Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol163.Issue1.Jan2000:

Incorporating observability thresholds of tumors into the

two-stage carcinogenesis model

Marjo V. Smith

a,*

, Christopher J. Portier

b a

Analytical Sciences, Inc., Suite 200, 2605 Meridian Parkway, Durham, NC 27713, USA b

National Institute of Environmental Health Sciences, Laboratory of Computational Biology and Risk Assessment, P.O. Box 12233, Research Triangle Park, NC 27709, USA

Received 19 October 1998; received in revised form 18 June 1999; accepted 30 July 1999

Abstract

This paper discusses a general way of incorporating the growth kinetics of malignant tumors with the two-stage carcinogenesis model. The model is presented using time-homogeneous rate parameters. In that case, the di€erential equations comprising the model are straightforward to solve using standard numerical techniques and software. An extension of the method to time-dependent rate parameters is included in Appendix A. Allowing the rate parameters to be time-dependent does incur computational cost. An ex-pression is given for the expected time without visible tumor, a generalization of the expected time to an observable tumor that includes the possibility of tumor regression. The model is illustrated using incidental liver tumor data in control rats from NTP rodent carcinogenicity studies, using linear birth±death kinetics of tumors combined with a non-absorbing detection limit. The approach is also shown to be potentially useful with tumor observability thresholds having more complicated features. Ó _{2000 Published by}

Elsevier Science Inc. All rights reserved.

Keywords:Rodent tumor analysis; Non-homogeneous coecients; Backward Kolmogorov equation

1. Introduction

The two-stage model of carcinogenesis developed by Moolgavkar and Venson [1] and Moo-lgavkar and Knudson [2] models the progression of normal cells to malignant cells via two stages. The ®rst stage follows an alteration of normal cells to precancerous cells or stage-one cells. The second stage follows the transformation of stage-one cells to malignant cells. The normal cell

*_{Corresponding author. Tel. +1-919 544 8500, ext.165; fax: +1-919 544 7507.} E-mail address:mvs@asciences.com (M.V. Smith).

0025-5564/00/$ - see front matter Ó _{2000 Published by Elsevier Science Inc. All rights reserved.}

population may be modeled as constant or as a linear birth and death process. The stage-one cell population is normally modeled as a linear birth and death process. Original applications of the model dealt with the malignant growth subsequent to the ®rst appearance of a malignant cell in a deterministic fashion, assuming a non-negative constant time interval to a tumor of observable size. Later work has called for more stochastic descriptions of the malignant growth process [3±5].

Exactly how the observability threshold should be modeled may be expected to di€er with tumor and circumstance. Tsodikov et al. [5] considered the progression of human breast cancer tumors to observable size. As human tumors are removed soon after they are observed, it made sense to model the tumor size threshold of observability as an absorbing, random barrier to the pure birth process of tumor progression. The NTP rodent carcinogenesis bioassays provide a very di€erent situation. As part of experimental protocol, some rats are sacri®ced before the experi-ment is completed, some die due to the presence of large tumors that may or may not be of the tumor type of interest, while others die accidentally. The studies last about two years, whereupon the surviving rats are sacri®ced. Regardless of how the rats died, tissues are sectioned and ex-amined microscopically for the presence of tumors. For some tumor types, e.g. liver tumors in rats, the appearance of malignant clones is suciently late that presence of the tumor may be assumed incidental to death. Since the tissues can only be examined after death, a tumor may have reached observable size some time before it is actually observed. It is possible that immune system response or increased apoptosis could shrink an observable tumor to below the observability threshold before death and tissue examination.

For tumor data of this type, the observability threshold may be assumed approximately con-stant but not absorbing, as the process does not stop when the microscopic observability threshold is reached. Slow growth of the tumors means that the time between the appearance of the ®rst malignant cell and a tumor of observable size may still have an appreciable impact, even though the microscopic observability threshold is much smaller than that for human cancers. The question of incorporating an observability threshold into models of this type has been addressed before [6], but the forward Kolmogorov equations (leading to partial di€erential equations) or ®ltered Poisson processes [7] were used. The theoretical development of these methods is attractive but not easily implemented using standard software.

