A cytokinetic model for heterogeneous mammalian cell populations

J. theor. Biol. (1974) 44, 49-90

A Cytokinetic Model for Heterogeneous Mammalian Cell Populations lI. Tritiated Thymidine Studies: T h e Per Cent Labeled Mitosis (PLM) Curve STANLEY E . SHACKNEY

Office of the Deputy Director, Division of Cancer Treatment, National Cancer Institute, Bethesda, Maryland 20014, U.S.A. (Received I March 1973, and in revised form 18 July 1973) A model is described for the simulation of tritiated thymidine radiotracer studies in mammalian cell populations, which takes into account intracycle age-specific variations in the rate of DNA synthesis and tritiated thymidine incorporation, and their relation to tritium detection characteristics by radioautographic methods. The per cent labeled mitosis curve is considered in detail. The model suggests that the per cent labeled mitosis curve alone does not contain sufficient information for the precise characterization of age structure and growth behavior in heterogeneous cell populations; however, in many instances certain features can be identified which permit the classification of population growth behavior in accordance with broad, but nonetheless useful criteria.

1. Introduction In an earlier paper (Shackney, 1973), a general model was proposed for the simulation of the growth behavior of mammalian cell populations as measured by simulated parameters. In this paper, we wish to extend the model to simulate studies employing tritiated thymidine as a radiotracer for D N A synthesis, and explore the validity of inferences that may be drawn from such studies with regard to the age structure of growing cell populations. In this paper we shall concern ourselves primarily with the per cent labeled mitosis (PLM) CUrVe.

2. Description of the Model (A) THE GENERAL MODEL

For the reader's convenience, the salient features of the general model are described here briefly: T.B.

49

4

50

S.E.

SHACKNEY

(a) Cell cycle time distribution characteristics Growing cell populations are characterized by a unimodal cell cycle time distribution whose mean and variance increase with increasing population size. The cell cycle time distribution function is intended to describe a set of constraints imposed on growing cells by characteristic interactions between the aggregate population and its environment.

Co) Maturation rate characteristics The growth of individual cells is further constrained by a rate controlled maturational sequence which must be completed between one division and the next. In keeping with earlier notation, we give the relative maturation age function as am (3, Tc), where Tc = cell cycle time, 0 <~ ~ ~< Tc, a, (O, Tc) = 0 , and a . (Tc, rc) = 1. This function characterizes relative maturational achievement as a function of time elapsed since last mitosis, and cell cycle time. It is assumed to be a monotonically increasing function as x increases from 0 to Tc. On differentiating am (3, Tc) with respect to 3, dm (3, Tc), the relative maturation rate function is obtained. If we choose DNA content as the parameter for maturationaI achievement, the immediate postmitotic or diploid nuelear D N A content corresponds to am (0, Tc), and the immediate premitotic or tetraploid value corresponds to am (Tc, Tc). Under these circumstances, the relative maturation rate function corresponds to the DNA synthesis rate function, where DNA synthesis rate is given as the fraction of the postmitotic DNA content reduplicated per unit time. Some of the properties of the relative maturation rate function are summarized briefly: (a) d~ (0, Tc) = d, (Tc, Tc) = 0; i.e., at mitosis the maturational process is complete, and the rate of maturation is zero. (b) dm (3, Tc) is positive everywhere on the open interval (3, Tc), increasing monotonically to a maximum, and then decreasing monotonically; i.e., all cells are maturing with time, but the ageing rate early and late in the cycle is relatively low.t It should be noted that ifa relative maturation rate function is t Attention is called to the restriction of the term maturation in this context to the cyclical growth process that every cell undergoes between divisions. This is not to be confused with differcntational maturation.

PLbl CURVES IN HETEROGENEOUS

Type I

lntermedtote

POPULATIONS

51

TyDo II

FIG. 1. DNA synthesis rate patterns in relation to cell cycle time lengthening. For explanation, see text. chosen such that early and late ageing rates approach 0, the discrete compartment model of Howard & Pelc (1953) becomes a special case of the present model. (c) With regard to the relation between DNA synthesis rate and cell cycle time lengthening, we shall consider three possibilities, as shown in Fig. 1. The rate of DNA synthesis may decrease at uniform rate throughout the cycle (type I, Fig. 1) the bulk of DNA synthesis may be restricted to the premitotic period (type II, Fig. 1), or the changes in DNA synthesis rate characteristics with cycle time lengthening may have intermediate characteristics. Of course, it is possible that all three patterns may occur in the same cell population. However, as we shall show, current evidence suggests that mammalian cell populations do indeed have distinctive DNA synthesis patterns which can be classified in accordance with the above schema. For certain purposes, we shall have use for alternate forms of the relative maturation age and relative maturation rate functions, C~m(~b, TC) and am (~P,Tc), where ~b = x/Tc, ~m (ok, Tc) = a m (~, Tc) and a m (4, Tc) = dm (% Tc). In this form, relative maturation age and relative maturation rate are characterized as functions of the fraction of the cycle time elapsed. The relative maturation rate function can be displayed conveniently in this form. Examples of type I, intermediate, and type II relative maturation rate functions are shown in Fig. 2. Furthermore, this form permits a more formal statement of the distinction between type I relative maturation rate functions, and other types, namely, that for type I functions

t~ m (d?, Tc) =0 dTc

52

S. E. $HACKNEY Maturation rate characteristics

0.5j

1.0

I-C

1.0

O"

0.5

(a)

(b)

(c)

Fro. 2. The D N A synthesis rate function, ~m (4, Tc). (a) Type I; (b) intermediate; (c) type IT. x-Axis (horizontal); relative position in the cycle; y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 60 hr. For explanation, see text.

and for intermediate and type II functions

a~. (¢, rc) OTc

<0.

(c) The death rate function The third major component of the model is the death rate function. Cell death, or loss of viability, is defined as the permanent loss of the capacity to divide, and is characterized by a death rate which is assumed to be independent of maturation age, but to increase with increasing cell cycle time. Lysis of non-viable cells and clearance of cell debris are assumed to be random processes. Equilibrium population size is achieved when cell birth rate and cell death rate are equal. For a more detailed exposition of the general model, the reader is referred to our earlier paper. (13) TRITIATEDTHYMIDINEINCORPORATION The rate of incorporation of tritiated thymidine into a given ceil is assumed to be proportional to the rate of DNA synthesis in that cell, and to the intracellular concentration of tritiated thymidine, all of which vary with time following pulse administration. Since absolute intracellular tritium concentrations are rarely measured directly, we shall define operationally a radioautographic unit of intracellular tritium concentration in the following way: In a cell that is synthesizing DNA at some reference rate, and is exposed to one radioautographic unit of tritiated thymidine for some reference time

PLM CURVES IN HETEROGENEOUS POPULATIONS

53

duration, the amount of tritium incorporated will be such that one fl disintegration will occur, on the average, over some reference emulsion exposure period. Since the rate of DNA synthesis is given as the fraction of the postmitotic DNA content synthesized per unit time, implicit in the definition of the radioautographic unit is some reference postmitotic DNA content. Correction for postmitotic DNA content is important for intraexperimental comparison of labeling intensity in cell lines of different ploidy. For simplicity, all simulations reported in this paper were performed on populations assumed to be of the same postmitotic DNA content. It should be noted that the radioautographic unit is a measure of tritium activity, and is proportional to the product of tritiated thymidine concentration and its specific activity. Reference values for the radioautographic unit used in the simulation studies described in this paper are as follows: (1) Reference relative DNA synthesis rate = 1. (2) Reference tritiated thymidine exposure duration = 5 min. (3) For simplicity, all simulations in this paper were performed at constant emulsion exposure duration, and constant section thickness. Attention is called to the relation, first proposed by Pelc (1963); relative exposure = tritiated thymidine dose x emulsion exposure time. Thus, for example, a twofold increase in administered dose at constant emulsion exposuret ime is equivalent to a twofold increase in emulsion exposure time at constant dose.

Iooc

8o0

6oo

== 4OO

200

0

I

2

3

4

5

T{me (hi'}

FIG. 3. Simulated intracellular trltiated thymidine pulse characteristics, patterned after the data of Chang & Looney (1965). Dose is given in radioautographic units.

54

S. E. SHACKNEY

With regard to in vivo tritiated thymidine pulse characteristics as a function of time, Staroscik, Jenkins & Mendelsohn (1964), and Chang & Looney (1965), have presented evidence that the bulk of tritiated thymidine uptake occurs within the first hour following injection in the mouse and rat respectively. Rubini, Cronkite, Bond & Fliedner (1960) have presented similar evidence in man. Chang & Looney (1965), have provided the most direct measurements of in rive intraceUular tritiated thymidinc pool concentrations as a function of time following injection in regenerating rat liver. They showed that peak levels were reached within the first half hour, but that measurable activity was still present three hours after injection. So far as we are aware, these data provide the best available quantitative description of in rive pulse characteristics at the tissue level. Accordingly, we shall use a standard pulse whose characteristics are shown in Fig. 3 for most of the simulation studies to be described in this paper. (C) THE SIMULATION OF POPULATION GROWTH IN THE PRESENCE OF TRITIATED THYMIDINE

The population is partitioned and distributed within a three dimensional array according to (a) cell cycle time class, (b) age within the cycle, for each cycle time class, and (c) cumulative tritium content. The movements of cells among the various compartments are summarized in Fig. 4. Within each cycle time class, cells move through progressive age class compartments. As the population grows, cells are also transferred from shorter to longer cycle

Increosing trltiated thymldine incorporation Advancing acJe within the cycle

~ Halving of Tritium content at mitosis

Reduction in acJJncjrote

FIG. 4, A summary of cell pathways in the model, x-axis--intracycle age class; y-axis-cell cycle time class; z-axis---cumulative tritium content class. For discussion, see text.

