Quantitative Theories of Oncogenesis

ADVANCES IN CANCER RESEARCH. VOL . 27

QUANTITATIVE THEORIES OF ONCOGENESIS Alice S . Whitternorel

.

Department of EnvironmentalMedicine. New York University Medical Center and Department of Statistics. Stanford University

I . Introduction .............................. ........................ I1. Expected Rates of Tumor Appearance .................................. I11. The Single Stage Theory of Iverson and Arley .......................... A . The Theory ....................................................... B. Examples and Application to Data .................................. IV. The Multicell Theory of Fisher and Hollomon .......................... A. Temporal Behavior of Observed Incidence Rates ..................... B. Concentration Dependence of Observed Incidence Rates ............. V. The Multistage Theory with Negligible Cell Loss ................ A. The Theory ....................................................... B. Application to Terminated Exposures ............................... VI. The Multistage Theory with Non-Negrigible Cell Loss .................. A. Radiation-Induced Tumors ......................................... B. Urethane-Induced Murine Lung Tumors ............................ VII . The Multistage Theory with Proliferative Advantage of Intermediate Cells .................................................... A . TheTheory . . . . . . . . ............................................ .......................... B . The Armitage and Doll Two-Stage ............................ C . The Fisher Theory .............. inated Exposures ........... D . Application of the Fisher Theory t E . Increased Numbers of Target ......................... VIII . Single Stage or Multistage Theory in Transformed Cell Types A. The Initiation-Promotion Phenomenon .............................. B. Growth of Clones of Transformed Cells ............................. C. Application to Initiation-Promotion Experiments ..................... IX. Implications for Dose-Response Relationships .......................... A . The Problem ...................................................... B. Radiation Carcinogenesis ........................................... X . Conclusion ......... ........... References ...........................................................

55 56 57 57 58 62 62 64 65 65 67 68 68 71 73 73 74 75 76 77 78 78 78 79 83 83 86 86 87

I . Introduction

The purpose of this chapter is twofold: to review the quantitative theories of the origin of neoplasms. and to discuss a new theory . This

On leave from Hunter College. CUNY. This work was supported by a grant from the Alfred P. Sloan Foundation to the SIAM Institute for Mathematics and Society and by a Rockefeller Foundation Fellowship in Environmental Affairs. 55

Copyright 0 1978 hy Academic Press. Inc. All rights of reproduction in any form resewed .

ISBN 0-12-006627-0

56

ALICE S . WHITTEMORE

theory synthesizes some of the others and describes aspects of initiation, promotion, and regression that have not been included previously. Armitage and Doll (1961) and Whittemore and Keller (1978) reviewed many of these theories &om a mathematical point of view. Here we discuss them without the use of mathematics. We shall examine them critically both qualitatively and quantitatively, by comparing their predictions with epidemiological and experimental observations. The theories relate the frequency and time of occurrence of tumors to the temporal and spatial distribution of oncogenic exposure. There are two main reasons for formulating such theories. One is to provide a framework for evaluating the consequences of proposed mechanisms of carcinogenesis. The other is to help determine allowable concentrations of known carcinogens in the environment, and to estimate the consequences of exceeding them. This is necessary because animal experiments must be done at concentrations high enough to cause some of the animals to develop tumors, while environmental concentrations must be low enough to produce very few tumors in man. Thus, apart from the great difficulties due to interspecies differences, animal experiments cannot be used directly to study low concentrations. Therefore some theory is needed to extrapolate the doseresponse relationships downward &om the high doses used in animal experiments to the low doses to be allowed in the environment. II. Expected Rates of Tumor Appearance

Oncogenesis is believed to involve two processes, transformation and growth. Transformation is the occurrence of one or more heritable changes in a cell which render it capable of generating a tumor. Growth is the process of multiplication by cell division whereby the clone of a transformed cell becomes a detectable tumor. Here the term transformed cell refers only to those altered cells that generate neoplasms, and not to their descendents. Cells which can be transformed to a state capable of generating neoplastic growth will be called target cells. A given tissue is assumed to contain some fixed number N of target cells, each of which has a chance of transformation which is unaffected by the tumorigenic fate of its neighbors. Then the expected rate of tumor occurrence in a tissue at time t is proportional to the number N of target cells. Thus it can be written Nr(t), where r(t) is the expected fraction of target cells which yield clinically detectable tumors per unit time. Similarly, the expected number of tumors which have appeared in the tissue by time t can be written NR(t). Here R(t) denotes the expected proportion of target cells which give rise to detectable tumors by time t .

QUANTITATIVE THEORIES OF ONCOGENESIS

57

The incidence rate is defined as the expected rate of tumor occurrence in tumor-fiee tissue. Under the above assumptions the incidence rate at time t can be shown to equal the expected rate Nr(t) of tumor occurrence in the tissue. Similarly the cumulative incidence at time t , defined as the sum of the incidence rates over all times prior to t , can be shown to equal the expected number NR(t) of tumors appearing in the tissue by time t . The time from initial exposure of a target cell to a carcinogen until it is transformed is called the transformation time. The growth time is the time required for a transformed cell to generate a clinically detectable tumor. All of the theories assume that the growth time is independent of the transformation time. The rate r(t) at which target cells yield detectable tumors can be expressed in terms of these two times. Therefore theoretical predictions for the transformation and growth times yield predictions for the incidence rate Nr(t). This is also true for the expected number NR(t) of tumors up to time t . Accordingly, we will present each theory by stating its assumptions about transformation and its assumptions about growth. These assumptions will imply the way in which r(t) and R(t) vary with time and with oncogenic exposure. The theoretical predictions of these quantities will be compared with empirical estimates of them. Ill. The Single Stage Theory of lverson and Arley

A. THE THEORY The earliest quantitative theory of carcinogenesis is that of Iverson and Arley (1950).This theory postulates that transformation of a target cell occurs as a single event, such as a somatic mutation or activation of an oncogenic virus. The rate 1 at which cells are transformed is assumed to be the sum of a spontaneous rate s of transformation in the absence of carcinogenic exposure, plus an induced rate due to exposure. The spontaneous rate s is assumed independent of time. The induced rate is taken to be proportional to the concentration c(t) of carcinogen at time t. The proportionality constant p , representing the transformation rate per unit of carcinogen, depends on the potency of the carcinogen and its metabolic derivatives, and on the sensitivity of the target cells. Since the concentration may vary with time, so may the rate 1 of transformation per cell. Thus we write

l(t) = s

+ pc(t).

(1) The rate of transformation of target cells in the tissue is l(t) times the number of unaltered target cells remaining in the tissue at time t. We

58

ALICE S. WHITTEMORE

assume that the rate Z(t) at which unaltered target cells are depleted is small at all times, and that the initial numberN of such cells is so large that depletion of target cells is negligible. Thus their number at all times is essentially N , and the rate of transformation in the tissue is NZ(t).The assumption of negligible target cell depletion is made in all the theories we consider. Iverson and Arley assumed that the clone of a transformed cell grows by the proliferation of its members, and that it is detected as a tumor when it contains some critical number a of cells. The probability per unit time that a cell in the clone divides is assumed to be a constant p. Since p is taken independent of carcinogen concentrations, carcinogenic exposure has no effect on the growth time of tumors. From these assumptions one can calculate the expected rate g(t) at which a transformed cell yields a detectable tumor at time t after transformation, and G(t), the proportion of transformed cells whose growth times are less than t . This theory ignores cell loss in a tumor due to necrosis, ignores the existence of nonproliferating tumor cells, and ignores variation in proliferation rates with time and with location within the tumor. However, it does provide for variation in tumor growth times, unlike the less plausible assumption of many theories that all tumors require a fixed time w for growth. The above assumptions for tumor growth and the expression (1)for the transformation rate l(t) complete the formulation of Iverson and Arley’s theory. Together they permit the calculation of the rate Nr(t) of tumor appearance and the expected number NR(t) of tumors in the tissue by time t . The theory involves five constants: s and p in (l),the growth constant p, the detectable size n, and the number N of target cells. In addition the concentration c(t) of carcinogen at the target cells must be specified.

B. EXAMPLESAND

APPLICATION TO

DATA

As a first example, let us suppose that c(t) is a constant c. Then (1) shows that the transformation rate I is the constants + p c . The rate r(t) of tumor appearance per target cell is in this case 1 times the proportion G(t) of all transformed cells whose growth times are less than t:

r(t) = lG(t) = sG(t) + pcG(t).