In this paper, an approach initiated by Portier et al. [8] is further developed to extend the standard two-stage carcinogenesis model to include a malignant growth process. The equations for the model are developed using `®rst step analysis' (see e.g. [9, pp. 79±88]), that also corre-sponds to the backward Kolmogorov equations. The formulation is ¯exible but simple, especially in the time-homogeneous case, and may be used with other models of observability thresholds (See Section 4). The backward Kolmogorov equations can be developed for time-dependent rate parameters as well [10,11], although there is a computational cost. The extension of the model developed in this paper to time-dependent rate parameters is given in Appendix A.

rats [13] is used to provide parameter values for the ®rst stages of the model. The paper closes with a discussion about using the equations with observability thresholds of di€erent types.

2. The model

2.1. General equations

Mathematically the two-stage model may be thought of as linked birth and death processes. The populations of normal cells and initiated or stage-one cells may each be assumed to be birth and death processes. Each malignant cell will initiate another process, describing the growth of that tumor. The system is assumed to be time-homogenous with cells acting independently. In

accordance with the usual development, in a small period of time_Dtfollowing any timet, only the

following events are assumed possible at the following probabilities:

For the growth process of the tumors, we will use:

The tumor is considered observable if it containsNor more cells. The following probabilities

are then de®ned:

W0(t)ˆPr(no tumor is visible attj1 normal cell, no stage-one cells, no malignant cells at 0). W1(t)ˆPr(no tumor is visible attjno normal cells, 1 stage-one cell, no malignant cells at 0). W2(t)ˆPr(no tumor is visible attjno normal cells, no stage-one cells, 1 malignant cell at 0).

If we can ®nd an expression for W0(t), then by the assumption of independence of cells, the

probability of a tumor being visible at timetcan be written as 1ÿW0…t†M;where the system starts

withM normal cells at time 0.

The strategy for computing W0(t) is to use the above assumptions to derive a system of two

di€erential equations, havingW0(t) as part of the solution. The technique, ®rst step analysis [9, pp.

79±88], is used to derive the di€erential equations over the interval ‰0;tŠ. Dividing the interval

‰0;t‡_DtŠ into the two subintervals ‰0;_Dt† and ‰_Dt;t‡_DtŠ allows using the conditional in the

Dtb2 birth of a malignant cell

Dtd2 death of a malignant cell

Dtb₀ birth of a normal cell from one normal cell

Dtd0 death or di€erentiation of a normal cell

Dtl₁ formation of a stage-one cell during a normal cell division

1ÿDt …b₀‡d0‡l1† nothing happens to a normal cell

Dtb1 birth of a stage-one cell

Dtd1 death or di€erentiation of a stage-one cell

Dtl₂ formation of a malignant cell during a stage-one cell division

de®nition ofW0(t) (set at time 0) together with the above assumptions over the interval‰0;Dt†. The

solution over the remaining interval‰Dt;t‡DtŠmay be written in terms of the solutions over‰0;tŠ

using time-homogeneity, since these two intervals have the same length. The backward Kolmogorov equations are derived using this same technique.

Beginning with a single normal cell at time 0, only four events are possible over the interval

‰0;_Dt†: (1) nothing might happen, so that there is only one normal cell at time_Dt, (2) the normal cell might divide into two normal cells, (3) the normal cell may die, or (4) the normal cell may divide into one normal and one stage-one cell. W0…t‡Dt†may be written as follows:

W0…t‡Dt† ˆPr…no tumor is visible att

‡_Dtj1 normal cell; no stage-one cells; no malignant cells at 0† ˆPr…no tumor is visible att

‡Dtj1 normal cell; no stage-one cells; no malignant cells at_Dt† Pr…1 normal cell; no stage-one cells; no malignant cells at

Dtj1 normal cell; no stage-one cells; no malignant cells at 0† ‡Pr…no tumor is visible att

‡_Dtj2 normal cells; no stage-one cell; no malignant cell at_Dt† Pr…2 normal cells; no stage-one cell; no malignant cell at

Dtj1 normal cell; no stage-one cells; no malignant cells at 0† ‡Pr…no tumor is visible att

‡Dtjno normal cells; no stage-one cells; no malignant cells at_Dt† Pr…no normal cells; no stage-one cells; no malignant cells at

Dtj1 normal cell; no stage-one cells; no malignant cells at 0† ‡Pr…no tumor is visible att

‡_Dtj1 normal cell; 1 stage-one cell; no malignant cells at_Dt† Pr…1 normal cell; 1 stage-one cell; no malignant cells at_Dt† ˆPr…no tumor is visible att

