J. theor. Biol. (1977) 68, 27-42

Mathematical Methods for Determining Cell DNA Synthesis Rate and Age Distribution Utilizbg Flow Microfluorometryt M. KIM$. School of Electrical Engineering,

Cornell University,

Ithaca,

New York 14853, U.S.A. AND SEYMOUR PERRY National Institutes

of Health, Bethesda, Maryland,

(Received 7 July 1976, and in revisedform

U.S.A.

3 December 1976)

Methods are developed for analyzing and interpreting DNA distributions of cell populations experimentally obtained by flow microfluorometric techniques. The first method derived can be used to compute the mean DNA synthesis rate and a transformation matrix relating a cell age distribution to its corresponding DNA distribution when these two distributions areiknown. This transformation matrix is used to compute the cell DNA distribution from a cell age distribution. The second part deals with computing the initial cell age distribution when several measurements of cell DNA distribution are available. This problem is shown to be equivalent to the observability problem for the discrete-time system representing the cell population. The observability condition in system theory and a computing method for determining the initial cell age distribution are discussed. An example using DNA distributions of Chinese hamster cells for the methods is presented. 1. IntrodlIction

Cell cycle and proliferation kinetic studies have played an important role in elucidating the dynamics of tumor cell populations (see Baserga, 1965; Cleaver, 1967; Lala, 1971; Skipper & Perry, 1970). The radioautographic t The research in this paper is supported in part by the Grant Number lROICA-16259, awarded by the National Cancer Institute. This paper was presented in part at the 1975 Winter Computer Simulation Conference, Sacramento, California, U.S.A., 18-19 Ikernher 1975. $ To whom all correspondence, requests for reprints and requests for the computer programs developed in this paper should be addressed. 21

28

M.

KIM

AND

S. PERRY

techniques (Cleaver, 1967) have been extensively used in experimental study of the cell cycle kinetics. Recently the introduction of flow microfluorometry (FMF) (VanDilla, Trujillo, Mullaney & Coulter, 1969; Steinkamp et al., 1973; Gohde, 1973) makes it possible to determine rapidly the DNA content per cell and to generate the DNA distribution for a large number of cells. The FMF technique has been increasingly employed in studying the cell cycle kinetics (see Kramer, Deavon, Crissman & VanDilla, 1972; Haanen, Wessels & Hillen, 1975; Tobey & Crissman, 1972). Quantitive methods for analyzing FMF data were reported (Baisch, Ghode & Linden, 1975; Dean & Jett, 1974). Kim, Bahrami & Woo (1974, 1975) developed mathematical representation of the relationship between cell age distribution and its cell DNA distribution with knowledge of the mean DNA synthesis rate. Usually the mean DNA synthesis rate of a cell population is not readily available. In this paper analytical methods will be derived for computing the mean DNA synthesis rate and the transformation matrix relating cell DNA and age distributions and computing the cell age distribution from FMF data. 2. Analytical Methods (A) MEAN

DNA

SYNTHESIS

RATE

AND

TRANSFORMATION

MATRIX

The problem is to compute the mean DNA synthesis rate of a cell population from an experimentally-determined cell DNA distribution when its corresponding cell age distribution is known or can be estimated. When solved, it should also give the transformation matrix relating cell age distribution to its corresponding cell DNA distribution. The experimental cell DNA vector Z(k) for experimental cell DNA distribution is defined. z”(k) = transpose of [Z,(k), . . . , Zi(k), . . . , z”,(k)],

(1)

where &(k) is the number of cells at DNA content state i at time k observed experimentally. Elements of Z(k) form the cell distribution according to the DNA content. Similarly, the cell age vectors 2(k) for proliferating cells and q(k) for nonproliferating cells are defined, 2(k) = transpose of [x,(k),

. . ., x,(k), . . ., x,,(k)],

q(k) = transpose of [q,(k), . . ., q&)1,

1

(2)

where xi(k) is the number of cells at age compartment i at time k. In order to derive an algorithm for computing the mean DNA synthesis rate of a cell population from experimental cell DNA distribution and cell age distributions, the following assumptions are made.