(2)

Here sG(t) is the spontaneous rate of tumor occurrence in the absence of the carcinogen, and pcG(t) is the induced rate due to concentration c of the carcinogen. If the spontaneous cell transformation rate s is

QUANTITATIVE THEORIES OF ONCOGENESIS

59

negligible, then b y (2) the rate Nr(t) = NpcG(t) of tumor appearance in the tissue at a given time t is proportional to carcinogen concentration c. It also follows that for t fixed, the expected number N R ( t ) of tumors per tissue is proportional to c. These results agree with Doll’s (1971) finding that the incidence rates of bronchial carcinoma among British physicians are proportional to smoking rate. This linear relationship between agestandardized incidence rates and average daily cigarette consumption is shown in Fig. 1. However the plots of cumulative incidence versus applied concentration of various carcinogenic agents shown in Fig. 2 exhibit a dose-response relationship that is more quadratic than linear. This disagreement with the Iverson-Arley theory indicates that the theory may be inappropriate for mouse skin carcinogenesis induced by benzpyrene and ultraviolet light. However the disagreement may be due to a discrepancy between applied carcinogen and actual concentration of the agent or its metabolites at and within the relevant target cells. The possible consequences of such a discrepancy will be discussed in Section IX,A. As a second example, suppose that a single dose C of carcinogen is administered at time t = 0. Then r(t) is the sum of the spontaneous rate sG(t) and an induced rate resulting from cells transformed at time zero by the carcinogen. The induced rate is pCg(t), where g(t) is the rate of

300

-

Age standardized incidence rate of bronchial carcinoma: 200 no. of -/lo5

man-mas 40050 -

I/

0 :

I

20 30 Cigarettes smoked/day 10

I

40

FIG.1. Incidence rate of bronchial carcinoma versus daily rate of cigarette smoking. The data points are from Doll (1971)and represent age-standardized incidence rates among men who started to smoke at ages 15 to 24 years and were not known to have changed their smoking habits. The straight line is eye-fitted to the data points.

60

ALICE S. WHJTTEMORE I A

5-

0

4-

e Cumulative tumor

m

3-

incidence

0

2-

1-

e-Le 0

c-

I

L

1

I

c

I

8

2

'

L

1

3

'

a

' 1

4

Dose rate of ultraviolet light, ergs/cm2/sec 54-

B

0

C

Cumulative 3tumor 2incidence

0

t -

appearance at t of tumors resulting from cells transformed at time zero. Thus we have

r(t) = sG(t) + p C g ( t ) .

(3)

More generally, suppose that exposure to a carcinogen is terminated at some time to.At times t greater than to,the rate r(t)of tumor appearance is again the sum of a spontaneous rate sG(t) plus a rate induced by the carcinogen. In this case the induced rate is due to the appearance at time t of tumors whose progenitors were transformed prior to to,and whose growth time exceeded t - to.Thus for large times t when growth periods exceeding t - to are unlikely, the tumor appearance rate should equal the spontaneous rate. This is in disagreement with the data of Doll and Hill (Doll, 1971; Doll and Peto, 1976) on inci-

QUANTITATIVE THEORIES OF ONCOGENESIS

61

dence rates of bronchial carcinoma among former cigarette smokers. These data suggest that the excess over nonsmokers’ rates remains constant for as long as 20 years after termination of smoking, although the growth period for human lung carcinomas has been estimated at only 5 to 7 years. The prediction that tumor rates revert to the spontaneous rate after exposure termination is also in disagreement with the experimental data of Lee (1975), discussed in Section V and shown in Fig. 7 and 11. In this experiment the backs of mice were painted with benzpyrene, and treatment was discontinued at various stopping times in different groups. Although the tumor growth period was estimated to be 10 weeks, the incidence rates do not return to the spontaneous rates within the animals’ lifetime. As a final example, let the concentration c(t) of carcinogen be arbitrary. We shall examine the total expected number of carcinogen induced tumors, and its dependence upon exposure pattern and intensity. It follows from (1)that the induced transformation rate for the tissue is N p c ( t ) . Thus the expected number of target cells ever transformed by the carcinogen is NpC, where C is the carcinogen concentration integrated over time. The ultimate expected number N R ( m ) of induced tumors is then the ultimate expected number NpC of carcinogen-transformed cells, multiplied by the fraction G ( m) of these transformed cells which ever grow to detectable tumors:

N R ( a ) = NpCG(m).

(4)

Note that the ultimate expected number of tumors per tissue given b y (4)depends only on the total integrated concentration C and not upon the pattern c ( t ) of exposure. Moreover it is proportional to C. If spontaneous incidence is negligible, these remarks apply to total tumor numbers. These results may be compared with plots of mean number of pulmonary adenomas per mouse versus total injected dose of urethane and versus pattern of exposure shown in Fig. 3 (White et al., 1967). It is evident from Fig. 3 that fractionation reduces the tumorigenicity of doses 0.5 mg/gm and greater, arid that the dose-response relationships have a quadratic component. Both of these results are inconsistent with the Iverson-Arley theory. As noted by Armitage and Doll (1961), fractionation forward in time can reduce tumor incidence at any observation time because cells transiormed later have less time to develop to detectability by the time of observation. However this cannot account for the results of Fig. 3, because White et al. found a single dose of urethane more effective evein when administered at the end of the fractionation period.

62

ALICE S. WHITTEMORE

40

Dose oubdividrd into n frocii

Mean number 30 of tumors per 20 mwse 10

' 0

04

08

12

16

20

Total applied dOJe (mglgrn body weight)

FIG. 3. Average number of pulmonary adenomas per mouse as a function of total injected dose of urethane (ethyl carbonate) measured in milligrams per gram weight of mouse for five patterns of administration: subdivision of dose into le), 2(0), 4(A), 8( x), and 1 q O ) portions. The injections were given at intervals of 2 days. Reprinted with permission from White eb al. (1967).

Other experiments of White, discussed in Section IX,A, indicate that concentrations of urethane (or a carcinogenic metabolite) at the target cells may vary nonlinearly with injected dose. Such variation could account for both of the inconsistencies noted between the IversonArley theory and the data of Fig. 3. IV. The Multicell Theory of Fisher and Holloman

A. TEMPORAL BEHAVIOROF OBSERVED INCIDENCE RATES

At about the time that Iverson and Arley published their theory, a number of investigators noted that the death rates for many forms of human cancer increased proportionately with the fifth or sixth power of age. Fisher and Holloman (1951)observed this for death from cancer of the stomach in women in the United States. Nordling (1953) found it for death from all forms of cancer in men in England, France, Norway, and the United States. Stocks (1953)observed it in data on death from stomach cancer in both men and women in England and Wales. Armitage and Doll (1954)found it for mortality rates due to cancer of different sites and for each sex in England and Wales. This relationship does not hold for all types of cancer; notable exceptions are childhood cancers and hormone-related cancers, such as cancer of the femaIe breast or male testes. However, it does hold for the majority of adult cancers. Examples of data exhibiting this proportionality of incidence rate to a power of age are shown in Fig. 4. To interpret this observation, it was assumed that the time from

63

QUANTITATIVE THEORIES OF ONCOGENESIS

I

1000

lncidsnw rats of nonmslanoma skin cancer 110. of c ~ / 1 0 man-yn. 5 '00 (log ralal

t ! f

l0O0

I

Gastric cancer mortality rate: no. of casa/l05 man-yn. lo -

(log scale)

10 1-

- - t

20 40 80 Age in years (log scale 1

4i

1oot

i

i

1'

i r' c

/

.l

20 40 80 Age in years (log scale)

FIG.4. Relationship between age anti incidence rates of (A) nonmelanoma skin cancer in Dallas white males for 1971-1972; (B) stomach cancer in Danish males for 19631967. Data points are from (A) Scotto et al. (1974), and (B) Clemmesen (1974). The straight lines, eye-fitted to the data, have slopes 5.66 and 5.47, respectively.

cancer detection until death was negligible compared to the time required for transformation and growth. It was also assumed that cancer fatality rates were independent of age. Therefore the incidence rate was assumed to be proportional to the mortality rate. Subsequent studies in which the age-specific incidence rates of cancer are recorded have confirmed this assumption. Furthermore the investigators assumed that the cancers were due to carcinogenic exposure that remained fairly constant from birth to death. Thus age was the same as the duration of exposure. Then the data show that the incidence rate is proportional to the fifth or sixth power of duration of exposure to a carcinogen of coristant concentration c. Two explanations of this power law relationship have been proposed. One, due to Fisher and Holloman, is that a certain number k (equal in this case to six or seven) of different cells have to be altered in a single tissue in order to forni a tumor. The other, due to Muller and Nordling, is that a single cell has to undergo k changes before it can generate a tumor. We shall first describe the consequences of the multicell theory of Fisher and Holloman. In the next section we shall treat the multistage theory of Mullev and Nordling.