‡_Dtj1 normal cell; no stage-one cells; no malignant cells at_Dt† …1ÿ_Dtb0

ÿDtd0ÿDtl1† ‡Pr…no tumor is visible att

‡_Dtj2 normal cells; no stage-one cell; no malignant cell at_Dt† _Dtb₀

‡Pr…no tumor is visible att

‡Dtjno normal cells; no stage-one cells; no malignant cells at_Dt† Dtd0

‡Pr…no tumor is visible att

‡Dtj1 normal cell; 1 stage-one cell; no malignant cells at_Dt† Dtl₁:

Using the above de®nitions forW0(t) andW1(t) and the assumptions of time homogeneity and

independence of cells,

W0…t‡Dt† ˆW0…t† …1ÿDtb0ÿDtd0ÿDtl1† ‡W 2

0…t† Dtb0‡1Dtd0‡W0

By subtractingW0(t) from both sides, taking the limit asDtgoes to zero, and rearranging terms,

dW0…t†=dtˆW02…t† b0ÿ ‰l1…1ÿW1…t†† ‡b0‡d0Š W0…t† ‡d0: …1†

In a similar way, the corresponding equation for W1…t† is

dW1…t†=dtˆW12…t† b1ÿ ‰l2…1ÿW2…t†† ‡b1‡d1Š W1…t† ‡d1: …2†

Clearly from the de®nitions,W0…0†W1…0† ˆ1.

These same equations (with W2…t† ˆ0), were derived by Portier et al. [8], using probability

generating functions.

2.2. Malignant growth kinetics

In order to solve the above di€erential equations for W0(t), an expression for 1ÿW2…t†, the

Pr…seeing a tumor attj1 malignant cell at 0†is needed. Note that up to this time, no assumptions about the growth kinetics of the tumor, other than time-homogeneity have been made. Each malignant cell is now assumed to progress as a time-homogeneous linear birth and death process. Bailey ([14, p. 94]) gives a closed form expression for the density of a birth±death process at timet, given a single individual at time t, for the case that b₂ 6ˆd2. In terms of the parameters de®ned

above fornP1,

Pr…tumor hasncells attj1 malignant cell at 0† ˆ …1ÿA†…1ÿB†Bnÿ1_{, where}

Aˆd2…e

…b2ÿd2†tÿ₁†

b2e…b2ÿd2†tÿd2

and

Bˆb2…e

…b2ÿd2†tÿ₁†

b₂e…b2ÿd2†tÿd₂ :

The Pr…seeing a tumor attj1 malignant cell at 0† is then

X1

nˆN

Pr…tumor of ncells attj1 malignant cell at 0† ˆ …1ÿA†…1ÿB†X

1

nˆN Bnÿ1:

Since forb2>d2, the expressionBremains less than 1 for all positivet, the formula for geometric

series can be used to write

Pr…seeing a tumor attj1 malignant cell at 0† ˆ…b2ÿd2†b

Nÿ1

2 ‰1ÿeÿ…b2ÿd2†tŠ Nÿ1

‰b₂ÿd2eÿ…b2ÿd2†tŠ

N :

Note that if b2ÿd2 and the observation times, t, are large, then the quantity exp…ÿ…b2ÿd2†t†

may be negligible. Then the probability of seeing a tumor at t can be approximated by

…b₂ÿd2†=b2. In that case, the model assumption of instant observability of a tumor at the creation

of a single malignant cell, would be a good approximation (as expected). The mutation rate l2

would simply include the factor, …b₂ÿd2†=b2. Note that the closer d2 is to b2, the bigger the

For the limiting case thatb2 ˆd2, Bailey gives

Pr…tumor hasncells attj1 malignant cell at 0† ˆ …b2t†

…nÿ1†

…1‡b₂t†…n‡1†; for n P1:

As before,

Pr…seeing a tumor attj1 malignant cell at time 0†

ˆX

1

nˆN

Pr…tumor hasncells attj1 malignant cell at 0† ˆ …b2t†

…Nÿ1†

…1‡b₂t†N:

For large t, this quantity approaches 0.

Substituting either of the above expressions for 1ÿW2…t†; would allow solving forW0(t), the

probability of not seeing a tumor from a single normal cell. Again, assuming the experiment starts withMnormal cells at time 0, the probability of observing a tumor at timetwould be 1ÿW0…t†M:

If the number of normal cells can be assumed to be approximately constant…b₀ˆd0ˆ0†, then

Eq. (1) reduces to a separable equation that can be solved in terms of an integral ofW1(t). In this

case, only the product of the number of normal cells,M, and the mutation rate for a single cell,l1,

is identi®able. Thus Eq. (1) may be written as

dW0…t†=dtˆ ÿMl1‰1ÿW1…t†Š W0…t†: …3†

Since the solution to this equation already includes the exponent,M, the solutionW0(t) would be

Pr…no visible tumor attjM normal cells; no stage-one cells; no malignant cells at 0†. The proba-bility of observing a tumor at time t would be 1ÿW0…t†.