PLM C U R V E S IN H E T E R O G E N E O U S P O P U L A T I O N S

55

time classes in order to satisfy the cell cycle time distribution function. As cells incorporate tritiated thymidine, they are further transferred to appropriate tritium content compartments. In the last age compartment, tritium content is halved, cell number is doubled, and cells are transferred to the first age compartment with the appropriate tritium content in the same cycle time class. The per cent labeled mitosis curve is obtained by examining the last age compartment in each cycle time class for all uptake classes at hourly intervals, simulated time. In the computation of the labeling index, all compartments are examined. The radioautographic transfer function used to convert tritium uptake]cell to observed grains/cell has been described separately (Shackney, unpublished manuscript). Additional details of the simulation procedure are given in the Appendix. 3. Behavior of the Model System (A) PRELIMINARY CONSIDERATIONS

In order to best describe the behavior of the model system under various conditions, we shall begin with the simplest case, namely that of young, rapidly growing, relatively homogeneous populations, and proceed to examine progressively more slowly growing, heterogeneous populations. We will show that in heterogeneous populations, precise characterization of population age structure at a given stage of growth is difficult, if not impossible, using the PLM curve alone. We will show further that the strengths of the per cent labeled mitosis curve technique in particular, and radioautographic methods in general, are that they are well suited to reveal trends in the changing growth characteristics of populations as they evolve through progressive stages of growth. The importance of PLM curve trend analysis may be appreciated from the schematic diagram shown in Fig. 5. The observation frame for mammalian cell populations is such that as one proceeds from smaller, more rapidly turning over populations to larger and more heterogeneous populations in larger hosts, the duration of the potential observation period increases, but the range of the growth curve accessible to direct observation and study becomes more restricted. In the study of human tumors, the amount and type of data that can be collected are restricted, the populations under study are heterogeneous, rendering data interpretation difficult, and subclinical and partially treated tumors are not subject to direct study at all. The cell cycle time distribution ranges given in the legend of Fig. 5 are to be taken as crude estimates. However, even a cursory evaluation of currently available kinetic data leaves little doubt that the effective range of the cell cycle

56

S. E. S H A C K N E Y I

0

(c)

I

2

3

4 5 6 7 Time (months)

8

9

fO

FI(). 5. The ~ ' o w t h curves and observation windows for (a) a rapidly proliferating mouse t u m o r such as L1210 leukemia; population size is observable over a range o f 105-108 cells, and the time frame for observation spans less than 2 weeks; (b) a solid t u m o r in the mouse.

The population size window spans 5 x 107--5 x 109 cells; the time frame spans about one month; and (c) a human tumor; the size window spans approximately 101°--1012 cells, and the time frame may range from several months to several years. Estimated ranges of cell cycle time distribution: (a) 8 to 40hr; (b) 10 to 100hr; (c) 12 to 600 hr.

time distribution increases with increasing host size. If it were possible to discern PLM curve trends that are related solely to changes in population age structure, such trends would cut across tissue and species distinctions, and would be most useful in reconstructing the age structure and growth behavior patterns of human tumors over a wide range of population sizes. (B) THE PLM CURVE IN RAPIDLY GROWING, RELATIVELY HOMOGENEOUS POPULATIONS

W~ shall take as our reference set of biological data the PLM curves obtained in embryonic mouse tail epithelium (Wimber, 1963; Wimber & Lamerton, 1965) and in very early (5 x 106 cells) mouse Ehrlich ascites tumor (Lala & Patt, 1966), shown in Fig. 6. These are two of the youngest in vivo populations that have been examined by the PLM curve technique. Two well-defined labeled mitotic waves and the beginnings of a third are observed. The first wave is more nearly trapezoidal in shape than the second, and it reaches a sustained plateau at I00~o mitotic labeling. In the case of early Ehrlich ascites tumor, the second wave approaches 100~ mitotic labeling; in the case of embryonic mouse tail epithelium the second wave is somewhat lower. In both instances, the trough between the first and second waves does not fall to zero, reaching values that lie in the 10 to 30~o range. We are unaware of any published PLM curve data in which two near-trapezoidal waves that reach or approach 100K labeling are separated by a trough that falls to zero.

PLM

CURVES

IN

HETEROGENEOUS

POPULATIONS

57

Embryonic mouse tall epithelium I00 80 :J; n

60

4O

/

20 I

0

i

I0

'20

:50 0

(of)

I

I

I0

20

:50

(o~) Time (hr)

Mouse

Ehrlich oscites tumo~

,/

IOO 80 I

w

60

(L

4O 2O

O

I

I0

20

50

(b) Tlme (hr)

FIG. 6. Experimental PLM curves in rapidly growing, homogeneous cell populations. (al) Embryonic mouse tail epithelium, redrawn from Wimber (1963). (a2) Embryonic mouse tail epithelium, redrawn from Wimber & Lamerton (1965). (b) Early mouse Ehrlich ascites tumor. Redrawn from Lala & Patt (1966). The near-trapezoidal shape o f the second wave, and the distinctness of the third wave are indicative of homogeneity in population growth characteristics. We find that we are able to simulate P L M curves with these features if and only if the range of the cell cycle time distribution is quite narrow. Two such simulations are shown in Figs 7 and 8. In both cases, the simulations were carried out in rapidly growing, late log phase populations. The range o f the cell cycle time distribution is 8 to 12 h, and the mean cell cycle time is 8.5 h. The relative maturation rate functions are of the intermediate type. The labeled mitotic waves are trapezoidal in shape; successive waves exhibit progressive rounding and reduction in labeling intensity. Maximum labeling intensity occurs in the midportion of the first wave. Because o f the halving of tritium content with cell division, mitoses with maximal grain counts are not well represented in the second wave.

58

S. E. SHACKNEY

An important feature of the simulated curves is the dependence of the shape and height of the labeled mitotic waves on the background threshold that is chosen to distinguish labeled cells from unlabeled ones. The data of Clarkson, Okhita, e t a & Fried (1967), demonstrate this feature for thresholds exceeding 5 grains/cell. Data obtained in our own laboratory (Shackney, Ford & Wittig, 1973) confirm this finding for a series of thresholds ranging from 1 to 100 grains/cell. In the case of embryonic mouse tail epithelium the background counting threshold was not reported. The simulated PLM curve obtained at a threshold of > 5 grains/cell in Fig. 7 is in good agreement with the data. In the case of early mouse Ehrlich ascites tumor, background threshold is stated as > 5 grains/cell; the simulated PLM curve in Fig. 8, thresholded at > 5 grains/cell, is representative of this population.

2000 r

0'4

1600~ ~- 0'3

12ooh

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I Iiiiiiii I I111111 I0 I00 I000 Cycle time (hr)

I 2

0

(a)

t I :3 4 Time(hr)

I 5

(b)

Maturation rate characferis)ics

I°°rltT~ ffY~

' f ~ ->1 Groins

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(c)

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!t/ lift I \ \\vii \'~?SGroi.,

2°IIIIJJ tt'/ \L/ o

IO

' OGro n,

\ i b,20G rains 20 3o Time (hr) (d)

FIe. 7. Simulated PLM curves in a late log phase homogeneous cell population. (a) cell cycle time distribution range: 8 to 12 hr, mean cell cycle time: 8.5 hr. Co) Tritiated thymidin¢ pulse characteristics. Abcissa scale range: 0 to 5 hr. Ordinate scale range: 0 to 2000 radioautographic units (RU). Maximum pulse height: 600 R U at I0 min. (c) The D N A synthesis rate function. Intermediate type. Moderate D N A synthesis rates early and late in the cycle (oompare with Fig. 8). x-axis (horizontal): relative position in/the cycle; y-axis (vertical): relative rate of D N A synthesis; z- axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 12 hr. (d) The threshulded PLM curves for thresholds > 1, > 2, > 3, > 4, > 5, > 10 and :> 20 grains/cell. Abcissa scale ranges from 0 to 30 hr.

PLM CURVES IN H E T E R O G E N E O U S P O P U L A T I O N S

59

5000-

(>4

4000 o.s

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-I

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I

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5

Maturation rate characteristics

I0¢~~-~ ^

/~ />1 Grains IFllr! / ! 1,,3Gra,ns IIII/I IIIIIIII HIVII | ~ > 4 G r a i n s Iff't~l.:,2Grains

N 11111IITil/II /IA l~>SGroins (c)

" lJl l ll l llV/ \ 2

illl If/

0

\ I~'11 ~kl// I0

.,~10 Grains

\~/ ~20G rains \~t.-..~ l ~ D G r a i n s 20 30 Time(hr) (d)

FIG. 8. Simulated PLM curves in a late log phase, homogeneous cell population with a peaked D N A synthesis rate function. (a) The cell cycle time distribution is identical to that shown in Fig. 7. (b) Tritiated thymidine pulse characteristics: ordinate scale range: 0--5000 RU. Maximum pulse height 1200 R U at 10 min. (e) The D N A synthesis rate function. Intermediate type. Low D N A synthesis rates early and late in the cycle. Compare with Fig. 7. Abcissa scale ranges from 0 to 30 hr. x-axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 12 hr. (d) The thresholded PLM curves for thresholds > 1, > 2, > 3, > 4, > 5, > 10, > 20, and > 50, grains/cells.

