The age incidence of female breast cancer— simple models and analysis of epidemiological patterns

The Age Incidence of Female Breast CancerSimple Models and Analysis of Epidemiological Patterns CHARLES DELISI Laboratory of Theoretical National Cancer Inrtiiute,

Biology,

Division

National

Insritu!es oj Health, Bethesda,

Received

29 April 1977; revised 7 Augwt

of Cancer

Maryland

Biology

and Diagnosis,

20014

I977

ABSTRACT We show transition malignant)

that

a three-stage

stochastic

probability from the intermediate stage leads to an equation which

model

of carcinogenesis

(hyperplastic) can be fitted

with

a discretized

stage to the final (clinically to a wide variety of age-inci-

dence curves for female breast cancer. This includes the inflected (so-called biphasic) incidences observed in many Western countries, as well as the peaked profiles observed in the Orient. Unlike previous mathematical theories which attribute biphasic incidence to a superposition of profiles for two etiologically distinct diseases, the theory presented here hypothesizes its biologic basis as change in ovarian functions accompanying menopause. This leads to an equation which fits the data equally well, though with fewer adjustable parameters. The extent to which the theory simulates the effect of various risk factors, including age at menarche, menopause and first birth, is briefly discussed.

I.

INTRODUCTION

The form and magnitude of the age-incidence profile of female breast cancer shows considerable geographic variability. The incidence in Western countries, studied either cross-sectionally or by cohort analysis, rises rapidly until the menopause, plateaus or diminishes slightly during the next few years (Clemmesen’s hook), then continues rising into old age [l]. In contrast, the incidence in many Oriental countries declines more or less continuously after attaining a menopausal peak whose magnitude is on average an order of ten smaller than in the West [2]. Neither the details of the profiles (e.g., Clemmesen’s hook) nor the reason(s) for the differences between Western and Oriental countries is understood, and there is apparently no quantitative theory which can simultaneously describe both, although a number of efforts have been made toward developing limited descriptions of certain characteristics. MATHEMATICAL

BIOSCIENCES

0 Elsevier North-Holland,

Inc., 1977

37,245-266

(1977)

245

CHARLES

246

DELISI

An early attempt at a semiquantitative description of breast-cancer age distributions in Western countries was based upon the observation that a semilogarithmic plot of incidence against age is approximately resolvable into two linear components intersecting at about the age of menopause, with the component of larger slope describing the premenopausal incidence [3]. The data were thus fitted by a four-parameter function (a slope and intercept for each component), and the pattern explained by assuming that menopause reduces the fraction of women at risk. While this hypothesis appeared reasonable, the description of distribution patterns by a superposition of two straight lines is nonetheless an approximation which describes neither Clemmesen’s hook, nor the gradually decreasing slopes of the incidence curves which become apparent upon closer examination of the data [4]. Description of details of this sort would thus appear to require more complicated functions. An alternative explanation which accounts for biphasic incidence was proposed by DeWaard et al. [S], who assumed that breast cancer is not a single disease, but consists of two etiologically distinct components occurring with different age distributions. This hypothesis was developed further by Hakama [6, 71, who was able to fit the incidence data in four Scandinavian countries by assuming that each type of breast cancer has a Gaussian age distribution. Although Hakama’s function fits the data well, it suffers from having six adjustable parameters, and consequently inferences drawn from it cannot be compelling. More recently Lee et al. [4] have studied the premenopausal slopes of incidence curves in a number of countries and have found a geographically independent decrease which terminates approximately at the menopause. They noted that if the decrease continued, it would lead to a breast-cancer incidence in old age lower than the observed value-a result which is surprising, since there is considerable evidence to indicate that menopause decreases breast-cancer incfdence [8-lo]. In this paper the theoretical problem of describing breast-cancer age-distribution patterns will be approached from a somewhat different point of view. I will show how the observations outlined above (viz., geographic variations in incidence profiles, Clemmesen’s hook, the characteristics of the incidence slopes prior and subsequent to the menopause, and the difficulty raised by Lee et al.) may be subsumed under a simple cellular model with four adjustable parameters. I will also briefly comment on the ability of the model to simulate the effects of timing of first birth and age at menarche on breast-cancer risk. II. Il. I.