64

ALICE S. WHITTEMORE

Let us suppose that the occurrence of a tumor in a tissue requires the presence of at least k altered cells, and that the fraction E of unaltered target cells which are changed per unit time is proportional to the concentration c of carcinogen:

1 =pc. (5) When p c is small and when the number k of altered cells is small compared to the number N of target cells in the tissue, it can be shown

that the rate at which the kth alteration occurs in the tissue is approximately proportional to

(Npc)ktk-l

(6)

If we assume that the time for growth to detectable tumor is negligible compared with the time required for the k alternations, then the expression (6) is proportional to the incidence rate Nr(t). With k = 6 or 7, the incidence rate would then be proportional to the fifth or sixth power of age and to the sixth or seventh power of concentration. This age dependence agrees with the observations noted at the beginning of this section. However, the concentration dependence is incompatible with experimental and epidemiological data.

B. CONCENTRATION DEPENDENCEOF OBSERVED INCIDENCE RATES There is evidence that incidence rates vary as the &st or second power of c, rather than as c6 or c’. For example, Doll (1971) analyzed the incidence of bronchial carcinoma in physicians in England and its relation to cigarette smoking. He found that the incidence rate was approximately proportional to the number of cigarettes smoked per day, and to the fourth power of the duration of smoking (see Fig. 1and 5). If we suppose that the number of cigarettes per day is a measure of the carcinogen concentration c , and let t denote duration of smoking, then his result suggests incidence rates are proportional to ct4. The result (6) requires k = 5 to give t4, but then the rates would be proportional to c5, which disagrees with the data. Furthermore Lee and O’Neill (1971) and Altshuler et al. (1971) determined the incidence rate of tumors on the skins of mice which had been painted repeatedly with various concentrations of the carcinogens benzpyrene and dibenzanthracene, respectively. In both cases the rate was found to be proportional to c2. In addition Lee and O’Neill found that it was approximately proportional to the cube of treatment time t. Now (6) requires k = 4 to give t 3 , but then it also yields that the incidence rate is proportional to c4, in disagreement with the data.

QUANTITATIVE THEORIES OF ONCOGENESIS

65

I

i

1000 -

1

i

500 -

Dose-standardized incidence rat0 of bronchial carcinoma no. of c a r e r ~ 1 ~ 5 100 man-yrs. (log scale) 50 -

7

-

: i

- [

10 10

I

20

1

1

1

,

40 60

FIG.5. Incidence rate of bronchial carcinoma, standardized for rate of cigarette sm6king, versus years of smoking. The data points are from Doll (1971) and represent incidence rates among men who started to smoke at ages 15 to 24 years and were not known to have changed their smoking habits. The straight line is eye-fitted to the data points.

The proportionality of the tumor incidence rate to such a high power of c can be avoided in the multicell theory b y assuming that there are a number of different types of alteration, and that at least one of each of k types of altered cells must be present in the tissue in order for a tumor to occur. If exactly n of the k alterations are induced by the carcinogen at rates proportional to its concentration c, then in (6) the term ( N ~ C ) ~ will be replaced b y a term p,-oportional to c". Here n can be any integer between 0 and k.

Nevertheless there remain some substantial objections to the multicell theory. One is the lack of specific assumptions about the spatial configuration of the altered cells in the tissue. Wright and Pet0 (1969) and Jones and Grendon (1975) have devised multicell theories incorporating such assumptions. However the main objection is the biological evidence for the monoclonicity of tumors. Thus w e shall describe the single cell multistage theory of carcinogenesis. V. The Multistage Thcory with Negligible Cell Loss

A. THE THEORY

According to the multistage theory of Muller (1951) and of Nordling (1953), a cell can generate a tusnor only after it has suffered a certain

66

ALICE S. WHITTEMORE

number, say k, of cellular events or changes. The quantitative consequences of this theory were derived by Stocks (1953) and by Armitage and Doll (1954). We assume that the k changes have different transition rates lxt), i = 0, 1, . . . , k - 1, given by

h(t) = SY + P d t ) ,

(7)

and that they must occur in the order 0, 1, . . . ,k - 1.A cell in stage i or an i-cell is one that has undergone the first i changes. Thus Zkt) represents the fi-action per unit time of i-cells which undergo their ( i + 1)st change. We also include the possibility of cell “death,” where death may be loss of ability to divide, with ddt) denoting the death rate for cells in stage i, i = 1, . , . , k - 1. These assumptions, which are represented schematically in Fig. 6, are a straightforward extension of the Iverson-Arley theory for transformation. In this section we shall assume that the loss of i-cells through death and through transition to stage i + 1 is negligible. If in addition the carcinogen concentration c(t) is constant at level c, then the transition rates I t are independent of time. I n this case the rate at which cells are transformed in the tissue is approximately proportional to NZ,,

. . . ,l!k-]tk-l.

If the growth time of a tumor is assumed to be a constant w, then the incidence rate is zero at times t less than w , and at times greater then w , it is approximately proportional to

Nl,,

. . . , l&l(t - wp-1.

(8)

An immediate consequence of (8)is that the logarithm of the incidence rate, when plotted against the logarithm of exposure duration t less w, forms a straight line of slope k - 1. Thus with w set equal to zero, this theory agrees with the power law relationships observed i n . Fig. 4. As was noted in the previous section, when exactlyn ofthe k changes

FIG.6. Schematic representationof the k-stage theory of transformation. Cells in the i* stage can either die at the rate d,(t) or be converted into cells in the (i stage at the rate ZQt). All target cells start as normal cells, in stage zero. Cells which reach stage k are transformed cells.

+ lr

QUANTITATIVE T:SEOFUES O F ONCOGENESIS

67

are induced by the carcinogen, then lo . . .l,+, and thus the incidence rate of (8) is proportional to c”, With n = 1,k = 5, and w = 0, (8) fits the cigarette data of Doll and Hill, as shown in Fig. 2 and 5. With n = 2, k = 4, and w = 10 weeks, it fits the mouse skin painting data of Lee and O’Neill. The quadratic dose-rate dependence of Lee and O’Neill’s cumulative incidence data is shown in Fig. 2C. Figures 2A and 2B show a quadratic dependence of cumulative incidence on dose rate of ultraviolet light and benzpyrene, respectively. The deviations from the quadratic relationship that occur at high incidence and high dose rates may reflect killing of transformed cells by the carcinogen.

B. APPLICATION TO

TERMIN4TED EXPOSURES

The multistage theory depirted in Fig. 6, with negligible cell loss, has been applied to the case in which the carcinogenic concentration is constant until some time to,after which it is zero. This is the case in the experiment of Lee (1975) in which the shaved backs of mice were painted periodically with bempyrene and other carcinogenic components of cigarette smoke. The treatment was discontinued at different stopping times in different groups of animals. Lee fit the multistage theory to these data, using maximum likelihood estimates of the relevant parameters. For the benzpyrene exposures, the estimates n = 2, k = 4, and w = 10 weeks were obtained. These results agree with the corresponding estimates of these parameters for the lifelong repeated benzpyrene treatment of Lee imd O’Neill (1971). Lee also found that benzpyrene appears to affect an early and a late change, with either p , or p , and either p 2 or p 3 approvimately zero. Evidence that p , is nonzero, i.e., that benzpyrene strongly affects the first change, is given by the experiment of Pet0 et al. (1975). They administered the same lifelong repeated benzpyrene treatment to mice starting at 10, 25, 40, and 55 weeks of age and found that tumor incidence after a given duration of treatment was no higher among the older mice. This suggests that benzpyrene acts on the first change and that other changes cannot tak,e place until this benzpyrene-related change is complete. Othenvisr older mice with more background exposure would have more one-cells available for the action of benzpyrene and thus a higher incidence of tumors. Lee’s terminated benzpyrerae data, and curves based on his estimates, are shown in Fig. 7. The curves fit the data well, despite the fact that the theory does not include possible retention of the benzpyrene or its metabolites in the #-issueafter termination of exposure, or variation in tumor growth times.

o.:l

68

ALICE S. WHITTEMORE 0

TUMOR INCIDENCE RATE (log scale) 0.01

*

c = 36 pg/wk

10

/i [/

---

50 200 10

c = 36 pg/wk = 25 wkr

to

50 200 10

c = 36 pglwk lo = 35 wks

200 10

50

C = 36 pg/wk treated for life

50

200

i

I

I TUMOR INCIDENCE RATE (log scale)

o,l I

I

I

0.01 10

50

ZOO 10

50 ZOO 10

50

I

200 10

50 200

TREATMENT TIME IN WEEKS -10 (log scale)

FIG. 7. Tumor incidence rate for benzpyrene painted mouse skin versus time (in weeks) since start of treatment, less 10 weeks. The concentration cft) of benzpyrene is given by c ( t )= c, 0 5 t 5 t oand c ( t ) = 0, t o< t, with different values of concentration c and stopping time to.The values are shown on each graph. The data points were obtained by the interval technique given by Eq. (3)of Hoe1 and Walburg (1972)from unpublished data of P. N. Lee. The solid lines are the incidence rates predicted by the multistage theory, with Lee’s maximum likelihood estimates of the parameters. His estimates of the parameters k and w of (8) are k = 4 and w = 10 weeks. With sland p1given by (7),i = 0, 1, 2, 3, he obtains p o = p 2 = 0. His values for bfc)= ( N / 6 )so (sl+pic) s2(s3 + p g ) are b(0)= 0.0125 X b(36)= 4.78 x and b(60)= 21.84 x His estimates of p&s, are 5.21 and 7.56 for the 36-pg and 60-pg exposure groups, respectively. The dashed lines indicate the predicted background rate.

