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Mathematical Biosciences 163 (2000) 75±89
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Incorporating observability thresholds of tumors into the
two-stage carcinogenesis model
Marjo V. Smith a,*, Christopher J. Portier b
a

b

Analytical Sciences, Inc., Suite 200, 2605 Meridian Parkway, Durham, NC 27713, USA
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 expression 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 Moolgavkar 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.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 4 8 - 6

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M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

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 experiment 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 examined 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 constant 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 corresponds 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.
To illustrate this approach, the time-homogeneous two-stage model is combined with tumor
progression modeled as a linear birth and death process with a non-absorbing observability

threshold. The resulting model is ®t to historical liver cancer incidence data from control groups
of male Fisher 344 rats from combined NTP carcinogenesis studies. The two-stage model when
applied to tumor incidence data is known to be non-identi®able even without the extension to
tumor progression [12]. Thus, an earlier simulation study of precancerous liver foci in male F344

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

77

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 Dt following any time t, only the
following events are assumed possible at the following probabilities:

Dt  b0
Dt  d0
Dt  l1
1 ÿ Dt  …b0 ‡ d0 ‡ l1 †

birth of a normal cell from one normal cell
death or di€erentiation of a normal cell
formation of a stage-one cell during a normal cell division
nothing happens to a normal cell

Dt  b1
Dt  d1
Dt  l2
1 ÿ Dt  …b1 ‡ d1 ‡ l2 †

birth of a stage-one cell
death or di€erentiation of a stage-one cell
formation of a malignant cell during a stage-one cell division
nothing happens to a stage-one cell

For the growth process of the tumors, we will use:
Dt  b2
Dt  d2

birth of a malignant cell
death of a malignant cell

The tumor is considered observable if it contains N or more cells. The following probabilities
are then de®ned:
W0 (t) ˆ Pr(no tumor is visible at tj1 normal cell, no stage-one cells, no malignant cells at 0).
W1 (t) ˆ Pr(no tumor is visible at tjno normal cells, 1 stage-one cell, no malignant cells at 0).
W2 (t) ˆ Pr(no tumor is visible at tjno 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 time t can be written as 1 ÿ W0 …t†M ; where the system starts
with M 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, having W0 (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

78

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

de®nition of W0 (t) (set at time 0) together with the above assumptions over the interval ‰0; Dt†. The
solution over the remaining interval ‰Dt; t ‡ DtŠ may 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† ˆ Pr…no tumor is visible at t
‡ Dtj1 normal cell; no stage-one cells; no malignant cells at 0†
ˆ Pr…no tumor is visible at t
‡ 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 at t
‡ 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 at t
‡ 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 at t
‡ 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†
ˆ Pr…no tumor is visible at t
‡ Dtj1 normal cell; no stage-one cells; no malignant cells at Dt†  …1 ÿ Dt  b 0
ÿ Dt  d0 ÿ Dt  l1 † ‡ Pr…no tumor is visible at t
‡ Dtj2 normal cells; no stage-one cell; no malignant cell at Dt†  Dt  b0
‡ Pr…no tumor is visible at t
‡ Dtjno normal cells; no stage-one cells; no malignant cells at Dt†  Dt  d0
‡ Pr…no tumor is visible at t
‡ Dtj1 normal cell; 1 stage-one cell; no malignant cells at Dt†  Dt  l1 :
Using the above de®nitions for W0 (t) and W1 (t) and the assumptions of time homogeneity and
independence of cells,
W0 …t ‡ Dt† ˆ W0 …t†  …1 ÿ Dt  b0 ÿ Dt  d0 ÿ Dt  l1 † ‡ W20 …t†  Dt  b0 ‡ 1  Dt  d0 ‡ W0

 W1  Dt  l1 :

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

79

By subtracting W0 (t) from both sides, taking the limit as Dt goes to zero, and rearranging terms,
dW0 …t†=dt ˆ W20 …t†  b0 ÿ ‰l1 …1 ÿ W1 …t†† ‡ b0 ‡ d0 Š  W0 …t† ‡ d0 :

…1†

In a similar way, the corresponding equation for W1 …t† is
dW1 …t†=dt ˆ W21 …t†  b1 ÿ ‰l2 …1 ÿ W2 …t†† ‡ b1 ‡ d1 Š  W1 …t† ‡ d1 :