Failure of the trough between the first and second waves to fall to zero has been attributed to growth rate heterogeneity (Wimber, 1963). However, on close examination, it becomes apparent that this explanation is not compatible with the configurations of the second and third labeled mitotic waves. In the simulations shown in Figs 7 and 8, the presence of a non-zero trough is due primarily to low levels of DNA synthesis early and late in the cycle. It is apparent from these simulations that a large family of different maturation rate functions can produce similar PLM curves under appropriate relative exposure conditions. For example, at moderate early and late DNA synthesis rates and moderate relative exposure (Fig. 7), trough height is comparable to that obtained at low early and late DNA synthesis rates and high relative exposure (Fig. 8). The feature that distinguishes the two simu-

~0

S . E . SHACKNEY

lotions most strikingly is the intensity o f mitotic labeling. When relative exposure can be estimated and/or when measurements of labeling intensity are available, the maturation rate function can be characterized somewhat more precisely. Thus, we m a y suppose that tritiated thymidine pulse height in an ascites t u m o r following intraperitoneal injection would be greater than that might be expected in fetal tissues following maternal injection. The supposition that the ascites t u m o r data were obtained at high relative exposure is supported by the observation that the median grain count/labeled cell was "heavy, above countable range" (Lala & Patt, 1965). Thus, the simulation shown in Fig. 8 coincides with the ascites t u m o r data both with respect to P L M curve shape and with respect to labeling intensity. Nonetheless, while the simulations shown in Figs 7 and 8 provide reasonably good representations o f embryonic mouse tail epithelium and early mouse Ehrlich ascites t u m o r respectively, these representations are by no means unique.

°' 1 Io.

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i

0.2 O. I

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0

IO00

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Maturation rote characteristics 80

(c)

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(d) Time (hr)

FIG. 9. Simulated PLM curves in an equilibrium size population. (a) Cell cycle time distribution range: 8 to 24 hr. Mean cell cycle time: 10 hr. (b) Tritiated thymidine pulse characteristics, ordinate scale range: 0 to 1000 RU; peak pulse height 600 RU at 10 min. (c) The DNA synthesis rate function. Intermediate type. x-Axis (horizontal): l"elative position in the cycle; y-axis (vertical): relative rate of DNA synthesis; z-axis (toward observer): cycle time class. There are 20 classes ranging from 8 to 24 hr. (d) The thresholded PLM curves for thresholds > 1, > 2, > 3, > 4, > 5, > 10 and > 20 grains/cell. Abcissa scale ranges from 0 to 30 hr.

PLM CURVES IN HETEROGENEOUS POPULATIONS

61

((3) THE EFFECTS OF RELATIVEEXPOSUREON THE PLM CURVE Figure 9 illustrates the P L M curve in a steady-state population at its equilibrium size, in which cell cycle time distribution ranges f r o m 8 t o 24 h, a n d the m e a n is 10 h. T h e m a t u r a t i o n rate function is o f the intermediate type, and the tritiated thymidine pulse peaks at 600 radioautographic units l0 rain following administration, as shown. T h e P L M curve c o m p u t e d f o r a tritiated thymidine pulse peaking at 300 radioautographic units is c o m p a r e d with that c o m p u t e d for a pulse peaking at 600 r a d i o a u t o g r a p h i c units in Fig. 10. F o r clarity, only b a c k g r o u n d thresholds exceeding 5 grains are shown. It is a p p a r e n t t h a t at low relative exposure, the first wave appears m o r e rounded, and the second wave is m u c h less p r o m i n e n t t h a n at m o d e r a t e relative exposure. Identical results would

I00

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800

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,

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(hi) Time (hr)

Fxo. 10. Comparison of simulated PLM curves at low and moderate relative exposure. Cell cycle time distribution characteristics and the DNA synthesis rate function are identical to those in Fig. 9. (al) Tritiated thymidine pulse characteristics at low relative exposure. Ordinate scale range: 0 to 1000 RU. Peak pulse height: 300 RU at 10 rain. (a2) The thresholded PLM curves at thresholds of > 5 and > 10 grains/cell at low relative exposure. In the first wave fewer than 40% of the labeled mitoses have grain counts exceeding 10 grains. Because of the halving of tritium content at motosis, fewer than 40 % of the mitoses comprising the second wave will have grain counts exceeding the 5 grain threshold. CoD Tritiated thymidine pulse characteristics at moderate relative exposure. Peak pulse height: 600 RU at 10 rain. (o2) The thresholded PLM curves at thresholds of > 5, > 10 and > 20 grains/cell at moderate relative exposure. The first wave is more intensely labeled, the second wave is more prominent, and third wave is discernible at the > 5 grain threshold.

62

S. E. SHACKNEY

obtain ff tritiated thymidine dosage were held constant and the emulsion exposure period were shortened. This population exhibits PLM curve characteristics that are similar to those that have been observed in mouse intestinal epithelium. This tissue was among the earliest examined by the PLM curve method, and has been studied extensively by a number of investigators. We have chosen three PLM curves from the literature (Fig. 11), all obtained in mice of approximately the same age, but at various relative exposures. Although the sparse documentation of experimental conditions and specimen processing conditions does not permit detailed comparisons among the data, the observed trends in PLM curve behavior as a function of relative exposure parallel those of the simulated data. The effects of high relative exposure are shown in Fig. 12. It is apparent that cells which were synthesizing DNA at low rates early and late in the cycle, in which tritiated thymidine incorporation was relatively low, and whose nuclear grain counts would have fallen below the background threshold at lower relative exposure, appear as progressively more heavily labeled cells with increasing relative exposure. As a consequence, the troughs between the peaks are nearly obliterated. Mouse intestinol epithelium I00 80 .J

60 40

20 0

"'1.

//

,AAI'll

I I0

I 20

.... 3O 0

(a)

I I0

I 20

30 0

. ,1....... I0

l 20

30

(c)

(b) Time (hr)

Trltioted thymldlne dose :

5 - 2 0 ( u s u a l l y 10) ~ C i per mouse

20/u.Ci per mouse

50~Ci

per mouse

Emuldon exposure time •

Bockground threshold

presumably I - 2 months*

not stated

not stored

28 cloys

4 gro/ns

not stated

FIG. 11. Experimental PLM curves in mouse intestinalepithelium. (a) Redrawn from Quastler & Sherman(1959).(b) RedrawnfromLesheret oL (1966).(c) RedrawnfromLesher, Fry & Kohn (1961). * Given in Hughes, et al. (1958).

PLM

CURVES IN H E T E R O G E N E O U S P O P U L A T I O N S

63

We have studied the effects of increasing relative exposure on the shape of the PLM curve in Sarcoma 180 in vitro (Shackney, et al., 1973). In these studies relative exposure was varied by changing emulsion exposure duration rather than tritiated thymidine dosage, in order to avoid any effects which might be attributed to radiation toxicity or thymidine toxicity at high dosage.

1600 60

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1200 8OO 400 0

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2

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(bl)

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(b=)

Time (hr) FIG. 12. Simulated PLM curves at high relative exposure. Cell cycle time distribution characteristics and the D N A synthetic rate function are identical to those in Fig. 9. (al) Tritiated thymidine pulse chaeacteristies at high relative exposure Ordinate scale range: 0 to 2000 RU. Peak pulse height 1200 R U at 10 rain. (a2) The thresholded PLM curves at thresholds > 5, > 10, > 20 and > 50 grains/cell. Note the prominent second and third waves, and the incresse in height of the trough. (bl) Tritiated thymidine pulse characteristics at very high relative exposure. Peak pulse height 1800 R U at 10 min. (b2) The thresholded PLM curves at thresholds of > 5, > 10, > 20 and > 50 grains/cell. Note increased labeling intensity of the labeled mitotic waves, and near-obliteration of the trough between the first and second wave.

The data, shown in Fig. 13, demonstrate that the shape of the PLM curve is indeed dependent on counting threshold and on relative exposure. The following features, present in the simulated curves, are exhibited by the data: (1) Maximum labeling intensity occurs in the midportion of the first wave. Mitoses with maximal grain counts are not well represented in the second wave, but mitoses whose grain counts fall between lower and upper limits

64

S. E. SHACKNEY PLM I00

o

80 2

6O

days 4 0

.>,o

20. 0 I00

~>1

.

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4 doy$

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e

40

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8

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i

_

_

~ ~ :i %iiiii!~i l i ~ i :~i ~ :

20 0 I00 80

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12 16 20 24 28 32 36 ~me (hi

FIO. 13. Comparison of experimental PLM curves in Sarcoma 180 in vitro at moderate and high relative exposure. Data are reproduced from Shaclmey et al. (1973) by permission. For discussion see text.

that are half of the effective lower and upper limits of the most heavily labeled cell class are well represented in the second wave. (2) With increasing relative exposure the second wave becomes more prominent. (3) With increasing relative exposure there is progressive obliteration of the trough between the first and second peaks.