VI. The Multistage Theory with Non-Negligible Cell Loss

A. RADIATION-INDUCED TUMORS

We shall now consider the multistage theory with “death” of intermediate cells. This theory has been applied to radiation carcinogenesis, in which there is evidence that radiation causes loss of cellular ability to divide as well as malignant transformation. [A detailed review of the difficulties inherent in quantitative analysis of radiation carcinogenesis is provided by Mayneord and Clarke ( 1975).] The theory has also been used by Neyman and Scott (1967), as discussed in Section VI,B. Cell death was first included by Burch (1960) in the two-stage

QUANTITATIVE TH1;ORIES OF ONCOGENESIS

69

theory to analyze radiation-indLced cancer. H e was able to get rough agreement of the theory with data on leukemia incidence in survivors of the atomic bomb explosion at Hiroshima. More recently, Marshall and Groer (1977) have included dosedependent cell death in the three-stage theory to analyze radiationinduced bone cancer. They assume that the death rates d i are equal and are proportional to carcinogen concentration: Here c(t) is the radiation dose late to a bone surface cell in rads per unit of time, and q is not small. ‘This assumption for ddt) leads to such depletion of target cells on the hone surface at high doses that tumors would be very unlikely to occur. However, osteosarcomas are observed at high doses. To account for this, target cells are assumed to be replenished at a large constant late. Then for dose rates less than one rad per day, depletion of targ3t cells can be neglected and their number remains essentiallyN. The theory for dose rates in this range is shown graphically in Fig. 8, The data for human and caninc: 226Raexposure at low doses indicates that tumor incidence varies as the square of dose. This follows from the theory if the transition rates for two cellular changes are proportional to dose. Therefore Marshall and Groer assumed that Further, although the average glowth of an osteosarcoma is estimated at seven years, tumors may appear 10 to 20 years after exposures as short as a month. This can be explained by the third change, which occurs independently of radiation at the constant rate 1,. Stage two cells are assumed to have lost their ability to stop dividing. However, they behave as normal cells uatil they receive a signal to initiate mitosis. The third change is such a signal to divide, so that 1, is related to the host’s rate of bone remodeling. We now assume that 1,t and pC(t) are small relative to unity, where

FIG.8. Schematic representation of the three-stage theory of transformation used by Marshall and Groer and by Neyman and Clcott. It is a special case of the k-stage theory of Fig. 6.

70

ALICE S. WHITTEMORE 2.5

20

'

Incidence rate of human osteosamoma 1 0 0.5

' 0

10 x) 30 40 50 60 70 Time since intake (years)

FIG.9. Incidence rate of human osteosarcoma versus time since intake for the highest 15intake levels of 228Raand 226Rashown in Table 11. Curves are based on Marshall and Groer's minimum chi-square fits to human osteosarcoma data of the three-stage theory with loss of cells in stages one and two. The parameter estimates are given in Table 11.

C ( t ) is the dose to the target cells integrated over time prior to t . We also assume that the growth time is a constant w . Resulting incidence rate curves are shown in Fig. 9. They agree well with human 226Radata in three respects: First, the incidence rate at time t depends only upon the total dose C(t - w ) up to time t - w , and not upon the pattern of administration. Second, at low doses the incidence rate is approxiTABLE I NUMBERS OF OBSERVED AND EXPECTED~ WITHIN

Injection level (pCi/kg body wt)

Number injected

0.00 0.01 0.02 0.06 0.17 0.31 0.95 2.80 8.00

44

Total:

DOGS WITH

OSTEOSARCOMA

17 YEARS AFTER INJECTION OF 22sRab

10 25 23

14 13

12 13 10 -

164

Observed

Expected

0 0 0 1 2 5 11 12 9

0.0

0.0 0.0 0.6 1.1 9.9 10.5 11.9 6.0

40

40.0

-

a Expected numbers were obtained by multiplying the incidence rate of the threestage theory by the number of animals at risk during 2-month intervals, and summing the product over all 2-month intervals in the 17-year period. The parameters p = 5.0 x lo-' per rad, q = 0.01 per rad, 2, = 0.1 per year, w = 2.5 yr and f (conversion factor relating intake level to dose rate to the bone in rads per year) = 350 radslyr per pCi/kg were determined to minimize the value of a x2 goodness of fit test, subject to the constraint that total observed and expected numbers of osteosarcomas be equal. From Marshall and Groer (1977).

QUANTITATIVE THEORIES OF ONCOGENESIS

71

TABLE I1 OBSERVED AND EXPECTED~ NUMBERSOF HUMANSWITH OSTEOSARCOMA WITHIN 70 YEARS AFTER INITIAL INTAKE OF 22ERaPLUS =%ab Intake level (pCikg bone wt)

Nuriber exposed

1 - 3.90 3.91- 5.48 5.49- 7.70 7.71- 10.81 10.82- 15.19 15.20- 21.35 21.36 30.00 30.01- 42.15 42.16 59.23 59.24- 83.23 83.24- 116.94 116.95-164.32 164.33-230.88 230.89-324.42 324.43-455.85 455.86-640.52 640.53-900.00

l!iO !!9

Total:

:\O

.7 ‘!O !!4 9 :!6 :I2 :I8 !7 15 7 18 16 3 3

--

4; 4

Observed

0 0 0 0 0

1 3 4 11 4 5 9 5 8 3 0 1

-

54

Expected

0.2 0.2 0.3 0.3 0.6 1.3 1.4 4.0 6.2 8.4 7.6 5.5 4.9 6.4 5.4 1.0 0.5 -

54.0

Expected numbers and parameters were obtained as in Table I, using one year per rad, rather than two month subintervals. The parameters are p = 4.7 x q = 0.01 per rad, Zz = 0.1 per year, w = 7 years andf = 15 raddyr per pCi/kg. From Marshall and Groer (1977).

mately proportional to the square of the dose C ( t - w). Third, for large doses, the incidence rate is approximately a constant, independent of time and dose. Observed and expected numbers of individuals with osteosarcoma following intake of 226Rain dogs and 226Raplus 228Rain humans, are shown in Tables I, and 11.

B. URETHANE-INDUCED MURCNELUNG TUMORS Shimkin and Polissar (1955)and others studied changes in the cell populations of mice’s lungs after a single injection of urethane. Examination of the lung tissue after sacrifice revealed clones of modified cells which were called hyperplastic foci. The number of such foci increased with time between injection and sacrifice, reached a maximum at about four weeks, and then declined. When a mouse was sacrificed after four weeks, pulmonary adenomas were found. Their

72

ALICE S. WHITTEMORE

number increased with time between injection and sacrifice. Because the question whether the hyperplastic foci were precursors of the tumors could not be answered by direct observation, a theoretical analysis was devised. According to the observations of White et aZ. (1967) as indicated in Fig. 3, the total number of tumors per mouse vanes as a quadratic function of the injected dose of urethane. Furthermore, as shown in Fig. 3, this number is decreased when a single dose is fractionated into several small subdoses. These observations contradict the IversonArley single stage theory discussed in Section 111. Therefore Neyman and Scott (1967) proposed a two-stage and a three-stage theory of cell transformation. They rejected the two-stage theory on the grounds that the resulting temporal behavior of the expected number of hyperplastic foci was inconsistent with the observed peak in their numbers at four weeks. Their three-stage theory is depicted in Fig. 8. Cells in stage zero are normal, cells in stage two are members of hyperplastic foci, and those in stage three are transformed cells. In the transition from stage one to stage two, a cell is assumed to produce two daughter cells in stage two by division. A possible biological basis for this assumption is that the one-cells have experienced a mutational event which is not expressed before mitosis. The rate 2dt) at which target cells undergo their first change is assumed proportional to carcinogen concentration c(t):

lo(t) = POdt).

(11)

However, the mitotic rate 1, of one-cells is taken to be a constant independent of carcinogenic treatment. The transition rate Z2(t) at which hyperfocal cells become malignant is given b y

12(t) = $ 2 + P24t).