ANALYSIS

OF

FMF

CELL

DNA

29

DATA

(1) The mean DNA synthesis rate is a function of the cell age within the cell cycle. A curve for the change in the DNA content of a single cell between successive age compartments within the S phase is assumed. Thus the DNA content of the cell must increase monotonically as it advances through the S phase of the cell cycle. (2) It is assumed that the dispersion due to instrumental noise and nonuniform staining is described by the Gaussian distribution. The probability

density function for the dispersion is given by gji(di) = (*n)t,“oj __--~-

exp [ - l/2 (“i,dy]

*

where dj is the true mean DNA content before the dispersion due to the artifacts and cJiis the DNA content of the ith DNA content compartment. The relationship between the true cell DNA content distribution and experimentally observed cell DNA content distribution is, Z(k) = Gz(k),

(4)

where L(k) is the r-dimensional true cell DNA content vector at time k. The element of row i and column j of matrix G is given by equation (3). The coefficient of variation is assumed to be the same for all DNA compartments. (3) Cells in a particular cell age compartment do not necessarily contain an identical DNA content. The DNA synthesis rate within a cell age compartment is assumed to be constant. This approximation can be improved by increasing the total number of cell age compartments. The relation between the true cell DNA distribution (without the dispersion due to instrumental and staining artifacts) and the cell age distribution at time k can be given z(k) = R,Z(k)+R,q(k),

where R, is the r x n transformation transformation matrix.

(5)

matrix

and R, is the TX m

When a true DNA distribution and its corresponding age distribution are known, the mean DNA synthesis rate and the transformation matrices R,, and R, in equation (5) and G in equation (4) can be determined. The cell age distribution is usually not experimentally observable and thus needs to be estimated, e.g. the asymptotic cell age distribution for an exponentially growing cell population is well known. If the cell population has noncycling cells, the asymptotic cell age distribution may be computed by using certain cell cycle kinetic models (Hahn, 1966; Kim et al., 1975).

30

M. KIM

AND

S. PERRY

The first step is to compute the true cell DNA distribution by using equations (3) and (4). If some parameters in the probability density function gji are not known, this can be estimated by using methods of nonlinear least squares fit, such as the Levinberg-Marquardt method developed for computer programs (see The IMSL Library, 1974). Once the true DNA distribution is determined, this and the cell age distribution can be used to compute the mean DNA synthesis rate and the elements of R, and R, as follows. As cells progress through the S phase, changes in the DNA content are determined by the DNA synthesis rate. Since the DNA synthesis rate within a cell age compartment is assumed to be constant, the curve for the DNA content vs. the compartment i can be represented as in Fig. 1. The flow diagram of an algorithm for determining

Mean cell age

Fro. 1. The mean contents for G1 and jth age compartment tions are determined constant within any

DNA content vs. the mean cell age of a cell population. The DNA Ga + M are 1 and 2 respectively. Certain fractions of cells in the are assigned to the (i-l), i, and (i + 1) DNA compartments. The fracas shown in the figure. Here the DNA synthesis rate is assumed to be age compartment but varies one age compartment to another.

the DNA synthesis rate and transformation matrix R is shown in Fig. 2. This algorithm is developed by computing the fraction of cells in the jth cell age compartment with different DNA contents corresponding to DNA compartments (i-l), i, (i+l) which is a function of the DNA synthesis rate. This algorithm also computes the elements of transformation matrix R,. Note that elements of the first and last row of R,, are neither 1 or 0 and these

ANALYSIS

,

OF

FMF

CELL

DNA

31

Q(i,j)=O/X ADNAcj,=ADNAIj)tz Q(i, jr =S/X,

$ S--S-D



4 0=0-s

s=xj

4

i=iti

j--j-Cl

N",

DATA

iMu?

i&Jyy

FIG.2. Flow diakram for computing the mean DNA synthesis rate from the true cell DNA content distribution and the cell age distribution. Subscript i indicates the ith DNA content compartment. Since only cells in the S phase are considered, i = 1 is the DNA content interval (1, 1 + Ap), i.e. the first DNA compartment for cells in the G1 phase is excluded. Subscript j indicates the jth cell age compartment in the S phase. 2, is the number of cells at the ith DNA content compartment. xI is the number of cells at the jth cell age compartment in the S phase. EDNA (j) is the amount of DNA synthesized during the jth cell age interval. i.e. the synthesisrate. NX is the total number of the cell age compartments in the S phase. MZ is the total number of the cell DNA content compartments between 1 (for G1) and 2 (for Ga + M). Q(i, j) represents the fraction of cells in the jth S phase age compartment assigned to the ith DNA compartmnet.