…2†

Clearly from the de®nitions, W0 …0†W1 …0† ˆ 1.
These same equations (with W2 …t† ˆ 0), 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 1 ÿ W2 …t†, the
Pr…seeing a tumor at tj1 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 time t,
given a single individual at time t, for the case that b2 6ˆ d2 . In terms of the parameters de®ned
above for n P 1,
Pr…tumor has n cells at tj1 malignant cell at 0† ˆ …1 ÿ A†…1 ÿ B†Bnÿ1 , where
Aˆ

d2 …e…b2 ÿd2 †t ÿ 1†
b2 e…b2 ÿd2 †t ÿ d2

Bˆ

b2 …e…b2 ÿd2 †t ÿ 1†
:
b2 e…b2 ÿd2 †t ÿ d2

and

The Pr…seeing a tumor at tj1 malignant cell at 0† is then
1
1
X
X
Pr …tumor of n cells at tj1 malignant cell at 0† ˆ …1 ÿ A†…1 ÿ B†
Bnÿ1 :
nˆN

nˆN

Since for b2 > d2 , the expression B remains less than 1 for all positive t, the formula for geometric
series can be used to write
Pr …seeing a tumor at tj1 malignant cell at 0† ˆ

…b2 ÿ d2 †bNÿ1
‰1 ÿ eÿ…b2 ÿd2 †t ŠNÿ1
2
:
N
‰b2 ÿ d2 eÿ…b2 ÿd2 †t Š

Note that if b2 ÿ d2 and the observation times, t, are large, then the quantity exp…ÿ…b2 ÿ d2 †t†
may be negligible. Then the probability of seeing a tumor at t can be approximated by
…b2 ÿ d2 †=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, …b2 ÿ d2 †=b2 . Note that the closer d2 is to b2 , the bigger the
impact on reducing the tumor incidence due to loss of malignant cells.

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M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

For the limiting case that b2 ˆ d2 , Bailey gives
Pr …tumor has n cells at tj1 malignant cell at 0† ˆ

…b2 t†…nÿ1†
…n‡1†

…1 ‡ b2 t†

;

for n P 1:

As before,
Pr…seeing a tumor at tj1 malignant cell at time 0†
1
X
…b t†…N ÿ1†
:
Pr…tumor has n cells at tj1 malignant cell at 0† ˆ 2
ˆ
…1 ‡ b2 t†N
nˆN
For large t, this quantity approaches 0.
Substituting either of the above expressions for 1 ÿ W2 …t†; would allow solving for W0 (t), the
probability of not seeing a tumor from a single normal cell. Again, assuming the experiment starts
M
with M normal cells at time 0, the probability of observing a tumor at time t would be 1 ÿ W0 …t† :
If the number of normal cells can be assumed to be approximately constant …b0 ˆ d0 ˆ 0†, then
Eq. (1) reduces to a separable equation that can be solved in terms of an integral of W1 (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†=dt ˆ ÿM  l1 ‰1 ÿ W1 …t†Š  W0 …t†:

…3†

Since the solution to this equation already includes the exponent, M, the solution W0 (t) would be
Pr…no visible tumor at tjM normal cells; no stage-one cells; no malignant cells at 0†. The probability of observing a tumor at time t would be 1 ÿ W0 …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 becomes visible. In that case, W0 …t†M ˆ Pr…no tumor is visible by time tjM normal cells; no stageone cells; no malignant cells at 0† can be written as Pr…T > tjM normal cells; no stage-one cells;
no malignant cells at 0†. Thus 1 ÿ W0 …t†M is the distribution function of T and the expected value
of T can be written as
Z 1
E…T † ˆ
W0 …t†M dt; provided the infinite integral exists:
0

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,

0 if there is a visible tumor at t
V …t† ˆ
1 if there is no visible tumor at t
M

may be de®ned.
Then, Pr…V …t† ˆ 1† ˆ W0 …t† . Now consider any ®nite time interval, ‰0; TfŠ. Let
R Tf
Tv…Tf† ˆ 0 V …t† dt. Tv…Tf† is a random variable de®ning the amount of time in ‰0; TfŠ without

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

81

visible tumor. Then the average amount of time in any given ®nite interval, ‰0; TfŠ without visible
tumor is
Z Tf
Z Tf
Z Tf
M
E‰V …t†Š dt ˆ
Pr ‰V …t† ˆ 1Š dt ˆ
W0 …t† dt:
E‰Tv…Tf†Š ˆ
0

0

0

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,
Z 1
Z Tf
M
M
W0 …t† dt ˆ
W0 …t† dt;
E…Tv† ˆ lim
Tf!1