( 12)

The death rates dl and d2 are constant, with d2 representing the excess of death over division rates for members of hyperplastic foci. According to (1l),all of the one-cells are produced by the urethane at or shortly after treatment. The continuous formation of hyperplastic foci at rate 22, by one-cell division offsets their net loss through death at rate d2 and their depletion to form tumors at rate Z2(t).Consequently this model predicts that the expected number of hyperplastic foci reaches a single maximum, in agreement with Shimkin and Polissar’s observations. However, if urethane is indeed eliminated from an animal’s system within 24 hours, then this version does not account for the quadratic dependence of total tumor numbers upon injected dose observed by White et al. (1967). Although the total expected number of

QUANTITATIVE THEORIES OF ONCOGENESIS

73

transformed cells per mouse docs contain a term proportional to total dose squared, the proportionality factor for the former is very small and probably cannot account for the observations. It corresponds to the occurrence of all three trailsitions in the short time (24 hours) during which the urethane is in the animal’s system. Experiments subsequent to the publication of Neyman and Scott’s theories have suggested that the quadratic dependence of tumor numbers on injected dose may be due to a quadratic relationship between actual dose to the target cells and injected dose. These experiments, discussed in Section IX,A, may account for the discrepancy between Neyman and Scott’s three-stagt: theory and the data. However the failure of subsequent investigations (White et al., 1970) to reproduce the hypercellular foci noted by Shimkin and Polissar casts doubt on the need for a three-stage theory for this tumor system. Alternate explanations for the decrease in tumor numbers with dose fractionation are discussed in Section IX,A. [f these explanations are valid, then urethane may indeed induce pulmonary adenomas in mice by a one: stage mechanism, as postulated by Iverson and Arley. VII. The Multistage Theory with Proliferative Advantage of Intermediate Cells

A. THE THEORY Although the multistage theory with k greater than 2 accounts for some of the data, there is no direct experimental evidence that cancer occurs in more than two stage:...This led both Armitage and Doll (1957) and Fisher (1958) to modify the multistage theory so that a twoor three-stage theory could explain the observed data. Their modifications give a selective ad1,antage to cells in intermediate stages which enable them to multiply more rapidly than normal cells, a possibility which had been mentioned by Platt (1955) and Muller (1951, p. 130). In the theory of the preceding sections only the cells in the final or kth stage were assumed to increase in expected number through proliferation. The assumption of a selective advantage for intermediate cells is supported by observations of hyperplasia, dysplasia, and other cellular abnormalities preceding malignant transformation. To describe these modifications, we shall now reconsider the multistage theory in a more general foim. To do so we suppose that a cell in stage i greater than zero produces a clone of similar cells, called an i-clone. Then we introduce the age x of an i-clone, which is just the

74

ALICE S. WHITTEMORE

time elapsed since its progenitor suffered its ith change. This theory assumes that whole clones can be eliminated by death, and that the occurrence of a transitional event in more than one cell of a clone is unlikely. The expected fraction per unit time of i-clones of age x in which further transition occurs is denoted by Zkx,t). These assumptions permit the calculation of the rate of transformation in the tissue, which is just the expected number of newly fornied k-clones in the tissue at time t . In the case when the transition and death rates are independent of age, the result agrees with that of the multistage theory described in Section V. Both Armitage and Doll (1957) and Fisher (1958) assumed that the death rate of i-clones of age x is negligible, and that the transition rate Zi(x,t) for intermediate clones is proportional to the size of the clone. They differed however in their assumptions about this size. B. THE ARMITAGE AND DOLL TWO-STAGETHEORY

The schematic diagram shown in Fig. 6 with k = 2 and with d, set equal to zero depicts this theory. Cells in stage zero are normal, those in stage one generate clones which are assumed to grow exponentially at some rate p, and clones in stage two are tumors. The assumption of exponential growth of one-clones means that the size of a one-clone of age x is ear. Here pllog 2 represents the expected number of one-cell doublings per unit time. The transition rate Zl(x,t) is taken to be proportional to one-clone size with proportionality factor Zl(t) depending on the carcinogen concentration at time t. Thus we may write

1 l(x,t) = Zl(t)ePf. (13) Armitage and Doll assumed that carcinogen concentration is a constant independent of time. Therefore the rate Zo of transition to stage one and the proportionality factor 1, on the right-hand side of (13)are constants. If tumor growth time is w , then the rate Nr(t) of tumor occurrence in the tissue is the rate at which two-clones are formed in the tissue at time t - w . Curves comparing human cancer mortality data with this predicted rate, for Armitage and Doll’s estimates of NZ, Z1, p and w = 2.5 years, are shown in Fig. 10. Unlike the incidence rate given according to the multistage theory by (8), the incidence rate predicted by this theory does not yield a straight line when plotted against exposure duration less w on a double logarithmic scale. Although the curves in Fig. 10 fit the data well, the assumption of exponential growth for one-clones leads to difficulties. Armitage and Doll’s estimates of /3 for cancer of the stomach, intestine, rectum, and

QUANTITATIVE THE:ORIES OF ONCOGENESIS 1000

75

A

Mortality rate: no. of deathdl06 man-yrs.

(log scale)

'k

4'0

;O

$5

u 25 30 40 50 65

Ace in years less 2 5 (log scale 1

FIG.10. Mortality rate in men ( X ) and women (0) for cancer of the stomach (A) and of the rectum (B) versus age in years less 2.5 years. The data were recorded in England in 1951-1955. The solid lines are theoretical curves predicted by the Armitage and Doll two-stage theory with proliferative advarctage of one-cells. The parameters of the theory were determined by eye-fitting the curves to the data points. Reprinted with permission from Armitage and Doll (1957).

pancreas are fairly uniform, with an average value of 0.13. This implies that the selective advantage for one-cells enables them to double only once every 5 years. Thus a one-clone would contain, on the average, only 16 cells after 20 years. Such clones, even if formed in an individual at birth, would not become detectable as hyperplasia or dysplasia in a human lifetime, since growth to detectable size (about 4 x lo6 cells) would take 110 years. C-msequently these clones cannot be interpreted as observable precancerous lesions. Fisher's variation of the theory, involving quadratic rather than exponential growth for intermediate clones, does not require such slow initial growth.

C. THE FISHERTHEORY We now consider the multistage theory with the size of intermediate clones proportional to the square of their age. This modification is motivated by the fact that the stem cells in the germinal layer of adult mammalian epithelium stay essentially constant in number, with half the daughter cells moving outward and dying without further division. Therefore the selective advantage of a nonmalignant clone of altered cells is likely to be small, and the clone might plausibly expand laterally like a circular disk at some slaw constant rate. The radius of such a clone of age x will be proportional to x, and its area proportional to x2. Thus Fisher's assumption that the transition rate ZXx,t) of intermediate clones of age x at time t is propcational to their size leads to Z,(x,t) = Zr(t)x2,

i = 1,

. . . , k - 1.

(14)

This theory is depicted schem,rtically in Fig. 6, with d i set equal to zero for all i, and with intermediate transition rates lAx,t) given by (14), i = 1, . . . ,k - 1. When the concentration of carcinogen is constant, 1,

76

ALICE S. WHITTEMORE

and the proportionality factors li(t) on the right-hand side of (14) are constant. I n this case the rate of transformation in the tissue is approximately proportional to the 3(k - 1)st power of exposure duration times the product lo, . . . , 1 k - l of all k transition rates. For constant growth time w and for duration times t greater than w , the incidence rate is then approximately proportional to Comparison of (15) with (8) shows that for constant exposures the consequences of this theory differ from those of the multistage theory only in the value of the power of time in the expression for the incidence rate. Here fewer stages are required to yield the observed high power of time dependence. A two-stage process with quadratic proliferation of one-cells results in the incidence rate increasing as the third power of time, in agreement with the mouse skin tumor data of Lee and O’Neill. A three-stage process with quadratic proliferation of one-cells and two-cells results in the incidence rate increasing as the sixth power of time, in agreement with some human cancer mortality data. This theory can be extended to allow a quadratic proliferative advantage to only some intermediate cells, such as those in the penultimate stage. In general, if n of the k - 1 intermediate clones grow in proportion to the square of their age, then the -incidence rate varies as the (2n k - 1)st power of exposure duration less growth time w . Thus a three-stage theory ( k = 3) with quadratic growth for two-clones ( n = 1)results in the incidence rate increasing as the fourth power of time, in agreement with the lung cancer data of Doll.