THE MODEL GENERAL

DESCRIPTION

There is considerable evidence to support the notion of tumor monoclonality [ 1 I], and consequently a tumor will be assumed to arise from a

THE AGE

INCIDENCE

OF FEMALE

BREAST

247

CANCER

single progenitor which evolves through a series of stages into a clinical malignancy. The interstage transition probabilities are expected to be complicated functions of host defenses [ 121, environmental factors [ 131, mitotic inhibitors [14], hormonal imbalances [ 131 and so forth, and each of these may therefore affect the frequency of observed malignancies. Moreover, there may be numerous processes involving both proliferation and differentiation within each stage, so that, for example, if the cells in a particular stage are undergoing any of a variety of membrane modifications and/or metabolic changes, the transition probability from that stage to the next would be interpreted as either a rate-limiting, or an effective, rate constant. A more detailed theory of carcinogenesis would add biological structure to the stages, sufficient to enable calculation of the interstage transition rates as functions of those for intrastage processes. However, detail at this level is unnecessary for the purpose of this paper, and I will instead adopt definitions solely in terms of parameters required to fit the data. The model from which the equation will be derived (Fig. 1) considers clones distributed among three compartments or stages. In order to fix ideas, the cells in each stage will be tentatively identified as normal epithelial cells (stage l), hyperplastic cells (stage 2) and transformed hyperplastic cells (stage 3). The introduction of two hyperplastic cell stages is not required by observed morphologies, but is consistent with the results of inducer-promoter experiments [16] which indicate that only a subset of hyperplastic nodules develop into malignancies upon exposure to a promoter. The interstage transition probabilities are specified as follows. Each normal cell has a probability P,(t) per unit time of entering stage 2. Once there, it may undergo clonal expansion, so that the number of cells in stage 2, even if none were to leave, could not be obtained simply by specifying

1

I

Ibl

FIG. 1. A schematic of the mathematical model (a) tentatively identified with a biological analogue (b). The average probability per clone per unit time that a cell in stage one will undergo transition to stage 2 is /I,(r), and the corresponding probability for transition from stage 2 to 3 is /3*(t). The model belongs to a more general class in which the tumor passes through (r > 1) hypexplastic stages before it can become a clinical malignancy (see Appendix C for details).

248

CHARLES

DeLISI

their rate of entry from stage 1. Complete specification of their number would evidently require investigating intrastage cellular kinetics and would thus introduce an increase in complexity and parameters which is not justified by the current level of information. I therefore introduce an average transition probability pz( t) per clone per unit time for a cell in stage 2 to enter stage 3. The average is over clone sizes. When a cell has entered stage 3, it may remain there inactive, or die, or after some delay involving clonal expansion, express itself clinically as a Tlpable malignancy. If the cell does not or cannot grow, for whatever rt..son, a clinical malignancy can only develop if another stage-2 cell enters stage 3 at a later time, and again this later entry may or may not develop. We will say that it is the nth entry which finally succeeds in growing to clinical size, where n (21) is to be determined by curve fitting. The model in this form predicts that on average, clinical malignancies should be accompanied by n - 1 transformed hyperplastic lesions.

11.2.

THE EQUATIONS

The quantity of interest is P(t) At, the probability that the nth entry into stage 3 occurs between time t and t+At. If P,_,(t) and P(t)At denote, respectively, the probability that n - 1 entries have occurred up to time t, and the conditional probability that the next entry occurs within the interval (t, r + At), then

P(t)At=

P,_, (r)p(t)At.

(1)

If x2(t) is the mean number of clones in stage 2 at time t and x3(t) the mean number that have entered stage 3 up to and including time t, then (Appendix A) (2) If the development is applicable to tumor growth, Eq. (2) must fit incidence data to within a multiplicative constant. Calling the constant PO, the observed incidence curve R(t) will be given by R(t)=Poe-“~x~-‘/?2(t)x,/(n-l)!.

(3)

The deterministic means x2 and xg are obtained by solving the differential equations corresponding to the model in Fig. 1. The interstage transition rates are assumed piecewise continuous, and may undergo abrupt changes at menopause. As a specific example, let & be a step function which changes discontinuously at the age of menopause (f,,,) from & to PZ < Pzi,

THE AGE INCIDENCE

and let /?r be constant.