0

0

the overall expected time without a visible tumor. Clearly E…Tv† ˆ E…T † in the case that an observable tumor cannot regress.
3. Illustrative example
3.1. Experimental protocols and results
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 carcinogenicity 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 mutation rate of l1 mutations per day, the normal cell division rate is assumed to be about the same as
the cited normal cell death rate, 1:2  10ÿ3 divisions/cell/day. The mutation rate per normal cell
division was given as 3:5  10ÿ8 mutations per division, so that the mutation rate of a normal cell is
3:5  10ÿ8 mutations=division  1:2  10ÿ3 divisions=day ˆ 4:2  10ÿ11 mutations/day.
Carthew et al. [15] give 15:2  108 as the number of normal hepatic cells, so that Eq. (3) is used
with Ml1 ˆ 0:06384 mutations of normal cells/day. Finally, Conolly and Kimball [13] give the
death rate for stage-one cells as 0.0153/day and the division rate for stage-one cells as 0.020/day, and
these are used for b1 and d1 in Eq. (2). This leaves b2 , d2 , l2 , and N to be determined by the data.

82

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

3.2. Estimation procedure
The observability threshold N, the second mutation rate l1 , and the malignant birth and death
rates b2 and d2 were 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 instrumental 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 set fti ; i ˆ 1; . . . ; mg, may denote the days on which at least one rat died, either by
sacri®ce or naturally. Let ci and di respectively refer to the numbers of animals that died on day ti ,
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 † ˆ Pr…no tumor is visible by time tjM normal cells; no stage-one or malignant cells at 0† can
be found for any i ˆ 1; . . . ; m. The likelihood expression may then be written as
m
Y
W0 …ti †di ‰1 ÿ W0 …ti †Šci ;
iˆ1

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 estimability problems. Both numerical and theoretical non-identi®ability were checked in the following 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
3 corresponding to an observability threshold, N ˆ 50, and a death±birth ratio, f ˆ 0:1. The exact

83

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89
Table 1
Loglikelihood values plus 391a
Ratios of death±birth rates of malignant cells, d2 =b2

Minimum cells in
observable clones

0.0

0.1

0.3

0.5

0.7

0.9

0.95

1.0

1
5
15
50
100
500
1000
10 000

ÿ0.2349
ÿ0.1577
ÿ0.1513
ÿ0.1487
ÿ0.1479
ÿ0.1469
ÿ0.1467
ÿ0.1462

ÿ0.2349
ÿ0.1585
ÿ0.1516
ÿ0.1488
ÿ0.1480
ÿ0.1470
ÿ0.1467
ÿ0.1462

ÿ0.2349
ÿ0.1606
ÿ0.1525
ÿ0.1492
ÿ0.1493
ÿ0.1471
ÿ0.1468
ÿ0.1463

ÿ0.2349
ÿ0.1637
ÿ0.1539
ÿ0.1498
ÿ0.1487
ÿ0.1473
ÿ0.1469
ÿ0.1463

ÿ0.2349
ÿ0.1674
ÿ0.1567
ÿ0.1510
ÿ0.1494
ÿ0.1476
ÿ0.1472
ÿ0.1464

ÿ0.2349
ÿ0.1566
ÿ0.1636
ÿ0.1555
ÿ0.1522
ÿ0.1486
ÿ0.1479
ÿ0.1467

ÿ0.2349
ÿ0.1507
ÿ0.1574
ÿ0.1600
ÿ0.1554
ÿ0.1497
ÿ0.1486
ÿ0.1469

ÿ0.2349
ÿ0.1448
ÿ0.1356
ÿ0.1328
ÿ0.1322
ÿ0.1318
ÿ0.1317
ÿ0.1317

a

The loglikelihood values re¯ect a loglikelihood surface without a relative maximum. Instead the loglikelihood increases in several directions.

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
1.89
2.92
4.02
4.68
6.18
6.82
8.95

Large or
small
2.23
3.76
5.34
6.21
8.35
9.27
12.31

Large or
small
2.47
4.59
6.87
8.13
11.07
12.36
16.48

Large or
small
2.41
6.12
9.88
11.99
16.89
18.95
26.03

Large or
small
1.64
7.93
19.59
25.97
40.63
46.98
68.43

Large or
small
1.48
6.51
26.20
39.26
68.72
81.34
123.25

Small

5
15
50
100
500
1000
10 000

Large or
small
1.97
3.18
4.37
5.10
6.77
7.48
9.70

1.34
5.04
18.01
36.53
184.75
369.74
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 3
Mutation rates to malignant cells times 107a
Minimum cells in
observable clones
1
5
15
50
100
500
1000
10 000
a