+

D. APPLICATIONOF EXPOSURES

THE FISHER

THEORYTO TERMINATED

We have noted that for constant exposures and for suitable values of the parameters, the age dependence of the incidence rates predicted by the Fisher theory is the same as that of the multistage theory. For example, the multistage theory with w = 10 weeks, k = 4 and with two of the four stages affected by benzpyrene accounts for the chronic mouse skin painting data of Lee and O’Neill discussed in Section V,A. The incidence rates given b y this theory are also predicted by the Fisher two-stage theory with w = 10 and with both stages affected b y benzpyrene. However the two theories differ in their consequences for nonconstant exposures, and it is therefore of interest to examine the Fisher theory applied to the terminated exposure data of Lee, discussed in Section V,B. Figure 11shows the incidence rates observed in

77

QUANTITATIVE THblORIES O F ONCOGENESIS

TUMOR INCIDENCE RATE 0.01 (log scale)

lo

r

10

- - - L

50 200

10

50 200

10

50 200

10

50 200

I

/

TUMOR

I

(log scale) c = 60 pg/wk

''Oo1

10

50 200

c = 60 ug/wk

10

50 200

c = 60 ug/wk

10

50 200

c = 60 po/wk

10

50 200

TREATMENT TIME IN WEEKS -10 (log scale)

FIG 11. The data of Fig. 7 compared with incidence rates given by the Fisher twostage theory, with w = 10 weeks. Fsx b(c)= N / 3 (so +poc)(sl+pic), the values were obtained by b(36) = 4.46 x and b(60)= 21.56 x b(0)= 0.0215 x equating total expected and observed numbers of tumor bearing mice in the control, 36and 60-pg lifetime exposure groups, respectively. The values for pds0 of 1.35 and 1.76 were obtained by equating total expected and observed numbers of tumor bearing animals in the 36- and 6@pg tenninatcd exposure groups, respectively. The dashed lines indicate the predicted background rate.

Lee's experiment versus those of the Fisher two-stage theory with w = 10 weeks and with both stages affected b y benzpyrene. Comparison of Fig. 7 and 11 indicates two differences between the incidence rates of the two theories. First, termination of treatment is followed by a more abrupt decrease in the Fisher rates than by those of the fourstage theory. Second, although the rates of both theories approach the background rate asymptotically d t e r treatment stops, those predicted by the Fisher theory do so mole slowly. Despite these differences, both theories provide acceptablt: fits to the data.

E. INCREASED NUMBERSOF TARGET CELLS Iverson (1954) has proposed another explanation for the pro-

portionality of human cancer incidence to such a high power of age. H e modified the single stage theory of Iverson and Arley by assuming that the expected number of normal target cells in the tissue increases

78

ALICE S. WHITTEMORE

with age according to a logistic equation. The resulting incidence rate fits the observed cancer mortality data for suitable values of the parameters involved. Nevertheless because of its ad hoc nature, this theory has not gained acceptance. There has also been evidence that young tissue is just as susceptible to neoplastic induction from a given treatment as older tissue. This evidence is clearly demonstrated by the recent experiment of Pet0 et al. (1975), discussed in Section V,B.

VIII. Single Stage or Multistage Theory with Variation in Transformed Cell Types

A. THE INITIATION-PROMOTION PHENOMENON In the early 1940s Berenblum, Mottram, and Rous discovered that exposures of rabbit and mouse skin to carcinogens at concentrations too low to produce any tumors will, when followed by repeated paintings of a noncarcinogenic substance called a promoter, yield large crops of tumors. It was found that the order of application is important: Few if any tumors result when promoter paintings are followed by application of carcinogen. Thus it is possible to divide the process of carcinogenesis in mouse skin into two stages: initiation, i.e., alteration of the cells in the tissue by a carcinogen, called the initiator, and promotion, i.e., development of the initiated cells to detectable tumors by the promoter. Because the quantitative theories described in the preceding sections do not postulate alteration of tumor growth rates by carcinogenic exposure, none of them can account for the regression of tumors that occurs after termination of promoter painting, or the lack of tumors that accompanies short exposures to high doses of promoter. The theory developed in this section generalizes many of the previously described theories, and, by including variable tumor growth which may be affected by exposure, it provides a bamework for examining initiation-promotion data.

B. GROWTHOF CLONES

OF

TRANSFORMED CELLS

We shall now assume that there are different kinds of transformed cells, indexed by a parameter a.This parameter indicates the “degree of malignancy” of the cell, which increases with a. Some theory of transformation is assumed to determine the rate at which a normal

QUANTITATIVE THEORIES OF ONCOGENESIS

79

target cell becomes a transfornied ceIl of type a. For example, the k-stage theory will do this if Zk-,(t)depends upon a. We next assume that each transformed cell of type a generates a clone of similar cells, which wc: shall call an a-clone. Let n(t,.r,a) be the number of cells at time t in an a-clone generated by a cell which was transformed at time 7 . Since a newly formed a-clone consists of one cell, n(t,t,a) equals one. We assume that each clone grows to a maximum size M. The clone’s rate of cellular increase or decrease per unit time is proportional to its size n reduced b y the factor (M - n)/M. This growth decelerating factor is the fractional part of its ultimate size to be contributed by remaining growth. The proportionality factor, denoted by u, and measured in cells gained or lost per clonal cell per unit time, is called the intrinsic growth rate of the clone. Both u and M increase with a and both depend upon the concentrations of the various agents to which the tissue is exposed. A clone is assumed to become a detectable tumor when it reaches some critical number of cells rn. Thus an a-clone transformed at 7 will become detectable at time t determined by (16)

n(t;a-,a)= rn.

To complete the formulation of the foregoing theory of growth, we must specify the critical number of cells m, the intrinsic growth rate u [ a , &t)l and the ultimate size M [ a , c ( t ) ] . With these specifications, the rate Nr(t) and expected amount N R ( t ) of tumor occurrence in the tissue can be deduced from the preceding assumptions. This theory generalizes the single or multistage theory with constant growth time by postulating instead a distribution of growth times corresponding to different transformed cell types.

c. APPLICATION TO

INITIATION-PROMOTION

EXPERIMENTS

We shall now apply a special case of the theory to analyze the effects of a single dose of initiator at time t = 0 and a promoter of concentration c. Although the promoter may play a role in transformation, we shall assume that only the initiator affects transformation and that only the promoter affects growth. We take the initiator’s effect to be virtually instantaneous, so that shortly after its application the tissue contains an expected numberf(a) Iif transformed cells of type a. The distribution f(a) of transformed cell types, shown hypothetically in Fig. 12, depends on the dose and type of initiator, as well as on the susceptibility of the host. A transformed cell of type a is assumed to generate a clone which

80

ALICE S . WHITTEMORE

EXPECTED NUMBER f ( a ) OF TRANSFORMED CELLS OF TYPE a

at

GROWTH PARAMETER a

Ultmot.

EXPECTED NUMBER f ( a ) OF TRANSFORMED CELL! OF TYPE a

expc1.d number

of

..... .

. . .,. . . -dl)

-u(d 0 GROWTH PARAMETER a

FIG. 12. A hypothetical distribution of transformed cell types in mouse epithelium shortly after initiation. Application of promoter at dose rate c increases the growth rate of an a-clone by an additive factor u(c). (A) The total number of tumors per mouse that have appeared by time t is the number of transformed cells whose parameters exceed a,, where a,+ u(c) is the growth rate of a clone formed at initiation and appearing as a visible tumor at time t . (B) The total number of tumors that ever appear with promoter application at does rate c is the number of transformed cells whose parameters exceed -u(c), as shown by the dotted area under the curve. Higher dose rates c', which increase growth rates by a greater factor u(c'), produce more tumors, as indicated by the striped area under the curve. Clones generated by transformed cells with negative parameters regress upon termination of promotion.

grows to size M > m at intrinsic rate a in the absence of promotion, and at rate a + u(c) with promotion at concentration c. Hence in this example clonal size M is independent of a and c, and u(a,c) = a

+ u(c),

with u ( 0 )= 0.

