequently it slows down (Andersson & Agrell, 1972). The doubling time of the population, Td, initially is about 8 hr. This indicates that the initial cell mean generation time of the proliferating cells is approximately 8 hr. If TO = 8 hr at I = 0 were to remain constant with aging of the tumor, that would be inconsistent with the much longer TO found by Baserga (1963) and Lala & Patt (1966) (see Table 2). TABLE
Duration
2
of different phases of EAT cells as reported in the literature
source Lala & Patt (1968) l-day Cday 7-day Baserga (1963)
To
TS
8 17 22 18
6 13 18 11
T G*+M
Td %I
Tca + M( %I
2 4 4 7
75 76.5 82 61
25 23.5 18 39
The growth fraction f for EAT population is close to 1 at the early age of the tumor, but with aging of the tumor it progressively declines (Table 1). For a 6-day old tumor f = 0.81 indicating that transition of cells from the proliferating group to the nonproliferating group up to day 6 has been small. Yet, the progressive decline in the rate of EAT population growth from day 0 to day 6 is quite pronounced (Andersson & Agrell, 1972). If this decline in the rate of growth is not due to transition of cells to the resting states, then it must be due to: (a) a prolongation of TO; and/or (b) occurrence of cell death. The procedure for determining the exact dependence of TO on age of the tumor or alternately on the size of the population, prior to day 6, is based on comparing the real time population growth with the biological time population growth. The following assumptions are made. (1) For t I 6 days transition of proliferating cells to resting states is small and can be neglected. (2) The probability of a proliferating cell dying in unit time AT, is d,(T). Probability d,(T) is the same for all proliferating cells. (3) The probability of a nonproliferating cell dying in unit time AT0 is d,(T). Probability dz(T) is the same for w1 and w2. (4) dl and d, are, in general, functions of 2, the total population. Motivation behind assumption 1 is that the growth fraction is close to 1 prior
CELL
CYCLE
KINETIC
449
MODEL
to day 6. Motivation behind assumptions 2 and 3 is that there is no indication of preferential age dependent cell death. Motivation behind assumption 4 is that the decrease in the growth fraction of EAT may be due to some inhibitor (Sama, 1973): Here it is assumed that the presence of the inhibitor is positively correlated with the total size of the population. In the model for EAT growth described by equations (1) and (2), quantities di and d2 are considered as parameters. Various equations for dl(Z) and d,(Z) are selected. Equations (1) and (2) are then solved for x(k) and w(k) by digital computer and Z(k) is obtained. Comparing Z(k) and Z(f) and utilizing the method described in Appendix A variation of the mean generation time TO with the total size of the tumor population is determined. A number of sets of the functions for d,, d,, and the rate of dead cell clearance were selected for simulation. In all cases the values of TO predicted increased monotonically with increasing size of the tumor, and were compared with the experimental values of TO obtained by L.ala & Patt (1968). The simulated results of the set of the functions which gave the best comparison with those reported by Lala & Patt (1968) are: l-day, the mean cell generation, TO, 8 hr (Lala & Patt, 1968) and 81.4hr (simulated); 4-day, T,, 17 hr (Lala & Patt, 1968) and 16 hr (simulated); 7-day, TO, 22 hr (Lala & Patt, 1968) and 24 hr (simulated). The relation between the mean generation time TO and the total population Z(h) is shown in Table 3. TABLE 3
Relation between the total population and mean generation time Total population
Z(h) (millions)
z-c4 4 kZ -c 32.4 32.4 IZ< 68.89 68.89 IZ< 117 11752-z looo
1oooIz
Mean
generation time TO(hr)
To = 8 TO = 0.52
log&Z)
-I- 7.68
To = 3 logm(Z) + 3.95 To = 11.13 log,,(Z) - 10.92 To = 17.71 log,,(Z) - 24.65 To =28-S
5. Resting Cells