can be determined by inspection because the elements of the first row are the coefficients for xi in the G, phase and those of the last row for xi in the G2 + M phases. Similarly, R, can be determined by inspection. (B) A METHOD

FOR DETERMINING

CELL

AGE DISTRIBUTION

It is useful to have knowledge of the cell age distribution of a cell population at any given time, since the cell age distribution describes the cell cycle kinetic state. An analytical method will be derived to determine the cell age distribution from experimental DNA distribution by using the cell DNA cell age relationship developed (equations (3), (4) and (5)) and represented by

32

M.

KIM

AND

S. PERRY

(Hahn, 1966; Kim et ul., 1974) l(k+l) = @,,jZ(k)+a&q(k), 0 + 1) = @m,W + (D,,q(k), >

(6)

where Q’s are the local state transition matrices expressing the transformation from 2 and q at time k to si and q at time k + 1. Now the problem is defined as follows. Given experimental distributions at time 0,. . . , S described by Z(O), Z(l), . . . , Z(F), one is required to determine the corresponding cell age distributions. The first step is to compute the true DNA distribution from experimental DNA distribution by using equations (3) and (4), as described in the previous section. The second step is to use the computed DNA vectors z’(O),. . ., Z(S) and equations (5) and (6) to determine the corresponding cell age vectors. For convenience the notation for equations (5) and (6) is simplified. z(k) = Rx(k),

(7)

x(k+ 1) = @x(k),

(8)

where x(k)=

[ii;],

Q=

[:%I,

R=[R,R,].

Here, x(k) = the (n +m) dimensional vector, @ = the (n +m) x (n +IM) local state transition matrix, and R = the r x (n +m) transformation matrix. Q, and R are assumed to be given. In this problem it is desired to determine at least x(0) from z(O),. . .,z(S), and, if possible, also x(l), . . . , x(S). Consider equation (7) with k = 0 z(0) = Rx(O).

(9)

Since the dimension r for z is not necessarily the same as the dimension (n+m) for x, we premultiply both sides of equation (9) by the transpose of the transformation matrix R. RTz(0) = R’Rx(0). Then (RTR) is (n +m) x (n +m) matrix and if (R’R) is invertible, is obtained, x(0) = (R*R) - ‘RT z(0).

(10) then x(0)

(11) x(0) is the linear least square solution of equation (9) (Dahlquist 8z Bjorck, 1974). The rank of the rectangular matrix R and the square matrix (R*R) is (n+m) if (RTR) is invertible. If the rank of R is less than (n +m) the additional z’s are included by using equations (7) and (8).

ANALYSIS

Where

OF

FMF

CELL

DNA

33

DATA

z(O) z(l)=@,,xtO), Id

(12)

z(k - 1)

O(k) =

R Rf& :

i-1

(13)

ROk-l

The minimum value of k for which the rank of 6(k) is (n +m) is the number of z’s required to determine x(O). If this k exists, the system described by equations (7) and (8) is defined to be observable and this k cannot exceed (n +m) (Padulo & Arbib, (1974). Then x(0) can be obtained

(14)

In system theory 8,,, is called the observability is the observability index. (C)

APPLICATIONS

OF TCHEBYCHEFF

POLYNOMIALS

matrix and the minimum

AND

LEAST

SQUARE

k

METHODS

Sometimes it is convenient to approximate the true DNA content vector, z, in equations (4) by a polynomial in which the channel number for DNA contents is the independent variable, as done by Dean & Jett (1974). Polynomial approximations, when properly selected, can smooth out measurement errors. We have found that the Tchebycheff polynomials are effective for approximating z. The Tchebycheff polynomials belong to a triangle family of orthogonal polynomials (Dahlquist & Bjorck, 1974). The elements of z can be expressed by the ith degree Tchebycheff polynomials,

z=

To(cl)Tltcl). TotG)Tl(G). .