(17)

Thus the effect of promotion is clonal growth acceleration either through increased proliferation, decreased cell loss, or both. To account for regression we assume that there are clones for which the growth rate a in the absence of promoter is negative. A clone with a deterministic growth rate a Iess than zero would decrease in size and tend to zero as time passes. Alternatively the growth of a clone might be a probabilistic birth-death process with a equal to the birth (proliferation) rate minus the death rate. In the latter case a clone might

QUANTITATIVE THII3ORIES OF ONCOGENESIS

81

grow, but its probability of becoming detectable would be very small. When promoter is applied, the growth rate is increased to a + u ( c ) with u(c) greater than 0. Consequently some clones can grow in the presence of the promoter while they would decrease in its absence. Let at be the parameter required for a clone formed at initiation to appear as a tumor at time t . Of course at depends on promoter concentration c. The expected number NR(t) of tumors that have appeared in the tissue by time t is precisely the total number of transformed cells in the tissue whose growth parameters exceed at. This is illustrated in Fig. 12A. The value of at can be determined from the fact that the number of cells in an at-clone lit time t is m. The value is

at = aft - u(c),

( 18)

where a is a constant depending on m and M . The ultimate expected number N R ( m ) of tumors that ever appear in the tissue is the total number of transformed cells in1 the tissue whose growth parameters exceed am,which by (18) is -u’c). This is because promotion at conr centration c increases all clonal growth rates b y the additive factor u(c). Thus all clones with parameters exceeding -u(c) have positive growth rates with such promotion, and will ultimately appear as tumors. This theory predicts that the total expected number N R ( m ) of tumors in a tissue increases with promoter concentration c, provided that u(c) increases with c. This is because higher promoter concentrations make positive the growth rates of clones with greater excess of death over proliferation rates. The situation is illustrated in Fig. 12B. Promoter concentration c’ is higher than c , and clones with parameters in the range from -u(c’) to -u(c) will appear as tumors when the concentration is c’, but not when it is c. We next examine the relationship between the expected numbers NR(t,c) and NR(t,c’) of tumors per tissue at two promoter concentrations c and c’, for which u(c’; is greater than u(c). The theory indicates that the number of tumors expected at time t with concentration c is simply the number expected with concentration c’ at the earlier time t/(1 b t ) , where b = [u(c’) - u(c)]/u, and a is the constant appearing in (18). This can be summai~-izedas

+

NR(t,c)= N R ( t / (1 + bt),c’). The relation (19)is compared in Fig. 13 with experimental data from the initiation-promotion experiments of Van Duuren et at., (1973). For a suitable value of b the lower two curves satisfy (19), and for another value of this quantity the uppei two curves satisfy it. Another test of

82

ALICE S. WHITTEMORE 18

15

AVERAGE NUMBER 10 OF TUMORS PER MOUSE 5

1

TIME SINCE START OF PROMOTION (WEEKS)

FIG. 13. The average number NR(t,c) of tumors per mouse as a function of time in weeks after the start of promotion at promoter dose rate c . The data points are from Van Duuren et al. (1973). A single dose of 5 pg of 7,12-dimethylbenz[a]anthracene(DMBA) was followed by thrice weekly applications of phorbol myristate acetate (PMA) at three different dose rates in micrograms per application: 0.5(0),2.5(.), %(A). The two upper curves, for which c’ = 25 and c = 2.5, satisfy (19) with b = 0.05/week for t < 1/ 0.05 weeks. The two lower curves, with c’ = 2.5 and c = 0.5, satisfy (19) with b = 0.06/ week for t < U0.06 weeks. The upper curve was obtained by a least squares fit of the hnctionf(x) = 17 [ l - exp{-B(x - 5) + C(x - 5)*}]to the data; the values B = 0.083/wk and C = 0.0021wk2 gave the best fit. The-arrows are at the levels NR(U0.06, 2.5) and NR( 110.05, 25), which would be the asymptotes of the curves NR(t, 0.5) and NR(t, 2.5), respectively, if the above relations held at these values o f t .

(19) is shown in Fig. 14 based on experimental data of Burns (1976). Again a suitable choise of the constant b makes the curves forc = 1 and c’ = 2.5 satisfy (19). These two comparisons of (19) with experimental data lend some support to this theory. We shall now show how the theory can account for regression, as shown in Fig. 14 b y the curve NR,(t) and the associated data points. When the concentration is suddenly decreased from c to 0, those clones with a in the range 0 > a > - d c ) have their growth rates reduced from the positive value a u(c) to the negative value a.Therefore, these clones start to decrease. Any of them which had already grown to be detectable tumors regress and ultimately become undetectable. This explains the gradual decrease of the curve N R d t ) in Fig. 14 after the promoter concentration is reduced to zero at t = 42 days. We have seen that the total expected number NR(w,c) of clones

+

QUANTITATIVE THEORIES OF ONCOGENESIS

AVERAGE NUMBER OF TUMORS PER MOUSE

83

lo

TIME SINCE START OF PROMOTION (DAYS)

FIG.14. The average number NR(t,c) 3f tumors per mouse in each of three groups, as a function of time in days after the start cif promotion. A single 25-pg dose of the initiator DMBA was followed by thrice weekly applications of the promoter PMA. The PMA was applied to each group at one of the following dose rates c(t) in micrograms per application: c(t) = l(A); c(t)= 2.5(.); c(t) = L5, 0< t < 42, c(t) = 0, 42 < t < 100, c(t) = 2.5, 100 < t < 172,c(t) = 10,172 < t < m, ( 0 )The thickness ofthe horizontal bar indicates the dose rate of promoter applied to the third group. The curves N R , and NR, satisfy (19) with b = 0.004Yday for t < l/0.0042 clays. The curve N R , was obtained by a least squares fit of the hnction f ( x ) = 12 I1 - exp{-B(x - 36)}] to the data; the value B = .OSl/day gave the best fit. The curve N R s represents a visual fit to the data for the third group. The data points are kom Ewns (1976).

which ever become detectable tumors increases with promoter concentration c, provided that u(c) increases with c. Thus higher concentrations of promoter produce more growing clones. This explains why the curve NR,(t) in Fig. 14 increases from one level to another when c is increased at t = 100 days and again at t = 172 days. It also explains another phenomenon observed ,experimentally but not shown in Fig. 14. The phenomenon is that telmination of promotion at higher concentrations produces a greater j+action of regressed tumors. This can be explained because higher coricentrations c increase the range -u(c) < a < 0 of clones which decrease without promotion. This and the above qualitative agreement between the theory and the data provide further support for the theory, and concludes our discussion of it. IX. Implications for Dose-Response Relationships

A. THE PROBLEM Estimates of human risk (lifetime probability of cancer) &om exposures to low doses of carcinogenic agents often involve the use of animal experiments conducted at doses sufficiently high to produce

84

ALICE S. WHITTEMORE

tumors in an appreciable fraction of the animals tested. There are two major sources of uncertainty in obtaining such estimates. The first concerns the mechanism relating animal risk at high doses to animal risk at low doses. The second is the role of interspecies differences in extrapolating from animal to human risk. Risk estimates at low doses based on experimental results at high doses are extremely sensitive to the shape of the curve relating risk to dose. Thus it is desirable to obtain such estimates using a doseresponse curve which is based on a biologically plausible theory. The multistage theory, which is supported b y some experimental and epidemiological evidence, has been proposed b y many investigators. For chronic exposures to chemical carcinogens, Guess and Crump (1977)use a procedure that includes the predictions of the multistage theory, with the number of dose-related stages determined by the data. Their method indicates that animal experiments are not likely to establish upper bounds for risk estimates that decrease with dose at a faster than linear rate. A serious difficulty with reliance on quantitative theories of cancer mechanisms to determine dose-response relationships is the possible discrepancy between applied dose and cellular concentrations. The relationship between tumor production and actual cellular exposure may be obscured b y nonlinearity between cellular .and applied concentrations, For example White (1972), using urethane labeled with radionuclides, estimated the internal dose to a mouse over the 24-hour period subsequent to injection of a single dose of urethane. Her estimates of internal exposure, measured in milligram-hours per gram of body weight, are shown plotted against injected dose in Fig. 15A. It is evident that these estimates of internal exposure are quadratic rather than linear functions of injected dose. Plots of average number of tumors per mouse versus injected dose and versus estimated internal exposure are shown in Fig. 15B and C, respectively. They indicate that while tumor numbers depend quadratically on injected dose in agreement with the observations shown in Fig. 3, they vary nearly in proportion to estimated internal exposure. Thus a theory tailored to fit the observed quadratic external-dose-response relationship is likely to be an inappropriate description of the tumor producing mechanism, and as such could seriously underestimate risk at low doses of urethane. The relationship between estimated dose and cellular concentration is even more tenuous in epidemiological studies. The difficulty suggests that future modeling efforts should emphasize incorporation of physiological, pharmacological, and biochemical information. The

QUANTITATIVE THECIRIES OF ONCOGENESIS 30 -

A

25 -

ii

20 -

Internal exposure (mg-hrs/gm) 15 (cumulative to 24 hrs)

o.o '

B 40 -

i

10 -

5-

85

7

7

I

I

Ic

Mean number of tumors per mouse

'(Id .I 0

5

10

Internal exposure (mg-hrslgrn)

FIG. 15. (A) Internal exposure versus injected dose in mice treated with ethyl carbonate (carbonyl-"C). Internal exposure, measured in milligram-hours per gram weight of mouse, was estimated by computing the areas under the curves giving amount of expired and eliminated I4Catom as a function of tinie after urethane injection, for times up to 24 hours. The calculations assume that at any instant the amount of unrecovered '*C atom is still in the animal. The quadratic relationship between estimated internal exposure and injected dose observed in (A) is supported by the plots of mean number of tumors per mouse versus injected dose (B) and versus c5stimated internal exposure (C). The straight lines in (B) and (C) connect extreme poinis. It is evident from (B) and (C) that tumor yield varies quadratically with injected dose and linearly with estimated internal exposure. Reprinted with permission from White (1972).

problem has received considerable attention for radiation carcinogenesis (e.g., Mayneord and Clarke, 19i5) and for chemotherapeutic agents (e.g., Bischoff, 1977). However there is need for study of the distribution, excretion, and metabolism of' chemical carcinogens for the purpose of establishing effective dose ;in experimental work. The model of Swartz and Spear (1975) for the metabolism of hydrocarbons in mouse skin carcinogenesis represents a slep in this direction.