In the previous section it was noted that prior to day 6 the retardation in the growth of EAT cell population is basically due to the prolongation of the ccl1 cycle duration of the proliferating cells and cell death. In this section transition of proliferating cells to the resting states is included and its effects on the growth of the population is studied. Appearance of the second peak in the experimental cell volume distribution with the aging of the tumor
450
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
(Andersson & Agrell, 1972) indicates that with the aging of the tumor there is a transition of the proliferating cells to the nonproliferating compartment and an accumulation of cells in the resting states. In the NCTC2472 ascites tumor, an accumulation of cells with a G2 DNA content also seemed to occur with increasing time after transplantation (Frindel, Valleron, Vassort & Tubiana, 1969). In the JB-I ascites tumor, it was shown that with increasing tumor age, there are first an accumulation of non-cycling cells with G, DNA content and in later stages also an accumulation of noncycling cells with Gz DNA content (Dombernowsky, Bichel & Hartman, 1974). The existence of noncycling cells with G, DNA content have been demonstrated in the sarcoma 180 ascites tumor and the Ehrlich ELD ascites tumor (DeCosse 8c Grelfant, 1968). In order to incorporate the effect of the resting cells in the model the following assumptions are made. (1) Transition of the proliferating cells to the resting states (denoted by G,, M + wl, w2) occur in an age specific manner, namely from age compartments 23 and 24. The probability of transition of a proliferating cell in compartment 23 to the resting state wi (also compartment 24 + w2) in AT, is denoted by F. (2) F is a function of the total cell population, i.e., F = F(Z). (3) The mean generation time of the proliferating cells increases according to the relation in Table 3. Motivation behind assumption 1 comes from consideration of the second peak in the experimental volume distribution of EAT (Andersson & Agrell, 1972). Assumption 2 is based on the experimental data indicating that the decrease in growth fraction of EAT cells is associated with crowding mediated by some type of inhibitor (Sarna, 1973). The increase in To in assumption 3 is consistent with the experimental data (Lala & Patt, 1968). The procedure for simulation of EAT cell growth is as follows. Initially certain functions for d,(Z), d,(Z), F(Z) and r(Z) are assumed and are used in computer simulation of the model. From this simulation the growth of the total population Z(T), the proliferating population Z,(T), and the nonproliferating population Z,(T) as a function of the biological time are obtained. Since the relationship between T and t is determined from assumption 3, the real time growth of the total population Z(t), the proliferating population Z,(t), and the nonproliferating population Z,(t) are calculated. If, in fact, functions d,(Z), d*(Z), F(Z) and r(Z) are properly selected then Z(t) will agree with the experimental data of Andersson & Agrell (1972), and the growthf(t) will agree with the values of experimental growth fraction shown in Table 1, otherwise those functions will be changed in a reasonable way until results consistent with experimental data are achieved. Again a number
CELL CYCLE
KINETIC
451
MODEL
of different sets of the functions for d,, d,, Fi and y are used for simulation. The set of the functions which gave a good comparison is: dl = 0+034(10g z)1*25 d 2 = 0.01 F = O-07~~)@s+“.03(&-)
The simulation
r= 1.0 results of the total population
0
2
4
6 Days
after
Z(t), the proliferating
8
IO
12
popula-
14
inoculohion
Fro. 3. Comparison of experimental data with the simulated growth of case 3. Z = total population, Zp = population in proliferating group, Z, ,= populationin nonprolifexating group, and the subscriptE indicatesexperimentalresuk (The experimentaldata from Andersson& Agrell, 1972.)