1.Toiwltcr~ = T,a,

* * T- ltclvi(cl) * * Ti- l(C,)T,(C,)

a0 a,

(13

. * * T- :;(c,)T& : Ii:1 6, (16)

34

M.

KIM

AND

S. PERRY

where Ci is the channel number and Ti(Cj) and aj are the polynomial and the coefficient of the ith degree term, respectively. Then computation of the true cell DNA distributed by using equations (3) and (4) can be represented as the following problem. Find a which minimizes (17)

IIGT,a--ZII

subject to the constraint, z = T,a 2 [O]. The constraint is introduced because all the elements of z must be nonnegative. This is a linear least square problem with inequality constraints. In order to solve this problem, we introduce the QR decomposition of the transformation matrix GTa in equation (17) and compute Qz, where Q is the orthogonal matrix in the QR decomposition (Lawson & Hanson, 1974). Then the problem can be solved by using the least distance programming method (Lawson & Hanson, 1974). Similarly, cell age distributions can be represented by the Tchebycheff polynomials as a function of the cell age which represents cell compartments in x and q. Again, computation of the cell age distribution from cell DNA distributions can be done by the least distance programming method, when the QR decomposition is introduced.

3. Experimental

Data for Analysis

FMF data for Chinese hamster cells (CHO cells) will be analyzed by the derived methods. Puck, Sanders & Peterson (1964), reported the cell generation time T of CHO cells to be 12.4 h while Kramer et al. (1972) said the generation time T to be 16.5 h and also T,, for the Gl phase and Ts for the S phase are functions of cell concentrations and culture conditions for FMF experiments. The CHO cell population grows exponentially and thus they are cycling cells. However, for other cell populations which include noncycling cells, the state vector q(k) for non-proliferating cells must be included. (A) DNA

SYNTHESIS

RATE

The cell DNA distribution of exponentially growing CHO cells in Fig. 3 (Kramer et al., 1972) is used to compute the mean cell DNA synthesis rate and the transformation matrix for cell age and cell DNA distributions. The mean cell cycle time of this CHO cell population is 16.5 h. Since the CHO cells considered here are exponentially growing, the asymptotic cell age distribution is (Hahn, 1966, Kim et al., 1974) XT= [N/pj

p-‘/”

)...) 2-‘/n )... 2-q.

(18)

ANALYSIS

OF

FMF

DNA

CELL

content

DNA

DATA

35

IceI I

3. The cell RNA distribution of CHO cells (from Kramer et al. 1972). 0, Expericomputer fit; - - - - , computer fit S distribution. mental data points; -, FIG.

Here N is the total cell population and n is the total number of jthe proliferating cell age compartments. The total number of the age compartments is chosen to be 16. The total number of the DNA content compartments is 80 which is the same as the maximum fraction number in the experimental cell DNA distribution. Using the nonlinear least square fit of the experimental cell DNA content distribution in Fig. 3 of step 1, one can determine the cell DNA distribution with instrumental artifacts. For this DNA data, the cells in the G1, S, and G,+M phases constitute 53.6x, 38.9x, and 7.5% of the total population, respectively. The same results were also obtained by Dean & Jett (1974). The fraction numbers corresponding to the DNA contents of cells in the G, and Gz phases (normalized DNA value of 1 and 2) are 36 and 72 respectively. The true cell DNA distribution of cells in the S phase (after eliminating the dispersion due to instrumental and staining artifacts) is shown in Fig. 4. This distribution excludes cells in the G, and Gz + M phases. From the given fraction of cells in the Gr, S, and Gz i-M phases and the given cell age distribution for exponentially growing cell population from

36

(a) I‘ M.