2.3. Average time to observable tumor

In the special case that malignant growth is irreversible or the observability threshold is

absorbing, a random variable, T can be de®ned as the time a malignant tumor ®rst

be-comes visible. In that case, W0…t†M ˆPr…no tumor is visible by timetjMnormal cells; no

stage-one cells; no malignant cells at 0† can be written as Pr…T >tjMnormal cells; no stage-one cells; no malignant cells at 0†. Thus 1ÿW0…t†M is the distribution function of Tand the expected value

ofT can be written as

E…T† ˆ

Z 1

0

W0…t†Mdt; provided the infinite integral exists:

In the case of reversible growth kinetics without an absorbing threshold, the time to tumor observability becomes less easily de®ned, as it is possible that a tumor may temporarily regress in size from larger than the observability threshold to smaller than the observability threshold. In that case, the random process,

V…t† ˆ 0 if there is a visible tumor att

1 if there is no visible tumor att

may be de®ned. Then, Pr…V…t† ˆ1† ˆW0…t† M

. Now consider any ®nite time interval,‰0;TfŠ. Let Tv…Tf† ˆRTf

visible tumor. Then the average amount of time in any given ®nite interval,‰0;TfŠwithout visible

The average total time without tumor may be found by taking the limit as Tf goes to in®nity, provided the limit exists. In that case,

E…Tv† ˆ lim

The methodology is illustrated by modeling liver tumors in male F344 rats. Historical control data sets from 60 NTP long-term carcinogenicity studies were combined to produce a tumor incidence data set for 3514 male F344 rats of which 86 had tumors of this type. The NTP car-cinogenicity studies only count the animals in which at least one tumor of a given type is observed. Thus no information on multiple tumors, tumor size or tumor lethality was available. Animals were regularly sacri®ced throughout the studies. The earliest death occurred at 11 days; the last death occurred at 736 days. After death, the liver tissue was sectioned and examined under a microscope for tumors. A tumor may be visible at approximately 50 cells, but only recognizable as a malignant tumor at several thousand cells. The term `observable' in this paper refers to being observable (and recognized) as a malignant tumor. The ®rst tumor was observed in a rat that died on the 457th day. Thus there is a very low incidence of liver tumors in this data set, with no early tumors observed.

Rat liver tumors are preceded by hepatic lesions, which may be identi®ed with stage-one cells. Precancerous foci are relatively common in rat livers, while malignant tumors are relatively rare. Parameter values cited by a hepatic lesion simulation study [13] were used for the ®rst part of the two-stage model, and the NTP incidence data used only to estimate the parameters involved with malignant growth. In this example, the number of normal liver cells is assumed to be approximately constant and suciently large that stochastic ¯uctuations are not important. In deriving a mu-tation rate ofl1mutations per day, the normal cell division rate is assumed to be about the same as

the cited normal cell death rate, 1:210ÿ3 _{divisions/cell/day. The mutation rate per normal cell}

division was given as 3:510ÿ8_{mutations per division, so that the mutation rate of a normal cell is}

3:510ÿ8_{mutations=division1:210}ÿ3_{divisions=dayˆ4:210}ÿ11_{mutations/day.}

Carthew et al. [15] give 15:2108_{as the number of normal hepatic cells, so that Eq. (3) is used}

with Ml₁ ˆ0:06384 mutations of normal cells/day. Finally, Conolly and Kimball [13] give the

3.2. Estimation procedure

The observability thresholdN, the second mutation ratel1, and the malignant birth and death

ratesb₂andd2were estimated by the maximum likelihood method. No mortality was modeled for

the animals themselves. In particular, it was assumed that none of the liver tumors were instru-mental in the death of an animal. The low overall incidence and especially the late onset of the liver tumors in this data set make this assumption plausible. In cases where this assumption would not be reasonable a more elaborate likelihood expression would have to be used (see e.g. [16]).