65

PLM C U R V E S IN H E T E R O G E N E O U S P O P U L A T I O N S (D) THE PLbl CURVE AND THE RELATIVE MATURATION RATE FUNCTION

The relative m a t u r a t i o n rate function assumes increasing importance in determining P L M curve characteristics in progressively m o r e heterogenous populations. T h e contributions o f the relative maturation rate function to P L M curve behavior can be appreciated best by examining the changes in the P L M curve t h a t occur with p o p u l a t i o n growth. W e shall consider a m o d e l population whose g r o w t h curve is shown in Fig. 14 and c o m p a r e the P L M curves obtained in late "exponential" growth and in early plateau phase growth.

0.4.

~.(c)

0.3 (al

I0 9

Total mass ~V~o qulvalents) tal ceils "~Vfoblecells

IOO

~OO

I0 I00 Cycle lime (hr)

I000

1

tO

o"

(b)

8

6 1 ~ ~ '13 r - - L L16L t ~ 19 - - ~ 22 25 Time (doys)

0'2 O-I I

FIG. 14. The simulated growth curve in a model population, with simulated cell cycle time distributions examined in late log phase and early plateau phases of growth. (a) The growth curves. Abscissa scale range: 30 days. Ordinate scale range: 6--10 (loglo (10e cells) logIo (101° cells). Lower curve: viable cells. Middle curve: total cells (viable and non-viable cells). Upper curve: total population weight in cell number equivalents assuming 109 cells per gram of tissue (total cells and cell debris). (b) The cell cycle time distribution at a population size of 1.5 × 108 cells. The cell cycle time distribution ranges from 8 to 80 hr, with a mean of 14 hr. (e) The cell cycle time distribution at 4 x 109 cells. Range: 8 to 80 hr, mean: 28 hr. (a) Type I maturation rate functions A relative m a t u r a t i o n rate function whose characteristics are such that the rate o f D N A synthesis decreases at a u n i f o r m rate at every point in the T.B.

5

66

S. E. S H A C K N E Y

Maturation rote characteristics

IO(X) 800

0"5

40o 2O0

(a)

~--J I

2

I I 3 4 (b)

I 5

SO01

0.4

801 0"3

601

0"2

401

&

0-1 I I c

Grains 201 ~ _ _ > 5 Jr>tO Grains /1>20 Grains 11tilJ||l||ldnll"-ll|BI I Illll|l I0 20 30 40 50 60 I000 0 I0 I00 :E ,-I (c2) (cl) n I001-

0"4

80}0-3 0.2

40

0"1

20

-

I IIIIIIII

A

I0

III

I IIII1[

10o

(dl) Cycle time (hr)

IOO0

/:>5 Grains I/ \ ~--./;>10 Groins J/~l "~..~l 1[:>20 Groins 0 I0 20 30 40 50 60 (d2) Time (hr)

FIG. 15. Simulated PLM curves in early and late growth for a population with a type I relative maturation rate function. (a) The DNA synthesis rate function is of type I. x-Axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of DNA synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 80 hr. Co) Tritiated thymidine pulse characteristics are identical to those in Fig. 3. (cl) The cell cycle time distribution in late log phase growth (see Fig. 14). Abscissa scale ranges from 0 to 60 hr. (c2) The thresholded PLM curves in late log phase growth for threshold of > 5, > 10 and > 20 grains/cell. (dl) The cell cycle time distribution in early plateau phase growth (sec Fig. 14). (d2) The thresholded PLM curves in early plateau phase growth for thresholds of > 5, > 10 and > 20 grains/cell. For discussion see text.

PLM CURVES IN HETEROGENEOUS POPULATIONS

67

Maturatim~ rate ¢haracteri=tics I-G: ,

0.4 0.3 0.2 0.1

(a)

--IllllllII~lllllI I III1111 IO

ID

=

I~-~ lOOm )

2000

eo

1600

1(30

I000

(b)

Cycle time (hr)

~

1200 8O0 20

40O

i

0

I/

I

2

5

4

5

(cl)

0

\

I0 2 0

~O.

, , > 5 Groins - ~ - J , > l 0 Groim 30 40 50

60

(c2)

2000 1600 1200 800 4O0

O

1

J

I

2

3

(dl)

1 4

5

0

J// I X..L I ~"'.',:::=,,=~)Grdns I0 2 0 3 0 4 0 5 0 6 0 (d2)

"lime (hr)

FIa. 16. Comparison of simulated PLM curves in plateau phase growth at moderate and high relative exposure for a population with a type I relative maturation rate function. (a) The DNA synthesis rate function is identical to that in Fig. 15(a). x-Axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of DNA synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 80 hr. Co)The cell cycle time distribution is identical to that in Fig. 15(dl). (cl) Tritiated thymidine pulse characteristics at moderate relative exposure. Ordinate scale range: 0 to 2000 RU. Peak pulse height 600 R.U at 10 min. (c2) The thresholded PLM curves are identical to those in Fig. 15(d2). (dl) Tritiated thymidine pulse characteristics at high relative exposure. Peak pulse height: 1200 RU at 10 rain. ((12) The thresholded PLM curves at thresholds of > 5, 10 and > 20 grains/cell. There is a moderate increase in peak height and labeling intensity, but there is no improvement in resolution of the first and second waves at high relative exposure.

68

s . E . SHACKNEY

cell cycle with increasing cell cycle time is one that we have designated as type I. Such a function, and the PLM curves associated with it in early and late stages of population growth, are shown in Fig. 15. The PLM curves are distinguished by the following characteristics: (1) The first wave broadens and shifts to the right with population growth. Peak height is reduced, and there is a pronounced reduction in labeling intensity in late stages of population growth. (2) The second wave is lower and much more diffuse than the first. (3) The trough between the first and second waves is obliterated in late stages of population growth. Peak height and wave labeling intensity depend on several factors, including relative exposure, the kurtosis, or peakedness, of the relative maturation rate function, and the variance in the cell cycle time distribution. Absolute peak height and wave labeling intensity vary directly with relative exposure. For a type I population in late stages of growth the second wave is not well defined, regardless of relative exposure (Fig. 16). PLM curve trends consistent with type I maturation rate functions are exemplified by the data of Simpson-Herren & Lloyd (1970) in mouse sarcoma 180, and the data of Frindel, Valleron, Vassort & Tubiana, (1969) in NCTC fibrosarcoma grown in ascitic form, as shown in Fig. 17. The data suggest that the former study was carried out at low relative exposure, and the latter at high relative exposure. When tritiated thymidine studies can be carried out only during late stages of growth, the presence e r a broad initial labeled mitotic wave and the absence of a clearly defined second wave constitute presumptive evidence that the population is heterogeneous and that it can be characterized by relative maturation rate function with type I features.

Co) Type H maturation rate functions A relative maturation rate function whose characteristics are such that the bulk of DNA synthesis is restricted to a fixed time interval prior to mitosis, is one which we have designated as type II. The simulated PLM curves associated with a type II population in early and late stages of growth are shown in Fig. 18. The PLM curve trends are summarized as follows: (I) The position and height of the first wave is relatively stable and is, for the most part, independent of population size. There may be some loss of labeling intensity in the first wave with population growth, but this tendency is not as pronounced as in the case of type I populations. (2) The second wave becomes broader and flatter with population growth. In contrast with type I populations, it is separated from the first wave by a

PLM CURVES IN H E T E R O G E N E O U S P O P U L A T I O N S Mouse sorcomo 180

~

I00

69

Mouse f Ibrosorcomo NCTC-2472 :iscitlc form)

Doy5

80 60

•

°°

40 20

I (ol)

I

0 .J

n

I

I

I I (b I )

//-\boy

I00 Day 0 80

\

60 40 20 0

i ~-.

m e =°

):

J2

.\

)

-~

r

t0 20 30 4 0 O

(o=)

I

I

I

I0 20 30 40

(b=) Time ( h r )

Flo. 17. Experimental PLM curves at different stages of growth which exhibit type I features. (al) and (a2) redrawn from Simpson-Herren & Lloyd (1970), showing PLM curves in mouse sarcoma 180 on days 5 and 8, respectively. Note the shift to the right and the reduction in height of the first wave, and the loss of a clearly defined second wave in late growth. (bl) and (b2) redrawn from Ffindel et al. (1969), showing mouse fibrosarcoma NTC-2472 grown in ascitic form, on days 4 and 12, respectively. Note shift to the fight and broadening of the first wave, and the loss of a clearly defined second wave in late growth.

trough which becomes deeper and broader in later stages of population growth. PLM curve behavior of this type is exhibited by hamster cheek pouch epithelium (Reiskin & Berry, 1968), as shown in Fig. 19. The cell cycle time distribution is somewhat broader than those that we are considering (compare with Fig. 27 below), but the PLM curve trends with population growth are similar to those observed in the model population. In general, the presence in the PLM curve of an early appearing, sharply defined peaked first wave which is followed by a deep trough in late stages of growth, constitutes presumptive evidence that the population has DNA synthesis characteristics consistent with type II relative maturation rate function. This pattern appears to occur with some regularity in a variety of epithelial tissues from different sources in different host species (rat esophagus, Leblond,

S. E. SHACKNEY

?0

Maturation rate characteristics

I000

f'01 0"5

800 600

GI

400 200

(a)

I

2

I 3

I 4

I 5

(b) t00~0,4

0'3

60 IF-

0"2

40 If" 20

O'I I IIIIl~lltlll IO

i

(cl)

J lilllll tOO IOOO

_ > 5 Grains ~ J / > l O Grains ~ L.->20 Grams I0 20 3o 4o 5o 60 / \

(c2)

-J O.