Ratios of death±birth rates of malignant cells, d2 =b2
0.0

0.1

0.3

0.5

0.7

0.9

0.95

4.91

5.46 or
4.91
9.7
9.9
10.1
10.1
10.1
10.1
10.1

7031 or
4.91
12.3
12.7
12.9
13.0
13.0
13.0
13.0

9082 or
4.91
17.1
17.8
18.0
18.1
18.2
18.2
18.3

16.4 or
4.91
28.8
29.3
29.9
30.1
30.3
30.4
30.4

49.1 or
4.91
66.9
87.5
88.2
89.3
90.6
90.8
90.8

98.2 or
4.91
83.1
165.1
174.3
176.5
180.3
181.1
182.3

8.7
9.0
9.0
9.1
9.1
9.1
9.1

1.0
4.91
102.8
347.7
1206
2432
12 240
24 510
245 100

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 4:91  10ÿ7 or …4:91=…1 ÿ d2 =b2 ††  10ÿ7 corresponding to two di€erent
solutions with very large or small malignant birth rates, respectively. See text.

84

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

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
of d2 =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 of d2 =b2 less 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
of d2 =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 ˆ 0) 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 4:91  10ÿ7 /
cell/day.
The case of a reversible observability threshold of one cell is more complicated. Note that in
this case l2 …1 ÿ W2 …t†† may be written as
l2

1ÿf
;
1 ÿ f  eÿb2 …1ÿf †t

where f ˆ d2 =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 that N ˆ 1, results in very
high (in fact unreasonable) estimates of b2 , so that the expression l2 …1 ÿ W2 …t†† can be approximated by l2 …1 ÿ f †. The estimate of this quantity is 4:91  10ÿ7 mutation/cell/day, as in the
original model described above. For each f, the corresponding estimates for l2 are
4:91  10ÿ7 =…1 ÿ f † mutations/cell/day, as shown in Table 3. Mathematically equivalent, and
therefore having exactly the same loglikelihood value, is the case that the estimated value of l2 is
very close to zero. In that case the expression l2 …1 ÿ W2 …t†† can be approximated by l2 , which is

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

85

then consistently estimated as 4:91  10ÿ7 mutation/cell/day as shown in Table 1. The two solutions are the result of ®tting the model under the condition that N ˆ 1 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 parameters 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 equations 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 time t, 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 1 ÿ W2 …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 1 ÿ W2 …t† for very large absorbing thresholds. The expression
ÿ

Nÿ1



N ÿ1 1 ÿ eÿ‰b2 ÿd2 Št
d2 ÿ
ÿ‰b2 ÿd2 Št
b2 ÿ d2 ‡
b2
1ÿe
N
N
…b2 ÿ d2 eÿ‰b2 ÿd2 Št †
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.
In the case that rodent studies similar to those of the example are used to examine tumors that
may be instrumental in death, a combination of absorbing and non-absorbing thresholds may be

86

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

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 say M cells. If M is large enough,
Saaty [19] gives indications for an approximation to the probability density of tumor size that
could be summed from N to M 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 (d2 ˆ 0),
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 as N ˆ cv, where v is the volume of a tumor
and c the concentration of tumor cells per unit volume. The authors held c constant and allowed
the observable volume, v, to be random with a density of p…v†. An expression for the probability of
observing
a tumor at time t can therefore be found by substituting the expression
R1
ÿb2 t cvÿ1
‰
1
ÿ
e
Š p…v† dv for 1 ÿ W2 …t†. Similar randomizations can be applied to the other
0
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 timedependent 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 of s, 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 0 6 s 6 t, the functions of the text
may be slightly rede®ned as follows:
W0 (s,t) ˆ Pr(no tumor visible at tjno normal cells, no stage-one cell, no malignant cell at s).
W1 (s,t) ˆ Pr(no tumor is visible at tjno normal cells, 1 stage-one cell, 1 malignant cell at s).
W2 (s,t) ˆ Pr(no tumor visible at tj1 normal cell, no stage-one cell, no malignant cell at s).
Then by the above de®nition,
W0 …s ÿ Ds; t†
ˆ Pr…no tumor is visible at tj1 normal cell; no stage-one or malignant cells at s ÿ Ds†
ˆ Pr…no tumor visible at tj1 normal cell; no stage-one or malignant cells at s†
 Pr…1 normal cell; no stage-one or malignant cells at sj1 normal cell; no stage-one or
malignant cells at s ÿ Ds†
‡ Pr…no tumor visible at tj2 normal cells; no stage-one or malignant cells at s†
 Pr…2 normal cells; no stage-one or malignant cells at sj1 normal cell; no stage-one or
malignant cells at s ÿ Ds†
‡ Pr…no tumor visible at tjno normal no stage-one; or malignant cells at s†