5ooo

=” 2

KIM

AND

S. PERRY

S phase cclIs

0

S phase cell in age ccmpartment

Cell CM ccntent fw only S phase krrmlized )

FIG. 4. The cell age distribution of S phase cells of an experimentally growing cell population. This age distribution is used for the CHO cells. (b) The cell DNA content distribution of the CHO cells in the S phase after eliminating dispersion due to instrument noise and staining artifact.

equation (18), one can compute the phase durations of the G,, S, and G, -I-M phases to be 7.6 h, 7-2 h, and 1 a7 h respectively. The cell age distribution of cells in the S phase is shown in Fig. 4(a). From the algorithm the cell DNA content and the DNA synthesis rate are computed and shown in Fig. 5. It is noted that the DNA synthesis rate is maximum slightly before the midpoint of the S phase. An exponentially growing cell population of L5178Y was studied for the DNA synthesis rate. The experimental cell DNA content distribution (Dean & Jett, 1974) was converted to the distribution without dispersion due to

sphlse

cell agehqe

cm-qrtment

s phase cell ape in qe comportment

FIG. 5. (a) The mean cell DNA content vs. the cell age computed by using the method derived. (b) The computed mean DNA synthesis rate of the CHO cells as a function of cell age.

ANALYSIS

OF

FMF

CELL

DNA

DATA

37

instrumental and staining noise which gave the fractions of cells in the G,, S, and G, + M phases to be 20-l %, 64.3 %, and 15.5 %. The cell cycle time used is 12 h. The cell DNA distribution for only the S phase and the computed DNA synthesis rate is shown in Fig. 6.

-

1

Cell DNA cadent bx only S phase hxnalized)

s phase cell age I” age ccmpartment

FIG. 6. (a) The cell DNA distribution of L5178Y cells in S phase after eliminating the dispersion due to instrument noise and staining artifacts. The experimental DNA distribution from which this was computed is from Dean & Jett, (1974). (b) The computed mean DNA synthesis rate of the L5178Y cells as a function of cell age.

(B) CELL

AGE DISTRIBUTION

Again, FMF data for CHO cells are used in this example of computing the corresponding cell age distributions from cell DNA distributions. The experimental DNA distributions of CHO cells (Kramer et al., 1972) were recorded at various times (2.5, 3.5, 7.5, 8.5, 12.0, 13 h) after release from thymidine block which is known to reduce the rate of DNA synthesis drastically. It is desired to compute the corresponding cell age distributions at as many time points as they can be computed. The true cell DNA distributions were computed by using equation (4). The optimum value of the coefficient of variations is found to be O-69. Also the transformation matrices R, and R, for R in equation (7) computed in the previous example are used. The rank of the transformation matrix R in equation (7) is found to be less than 16 (= the dimension of x(k)), thus (R*R) in equations (10) and (11) is not invertible. This means that one cannot determine x(0) only from z(0) and additional z’s at time k > 0 are required. Also the local state transition matrix ip in equation (8) is required. Since all the CHO cells are

38

M.

KIM

AND

S. PERRY

cycling cells, Q, can be shown to be (Hahn, 1966)

P

2a

1 -a-j?) @=

;

:::

0 0

i

(1-a-B) a

... .. . .. .

6 0

0 0

::: . ..

)-

2(1-a--/3: 2a 0 0

/i W--a--8>

where a is the probability that cells in an age compartment advance two compartments after a unit time, and B is the probability that cells in an age compartment do not advance to the next compartment after a unit time. a and j are assumed to be the same since the probability distribution of the cell cycle time can be assumed to be Gaussian. Then only single parameter needs to be determined for @. a must be between 0 and 0.5. Since a is not known a priori, various values of from 0.05 to O-5 are used. For these values, the rank of the observability matrix tIC2)in equation (13) is computed to be 16. Thus equation (14) for x(0) exists, i.e. the following equation can be solved for x(0).

Therefore, for this system only z(0) and z(1) are required to solve for x(0). Since at time 0 the cells!are released from thymidine block, all the cells are assumed to be at the last age compartment forithe Gl phase. Among various a’s the best fit is given by a = O-15. The computed cell age distributions at time 2.5, 7.5, and 12 h are shown in Fig. 7. Each cell age distribution is

12 h

2*5t

I

--I

G,

L

S G2+M

FIG. 7. The computed cell age distribution release from thymidine block.

L

G, SG2+M Cell age

G2+

of CHO cells at time 23, 7.5, and 12 h after

ANALYSIS

OF

FMF

CELL

DNA

39

DATA

determined from two cell DNA distributions at the corresponding time and 1 h later. Note that the cells are still reasonably synchronized at 2-5 h after the release from thymidine block but become more unsynchronized from 7.5 h to 12 h. The computed true DNA distributions (without instrumental artifacts) are shown in Fig. 8. The cell DNA distributions computed by using the cell age distributions in Fig. 7 and equations (7) and (8) are plotted in Fig. 9 with the experimental data superimposed.