tion Z,(t), and the nonproliierating population Z,(t) are compared with the experimental results (Andersson & Agrell, 1972) in Fig. 3. Up to now parameters d,(Z), d,(Z) and F(Z) have been expressed in units of cells per ATO, but in experimental cases rate of cell death is expressed in cells per unit time (e.g., per hour). In order to obtain the rate of proliferating cell death d:(a, the rate of G2, M-arrested cell death d;(Z), and the rate of
452
hi.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
transition of proliferating cells to the resting states F*(Z), are all expressed in real time, the following relationships are used. d:(Z) = d,(Z) n cells/hr T,(Z) df(Z) = d,(Z) &)
cells/h1
0
F*(Z) = F(Z) n cells/hr (16) To(Z) where T,(Z) is the mean generation time of the proliferating cells expressed in hours at the time that the total population is Z. Using equations (14), (15) and (16) the function for T,(Z), d:(Z) and F*(Z) is calculated. From this result, it was seen that the rate of proliferating cell death for a tumor of 150 million cells is O-0165 cells@, and the rate of nonproliferating cell death is 0.0175 cells/hr. The overall rate of cell death is O-0165 cells/hr. Lala & Patt (1968) interpreting their experimental findings, have arrived at an overall cell death rate of 0.013 cells/hr for a similar tumor. For a tumor of 600 million cells, values for dy and df are 0.014 and 0.01 cells/hr respectively, and the rate of overall cell death is 0.014 cells/hr overall cell death rate reported for a similar tumor (Lala & Patt, 1968). It should be noted here that if function d, was selected such that d;(Z) had a much lower value for low values of Z, the growth of the population would have changed very slightly. This is due to the fact that for low Z the number G,, M-arrested cells is only a small percentage of the total population. 6. Cell Age Distribution
of EAT Population
As it was established earlier, the growth of EAT population is not exponential. It is expected then, that the cell age distribution of cells in this population change with aging of the tumor. In this section the simulated cell age distribution is presented. It should be noted that the age refers to the biological age. It was assumed that at the time of tumor transplantation (day 0) the cell age distribution was that of an exponentially growing culture. The results of simulation shown in Fig. 4 shows that with aging of the tumor, the cell age distribution of proliferating cells starts from exponential distribution and gradually approaches uniform distribution. The overall cell distribution of the population can be obtained by including the age distribution of G,, M-arrested cells. The latter appears as a peak in the latter part of cell age distribution.
CELL
Day
I
CYCLE
KINETIC
4
6
i
T.
0 Cell
9
3.2
0
age distribution
FIG. 4. Time course behavior of cell age distribution. M-arrested cells.
7. Cell Size Dlstrlbution
453
MODEL
-,
proliferating
cells; ---,
GP,
of EAT Cell Popdation
In this section time course behavior of cell size distribution of EAT cells is determined. Also the change in the mean size and mean volume of EAT cell population is studied. Cell size distribution is obtained by the method described in section 2. In this method, which is based on performing a linear transformation on the age vector x(k) and w(k), it is assumed that the mean cell size of a newborn cell is normalized to 1 and the mean cell size just before division is 2 and that the size distribution of cells of any particular age is Gaussian with a coefficient of variation of 11.5 %. The curve for size growth versus age compartment in Fig. 2 (Scherbaum & Rasch, 1957) is used which gives good comparison with the experimental data (Andersson & Agrell, 1972). The simulated cell size distributions obtained for cells in 1, 4, 6, 10, 11, 12 and 13-day old tumors are shown in Fig. 5. The experimental results (solid circle) are shown in the figure. As it can be seen from this figure, there is an increase in the relative number of larger cells with aging of the tumor as manifested by the appearance of the second peak in the size distribution with
454
M.
KIM,
K.
BAHRAMI
AND
K.
0I 2I 2 Ceil
B.
WOO
2
I
2
size (normalized)
FIG. 5. Time course behavior of simulated (solid line) and experimental (solid circle) cell size distribution of EAT population obtained by computer simulation. The experimental points (solid circle) are from Andersson & Agrell (1972).
ii 0’12 i 008 B cr’ 004 0
I
I
I
I
I
I
2
4
6
8
IO
I2
Days after
inoculation
FIG. 6. Simulated (solid line) and experimental (solid circle) graph of mean cell volume versus days after inoculation. The experimental points from Andersson & Agrell(l972).
CELL
CYCLE
KINETIC
455
MODEL
the aging of the tumor. The changes in the average volume of cells in EAT population with aging of the tumor are shown in Fig. 6. The simulated cell size distribution, shown in Fig. 5 compares favorably with the experimental results, However, the experimenta values of days 12 and 13 at the larger cell size part are higher than the simulated values. This may be due to the fact that dormant cells in the resting state’may continue to grow in size while the model assumes that the size of dormant cells remain constant. 8. Ceii DNA Distribution
of EAT
In this section the time course behavior of cell DNA distribution and also the change in DNA content per cell of EAT population are presented. The method for obtaining cell DNA distribution is based on performing a linear transformation on age vectors x(k) and w(k) discussed in section 2. Three different curves for DNA content of a cell according to its age are assumed (Fig. 7). Duration of S-phase is considered to last 314 of the generation time.
2.0 a ‘Z::
I.8 -
I
k-ssG~,+M+ Age comportment FIG.