OF FEMALE BREAST CANCER

Then (Appendix

249

B)

x2= p,xa[ (A - C)e-P22’+

Re-81’],

(4)

and x,=L%,

I

‘“? /0

x2

dt

+

P22

J

x2

(5)

4

‘PI

where A _ exp( Pz2tm -

ht,) - exp( P22L - P2lL) 9 P21-PI

and

CE

exp( P22L -Pit,) P22-PI

’

x,, being the number of cells at risk at t = 0, which, unless otherwise stated, is taken to be the age of menarche. Equations (4) and (5) become especially simple if the time scales for processes involved in the development of a clinical malignancy (viz., l/p, and 1//12) are long compared to the average human life span (which is what one might expect for a disease with a low frequency of occurrence). Specifically, by neglecting terms higher than second order in t, one has that (Appendix B)

)cw

X2=P,XOt-PIXO(P22+PI)t2/2+PIXO(P22-P21

(6)

and x3=(P2I-P22)PIxotrtr/2+P,P22xo2/2.

Substituting R(t)=R0exp[

(7)

Eqs. (6) and (7) into Eq. (3),

[(YI-Y2)t~+Y2t2]n-‘Y2t/(~-11)!,

-(yl-y2)ti-y2t2]

(8)

where Ro=2Po, YI -xoP,

P21/Z

250

CHARLES DELISI

and

t-c t,,

-VI7 Y2=

XOPl

i

P22

2

’

t>t,.

The short-time approximation leading to Eq. (8) reduces the number of stages in the model by yielding a single effective rate constant for the transition from a normal epithelial cell to some hyperplastic intermediate. In a sense the equation thus corresponds to the simplest possible model+ne in which there is only a single effective intermediate stage spanning the enormous and complex gap between a normal cell and a clinical malignancy. The transition to the intermediate occurs continuously (with rate constant y,, prior to menopause) and would thus correspond to situations in which there are a relatively large number of events leading to hyperplasia, whereas the transition from the intermediate to a clinical malignancy is, in keeping with the assumed stochastic nature of the process, discretized by the adjustable parameter n. It should also be noticed that as y, t2 becomes progressively smaller, R approaches t2”-’ asymptotically, so that previous expressions for fitting various forms of epithelial cancers [18-201 follow as a limiting case. III.

RESULTS

Attempts to understand epidemiological patterns in terms of biological processes, especially as they relate to endocrine influences, have motivated a number of studies on the effects of pregnancy, menarche and menopause on breast-cancer risk. A viable model of breast cancer must therefore lead to an equation which is sufficiently plastic to simulate the effects of these influences, as well as the average properties of incidence profiles. The philosophy of the theory presented here is to try to incorporate the multiplicity of factors involved in the etiology of breast cancer within some small number of parameters. This allows the development of the broad outlines of a model and leads to a relatively simple expression for age incidence, thus allowing some assessment of the validity of the model. The objective of this section is to try to gain insight into its limitations by studying the extent to which parameter changes simulate the effects of various risk factors on observed incidence, and the extent to which incidence data may be fitted by Eq. (8). III.1.

AGE DISTRIBUTION

PROFILES

The derivation of Eq. (8) assumes a population of initially normal mammary epithelial cells which are subject to transformation, with some

THE AGE INCIDENCE

OF FEMALE

BREAST

251

CANCER

small probability, into hyperplastic lesions and subsequently into a clinical malignancy. Since all cells are assumed to be normal initially, then the probability that a time t (chronological age minus age at menarche) will have elapsed before a clinical malignancy is observed will, at least initially, increase with time. It is evident, however, that since this probability cannot exceed unity, the incidence curve must eventually peak when the majority of the at-risk population has contracted breast cancer, and then decline toward zero. This sort of behavior is characteristic of incidence profiles among Asian populations [2] and is well described by Eq. (8) (Fig. 2). According to the theory, the main difference between the profiles of Asian populations and those of Western countries is that in the latter, the peak probability occurs at a much later age. Thus, for example, Eq. (8) also fits the data for Scandinavian countries as shown in Figs. 336, though theory predicts the curves must eventually peak, just as in Japan. This predicted peak in Western countries also follows from Hakama’s theory [6]. As an indication of the goodness of fit of these theoretical profiles to the data, x2 values of 3.62, 3.63, 6.91, 2.88 and 1.67 were obtained for Japan, Sweden, Norway, Denmark and Finland, respectively. With five degrees of loo0 F

r 1'

x I

JAPAN

I

I

I

I

I

30

40

50

60

70

AGE lvearsl

FIG. 2. Number of cases per 100,000 females in Miyagi, Japan, 1962-1964 (@), and the distribution obtained using Eq. (8) (solid line) with the parameters listed in Table 1 (data from Doll, Muir and Waterhouse [2]). Events prior to menarche (taken as age 12) were neglected, and menopause (r,,,) was taken to be 48.

CHARLES

252

11

I

!