I00 f('~

0'4 0'3

60

0"2

4O

0'1

20

I llllllllJllJllll|ll~lll I IIIII IO IOO IOO0 (dl) Cycle time (hr)

~ / > 5 Grains II ~, ~ / ~ ' ~ / , , > 1 0 Grains IJr,~ I~.=~,-.---I~I~>20 Grains 0 I0 20 30 40 50 60 (d2) Time (hr)

FIG. 18. Simulated PLM curves in early and late growth for a population with a type H relative maturation rate function (a), x-axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of DNA synthesis; z-axis (toward observer): cycle time class. There are 20 classes ranging from 8 to 80 hr. Tritiated thymidine pulse characteristics, (b), and the cell cycle time distributions in early and late growth (el and dl, respectively) are identical to those in Fig. 15. (c2) and (d2) show the thresholded PLM curves at thresholds of > 5, > 10 and > 20 graln~!cell in early and late growth, respectively. Note that the trough deepens and widens with progressive growth while the first wave remains fixed in position.

PLM CURVES

IN HETEROGENEOUS

71

POPULATIONS

Homste¢ cheek pouch

I00

Newborn

&

Young (26 day)

~ L ~ I P Adult

t 0.

ie

0

I0

20:30

Ca)

40 0

10

20

30

40

(b)

50

60 70

80 0

I0

20

30

40

(c)

Time (hr)

FIG. 19. Experimental PLM curvesshowingtype II features. Data redrawn from Reiskin & Berry (1968).The first waveremains fixedin position; with progressivegrowth the trough deepens and widens.

Greulieh & Pereira (1964); mouse dorsal epidermis, Iversen, Bjerknes & Devik (1968); hamster cheek pouch, Reiskin & Berry (1968); mouse cornea, Fry & Weber (1969)). (c) Intermediate maturation rate functions The PLM curves in early and late growth that are associated with a population whose relative maturation rate function characteristics are intermediate between type I and type II, are shown in Fig. 20. As might be expected, these PLM curves exhibit features intermediate between those found in type I and those found in type II populations. In summary: (1) the first ~vave broadens and shifts somewhat with population growth, but this tendency is not as pronounced as in the case of type I populations; (2) the second wave becomes broader and flatter with population growth; (3) the trough between the first and second wave is preserved with population growth, but in late stages of growth, it is not as deep nor as wide as in the case of type II populations. In populations with intermediate relative maturation rate functions, the second wave tends to remain recognizable and distinct even in late stages of population growth, and an increase in relative exposure accentuates its presence (Fig. 21). At low thresholds, the second wave may be quite prominent. PLM curves in late stages of growth consistent with intermediate relative maturation rate function characteristics are well represented in the kinetics literature. It is of some interest that cancers arising in epithelial tissues tend to exhibit intermediate features (Fig. 22).

72

S. E. SHACKNEY Maturation rate ¢horacteristics 1.0

I000

200

0

(a)

I

i

3

2

I

(bj

4'

I

5

I00 0.4 0.3

6O

0"2

4O

&

0'I

IO

/>5

20

I Illtllltlllll|llh~lllllt

I I11111

I00

I000

(cl)

Grain=

/,, ~ ~ \~.>10 Grain=l r\/,, ~ /~>L~O Grain=l 0 I0 20 30 40 50 6 0 (c2)

I00 I0.4 0.3

6O

0.2 0.1

4O

-

& I I IIIIIII

I0

20 III I I I lib

I00

(dl) Cycle time (Iv)

I000

O

U/

\ x,~

¢1

I\

~

~5 ~J~>lO • ~

Groins Grains

i~>20Grains

IO 20 3 0 4 0 5 O

60

(d2)

Time (hr)

FXG. 20. Simulated PLM curves in early and late growth for a population with an inter° mediate relative maturation rate function (a), x-axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 80 hr. Tritiated thymidine pulse characteristics, (b), and the cell cycle time distributions in early and late growth (el and dl, respectively) are identical to those in Fig. 15. (a2) and (d2) show the thresholded PLM curves at thresholds of > 5, > 10 and > 20 grains/cell, in early and late growth, respectively. PLM curve features are intermediate between those found in type I and type II populations.

73

PLM CURVES IN H E T E R O G E N E O U S P O P U L A T I O N S Maturation rate characteristics l,Oi. . . . . 0-3-

0'5

~ O,2r0'I I IIIIIIIlatllllllllhJIII I II[lllll I0 I00 I000

(o)

(b) Cycle l i m e ( h r )

1oo

2000 1600 1200

6O

800

40 20

_:)5 Greins / \ v ~ j ~ > l O Grains /_I\ I I "~~..>20 )Grains 0 IO 20 30 40 50 6(3

400 i

0

I

I

2

3

I

4

5 s:

(cl)

1600

(c2)

.J n

,oorr

I

>3 Grains

60

1200 L 800 400

t

o

!

2

1

3 (dt)

V/

I

4

5

0

×>,o Gra

i\ ) ) "C-P>eOGr~ I0 20 30 40 50 60 (d2)

,

Time (hr) FIO. 21. Comparison of simulated PLM curves in plateau phase growth of moderate and high relative exposure for a population with an intermediate relative maturation rate function (a), x-axis (horizontal): relative position in the cycle, y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 8 to 80 hr. The cell cycle time distribution, (b), and tritiated thymidine pulse characteristics at moderate relative exposure (cl) and high relative exposure (dl) are identical to the corresponding illustrations in Fig. 16. (c2) The thresholded PLM curves are identical to those in Fig. 20(d2). (d2) The thresholded PLM curves at high relative exposure for thresholds of > 3, > 4, > 5, > 10 and > 20, grains/cell. The second wave is distinct, and may be quite prominent in late stages of growth, especially at high relative exposure and low threshold.

74

S, E, S H A C K N E Y Human epidermoid carcinoma

60

"°

4o mb/ If"

0

~ /~.__________ V . , . ,, . . . . J .

i"

I

I

1

[

[

I

I

I

. I

1

10 ~'0 30 40 50 60 70 80 90 I00110 120

:E

(a) I00 [

.

I

0

.

.

(b) .

I

I

I

=

¢

)

I

)

I0 20 30 40 50 60 70 80 90 I00110120 0

1

I

I

I

|

I

I

I

l

I0 20 30 40 50 60 70 80 90100 li0120

(c)

(d) Time (hr)

F'z~. 22. Experimental PLM curves showing features consistent with intermediate D N A synthesis rate functions. (a) and Co) redrawn from Frindel et al. (1968), cases 3 and 4.

(c) and (d) redrawn from Ter-zet al. (1971), cases 4 and 5. (E) THE PLM CURVE IN RELATION TO THE VARIANCE IN THE CELL CYCLE TIME DISTRIBUTION

We have, so far, considered a cell cycle time distribution in late growth in which the variance was not great, in order to focus on those PLM curve features which are characteristic of the relative maturation rate function. With increasing variance in the cell cycle time distribution, the features of the PLM curve that are useful in establishing relative maturation rate characteristics become progressively less distinctive at any given relative exposure. The distinctiveness of these features is partially restored at higher relative exposure, as illustrated in Fig. 23. Both populations have the same intermediate relative maturation rate function and the same mean cell cycle time, ditfefing only in the variance of the cell cycle time distribution. At moderate relative exposure, for the population in which the variance in the ~11 cycle time distribution is greater, the PLM curve exhibits features that may not be distinguishable from those associated with a type I relative maturation rate function. At higher relative exposure, the trough and second wave characteristics of aft intermediate relative maturation rate function are more apparent. This type of PLM c u r ~ behavior is exhibited by human maliguant melanoma. The data of Shirs~wa, Luce, Tannock & Frei (1970) are shown

PLM CURVES IN HETEROGENEOUS

75

POPULATIONS

Maturation rote characteristics I

0.4 ~ 0.3

0.5

0.1

(a)

A

I IIIIIIIl..r[llUlllllllll

I0

I I IJiltii

I II111111

I00

(bt)

I000

A

iO

Cycle time (hr)

It

I I IIIIIII

IO0

iO(O)

(b2)

IO0

2000 1600 1200

60

800

40

400

\

0

20 =

I

I

2

3

I

4

5

{c)

0 .J O.