7. Increase in average per cell DNA content for cells in EAT population.
For each of these assumptions simulated (predicted) cell DNA distributions are obtained. Cell DNA distributions for the rate of DNA synthesis that is large close to the end of the S-phase (assumption 3) are shown in Fig. 9. For each of these three assumptions, the variation of mean DNA content/cell with the aging of the tumor is calculated (Fig. 8). In all cases, the mean DNA content/cell increases monotonically with aging of the tumor. Figure 8 indicates that the results obtained with assumption 3 are close to the experimental observations.
456
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
2.0 .
Assumptions
I ond 2
8 !i 4 IO
0
I 2
I
I
I
I
I
4
6
8
IO
12
Days
after
inoculation
FIG. 8. Increase in averageper cell DNA content for cells in EAT population. The simulated curves are solid line. The experimental points are solid circles from Andersson & Agrell (1972).
Assun,pt
ion
3
2
I DNA
FIG.
2 content
I
21
21
21
2
(normalized)
9. Cell DNA distribution for Assumption 3.
9. Discussion
From the computer simulation of the model it may be concluded that the retardation in the rate of growth of EAT population is due to a combination of several factors. In the early stages of tumor growth, when the cell population has not reached 600 million cells, the primary reason for retardation of the rate of growth is the prolongation of the mean generation time of the proliferating cells. The prolongation of the mean generation time of the
CELL
CYCLE
KINETIC
MODEL
457
proliferating cells with increasing population is slow at the beginning of the tumor transplantation. This explains the exponential behavior of the population growth- at the time of tumor transplantation. The mitotic cycle of 8.4 hr is predicted for this stage of growth. Subsequently, at a tumor population of about 200 million cells the increase In the mean generation time of the proliferating cells becomes rapid. This corresponds to a pronounced decrease in the rate of tumor growth that occurs in experimental tumors at this stage of growth. On the whole the mean generation time of the proliferating cells does not remain constant as reported by Baserga (1963), but it increases (as reported by Lala & Patt, 1968) from 8.4 hr in the early stages of the tumor to 28 hr in the final stages of the tumor. At the time of the tumor transplantation the transition of cells to the resting states is very slow, and the growth fraution is very close to 1. With increasing of the size of the tumor, the transition of proliferating cells to the resting states increases, but even at a 600 million cell tumor the effect of this transition in retardation of the rate of growth of EAT is slight compared to the effect of the prolongation of the mean generation time of the proliferating cells. When the total population reaches 900-1000 million cells then the transition of proliferating cells to the resting states increases very sharply. This has the effect of reducing the growth fraction rapidly, and also altering the cell age and size and DNA distributions significantly. The rate of growth of population at any time, then, is a function of the total cell population. The rate of cell death from the proliferatingipopulation is small for early stages of the tumor, but it is higher and approaches a constant rate for older tumor. The rate of cell death in the nonprolifbrating group prior to day 6 does not effect the growth of the total population appreciably, since prior to day 6 the nonproliferating population is a small part of the total population. The rate of nonproliferating cell death from population of 600 million cells on does not change appreciably. The rate of the, overall cell death determined from the model are generally in good agreement with the experimental data. The effect of cells in Gz, M-arrest on the cell $ize distribution is the appearance of a second peak in the size distribution with the aging of the tumor, and also, increasing mean cell volume of cells in the tumor. It may be concluded from this study that the process of transition of proliferating cells to resting states depends on the, sire of the population of the tumor. The latter in turn depends on the change in growth media and the increase in some type of growth inhibitor brought about by aging of the tumor. If, in fact, the growth media is to be altered by addition or removal of certain chemicals, without changing the cell population, then the rate of transition of cells to resting states must accordingly be modified. This modification could easily be introduced in the model.
458
M.
KIM,
K.
BAHRAMI
AND
K.
B.