I

I

I

30

40

50

60

70

AGE

FIG. 3.

(wars)

See caption

to Fig. 2.

loo0 F

AGE

FIG. 4.

(years1

See caption

to Fig. 2.

DELISI

THE AGE INCIDENCE

253

OF FEMALE BREAST CANCER

DENMARK

t ::

30

40

50

60

70

AGE Iyears)

FIG. 5.

See caption to Fig. 2.

FINLAND

1

I 30

I 40

I 53

I 60

AGE (years)

FIG. 6.

See caption to Fig. 2.

I 70

254

CHARLES

DELISI

freedom xi5 = 11.1, so that there appears to be no reason to reject the hypothesis that the theoretical profiles fit the data. These results follow from an equation which implicitly takes breast cancer to be a single disease. The hook follows from an assumed permanent change in ovarian functions at menopause which is parametrized by a rate constant for transition between hyperplastic stages. One of the criticisms of the hypothesis that changes at menopause are responsible for biphasic incidence profiles has been the notion that a permanent change would not lead to only a transient decline in incidence. In fact, the model shows that this criticism need not be valid. It is evident from Table 1 that the form of incidence profile for Japan differs from those for Scandinavia primarily as a consequence of a larger y, and a smaller I&. The former parameter, it will be recalled, is an effective interstage transition rate constant, and a larger value implies that cancers should develop more quickly. The theory in its present form does not predict the physiological basis of the increased rate, which may be the result of any combination of factors, including endocrinological, immunological, environmental (including diet), and so forth. A somewhat surprising aspect of the results, however, is that although tumor development is predicted to be faster in the Japanese population, the incidence is known to be lower. A biologic basis for such a result is not evident, but it might be explained by taking account of genetic variations within a population. As the simplest example, if only a fraction of the population of a country is at risk, then the simultaneous occurrence of a lower overall incidence and a higher interstage transition rate in Japan would follow if the at-risk fraction were smaller than in Scandinavia, but had a greater predisposition than the corresponding at-risk fraction in Scandinavia toward developing breast cancer. This point of view is illustrated by the parameters in Table 1, which predict that the at-risk fraction of the population (I&/2) in Japan is 0.01, or about an order of magnitude smaller than in Scandinavia. How should a result of this magnitude be interpreted? First, the introduction into the theory of an at-risk fraction of the population should TABLE 1 Parameters Determined by a Nonlinear Least-Squares Regression Analysis [24] of Observed Incidence Profiles (Figs. 2-6) Sweden

Norway

Finland

0.22 IO.07

0.15*0.05

0.13?0.02

0.15 2 0.02

5.81?0.90

7.16?

7.83 + 0.83

5.1820.57

Denmark R, (year-

‘)

y, (10m4 year-‘) ~2 bearn

‘)

1.02

0.60y, +0.03y,

0.73y, zO.O3y,

0.65y,-tO.O3y,

0.7Oy, kO.O3y,

2.62kO.17

2.7320.16

2.76kO.13

2.26kO.10

Japan 0.02 2 0.002 13.25?

1.20

0.65~,+0.03~, 2.30~0.19

THE AGE INCIDENCE OF FEMALE BREAST CANCER

255

not necessarily be taken to mean that only a portion of the population is susceptible to breast cancer. Everyone may be susceptible, but genetic predispositions are known to exist, so some people are more susceptible than others. The theory simplifies the range of genetic variation by allowing only two categories. The meaning of the nonsusceptible category is simply that wihin this group, death from other causes is much more likely to occur prior to contraction of breast cancer than it is for the susceptible population. Thus R,/2 (or P,) should be interpreted as an apparent at-risk fraction. The fraction of the population which is apparently not at risk arises from a relatively low predisposition for cancer with consequent masking of susceptibility due to death from other causes. According to the above remarks, an at-risk fraction less than one may be attributed to genetic variations within a population. However, the reasons for the variations in PO among different populations must be more complicated. Specifically the differences between Scandinavia and Japan is unlikely to be due to genetics alone, since within a generation or two, the descendents of Japanese woman who had migrated to the United States have incidences approaching local Western values [ 131. It would thus appear that there is an interaction between genetics and environment, so that a particular environment may act to catalyze the expression of a genetic predisposition toward or away from susceptibility. As a minor variation on the above development in which a population is heterogeneous with respect to susceptibility, one can consider the consequences of taking the entire population equally at risk. In this case it turns out that in order to fit the Scandinavian profiles, both yi and n must decrease at menopause (i.e., the incidence equation still has four adjustable parameters), and for suitable choices, data fits are obtained which are as good as those shown in Figs. 3-6, and sometimes better. For example, with y, = 1.77 X 10e4 year-‘, y2=0.4y,, n(tt,)=1.77, the sum of the squares of the deviations of the calculated from the observed data for Finland is 4.6, whereas for the result in Fig. 6 it is 6.1. A difficulty arises, however, when one attempts to fit the Japanese profile. In this case, in order to fit the rising portion of the curve, y, must be chosen considerably smaller than the value given in Table 1. This leads to a peak in the curve which occurs well after age one hundred, and the profile takes on the appearance of those in the West. A local peak occurring at about age fifty can still be obtained, of course, by choosing &(t) as some sufficiently complicated function of time and by making its form geographically dependent. Although there is no reason to rule out this approach a priori, the model would lose some of its simplicity and generality and begin to assume a more ad hoc character.