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Models that include pharmacological and biochemical information would also be useful in reducing uncertainty due to interspecies differences. Furthermore their predictions for changes in internal exposure with different dosage schedules may explain anomalous experimental findings concerning the reducing and enhancing effects of dose fractionation. Klonecki ( 1976) has proposed another explanation for changes in tumorigenicity with fractionation: Changes occur because initial applications of fractionated dose alter the number of target cells in susceptible phases of the cell cycle.

B.

RADIATION CARCINOGENESIS

A full discussion of the dose-response curve and the effects of dose fractionation and protraction for radiation carcinogenesis is beyond the scope of this paper. Reviews of the subject are provided by Brown (1976), Shellabarger (1976), and the National Council on Radiation Protection and Measurements (1975). The following summary points of these reports are of interest. For low energy transfer (LET) radiation (x-rays and gamma rays), the form of the dose-response relationship varies widely depending on species and tumor site. Moreover, dose fractionation and protraction generally reduce tumor risk. For high LET radiation (neutrons) the dose-response relationship appears to be linear, and lengthening the exposure time of a given dose does not appear to reduce tumor yield. The differences between high and low LET radiation have been studied in terms of relative biological effectiveness (RBE), which is the ratio of low L E T radiation dose to high LET radiation dose for doses yielding equal tumorigenic effect. Experimental results indicate that RBE increases with decreasing neutron dose. All of these findings suggest that the carcinogenic effects of low LET radiation may require more than one radiation related event, while high LET radiation may induce tumors b y a one-stage mechanism. However there are human data indicating a linear dose-response curve for low LET radiation. X. Conclusion

Some of the defects of the theories discussed in the preceding sections are quite evident. For example the crucial distinction between benign and malignant tumors is ignored by many of them. Furthermore, none of them has included the possibility of cell repair or the action of the host’s immune system. Many of the assumptions underlying the theories represent extremely simplified versions of reality. An example of this is the as-

QUANTITATIVE TI-iEOFUES OF ONCOGENESIS

87

sumption that the target cells of a tissue are fixed in number and independent of each other with respect to likelihood of transformation. In addition, the sensitiviiy of target cells to transformation, as measured by the constants s i and p i of (7), is assumed constant. However, it may well vary with cell cycle time, with location in the tissue, and with hormone balance, level of immune response, and other factors in the cellular environment. Although we have presented the theories as deterministic, all of them were formulated as descriptions of the probabilities of transformation and tumor appearance. The quantities described here are the expected values of tumor rates and numbers. Most of the theories have emphasized these expected values and have not stressed variances and other statistics of the tumor incidence rates. To extract additional information fiom these statistics it will be necessary to consider two kinds of variation. One results from the stochastic processes within each individual, which have been treated in the present theories. The other is due to variation from one individual to another because of differences in susceptibility. It would be desirable to separate these two kinds of variation, and to determine what part of the observed variation each contributes. There is no reason why one thieory should apply to all tumor sites, to all agents, and to all species. It :eems likely that many different mechanisms are operating, and the theories proposed to describe them should reflect this variety. Despite all of their limitations, the single and multistage theories provide a flexible, broad, and biologically plausible fiamework in which to examine the gross behavior of oncogenesis data. They also providle a base for the development of investigations concerning the complicoating factors mentioned above, so that underlying mechanisms can b e elucidated. ACKNOWLEDGMENTS I wish to thank Norton Nelson for sug,jesting this review, and Bernard Altshuler and Joseph B. Keller for many helpful discussions related to it. REFERENCES Altshuler, B., Klassen, W., Troll, W., and Orris, L. (1971).Proc. Am. Assoc. Cancer Res. 12, 49. Armitage, P., and Doll, R. (1954). Br. J . ICancer 8, 1-12. Armitage, P., and Doll, R. (1957). B r . J . 17ancer 11, 161-169. Armitage, P., and Doll, R. (1961). In “Pioceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Prot8ability” (J. Neyman, ed.), Vol. IV, pp. 19-38. Univ. of California Press, Berkeley. Bischoff, K. B. (1977). In ‘‘Environments1 Health: Quantitative Methods” (A. Whittemore, ed.), pp. 3-12. SOC. Itid. Appl Muth., Philadelphia, Penn\ylvania.

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Blum, H. F. (1959). “Carcinogenesis by Ultraviolet Light.” Princeton Univ. Press, Princeton, New Jersey. Brown, J. M. (1976). Health Phys. 31, 231-245. Burch, P. R. J. (1960).Nature (London) 185, 135-142. Burns, F., Vanderlaan, M., Sivak, A., and Albert, R. E. (1976). Cancer Res. 36, 14221427. Clemmesen, J. (1974).Acta. Pathol. Microbiol. Scand. S u p p l . 247, 1-255. Doll, R. (1971).J. R . Statistic. SOC. 134, 133-166. Doll, R., and Peto, R. (1976). Br. Med. J. 2, 1525-1536. Fisher, J. C. (1958). Nature (London) 181,651-652. Fisher, J. C., and Holloman, J. H. (1951). Cancer 4 , 9 1 6 9 1 8 . Guess, H. A., and Crump, K. S. ( 1977).In “Environmental Health: Quantitative Methods (A. Whittemore, ed.), pp. 13-30. SOC.Itid.A p p l . Math., Philadelphia, Pennsylvania. Hoel, D. G., and Walburg, H. E., Jr. (1972).J.Natl. Cancer Inst. 49,361-372. Iverson, S. (1954). Br. /. Cancer 8,575-584. Iverson, S., and Arley, N. (1950).Acta Pathol. Microbiol. Scand. 27, 773-803. Jones, H. B., and Grendon, A. (1975). Food Cosmet. Toxicol. 13,251-268. Klonecki, W. (1976).Zastosow. Mat. A p p l . Math. 15, 163-186. Lee, P. N. (1975). Tob. Res. Counc. Rev. Act. 1970-1974 (London), pp. 28-32. Lee, P. N., and O’Neill, J. A. (1971). Br. J. Cancer 25, 759-770. Marshall, J. H., and Groer, P. G. (1977). Radiat. Res. 71, 149-192. Mayneord, W., and Clarke, R. H. (1975). Br. J. Radiol. S u p p l . 12, 1-112. Muller, H. J. (1951). Sci. Prog. 7, 93-493. National Council on Radiation Protection and Measurements (1975). “Review of the Current State of Radiation Protection Philosophy,” Rep. No. 43. NCRPM, Washington, D.C. Neyman, J., and Scott, E. (1967).I n “Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability” (L. M. Le Cam and J. Neyman, eds.), Vol. IV, pp. 745-776. Univ. of California Press, Berkeley. Nordling, C. 0. (1953). Br. J. Cancer 7,68-72. Peto, R., Roe, F. J. C., Lee, P. N., Levy, L., and Clack, J. (1975). Br. J. Cancer 32, 41 1-426. Platt, R. (1955). Lancet 1,867. Scotto, J., Kopf, A. W., and Urbach, F. (1974). Cancer 34, 1333-1338. Shellabarger, C. J. (1976). Cancer 37, 1090-1096. Shimkin, M. B., and Polissar, h4. J. (1955).J. Natl. Cancer Inst. 16, 75-93. Stocks, P. (1953).Br. J . Cancer 7,407-417. Swartz, J., and Spear, C. (1975). Math. Biosci. 26, 19-39. Van Duuren, B. L., Sivak, A., Segal, A., Seidman, I., and Katz, C. (1973).Cancer Res. 33, 2 166-2172. White, M. (1972). I n “Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability” (L. M. Le Cam, J. Neyman, and E. L. Scott, eds.), Vol. IV, pp. 287-307. Univ. of California Press, Berkeley. White, M., Grendon, A,, and Jones, H. B. (1967).In “Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability” (L. M. Le Cam and J. Neyman, eds.), Vol. IV, pp. 721-743. Univ. of California Press, Berkeley. White, M., Grendon, A., and Jones, H. B. (1970).Cancer Res. 30, 1030-1036. Whittemore, A., and Keller, J. B. (1978).SOC. Ind. A p p l . Math., Reu. 20, 1-30. Wright, J. K., and Peto, R. (1969). Br. J. Cancer 23,547-553.