With the above assumption, no distinction between natural deaths and sacri®ces needs to be made, and the setfti; iˆ1;. . .;mg, may denote the days on which at least one rat died, either by

sacri®ce or naturally. Letcianddirespectively refer to the numbers of animals that died on dayti,

having or not having observable liver tumors. For any given set of parameter values, Eqs. (2) and (3) can be numerically solved by any di€erential equation solver. For this paper the estimation was done using Fortran IMSL subroutines, with Matlab used to check the results [17,18]. Thus

W0…ti† ˆPr…no tumor is visible by timetjMnormal cells; no stage-one or malignant cells at 0† can

be found for anyiˆ1;. . .;m. The likelihood expression may then be written as

Ym

iˆ1

W0…ti†di‰1ÿW0…ti†Šci;

though the logarithm of this expression was actually maximized. The IMSL routine, DIVPAG was used to solve the di€erential equations and the routine, DBCONF, was used to optimize the loglikelihood.

The data was ®t to the model by ®xing the ratio,d2/b2, successively at 0.0, 0.1, 0.3, 0.5, 0.7, 0.9,

0.95, 1.0 and the observability threshold, N, at 1, 5, 15, 50, 100, 500, 1000, and 10 000 cells. The

maximum likelihood estimates for b2 and l2 were found for each such combination.

3.3. Results

The values of the loglikelihood expressions are given in Table 1. The values shown do not di€er statistically from each other, but do vary in a consistent way across the combinations of death±

birth ratios and observability thresholds. Corresponding estimated values for b2 and l2 are

presented in Tables 2 and 3.

Examination of Table 1, shows the optimization surface to be very shallow, with no relative maximum. The relative ¯atness of the loglikelihood surface indicates some identi®ability or es-timability problems. Both numerical and theoretical non-identi®ability were checked in the fol-lowing way. First, the numerical integration was carefully checked by two di€erent di€erential equation packages. By adjusting tolerance levels, the solutions to the di€erential equations could be made to agree on 7±10 signi®cant digits. Less accurate tolerance levels led to only minor di€erences in the table entries and no qualitative di€erences.

The con®dence in the numerical solutions to the di€erential equations meant that these equations could be used to check theoretical non-identi®ability. The probability of observing a tumor was computed using the equations, using the estimated parameter values from Tables 2 and

Table 3

Mutation rates to malignant cells times 107a

Minimum cells in observable clones

Ratios of death±birth rates of malignant cells,d2=b2

0.0 0.1 0.3 0.5 0.7 0.9 0.95 1.0 50 9.0 10.1 12.9 18.0 29.9 88.2 174.3 1206 100 9.1 10.1 13.0 18.1 30.1 89.3 176.5 2432 500 9.1 10.1 13.0 18.2 30.3 90.6 180.3 12 240 1000 9.1 10.1 13.0 18.2 30.4 90.8 181.1 24 510 10 000 9.1 10.1 13.0 18.3 30.4 90.8 182.3 245 100

a

The mutation rates to malignant cells increase sharply with increasing death rates. In the case that a single cell is observable, the mutation rate may equal either 4:9110ÿ7_{or…4:91=…1ÿd}

2=b2†† 10ÿ7corresponding to two di€erent

solutions with very large or small malignant birth rates, respectively. See text. Table 2

Birth rates of malignant cells times 100a

Minimum cells in observable clones

Ratios of death±birth rates of malignant cells,d2=b2

0.0 0.1 0.3 0.5 0.7 0.9 0.95 1.0 1 N/A Large or Large or Large or Large or Large or Large or Small

small small small small small small

5 1.89 1.97 2.23 2.47 2.41 1.64 1.48 1.34 15 2.92 3.18 3.76 4.59 6.12 7.93 6.51 5.04 50 4.02 4.37 5.34 6.87 9.88 19.59 26.20 18.01 100 4.68 5.10 6.21 8.13 11.99 25.97 39.26 36.53 500 6.18 6.77 8.35 11.07 16.89 40.63 68.72 184.75 1000 6.82 7.48 9.27 12.36 18.95 46.98 81.34 369.74 10 000 8.95 9.70 12.31 16.48 26.03 68.43 123.25 370.73

a_{The single cell observability threshold allows two solutions with equal likelihood values corresponding to unnaturally}

large or small malignant cell birth rates. See text. Table 1

Loglikelihood values plus 391a

Minimum cells in observable clones

Ratios of death±birth rates of malignant cells,d2=b2

0.0 0.1 0.3 0.5 0.7 0.9 0.95 1.0

a_{The loglikelihood values re¯ect a loglikelihood surface without a relative maximum. Instead the loglikelihood}

proportion of tumor bearing animals could then be computed at a set of time points. This arti®cial data set was then used exactly as the data set of the illustrative example to estimate the parameter values by optimizing the loglikelihood. A shallow likelihood surfaces similar to the one shown in Table 1 was constructed, but with a clear relative maximum at the parameter values used to construct the surface. Under true theoretical non-identi®ability, such a relative maximum would not likely appear at the right spot. Thus there does not seem to be true theoretical non-identi®-ability.