Ik”,lfJ=!,

15 000

0

22

44

22 CNA

44

22

44

confenl

FIG. 8. The computed true cell DNA distributions (without instrumental and staining artifacts) of CHO cells at time 25, 75, and 12 h after release from thymidine block.

. iii I 4

12h

I

ad, 4 22 DNA

content

44 (Fation

22

44

IX).)

FIG. 9. The computed cell DNA distributions of CHO cells at time 2.5, 7.5, and 12 h after release from thymidine block. The experimental data are from Kramer et al. (1972). 0, Experimental point; -, computed.

4. Discussion Quantitative methods are derived for determining certain cell cycle kinetic parameters from FMF DNA data. The first algorithm demonstrated to be

40

M. KIM AND

S. PERRY

capable of determining the mean DNA synthesis rate of a cell population from a FMF cell DNA distribution and its corresponding cell age distribution in addition to calculating the fraction of cells in the G,, S, and G2 +M phases. It also computes the matrix which transforms a cell age distribution at time k into the corresponding cell DNA distribution at time k. The cell age distribution of a cell population cannot be usually obtained by direct experimental measurements. Therefore, the cell age distribution is needed to be estimated for the algorithm. For an exponentially growing population the asymptotic cell age distribution is known. For a cell population comprising cycling and noncycling cells, one can use the growth fraction and other cell cycle kinetic information to compute the asymptotic cell age distribution (Kim et al., 1974). In the algorithm the cell DNA and age distributions are represented by the cell DNA content and age vectors which are discrete representations of the distributions. Accuracy can be improved by increasing the number of compartments representing the discrete DNA contents and cell age. The transformation matrices R, and R, in equation (5) determines the cell DNA content distribution from a cell age distribution. This relation is true as long as the mean DNA synthesis rate remains unchanged. If the DNA synthesis rate is changed, the new transformation matrices corresponding to the new synthesis will have to be computed. If there exist variations in the actual DNA content of cells at a particular cell age compartment, these variations need to be represented in transformation matrices R, and R,. Then the mean DNA synthesis rate and transformation matrices G, R,, and R, can be more conveniently determined simultaneously by using methods of nonlinear least square fit. In the second part of this paper it is shown that FMF DNA data can be used to compute the cell age distribution at a particular time considered. If the rank of the transformation matrix R is (n+m), (RTR) in equation (10) is invertible. Thus equation (10) is solvable for x(0). That is, in order to compute the cell age distribution at timej, only the corresponding cell DNA distribution at time j is required. Let us examine R = [R, R,] in equations (5) and (7). R, for the cycling cell age vector x(t) has the form,

Rn= lnp,l 1 ns j Y

'0'

/

1

n(cz+~) 0

I Rs I I !’ l...l

I the number of age compartments in G,, n, = the number of where nG1 = age compartments in S, ~(o*+~) = the number of age compartments in

ANALYSIS

OF

FMF

CELL

DNA

DATA

41

G2 +M, R, = the (r x ns) matrix determined by the algorithm with the DNA synthesis rate. The first no, columns of R,,has only one linearly independent column and the last n(02+M) column also has one. Therefore, the maximum number of linearly independent columns of R, is possibly (n,+2) or in R the maximum is (nsf2+nt). If no,and Ito2+M are not one, the rank of R is less than (n + m). Then one needs additional DNA distributions at timej+ 1, j+ 2,. . . ,j+ k - 1, until the rank of the observability matrix m’(k) in equation (13) becomes (n+m). In equation (14) we need the state transition matrix 0 to compute the cell age distribution. It is computationally desirable to solve equation (12) by using a nonnegative least square method (Lawson & Hanson, 1974) than to solve equation (14) since the cell age vector x(j) is nonnegative. If FMF DNA distributions are not measured consecutively, e.g. at time O&10, then the observability matrix is

R Oc3)= RQ5 . R@'O

[1

If the rank of o(3) is (n + m), then the following equation is to be solved for x(O).