•

Groins ~ _ ~ % / : > 5 Groins ~ , ~ 5 / ~ l O Groins .//::>10 Grains 1,>20 Groins l I 1~-:>20 Grains 0 I0 20 30 40 50 60 I0 20 30 40 50 60 (c2) (c))

2OO0

I00

1600

80

1200 I~

60 I 40 ~ / > 5 Groins 20 t ~, ~ f " ~ _>10 Groins I IL I I ~i-~//>20 G~ns 0 I0 20 30 40 50 60 0 (dl) Time (hr)

40O I

0

I

I

3

2

(d)

I

4

5

~'~/>5

Grains

10 20 30 40 50 6O (d2)

Fie. 23. Comparison of PLM curves in populations with an intermediate D N A synthesis rate function, (a), at different cell cycle time variances and at moderate and high relative exposure, x-axis (horizontal): relative postion in the cycle; y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer); cycle time class. There are 20 classes, ranging from 8 to 80 hr. ('ol) The cell cycle time distribution is identical to that in Fig. 14(c). (I02) The cell cycle time distribution has a range of 8-80 hr, and a mean of 28 hr, as in (bl), but the variance is greater. At moderate relative exposure, (c) where a peak of 600 RU is reached at 10 rain, and with moderate cell cycle time variance, the thresholded PLM curves exhibit easily recognized intermediate features, (el). With increasing cell cycle time variance, these distinctive features are lost, and differentiation from type I features become difficult (c2). At high relative exposure, (d), the distinctiveness of intermediate D N A synthesis characteristics is accentuated at moderate cell cycle time variance (dl), and partially restored when cell cycle time variance is great (d2).

76

S. E. S H A C K N E Y Human melanoma I 0 0 | (a)

O|

I

I

.I

I

I

I

I

I

1

I

I

I

I

I

1

I

1

I

\_I /

0

I

I

40

I

I

80

I

120

I

160

200

240

280

32.0

Time (hrl

Fie. 24. Experimental PLM curves illustrating the effects of relative exposure in the presence of a cell cycletime distributionwitha large variance. (a) and (b) human malignant melanoma. Redrawnfrom Shirakawaet aL (1970). Relativeexposure is high, and features consistent with an intermediateDNA synthesisfunctionare clearlydemonstrated. in Fig. 24; those ofTerz, Curutchet & Lawrence (1971) and Young & DeVita (1970) are shown in Fig. 25. Pertinent data relating to relative exposure are summarized in Table 1. It can be estimated that the PLM curves of Shirakawa et al. (1970) were obtained at a relative exposure that was 1.5 to 2 times that of the data of Terz et aL and 4 to 8 times that of the data of Young and DeVita. It is apparent that at high relative exposure, malignant melanoma extfibits distinctive intermediate relative maturation rate function characteristics, and that with decreasing relative exposure, the distinctiveness of these characteristics is lost progressively. This example illustrates the importance of adequate documentation of tritiated thymidine dosage, route of administration, emulsion exposure time, background threshold, and labeling intensity. For example, Young & DeVita administered their tritiated thymidine by intratumor injection. Although local tritiated thymidine concentration may have been high, its availability time was presumably quite short, and net incorporation relatively low. Thus, even at long emulsion exposure times, relative exposure remained low in comparison with that obtained following intravenous injection. In the case of type II relative maturationt rate functions, with increasing variance in cell cycle time, the trough between the first and second wave tends to become narrower and less deep, and the distinction between type II and intermediate relative maturation rate functions becomes more difficult.

0

I

1

I

I

r

I

I

I

I

I

i

I

I

i

Time (hr}

(b)

I I0

(d}

I

I

I

I

I

I

I I i i 20 30 40 50 60 70

:2',

I I I I I I I I I I I I I I I I0 20 30 4.0 50 60 70 80 90 I00 I10 120 130140150160 0

(c}

I

-/\ -/

Human melanoma

FIG. 25. Experimental ELM curves obtained in human malignant melanoma at low relative exposure. (a) Redrawn from Terz et al. (1971). (b)-(d) Redrawn from Young & DeVita (1970). Features consistent with an intermediate D N A synthesis rate function are not clearly discernible. Compare with Fig. 24.

0

20

40

60

8O

a. I00

,_I

20

40

60

80

I00

78

S. E. S H A C K N E Y

TABLE 1

Investigators

Shirakawa et al.

(197o)

Terz et al.

0971)

Young & DeVita

097o)

Tritiated thymidine dosage

8.5-10 mCi/ patient

0.1 ~tCi/g body weight (estimated 5-7 mCi/ patient)

20 ttCi/tumor nodule

Route of admlnls-

intravenous

intravenous

intratumor

Emulsion exposure time

30-45 days

3-4 weeks

8 weeks

Background threshold

~> 2 grains

~ 2 grains

:> I grain

Maximum labeling

,-., 70 grains/cell

not stated

10-15 grains/cell

stration

intensity

(F) THE EFFECTS OF TRITIATED THYMIDINE PULSE CHARACTERISTICS ON THE PLM CURVE

In the simulations we have considered so far, pulse shape has been held constant. However, the pharamacologic distribution of tritiated thymidine may be expected to differ from tissue to tissue for a given route of administration (Hughes et al., 1958), and pulse characteristics for a given tissue may not be the same for different routes of administration. In corneal tissue, for example, topical administration followed by a cold thymidine chase may be expected to produce a high, narrow pulse. Since corneal tissue is avascular, parenteral administration may be expected to produce a lower, broader tissue level pulse. The data of Fry & Weber 0969) in mouse cornea are shown in Fig. 26. When topical and intraperitoneal routes of tritiated thymidine are compared, it is evident that following intraperitoneal administration the first wave is wider and the second wave is more prominent. The PLM curve in mouse cornea exhibits features consistent with a type II relative maturation rate function in a population whose cell cycle times are distributed about long cycle time values. The PLM curve simulations for such a population, exposed to a narrow peaked pulse, and to a lower, more diffuse pulse are shown in Fig. 27. The simulated curves exhibit features comparable to those found in the experimental data.

PLM CURVES IN HETEROGENEOUS P O P U L A T I O N S

79

Mouse cornea

lool- i" 60

(a ~l !

topically

40

_J

2~1I, ~, , ,ooI-r.-\ 4°1-1 oL;

I

I0

(b)l

c3.1T0R ~

20

7,'" 1

,~--~'~

Ii

./"" I /I

I

30 50 Time (hr)

/1

-./ I

"1

70

I

I

|

90

Fro. 26. Experimental PLM curves obtained in mouse cornea following topical tritiated thymidine administration, (a), and intraperitoneally administered tritiated thymidine Co). Data redrawn from Fry & Weber (1969).

It is tempting to ascribe sustained labeling late in the PLM curve to late reutilization of tritium released from cells which had incorporated tritiated thymidine at the time of pulse exposure and which subsequently lysed. However, the amounts of tritium released from dying cells must be quite small relative to the initial dose, and the concentration and dose rate of tritium available for such reutilization must be quite low. The model suggests that the late appearance of prominent labeling in the PLM curve can be attributed more appropriately to prolonged availability during the initial pulse.

(G) SOME PROBLEMS IN THE CHARACTERIZATION OF HUMAN CELL POPULATIONS

As indicated earlier, the study of clinical material is constrained by several factors. The types of data and the frequency of collection are limited by considerations of tumor accessibility and patient comfort. Thus, for example, the nadir of the trough may be missed, or a broad second wave may be poorly represented, and the differentiation between intermediate and type I or type H DNA synthesis characteristics may prove to be difficult. Furthermore, concern over radiation toxicity dictates that the dosage of tritiated thymidine administered to patients he kept as low as possible. At low relative exposure, the information content of individual radioautographs

80

S.

E. S H A C K N E Y

Maturation rate characteristics 1.0 !

(>9

I

O~ Oq ~ o.6

0.5

g~ ~0-3 02

•

u_

0.1 0

(a)

5000 -

I00 I

4000 -

801

2000~

6oj 40i

!000

20

3000

0

I

I

I

i

I

2

3 (C~)

4

ID

IIIIIIIII

|lMIIIIk

i lllillU

I0 I00 I000 Cycle time (hr) (b)

/>SGrains /./v'~-.~.l 0 Grai ns I i ~ Q GIains 0 10 30 50 70 90 I10 (C21

.J O-

{3

5000

IO0~

.....

4000 3000 2000 I000 0

I

I

I

I

2

3

4

(dl)

0 10 30 50 70 90 I10 (d2)

FIG. 27. Simulated PLM curves illustrating the effects of pulse characteristics on a population with a type II D N A synthesis rate function, (a), at equilibrium population size, x-axis (horizontal): relative position in the cycle; y-axis (vertical): relative rate of D N A synthesis; z-axis (toward observer): cycle time class. There are 20 classes, ranging from 12 to 120 hr. (b) The cell cycle time distribution ranges from 12 to 120 hr, with a mean of 72 hr. (ci) Tritiated thymidine pulse characteristics. A narrow peaked pulse. Ordinate scale range: 0 to 5000 R U ; peak height of 2400 R U reached at 10 rain. (c2) The threshold PLM curves, at thresholds > 5, > 10 and > 20 grains/cell show a narrow first wave, a deep, broad trough, and 'a low, broad second wave. (dl) Tritiated thymidine pulse characteristics. A broad pulse. Peak height of 1200 R U reached at 20 rain (d2) The thresholded PLM curves at thresholds of > 5, > 10 and > 20 grains/cell. The first wave is broader than in (c2), and the second wave is more prominent. The irregularities in the curves are computational artifacts (see Appendix).