CHARLES

256 111.2.

CHARACTERISTICS

OF THE INCIDENCE

DELISI

SLOPE

The time rate of change of breast-cancer incidence provides additional insight into the parameters affecting the dynamics of carcinogenesis and the requirements for a viable model. According to Eq. (8) the slope of a semilogarithmic plot of incidence against age is

dlnR _ dt

-2y2t+++

2(n - l)Yd (Yl - YX + Y# ’

t#t,.

(9)

Thus, according to the theory, the slope is determined by the difference between two time-dependent terms and is a function of n, y, and yz. For parameter values similar to those shown in Table 1, the positive terms in Eq. (9) dominate the sum but decrease as age increases, whereas the negative term increases (in absolute value). The slope is therefore positive at early time, but decreases continuously until the menopause. At this “point”, the effective interstage transition probability drops precipitously, and this not only reduces the incidence, but increases the slope. The increase in the latter, however, still leaves the average postmenopausal slope smaller than its average premenopausal value (Figs. 3-6). These aspects of the theory are in accord with observation and subsume the earlier analyses of Lilienfeld and Johnson [3] and Lee et al. [4]. In fact, the latter authors conclude that the premenopausal slope is linearly decreasing, whereas the theory predicts that it decreases with some degree of curvature. The curvature however is slight, and even if present could not be reliably distinguished from linearity because of the magnitude of the errors involved in estimating the slope. As a rough indication of how well the theory predicts the estimated slopes (i.e., using the parameter values in Table 1 with no further adjustment) I have plotted the results for Sweden and Japan (the countries for which the two largest values of x2 were obtained) in Figs. 7 and 8. Although the conformity of Eq. (8) to the several types of observed age-distribution patterns suggests a certain degree of utility for fitting incidence data, it does not of course provide a test of the validity of the model. In order to gain further insight in the applicability of Eq. (8), I will briefly consider the extent to which it simulates, in broad outline, the effects of menarche, menopause and pregnancy on breast-cancer risk.

111.3. MENOPAUSE The theory provides a simple means for predicting incidence in the absence of change at menopause, and thus for assessing its effect, by keeping the parameters time-independent. As seen in Fig. 9, the hypothetical incidence is considerably higher than its observed value. There are two opposing effects which contribute to the magnitude of the predicted value. One is

THE AGE

INCIDENCE

OF FEMALE

BREAST

30.0

CANCER

257

JAPAN

. ;.‘\t

. z x Ni > 2 10.0 E z ; 5.0

‘.

s 4

0.0

‘I_

=.< -5.0 1

.

20.0 :

I

I

27

37

I

I

57

67

I

t

47 (years)

FIG. 7. The slope predicted by Eq. (9) [solid line] using the parameter 1. (0) represents a slope estimate from the observed incidence in Japan.

values in Table

SWEDEN 25.0

1

I

I

I

I

27

37

47

57

67

t Ivearsl FIG. 8.

See Fig. 7.