WOO
In the model used, it was assumed that the transition of proliferating cells to resting states occurs from proliferating cells in age compartments 23 and 24. The assumption was motivated by experimental data as well as the experimental cell volume distributions. If, in fact, the transition of proliferating cells to the resting states occurred somewhere else in the cell cycle, again the discrete-time model can easily be modified to accommodate these changes. As it is already evidenced, in the model used here for simulation of kinetics of EAT population, which is nonlinear, there are several parameters that are selected simultaneously. Since these parameters are interacting and the model is nonlinear, it may be very difficult, if not impossible, to find meaningful analytical relationships between them. In this study a general quantitative model for the dynamics of the cell cycle kinetics and proliferation is developed and it is used to analyze and simulate the cell cycle kinetics of Ehrlich ascites tumor and to predict various parameters of the cell cycle kinetics. The authors wish to express their thanks to Dr Seymour Perry, Deputy Director, Division of Cancer Treatment, National Cancer Institute, for support and interest in this work. REFERENCES ANDERSWN, G. K. A. & AGRELL, I. P. S. (1972). Virchows Arch. A&. BASERGA, R. (1963). Arch. Path. 75, 156. BASEKGA, R. (1965). Cancer Res. 25,581. BELL, G. I. &ANDERSON, E. C. (1969). Biophys. J. 4, 329. CLEAVER, J. E. (1967). Thymidine Metabolism and Cell Kinetics.
CeilPurh.
11, 1.
Amsterdam: North
Holland Publishing Co. DE-E, J. J. & GELFANT, S. (1968). Science, N. Y. 162, 698. DOMBERNOWSKY, P., BICHEL, P. & HARTMAN, N. R. (1974). Cell Tissue Kinetics 7,47. FRIED, J. (1970). Muth. Biosci. 8,379. FRINDEL, E., VALLERON, A. J., VASWRT, F. & TUBIANA, M. (1969). CeN Tissue Kinetics 2,51. HAHN, G. (1966). Biophys. J. 6, 275. HAHN, G. (1970). Math. Biosci. 6, 295. KIM, M., BAHRAMI, K. & Woo, K. B. (1973). Proc. 26th A. Co& Engng Med. Biol. 15,72. LALA, P. K. (1971). In Methods in Cuncer Research (H. Busch, ed.). New York: Academic LALA, P. K. (1972). Cancer 29,261. LALA, P. K. & PATT, H. (1968). Cell Tissue Kinetics 1, 137. LE~O~ITZ, J. L. & RLJBINOW. S. I. (1969). J. theor. Biof. 23,99. OMINE, M. & PERRY, S. (1972). J. natn. Cancer Inst. 48, 697. PAN, H. M. & STRAUBE, R. L. (1956). Ann. N. Y. Acad. Sci. 63,728. SARNA, G. P. (1973). In press. SCHERBAUM, 0. & RANCH, G. (1957). Actu path. microbiof. scand. 41, 161. SHACKNEY, S. E. (1973). J. theor. Biol. 38,305. STEEL, G. G. (1968). Cell Tissue Kinetics 1, 193. STEWART, P. G. & HAHN, G. (1971). Cell Tissue Kinetics 4,279. TAKAHASHI,M. (1968). J. theor. Biol. 18,202. TOBEY, R. A. L CRISSMAN, H. A. (1972). Cuncer Res. 32,2726.
CELL
CYCLE
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MODEL
APPENDIX
Relationship between Proliferating Cells Mean Generation tie Age of lhmor
and
In this appendix, comparing Z(T) with the experimental Z(t), a relationship between proliferating cell mean generation time To and age of tumor is developed. It is required that the simulated growth Z(T) = Z(kAT,) = Z(k) coincide with the experimental growth Z(r). For any k = k,, the quantities Z(k,) and Z(k, + 1) are known from simulation of the model. Since Z(t) is monotonically increasing in t then there exists t, and ts, both unique and t, > t, such that z(h) = Z&) and Z(t,) = Z(k, + 1) Denote the rate of increase in Z(t) at t = t1 by r,(t,). For t, close to t,, using a lirst order approximation, the rate of growth is
On the other hand the rate of growth of Z(T) for T = k, AT, is
r,tWTJ
=
Ztki +l)--Z(h)
AT, Coincidence of Z(t) and Z(T) requires that
*
(442)
r,(h) = r,Uw%). (A3) Equations (Al), (AZ) and (A3) imply s AT, = (tz - tJ. (A4) From equation (A4) and the definition of AT, it follows that T,(t,) = nAT,(tr) = n(ts-ti). (A% Equation (A5) expresses the proliferating cells mean generation time at time tr in terms of f, and t2 where t, and tz correspond to discrete times kl and kl + 1, respectively.