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may be correspondingly low, and differential PLM curve features may be difficult to discern. Some of the problems of PLM curve characterization are illustrated in carcinoma of the breast. The data of Young & De¥ita (1970) and Terz et al. (1971) are shown in Fig. 28. The presence of a broad, late peaking first wave, and the absence of a clearly defined second wave are suggestive of type I DNA synthesis characteristics. However, with the exception of one case [Fig. 28(d)], relative

Human br~s! carcinoma

'°°/(o)

.......... I L,~).

I 1(o).

1\ o ~'o I00.

O

\.../,.... o ,o ~o ~o ~o 'o ,'o ~o ~'o do ~o ~o

'.

" ...... I

I0 20 30 40 50 0

I0 20 30 40 50 60 70 80

Time (hr)

FIG.28. ExperimentalPLM curveswhichillustratethe difficultiesin obtainingand interpreting human data. (a)-(c) Redrawn from Young & DeVita (1970). (d) and (e) Redrawn from Terz et al. (1971), cases 2 and 3.

exposure was low (mean grain counts of 10 to 15 grains/cell), and an intermediate DNA sysnthesis rate function cannot be ruled out [see section 3(d)]. In addition, for the cases of Terz et al. [Fig. 28(d), (e)] the paucity of data beyond 24 h precludes the identification and characterization of a possible second wave. This example underscores the need for the pooling of data from different sources for the characterization of human cell populations. For proper comparisons of such data, it is important that experimental conditions and specimen processing conditions be amply documented at the very least. While it is important to compare data obtained at different relative exposures, it would be highly desirable to introduce some degree of uniformity into radioautographic procedures by establishing certain methodologic standards. T.B.

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4. Discussion

The model described in this paper has successfully simulated a wide variety of PLM curves, ranging from those obtained in very young populations with near-trapezoidal labeled mitotic waves, to those obtained in heterogeneous populations in late stages of growth. The model departs from currently accepted models in several respects, perhaps the most important of which is abandonment of the concept of discreteness of the conventional cycle phases. A brief review and critique of pertinent aspects of the origin, development, and consequences of this concept follows. (A) THE DISCRETE CYCLE PHASE MODEL AND ITS CONSEQUENCES

In 195t, Howard &Pelc showed by radioautographic methods that acid insoluble nuclear 32p incorporation could be demonstrated in some but not all interphase Viciafaba meristem cells within the first few hours of 32p exposure, and that mitotic labeling lagged behind interphase labeling by several hours. They concluded that 32p incorporation into DNA and, hence, active DNA synthesis, occur during only part of interphase. Shortly thereafter (Howard & Pelc, 1953), these conclusions were formalized in a model which divided the cell cycle into discrete phases, GI, S (synthesis), G=, and D (M), where DNA synthesis was restricted exclusively to S phase. The model represented an idealization of the data, as the authors were careful to point out. Discrepancies between the model and the data were attributed primarily to population heterogeneity, i.e. to variability of cycle time and cycle phase durations, to the presence of differentiated cells in the proliferating population, and to the consequences of grain count halving in lightly labeled cells. The authors did not, however, consider the possible role of variation in the rate of DNA synthesis in individual cells as they progress through the cycle. The enthusiastic acceptance of the discrete cycle phase model had several important long term consequences. It initiated a long line of conceptual development that was based on rigid compartmentalization implying qualitative rather than quantitative intracycle age-state differences, and implied discrete all-or-none state transitions as opposed to rate modulation of continuous growth processes. The first PLM curve employing tritiated thymidine was described by Quastler & Sherman (1959) in mouse intestinal epithelium [see Fig. 11(a)], and interpreted in accordance with the discrete cycle phase model. Ideally, two separate trapezoidal labeled mitotic waves were expected. Departures from the ideal were ascribed to cycle phase variability. In particular, the absence of a prominent second wave was attributed to a highly variable GI phase. Variability in the rate of DNA synthesis as cells progress through the

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cycle, with corresponding differences in efficiency of DNA synthesis detection, were not considered. The interpretation of Quastler and Sherman, then, implicity assumed that the detection of DNA synthesis by radioautographic methods could be treated as an all or none phenomenon. Thus, the shape of the PLM curve was implicitly assumed to be independent of relative exposure, such that the durations of the cycle phases and their variability could be inferred from the data. Early graphical methods for estimating cycle time and cycle phase durations (Painter & Drew, 1959; Wimber, 1963) were based on the hypothetical bitrapezoidal PLM curve expected in an ideal population with invariant, discrete cycle phases. Although no real PLM curves exhibited such features, deviations from the ideal were attributed to population heterogeneity, and it was recognized that estimates of cycle time and cycle phase durations obtained by such graphical methods represented approximate average values. An assumption which has dominated the analysis of cell kinetics data from the beginning has been that the proportion of labeled cells following pulse exposure is equal to the proportion of the cell cycle devoted to DNA synthesis (Howard & Pelc, 1953). That is, expected number of labeled cells total cells

Ts Tc

[More recently, a correction factor has been added to account for the nonuniform distribution of cells throughout the cycle (Steel, 1968).] This ratio has been taken to represent the expected labeling index for the population. Although it has been recognized that graphical estimates of average cell cycle time and cycle phase durations may be crude at best in herogeneous cell populations, it has not been widely appreciated that the ratio, Ts/Tc, is also an estimated average value which is no less crude than its component measurements. For subpopulations with cycle times much longer than the estimated average, the ratio, observable Ts/Tc, is much lower than the expected average labeling index. The discrepancy between the expected average labeling index and the observed labeling index led to the development of another conceptual compartmental dichotomy, namely that of the growth fraction (Mendelsohn, 1962). The growth fraction calculation, in essence, assigns portions of those subpopulations with cell cyle times longer than the estimated average, for which observable Ts/Tc is less than the expected average labelling index, to a nonproliferating pool [or a Go state (Quastler, 1963)]. The relaxation of the rigid compartment boundaries imposed by earlier interpretations of radiotracer kinetic studies appears to carry with it the penalty of increased complexity of data analysis and interpretation. To be

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sure, models should be no more complex than necessary. On the other hand, over-simplification may generate misleading conclusions and lines of investigation that might be nonproductive. The effects of compartmentalization of cytokinetic proliferative behavior into discrete states can be illustrated in several investigative areas of current interest: (I) Several computer programs based on the discrete cycle phase model have been developed for the extraction of cell cycle time and cycle phase distributions from the shape of the PLM curve (Barter, 1970; Hartmann & Pedersen, 1970; MacDonald, 1970; Steel & Hanes, 1971; Takahashi, Hogg & Mendelsohn, 1971). The tacit assumptions are that DNA synthesis rate is constant throughout the S phase, and that the detection of D N A synthesis is an all or none phenomenon. This, in turn, would imply that the shape of the PLM curve and the kinetic parameters derived from it are, for all intents and purposes, independent of relative exposure and background counting threshold. As data available in rodent intestinal epithelium suggest, and as our own experimental studies (Shaekney et al., 1973) clearly show, this is not the ease. It is apparent, then, that failure to consider intraeyele variation in DNA synthesis rate has resulted in the overly optimistic expectation that the PLM curve alone contains sufficient information for the characterization of the age structure of heterogeneous proliferating cell populations. (2) The division of cell populations into discrete pools of proliferating and nonproliferating cells has led to the postulation of discrete G~ --} 0, Go --} GI, (72 ~ Go, and Go--} G2 transitions. However, there is no distinguishing between long G1 and Go, and long G2 and Go cells on experimental grounds, so that the existence of discrete proliferative-non-proliferative transitions must remain a matter of conjecture. The discrete cycle phase model has prompted the postulation of control mechanisms of replication that involve the temporally discrete elaboration of biochemical signals or trigger substances which govern the transition from one phase to the next. To our knowledge, no such phase transition mediators have been identified. On conceptual grounds, the postulation of such mediators merely begs the question of replicative control, since mechanisms which govern their elaboration must be postulated in turn. If it is assumed that in each cycle phase some discrete critical step or steps in the maturational sequence must be completed before a cell can progress to the next phase, then multiple discrete control loops, each with its own sensor and effector ares must be postulated. Finally, mechanisms for co-ordinating controls within each cell and among cells must also be postulated. In a continuous model, the maturational sequence may be viewed as a system of interdependent variable rate processes, of which only one, DNA