258

CHARLES

DELISI

that menopause interrupts a decline in slope-an effect which contributes to decreased risk in the absence of menopause. The other is that menopause interrupts increasing incidence-an effect which leads to greater risk. Evidently, for the parameters which fit the observed incidence data, the second effect dominates. This theoretical result is what one would expect on the basis of observations that late menopause significantly increases breast-cancer risk [8]. It is also consistent with the observation that early menopause, either natural or induced, decreases breast-cancer risk. The prediction of this latter observation is shown in Fig. 10, which also indicates that (i) the extent of protection increases as age at menopause decreases, and (ii) the extent of protection declines as age beyond the menopause increases. The first of these predictions is borne out by observation [9]; the second requires confirmation. The effect of timing of menopause on breast-cancer risk may actually be considerably more complicated than indicated by Fig. 10, which was obtained only by changing t,,, and which therefore represents the simplest possibility. Thus the theory predicts that early menopause and its concomitant ovarian changes (parametrized by y, - yZ) are in themselves sufficient

1’ FIG. 9. Finland

I

I

I

I

I

30

40

50 AGE (years!

60

m

Comparison of predicted postmenopausal ) and without (_ _ _ _ _ _) a change

with (

female breast at menopause.

cancer

incidence

in

THE AGE

INCIDENCE

OF FEMALE

BREAST

259

CANCER

FINLAND

4t 2 i 11

I

I

I

I

I

30

40

50

60

70

AGE (years)

FIG. 10.

Effect of early menopause

according to Eq. (8) using the parameters curve) and 38 (lower curve).

on the age-incidence in Table

curve for Finland

1, with fm =48 (upper

to afford protection; there need not be any additional magnitude of y, - yz (i.e., y, - y2 need not be a function because y, - yz scales nonlinearly with time. 111.4.

predicted

curve), 43 (center

change in the of r,,,). This is

PARITY

There is considerable evidence linking late first birth to increased breast-cancer risk and indications that the effect is most pronounced in old age [21]. These observations led to predictions that the age of onset of breast cancer in women first-parous relatively late in life should be earlier than average. However, the reverse has been found, and it now appears that late first birth not only increases risk, but also increases the age at onset

PI. There are a number of possible explanations for these observations, one of which is that the effect of first pregnancy on breast epithelial cells depends upon their state of health, stimulating further transformation of lesions already present, but retarding transformation of normal cells [ 131. In terms of the model presented here, the latter effect would be interpreted as

CHARLES DELISI

260

a decrease in y,, while the former could be interpreted as a decrease in n. If this explanation is compatible with Eq. (8) one would expect that on an incidence vs. age plot, these variations should lead to a simultaneous decrease in the incidence and increase in the (positive) slope. I have not been able to simulate such an effect, although I also have not proved its impossibility with regard to the model. It appears, however, that at the very least, the model is restricted with respect to this particular explanation. An alternative explanation resides in the possibility that the effect of first birth may be quafitatiuely dependent upon the genetic constitution of the individual, affecting different women in different ways. Specifically, if there are two populations-a larger one in which risk is decreased by early first birth, and a smaller one in which it is increased (e.g., by increasing the effective transition ratefithe observed effect of early first birth for the combined populations will be protective. However, since those women who do contract breast cancer belong primarily to the second population, the observed age of onset will be earlier.

IV. IV.1.

DISCUSSION TIME

DELAYS

The theory has been formulated without incorporation of time delays, but requires them at several points. To begin with, the time of arrival of the nth entry into the third compartment is not the same as the time of diagnosis of a clinical malignancy: the arrival is only supposed to satisfy a necessary and sufficient condition for subsequent development of a tumor. Moreover, the delay introduced by this distinction is expected to have two components, one being the interval between the nth entry and the formation of an observable growth, and the other being the interval between this last event and the time of official diagnosis (the latter would actually be a distribution of delay times, as might the former). Because the magnitudes of these delays are not well known, there seems little to gain by explicitly incorporating them into the equations, although that can easily be done as more information becomes available. It is anticipated that the primary effect of this omission will be change in the least-squares parameters (Table 1). For example, with a two-year delay, a least-squares fit of the Finland data comparable to that shown in Fig. 10 results in R,=0.197?0.03 year-‘, y, =(4.34*0.62)x 10m4 year-‘, yz=0.7y, and n=2.02+0.11. In close connection with the effect of this delay is the effect of variations in age at menarche. Since I have assumed that the processes responsible for breast-cancer initiation and development begin at menarche, earlier menarche is equivalent to a shift in the time origin to an earlier age. With other parameters held constant, this leads, prior to peak incidence, to a reciprocal relationship between age at menarche and incidence [ 131.