In the absence of numerical or theoretical non-identi®ability, it seems likely that the very shallow likelihood surface found was generated by the particular pattern of the data set used to compute that likelihood. The data set used had very few tumor bearing animals and most of those there were detected quite late. Such a pattern of observed tumors could be explained either by higher or lower detection thresholds, with corresponding and partially compensating malignant birth rates.

Examination of the results in Tables 1±3, shows the likelihood surface increases with increasing threshold size. For a given threshold size the likelihood increases for both smaller (towards 0.0) and larger (towards 1.0) ratios of malignant death±birth rates (d2=b2). However, the larger values

ofd2=b2 require much higher (non-biological) birth and mutation rates for threshold sizes above

100 cells (See Tables 2 and 3). For the lower values of d2=b2, the birth and mutation rates stay

plausible for much higher observation thresholds. In fact, some of the birth rates and mutation rates corresponding to values ofd2=b2less than 0.5 for an observation threshold of 10 000 cells are

seen as still plausible. However, there is no longer any distinction in likelihood between the values ofd2=b2. These results are reasonable, given that tumors are microscopically identi®ed at several

thousand cells. The indication is that the death±birth ratio is relatively small, based on the plausibility of values in Table 2.

The case that a single cell is observable and the death rate is zero (ratio,d2=b2 ˆ0) is equivalent

to the original model with a zero length time interval between the ®rst malignant cell and the observability threshold. In that case, the expression for the probability of an observable tumor at t, given a single malignant cell at 0, reduces to 1. The model is then independent of the birth rate

b2. In that case, the optimal estimated value for the mutation rate to malignancy is 4:9110ÿ7/

cell/day.

The case of a reversible observability threshold of one cell is more complicated. Note that in this casel₂…1ÿW2…t††may be written as

l₂ 1ÿf

1ÿf eÿb2…1ÿf†t;

wheref ˆd2=b2 and is assumed to be ®xed at one of the levels of Table 1. The assumption that a

single malignant cell is equivalent to an observable tumor assumes a very aggressive tumor with a

very high birth rate. Fitting the model to the data under the condition thatN ˆ1, results in very

high (in fact unreasonable) estimates of b₂, so that the expression l₂…1ÿW2…t††can be

approx-imated by l₂…1ÿf†. The estimate of this quantity is 4:9110ÿ7 _{mutation/cell/day, as in the}

original model described above. For each f, the corresponding estimates for l₂ are

4:9110ÿ7_{=…1ÿf† mutations/cell/day, as shown in Table 3. Mathematically equivalent, and}

then consistently estimated as 4:9110ÿ7 _{mutation/cell/day as shown in Table 1. The two}

so-lutions are the result of ®tting the model under the condition thatNˆ1 and that the death±birth

ratio is ®xed.

The expected time without visible tumor within the duration of the experiment (736 days) was about 730 days for all combinations of threshold size and values of…d2=b2†. The expected time

without visible tumor overall was about 1300 days for all cases, much longer than the life of the animal, and a reasonable result from a data set with so few tumor bearing animals and no modeled mortality.

4. Discussion

In this paper, a framework of two di€erential equations with a solution that leads to the direct computation of the probability of observing a tumor at a given time has been presented for the time-homogeneous case. Using time-independent rate parameters is clearly not ideal biologically, but does signi®cantly simplify the numerical computation of the model, and thereby also its application to real data sets. Furthermore, frequently not enough is known about the rate pa-rameters to adequately model them as functions of time. For these reasons, the same model for the case of time-dependent rate parameters has been developed in Appendix A only. The equa-tions derived in Appendix A are very similar to those presented in the text, but are not easily numerically integrated except in one (quite useful) special case. In addition the equations must be numerically integrated backwards, from each observation time point back to zero, increasing the computational burden of the method.

The equations of the model presented in the text may be used with other types of observability threshold and even other growth kinetics of tumors. The only requirement is that an expression be

found for the probability of observing a tumor at timet, given one malignant cell at time 0. The

equations work well with the usual assumption that the tumor growth follows a linear birth and death process. Under those conditions, and with the assumption of fast growing tumors, the traditional two-stage model is recovered.