It is found that if the interval between one measurement and the next is large, the next measurement should be timed accurately so that the interval becomes integer multiples of the unit time. Otherwise inaccuracy results in the computation. In the CHO cell population simulated, the state transition matrix CDwas not known but was determined in the process of computing the cell age distributions and the parameter CI which gave the best fit. The parameter rx can be regarded as the rate of unsynchronization. Recently there has been a rapid increase in FMF study of antitumor agents (see Haanen et al., 1975; Schumann, Ghode & Buckner, 1975; Tobey & Crissman, 1972; Yataganas, Strife, Perez & Clarkson, 1974). The knowledge of cell age distributions computed by the method can be potentially very useful in the study of the effects of antitumor agents on tumor populations. The cell age distributions before and after the application of a drug can be used in conjunction with the state equation (6) and additional vector for sterilized cells due to chemotherapy to study analytically the sensitivity of a phase in the cell cycle to the drug.

42

M.

KIM

AND

S. PERRY

REFERENCES BAISCH, H., GOHDE, W. & LINDEN, W. (1975). Radiat. Environ. Biophys. 12, 31. BASEROA, R. (1965). Cancer Res. 25, 581. CLEAVER, J. E. (1967). In Frontiers Biol. 6, Chs. 4, 5, 6. DAHL~UIST, G. & BJORCK, A. N. (1974). Numerical Methods, p. 137. Englewood Cliffs, N. J. : Prentice Hall. DEAN, P. N. & Je-rr, W. M., (1974). J. Cell Biol. 60, 523. GOHDE, W. (1973). In Fluorescence Techniques in CellBiology, p. 79. (A. Thaer, & M. Servetz eds), Berlin and New York: Springer. HAANEN, C., WESSELS,H. & HILLEN, J., (1975). Proceedings of 1st International Symposium on Pulse-Cytophotometry. Ghent, Belgium: European Press. HAHN, G. M. (1966). Biophys. J. 6, 275. The ZMSL Library, (1974). Vols 1 and 2. Houston, Texas: International Mathematical and Statistical Libraries, Inc. KIM, M., BAHRAMI, K. & WOO, K. B. (1974). IEEE Trans. biomed. Engng. 21, 387. KIM, M., BAHRAMI, K. & Woo, K. B. (1975). J. theor. Biol. 50, 437. KRAMER, R. M., DEAVEN, L. L., CRISSMAN, H. A. & VANDILLA, M. A. (1972). Adv. Cell

molec. Biol. 2, 47.

LALA, P. K., (1971). Meth. Cancer Res. 6, 3. LAWSON, C. & HANSON, R. (1974). Solving Least Square Problems, p. 158. Englewood Cliffs N. J.: Prentice Hall. PADULO, L. & ARBIB, M. (1974). System Theory, p. 254. Philadelphia: W. B. Saunders. PUCK, E. T., SANDERS, P. & PETERSON, D. (1964). Biophys. J. 4, 441. S~~JMANN, J., GOHDE, W. & BUCHNER, T. (1975). Proceedings of 2nd International Symposium on Pulse-Cytophotometry. Ghent, Belgium: European Press. SKIPPER, H. & PERRY, S. (1970). Cancer Res. 30, 1883. STEINKAMP, J. A., FUL\KYLER, M. J., COULTER, J. R., HIEBERT, R. D., HORNEY, J. L. & MULLANEY, P. R. (1973). Rev. Sci. Instrum. 44, 1301. TOBEY, R. A. & CRISSMAN, H. A. (1972). Cancer Res. 32, 2726. VANDILLA, M. A., TRUJILLO, T. T., MULLANEY, P. R. & COULTER, J. R. (1969). Science, N. Y. 163, 1213. YATAGANAS, X., STRIFE, A., PEREZ, A. & CLARKSON, B. (1974). Cancer Res. 34,2795.

Mathematical methods for determining cell DNA synthesis rate and age distribution utilizing flow microfluorometry.

J. theor. Biol. (1977) 68, 27-42 Mathematical Methods for Determining Cell DNA Synthesis Rate and Age Distribution Utilizbg Flow Microfluorometryt M...
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