THE AGE INCIDENCE

OF FEMALE

BREAST

261

CANCER

Another delay which is expected to be present and which has not been explicitly incorporated is that due to changes at menopause. As indicated earlier, the break in the incidence curve observed in a high-risk country, which may be a plateau followed by an increase, or an actual dip followed by an increase, is generally attributed to menopausal changes. One explanation involves hypothesizing two etiological types of breast cancer, one premenopausal and ovary-related, and the other postmenopausal and related to adrenal steroids [l]. The break then occurs as the result of displacement between the two distributions [6]. The other hypothesis, the one adopted here, is that there is primarily one etiologic type of breast cancer, whose growth, however, is influenced by hormonal changes, presumably ovarian, at menopause. (For a review of the evidence see Ref. [23]). In this case it is evident that there ought to be some delay between onset of menopause and the observed effect on tumor incidence. If any hypothesis linking menopausal changes to biphasic breast-cancer incidence is correct, however, the delay cannot be more than a few years, since the break in the curve seems to occur shortly after the average age of menopause (21. The reason for this apparently small delay is not obvious. However, it would be consistent with the assumption that an effect of menopause is to retard transformation of advanced hyperplasia into clinical malignancy, rather than to retard transformation of normal cells into hyperplastic ones. Again, however, the primary effect of incorporating such a delay is a small change in the magnitudes of the parameters obtained from the least-squares analyses. IV.2.

VARIATIONS

ON A MODEL

Figure 1 represents one of a class of models which has r > 1 stages and n > 1 entries into the final stage as a requirement for growth to a clinical malignancy. As shown in Appendix C, the initial (early stage) slope of an incidence curve does not allow determination of both n and r, so that from this portion of the curve one cannot distinguish between a model which has n=n* (>l) entries and r=r* stages, and one which has r=r* + n* - 1 stages and n* = 1 entries. However, as noted earlier, the slope of an incidence curve is not constant, and models become distinguishable at later ages (240 years). Several models were attempted during the course of this work, and the one presented gave the best fit with the smallest number of parameters, as measured by the sum of the squares of the deviations from the observed points. IV.3.

INTRASTAGE

KINETICS

AND

THE EFFECT

OF DEATH

TERMS

The equations have been formulated in a way which allows only transitions between compartments without explicit terms for loss due to death. In keeping with the meaning of the transition rate constants proposed in Sec. 11.1, this formulation implies that the clones are fit for survival; i.e., birth

262

CHARLES DELISI

rate exceeds death rate, although perhaps by only a small amount. It is evident that whatever the details of the intrastage kinetics, there will be constraints imposed by some as yet to be determined requirements that the size of hyperplastic lesions remain within prescribed limits. Meeting these requirements will likely involve growth on a time scale which is long compared to the average human life span, and a rate constant which may itself be a function of time. The above notions are inherently deterministic and consequently cannot be applicable to stage 3, the stochastic nature of which constitutes the essence of the model. In fact, clonal death has been implicitly allowed in stage 3, since the first entry into it does not always lead to a clinical malignancy. The model predicts that it is the nth entry which grows, where n has been determined as somewhat larger than 2. Previous entries are presumably checked or eliminated by host defenses or growth control mechanisms, whose effects are expected to play a discernable role when small numbers are involved. Unfortunately there is no obvious biological interpretation of n; i.e., it is not clear why the nth entry should succeed while previous entries have failed. Conceivably, although early entries do not grow, they might induce irreversible biological changes in host control mechanisms which create a locally fertile environment for growth of later clones. In any case, since n is noninteger, the population must, as one might expect, be heterogeneous with respect to it.

APPENDIX

A.

SOLUTIONS

Our interest is in and including time artifice, irreversible clones have entered

TO THE STOCHASTIC

PROBLEM

the number of clones which have entered stage 3 up to t. This can be calculated most readily by using, as an entry into stage 3. Let Z’,(t) be the probability that n stage 3 up to and including time t. Then

P,(t+At)=P,(t)[l-x,(t)p,(t)At]+P,-,(t)x,(t)p,(l)At,

(A.1)

or

~=x,(t)p*(r)[r,-,(f)--P.(z)], PO(0) = 1;

This is a standard

differential

P,(O)=O, difference

(A.21

n>O. equation

whose solution

is

THE AGE INCIDENCE

OF FEMALE

The integral can be identified Appendix B). Therefore

If P(t)At is the probability (t, t + Al), then

BREAST CANCER

263

with the mean number

of cells in stage 3 (see

that the nth entry occurs during

P(f)Ar=P,_,

APPENDIX

B.

The differential

SOLUTION equations

(t)&(t)x2(t)At.

FOR THE DETERMINISTIC corresponding

dx, (0 = -& dt

=

&

(t)x,(t)r

-P(r),

-

P2

P.1) P.2)

(9x2(&

(‘)

dt

Integrating

P (q

MEANS

to the model in Fig. 1 are

dx,(t) dt

the interval

P.3)

=/32(+2(t).