In the text tumor growth and observability were assumed to follow a linear birth and death process with a non-absorbing threshold. This allowed the use of a formula for the probability

density of tumor size to ®nd an expression for 1ÿW2…t†. Combining reversible tumor kinetics

with an absorbing observability threshold is much more dicult. Saaty [19] can be used to derive

an approximate expression for 1ÿW2…t† for very large absorbing thresholds. The expression

b₂

is very similar to that for the non-absorbing threshold case, adding only the third term in the ®rst factor. For absorbing observability thresholds of more moderate size, Sherman and Portier [20], using the probability generating function approach to deriving the equations, append a system of equations corresponding to malignant clones of increasing size to the above derived system.

used. The microscopic tissue examination could detect tumors with a moderate number of cells, a

non-absorbing observability threshold say N. Tumors that a€ect the mortality of the rodent

would typically be much larger, with an absorbing endpoint of sayMcells. IfMis large enough,

Saaty [19] gives indications for an approximation to the probability density of tumor size that

could be summed fromNtoM in a similar way to the example in this paper. In such a case, the

likelihood expression would also need to be changed to re¯ect the possibility of a tumor a€ecting mortality [16].

Note that if the growth kinetics of the tumor are approximated by a pure birth process (d2ˆ0),

that there is no distinction between an absorbing barrier and a non-absorbing one. Thus the observability threshold of human breast cancer described as an absorbing but random barrier by Tsodikov et al. [5], can also be described using the equations in this paper. In the Tsodikov paper,

the number of cells in an observable tumor is written asN ˆcv, wherevis the volume of a tumor

andcthe concentration of tumor cells per unit volume. The authors heldcconstant and allowed

the observable volume,v, to be random with a density ofp…v†. An expression for the probability of

observing a tumor at time t can therefore be found by substituting the expression

R1 0 1ÿe

ÿb2t

‰ Šcvÿ1p…v†dv for 1ÿW2…t†. Similar randomizations can be applied to the other

observability thresholds.

Appendix A

The backward Kolmogorov approach can be used to deal with time-dependent rate parameters, as was made clear in [10]. The diculty is that in developing the backward equations with time-dependent rates, the ®nal, right-hand endpoint of the time interval must be kept constant, so that the equations are written in terms of the left hand endpoint, or initial time point. Thus in [11], Eq. (5) is written in terms ofs, the left-hand end point. The cost of using this method then, is that the entire system of di€erential equations must be numerically solved backwards from each ®nal time point of interest to the initial time.

The above approach is applied to the problem of the text. If 06s6t, the functions of the text may be slightly rede®ned as follows:

W0(s,t)ˆPr(no tumor visible at tjno normal cells, no stage-one cell, no malignant cell at s). W1(s,t)ˆPr(no tumor is visible attjno normal cells, 1 stage-one cell, 1 malignant cell ats). W2(s,t)ˆPr(no tumor visible at tj1 normal cell, no stage-one cell, no malignant cell ats).

Then by the above de®nition,

W0…sÿDs;t†

ˆPr…no tumor is visible attj1 normal cell; no stage-one or malignant cells atsÿDs† ˆPr…no tumor visible attj1 normal cell; no stage-one or malignant cells ats†

Pr…1 normal cell; no stage-one or malignant cells atsj1 normal cell; no stage-one or malignant cells atsÿDs†

‡Pr…no tumor visible attj2 normal cells; no stage-one or malignant cells ats†

Pr…2 normal cells; no stage-one or malignant cells atsj1 normal cell; no stage-one or malignant cells atsÿDs†

Pr…no normal cells; stage-one; or malignant cells atsj1 normal cell; no stage-one or

malignant cells atsÿ_Ds†

‡Pr…no tumor visible attj1 normal cell; 1 stage-one cell; no malignant cells ats† Pr…1 normal; 1 stage-1; no malignant cells atsj1 normal; no stage-one or

malignant cells atsÿ_Ds†

ˆPr…no tumor visible attj1 normal; no stage-one or malignant cells ats† …1ÿb₀…sÿDs† Dsÿd0…sÿDs† Dsÿl1…sÿDs† Ds†

‡Pr…no tumor visible attj2 normal; no stage-one or malignant cells ats† b₀…sÿ_Ds† _Ds

‡Pr…no tumor visible attjno normal; stage-one; or malignant cells ats† d0…sÿDs† Ds

‡Pr…no tumor visible attj1 normal cell; 1 stage-one cell; no malignant cell ats†

l₁…sÿDs† Ds:

Using only the assumption of independence of cells, the right-hand side may be written as

W0…s;t† ‰1ÿb0…sÿDs† Dsÿd0…sÿDs† Dsÿl1…sÿDs† DsŠ ‡W 2

0…s;t† b0…sÿDs† Ds

‡d0…sÿDs† Ds‡W0…s;t† W1…s;t† l1…sÿDs† Ds:

Similar to the development in the text then, the terms of the equation can be re-arranged so that

W0…s

‰ ÿ_Ds;t† ÿW0…s;t†Š=Dsˆ ÿ‰b0…sÿDs† ÿd0…sÿDs† ÿl1…sÿDs†Š W0…s;t† ‡W20…s;t†

b₀…sÿ_Ds† ‡d0…sÿDs† ‡W0…s;t† W1…s;t† l1…sÿDs†

As usual,_Dsapproaches 0 from above on both sides of the equation, and with the assumption that

the rate parameters are left continuous, the following ordinary di€erential equation is formed:

dW0…s;t†=dsˆ ÿW2₀…s;t† b0…s† ‡ ‰l1…s† …1ÿW1…s;t†† ‡b0…s† ‡d0…s†Š W0…s;t† ÿd0…s;t†:

Note that the derivative is taken at the left-hand (that is the initial) time point. Clearly, there is now no a priori information at the point,sˆ0, sinceW0…0;t†is the solution. Instead, there is a

boundary condition at the right-hand endpoint, since from the de®nition, W0…t;t;† ˆ1. The

following di€erential equation forW1…s;t† is derived in the same way:

dW1…s;t†=dsˆ ÿW12…s;t† b1…s† ‡ ‰l2…s† …1ÿW2…s;t†† ‡b1…s† ‡d1…s†Š W1…s;t†

ÿd1…s;t† withW1…t;t† ˆ1:

The development of an expression for 1ÿW2…s;t†, the Pr(at least one visible tumor at tj1

ma-lignant cell ats) is completely similar to the development in the text, but with expressionsAandB that are now functions ofsas well ast. Adapting Bailey [14, p. 110] and further,A…s;t†andB…s;t†

can be written in terms of the integral, I…s;t†, where

I…s;t† ˆ

Z t

s

b₂…u†exp

Z t

u

b₂…s†d2…s†ds

Then

A…s;t† ˆ1ÿexpf

Rt

sb2…u† ÿd2…u†dug

1‡I…s;t†

and B…s;t† ˆI…s;t†=…1‡I…s;t††. As in the main text then, the Pr(at least one visible tumor attj1 malignant cell ats) is

X1

nˆN

Pr…tumor of n cells at tj1 malignant cell at 0† ˆ …1ÿA†…1ÿB†X

1

nˆN Bnÿ1:

B…s;t†is necessarily less than 1 ift>s, so that the geometric series formula may again be applied. Finally for the case of time-dependent rate parameters,

Pr…a visible tumor attj1 malignant cell ats† ˆ expf

Rt

sb2…u† ÿd2…u†dug

1‡I…s;t†

I…s;t†

1‡I…s;t†

Nÿ1

ˆ1ÿW2…s;t†:

The two equations derived above must again be solved numerically. Most software used to solve di€erential equations allows stepping backwards from the right-hand endpoint (that has the boundary condition) to the left-hand endpoint. Frequently this is done by labeling the right-hand endpoint the `initial point' and the left-hand endpoint, the `®nal point'. In other words, for the software, the `initial' point is larger than the `®nal' point.

One ®nal numerical problem remains, and that is evaluating the integralI…s;t†. Although there are numerical quadrature programs and subroutines, they are not always easily used in con-junction with numerical solving routines for di€erential equations. Hence this approach is only recommended in the case that the integralI…s;t† admits of an analytical solution. Note that the integral does have such an analytical solution whenever the following two conditions hold: 1. The malignant cell birth rate is analytically integrable and

2. the ratio of the malignant cell death rate to the malignant cell birth rate is piecewise constant. In the above special case, the two di€erential equations may be written explicitly and solved using only a numerical ordinary di€erential equation solver.

Note that the special case can be quite useful. The malignant cell birth rate can be any kind of polynomial or simple trigonometric function, for example. The ratio of malignant birth rate to death rate needs to be only piecewise constant, so that the ratio itself may change at intervals. Also there are no conditions beyond left continuity on any rate parameters other than those governing malignant growth. In fact, these conditions include most actual applications of time-dependent rate parameters found in the literature (e.g., [7]).

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