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

87

 Pr…no normal cells; stage-one; or malignant cells at sj1 normal cell; no stage-one or
malignant cells at s ÿ Ds†
‡ Pr…no tumor visible at tj1 normal cell; 1 stage-one cell; no malignant cells at s†
 Pr…1 normal; 1 stage-1; no malignant cells at sj1 normal; no stage-one or
malignant cells at s ÿ Ds†
ˆ Pr…no tumor visible at tj1 normal; no stage-one or malignant cells at s†
 …1 ÿ b0 …s ÿ Ds†  Ds ÿ d0 …s ÿ Ds†  Ds ÿ l1 …s ÿ Ds†  Ds†
‡ Pr…no tumor visible at tj2 normal; no stage-one or malignant cells at s†
 b0 …s ÿ Ds†  Ds
‡ Pr…no tumor visible at tjno normal; stage-one; or malignant cells at s†
 d0 …s ÿ Ds†  Ds
‡ Pr…no tumor visible at tj1 normal cell; 1 stage-one cell; no malignant cell at s†
 l1 …s ÿ Ds†  Ds:
Using only the assumption of independence of cells, the right-hand side may be written as
W0 …s; t†  ‰1 ÿ b0 …s ÿ Ds†  Ds ÿ d0 …s ÿ Ds†  Ds ÿ l1 …s ÿ Ds†  DsŠ ‡ W20 …s; t†  b0 …s ÿ Ds†  Ds
‡ d0 …s ÿ Ds†  Ds ‡ W0 …s; t†  W1 …s; t†  l1 …s ÿ Ds†  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† ÿ W0 …s; t†Š=Ds ˆ ÿ‰b0 …s ÿ Ds† ÿ d0 …s ÿ Ds† ÿ l1 …s ÿ Ds†Š  W0 …s; t† ‡ W20 …s; t†
 b0 …s ÿ Ds† ‡ d0 …s ÿ Ds† ‡ W0 …s; t†  W1 …s; t†  l1 …s ÿ Ds†
As usual, Ds approaches 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†=ds ˆ ÿW20 …s; t†  b0 …s† ‡ ‰l1 …s†  …1 ÿ W1 …s; t†† ‡ b0 …s† ‡ d0 …s†Š  W0 …s; t† ÿ d0 …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, s ˆ 0, since W0 …0; t† is the solution. Instead, there is a
boundary condition at the right-hand endpoint, since from the de®nition, W0 …t; t; † ˆ 1. The
following di€erential equation for W1 …s; t† is derived in the same way:
dW1 …s; t†=ds ˆ ÿW21 …s; t†  b1 …s† ‡ ‰l2 …s†  …1 ÿ W2 …s; t†† ‡ b1 …s† ‡ d1 …s†Š  W1 …s; t†
ÿ d1 …s; t† with W1 …t; t† ˆ 1:
The development of an expression for 1 ÿ W2 …s; t†, the Pr(at least one visible tumor at tj1 malignant cell at s) is completely similar to the development in the text, but with expressions A and B
that are now functions of s as well as t. Adapting Bailey [14, p. 110] and further, A…s; t† and B…s; t†
can be written in terms of the integral, I…s; t†, where
Z t

Z t
I…s; t† ˆ
b2 …s†d2 …s† ds du:
b2 …u† exp
s

u

88

M.V. Smith, C.J. Portier / Mathematical Biosciences 163 (2000) 75±89

Then
A…s; t† ˆ 1 ÿ

expf

Rt
s

b2 …u† ÿ d2 …u† dug
1 ‡ I…s; t†

and B…s; t† ˆ I…s; t†=…1 ‡ I…s; t††. As in the main text then, the Pr(at least one visible tumor at tj1
malignant cell at s) is
1
X

Pr…tumor of n cells at t j1 malignant cell at 0† ˆ …1 ÿ A†…1 ÿ B†

nˆN

1
X

Bnÿ1 :

nˆN

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

Nÿ1
expf s b2 …u† ÿ d2 …u† dug
I…s; t†
Pr…a visible tumor at tj1 malignant cell at s† ˆ
1 ‡ I…s; t†
1 ‡ I…s; t†
ˆ 1 ÿ W2 …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 integral I…s; t†. Although there
are numerical quadrature programs and subroutines, they are not always easily used in conjunction with numerical solving routines for di€erential equations. Hence this approach is only
recommended in the case that the integral I…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 timedependent rate parameters found in the literature (e.g., [7]).

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