Eq. (B.l), x,(t)=x,-

I,‘P(+k

(B.4)

From Eq. (B.2) $ [ x2(t)c{@~(T)d’] = /3(t)e182(7)h, x2(t)eP2(r)=

/0

$(~)&‘)dq

and thus x2(r)=e-~2(‘)

s

o'@(~)epz(T)dr,

P-5)

264

CHARLES

Substituting

DELISI

Eq. (B.5) into (B.3) and integrating,

x3(t)

=

Jgis2

(7)r-mp

(t’)e~~(“)df’dT.

(B.6)

If j3, is constant and &(t) is a step function which changes discontinuously at t = f, from pz2 to &,, Eq. (B.5) integrates to x*(t) = p,xe[ (A - C)e-P22’+

BeW81’],

(B.7)

where AE

B=

exp( P22h- Pd,) - exp(P22L- PZIM

P2,--PI

>

b2*L p,

c E exp(

’

P22fm - PI fm) P22-PI

.

If the time scales for processes involved in tumor development are long compared to the average human life span, the exponentials can be approximated by an expansion which retains terms up to second order in time. Then

xo-x,(t)“XoP,f-XoP:f2/2,

P.8)

x2(t)~P,x,(P22-P2,)~~/2+P,XO~-P,XO(P22+P,)~2/2

(B.9)

and

~~(+=x~-x,(r)-~~(t,zz(

APPENDIX

C.

iij21;P22)&x,,~;+

VARIATIONS

Pi’;xof2.

(B.lO)

ON THE MODEL

In order to illustrate the type of effect which the number of stages and/or the number of arrivals in the final stage has on the form of the incidence profile, we consider the more general system i, = - P,x,, X,=P,x,-P2x2, (C.1)

&=Pr-,xr-I-P,-%.

THE AGE INCIDENCE

For algebraic

OF FEMALE

simplicity

BREAST

265

CANCER

the fi (1 < j < r) will be treated as constant.

Then

x, = x,,e-81’,

(C.2) r-l Xr”

XI0

fl @/(r-l)! j=l

i

1

tr--l=q-‘.

With the number of clones in stage r Poisson-distributed, and with n, entries on the average a necessary and sufficient condition for occurrence of a clinical malignancy, we have p,(t)=

e-W_’

(~,t’-‘)“‘-‘~~t’-Z/(n,

- l)!

(C.3)

and

64) The slope at early time is determined primarily by the positive term. From this portion of the curve alone, it is not possible to distinguish models for which r and rr, are distinct, from those having just one or the other of these parameters. However, at later time the negative term in Eq. (C.4) becomes important, and both r and n, are in principle uniquely determined. A number of different models were tried; the one presented in the text gave the best overall agreement with observation. I am indebted to Dr. Lance Liotta for a number of thought-provoking and informative discussions, and for reading and commenting upon the manuscript prior to publication. As with several of my previous works, this manuscript has also benefited greatly from a careful and penetrating reading by Dr. George Bell. Finally, the referee’s thoughtful comments are appreciated. REFERENCES J. Clemmesen, Statistical studies in malignant neoplasms. I. Review and results, Acta Par/ml. Microbial. Scnnd. 174, 249-276 (1965). R. Doll, C. Muir and J. Waterhouse, @Is.), Cancer Incidence in Fioe Continents, Springer, Berlin, 1972. A. Lihenfeld and E. Johnson, The age distribution in female breast and genital cancers, Cancer 8, 875-882 (1955). J. A. H. Lee, P. G. Chin, W. A. Kukull, R. S. Tompkins and F. W. Ahson, Relationship of age to incidence of breast cancer in young women, J. Nut/. Cancer

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CHARLES

266

8 9 10

DELISI

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M. Feinleib, Breast cancer and artificial menopause: a cohort study, J. Nafl. Cancer Inst. 41, 315-329 (1968). J. M. Friedman and P. J. Fialkow, Cell marker studies of human tumorigenesis, Transplant. Rev. 28, 17-33 (1975). F. M. Bumet, Immunological surveillance in neoplasia, Transplanf. Reu. 7, 3-25. B. MacMahon, P. Cole and J. Brown, Etiology of human breast cancer: a review, J.

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T. J. Bailey, The Elements of Stochastic Processes with Applications to the Natural Sciences, Wiley, New York, 1964. P. J. Cook, R. Doll and S. A. Fellingham, A mathematical model for the age

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