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replication, need be postulated as having to be completed (during the "G2" period) in order that mitosis can occur. Control of replication rate GaB be exerted by multiple factors, either specific (hormones, drugs) or non-specific ( 0 2 tension, nutrient availability, pH, the accumulation of toxic products of metabolism), at any point in the cycle, the final common pathway being a limitation of DNA synthetic rate. (3) Under steady-state conditions, certain equivalences exist between the discrete phase model and the continuous one. The S phase of the discrete model corresponds with the period of maximal DNA synthetic rate in the continuous model, and the non-proliferative pool of the discrete model corresponds with a segment of the cycle time distribution which spans long cell cycle times in the continuous model. Equivalence between discrete and continuous model descriptions of dynamic changes in population age structure is possible only under a restricted set of circumstances. The changes that occur with growth in a population with a type II DNA synthetic rate function may be described equivalently in discrete terms as a lengthening of the G~ phase. However, either description implies PLM curves in which the first wave remains fixed in height and position with population growth [see section 3(D),(b)], and, as we have seen [sections 3(D)(a), 3(D)(C)], this type o f PLM curve behavior is by no means universal. The growth dependent changes in age structure in intermediate and type I populations have no sarisfactory equivalent descriptions in discrete model terminology, since the discrete models concern themselves with the duration of D N A synthesis but not its intensity. There is even greater divergence between the continuous and discrete representations of population age structure changes following sudden perturbation by high dose irradiation or the administration of cytotoxic drugs. For example, age specific effects of radiation, interpreted within the context of the discrete cycle phase model, may be listed as G1 block, maximal cell kill at the Gt-S interface, depression of DNA synthesis, G, block, and mitotic delay (i.e. cycle time lengthening) (see Tolmach, Weiss & Hopwood, 1971). Interpreted within the context of a continuous model, all of these effects can be collected and subsumed under a single rubric, namely, a reduction in maturation rate, the magnitude of which is dependent on the initial agespecific rate at the time of exposure. A more detailed description of simulated radiation and drug interactions in model populations such as those described in this paper will be given separately. (13) THE USESAND LIMITATIONSOF TRITIATEDTHYMIDINETRACERSTUDIESIN MAN The present model suggests that PLM curves obtained in human cell populations can provide only crude estimates of the mean, variance, and range

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of the cell cycle time distribution, and that inferences regarding DNA synthesis characteristics are limited at best to the distinguishing of patterns which we have designated as type I, intermediate, and type II. Under some circumstances, even these broad distinctions may be difficult, as when population heterogeneity is great, relative radioautographic exposure is low, and when clinical considerations preclude extensive collection of data in any given patient. Undoubtedly, the elucidation of kinetic similarities and differences among normal and neoplastic human cell populations will require the piecing together of data from many sources. Extensive documentation of experimental conditions and specimen processing conditions is essential for proper evaluation of data comparability. Standardization of radioautographic procedures in order to maximize the comparability of data from different sources would be highly desirable. The usefulness of defining broad distinctions between normal and neoplastic human cell populations, and among different neoplastic cell types cannot be overemphasized. For example, differences in response to "cycle stage specific" drugs by type I, intermediate, and type II populations can be exploited to maximize tumor response and minimize host toxicity with appropriately designed therapeutic regimens. If critical host tissues and various tumor types could be shown to exhibit stable and distinctive differences in maturation rate function characteristics, then the problems of individualization of optimal drug regimens in the absence of extensive kinetic data in every patient, are simplified considerably. In this regard, the observation that normal epithelial tissues tend to exhibit type II features, while neoplastic populations derived from epithelial structures tend to exhibit intermediate D N A synthesis characteristics, [sections 3(D)(b),(c)] is of considerable interest. A much more extensive data base will be required to test the validity of such a generalization. It may be fortuitous that the best examples of type I functions that we were able to find were classified as sarcomas. Nonetheless, the possibility that ontogenic and/or histologie classifications may be correlated with DNA synthetic patterns is an intriguing one. There are other parameters in addition to the PLM curve which may be quite useful in characterizing the growth behavior of human cell populations. In this regard, we call attention to the fact that the PLM curve is but one of the data types which can be simulated by the present model. In the same model populations, and under the same simulated pulse labeling conditions, the following additional parameters can be obtained: (1) the 1 h labeling index; (2) the intermitotic grain count distribution (i.e. the rnultithresholded labeling index at 1 h);

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(3) grain count halving times (i.e. the multithresholded labeling index as a function of time); (4) the mitotic index; (5) the DNA content distribution; (6) the grain count distribution as a function of DNA content; (7) counts per minute as a function of D N A content (simulation of liquid scintillation counting studies); and (8) loss of label from the population as a function of time. In addition, changes in the labeling index and intermitotic grain count distribution can be studied under simulated long-term tritiated thymidine infusion conditions, or with multiple injections. A detailed description of these parameters and their correlations will be given separately. For the present discussion, we would point out that each of these parameters has its own advantages in illuminating various facets of population age structure in late stages of growth. It is suggested that a reconstruction of the age structure and kinetic behavior of heterogeneous human populations can be accomplished by a judicious, and, at present, artful analysis of multiple kinetic parameters. With regard to the feasibility of multiple parameter studies in man, it should be noted that many of the parameters listed above (numbers I-4) can be measured in the same samples used to obtain PLM curve data. Recent advances in flow microfluorometry instrumentation have made it possible to measure cell DNA content rapidly and accurately in large numbers of cells. With wider availability of such instruments, and with improvements in cell dispersion methods, the D N A content distribution could easily become one of the medical oncologist's most readily measured parameters of population age structure. Recent developments in the instrumentation of ceil sorting promise to make combined stiadies of DNA content and radioautography no more cumbersome than radioautographic studies alone. The advantages of combining these parameters in clinical studies derive from the fact that DNA content classes represent multiple simultaneous "windows" on the cell cycle. Combined studies, then, could provide at least as much information as the PLM curve with its single mitotic "window", but from fewer biopsy samples.

5. Summary and Conclusions The model presented in this paper has simulated successfully PLM curves from a spectrum of populations which ranges from young, rapidly growing mouse tissues to late stage, slowly growing human cancers. Unlike earlier models, the present model accounts for the effects of intracycle age-speeitic variations in DNA synthesis rate and tritiated thymidine incorporation in

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rolation to radioautographic tritium detection characteristics. The model suggests that the effects of age-specific variations in D N A synthesis rate are formidable indeed, and that they preclude precise characterization of population age structure and growth characteristics from the PLM curve alone. The PLM curve does, however, permit the estimation of population heterogeneity to some degree, and, in many instances, permits the identification and classification of DNA synthesis rate patterns and their characteristics changes over a wide range of population sizes. The PLM curve, in conjunction with other kinetic parameters, may provide sufficient information for the elucidation of kinetic differences between normal and neoplastic tissues and among neoplastic populations, which may be exploitable therapeutically. Data currently available in the literature suggest that such exploitable kinetic differences do, in fact, exist between normal and neoplastic cell populations. Since background threshold and radioautographic relative exposure play an important role in determining PLM curve characteristics, experimental conditions and specimen processing conditions should be amply documented when PLM curves are reported, and some measure of labelling intensity should also be given. Standardization of radioautographic methodology would be highly desirable. The author wishes to thank Dr Seymour Perry for his advice and support of this work, and for his critical review of the manuscript.

APPENDIX

1. The Simulation of Intracycle-aging In our earlier paper (Shackney, 1973), two methods were described for transfering cells from one compartment to the next. By one method, each ceil cycle time class is divided into compartments of equal maturational achievement. Transit time through each compartment is obtained from the relative maturation age and relative maturation rate functions. Cell growth is simulated by choosing some simulation time interval, At, which is shorter than the shortest compartment transit time, and transferring the cell fraction, At/compartment transit time, from each compartmentto the next. This method provides for the smooth flow of cells through all age compartments in all cycle time classes. Furthermore, transfer of cells to comparable age compartments across cycle time classes is a straightforward and efficient computational procedure. This method is ideally suited for growth curve simulations. However, its chief drawback with regard to PLM

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curve simulations is that there is no way to distinguish ceils by their residence time within a given compartment; PLM curve simulations require that compartment transit time (mean compartment residence time) and actual compartment residence time must coincide for all cells if these ceils are to pass the mitotic "window" in proper temporal sequence. The second method for simulating intra-cycle aging is to divide each cycle time class into compartments of equal residence time and transfer the entire contents of each compartment within a cycle time class when simulated clock time intervals equal or exceed compartment residence time for that cycle time class. This is the computational procedure used in our simulations. Its chief advantage is that it provides for orderly temporal sequential progression of ceils through the cycle. However, it does have several disadvantages: Transfer of ceils across cycle time classes to compartments of comparable maturational achievement requires a cumbersome and time consuming computation. In long cycle time classes, compartment transit times are relatively long, and cells "lurch" through the cycle, unless the number of age compartments is large. For the simulations presented in this paper there were 40 age compartments per cycle time class, 20 cycle time clases, and 17 tritium uptake classes per age compartment. Despite the large number of age class compartments, evidence of cells, "lurching" through the cycle, manifested as oscillations in the PLM curve, becomes prominent under the following circumstances: (1) When the range of cell cycle time distribution is broad, and mean cell cycle time is long, (2) When the relative maturation rate function is either type II, leptokurtotic, or both, and (3) When relative exposure is high and/or the tritium pulse is narrow (see Fig. 27, for example). These oscillations are computational artifacts which can be reduced or eliminated if the number of compartments is increased further. 2. The Duration of Mitosis

We have found that the width of the mitotic window does not materially affect PLM curve characteristics, provided that it is much less than Tc. Metaphase and anaphase durations, which are the mitotic subphases most commonly examined in PLM curve studies, tend to be considerably less variable than prophase durations. In the simulations presented in this paper a mitotic window of constant size, equal to 1/40 of the minimum cell cycle time, is assumed, and the contents of mitotic age compartments of all cycle time classes greater than Tcml,imum are weighted accordingly.

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