Distribution of the Number of Clonogenic Tumor Cells Surviving Fractionated Irradiation
LEONID G. HANIN
Department of Mathematics
IdahoStateUniversity
Pocatello, ID83209-8085
USA
and
Department of Biostatistics and Computational Biology
University of Rochester
601 Elmwood Avenue
Box 630, Rochester, NY14642
USA
Abstract: - Iterated birth and death Markov process is defined as an n-fold iteration of a birth and death Markov process describing kinetics of certain population combined with random killing of individuals in the population at given times with given survival probabilities. A long-standing problem of computing the distribution of the number of clonogenic tumor cells surviving an arbitrary fractionated radiation schedule is solved within the framework of iterated birth and death Markov process. It is shown that, for any initial population size, the size of the population at arbitrary time follows a generalized negative binomial distribution, and an explicit computationally feasible formula for the latter is obtained.
Key-Words: - Clonogenic tumor cell; Fractionated irradiation; Generalized geometric distribution; Generalized negative binomial distribution; Iterated birth and death Markov process; Probability generating function.
1 Introduction
The present work solves, under realistic biological assumptions, the following long-standing problem in radiation oncology, see e.g. [1]: To find the distribution of the number of clonogenic tumor cells surviving a given schedule of fractionated irradiation. Solving this problem is critical for developing quantitative methods of assessment of the efficacy of radiation cancer treatment and designing optimal schedules of cancer radiotherapy. The clonogenic model of post-treatment tumor development introduced in [2] provides a link between the distribution of the number of surviving clonogenic cells and the distribution of the observed time to tumor recurrence. This makes solving the problem at hand a crucial step in developing biologically motivated models of post-treatment survival. For an at length discussion of biomedical significance of the above problem and previously developed approaches to its solution, see [3-5] and references therein.
We will proceed from the following widely accepted model of tumor population kinetics, see e.g. [6, 7]. A tumor initially comprising a non-random number i of clonogenic cells is exposed to a fractionated radiation schedule consisting of n instantaneously delivered doses administered at times , where . It is assumed that every clonogen survives exposure to the dose with the same probability , , given that it survived the previous exposures, and independently of other clonogens. Observe that it is not necessarily supposed that all tumor cells are clonogenic. We also assume that irradiated tumor cells are killed instantaneously at the moment of exposure to radiation. According to a convention commonly accepted in radiation biology, a tumor cell is killed if it is incapable of producing a viable clone. Before, between and after the exposures, surviving clonogens proliferate and die spontaneously independently of each other with time-dependent birth rate and spontaneous death rate . Finally, it is assumed that spontaneous cell death between radiation exposures is an instantaneous event, and that both descendants of a clonogenic cell are clonogenic.
According to the model, the size of the population of clonogenic cells between exposures follows a birth and death Markov process, and at the moment of exposure to dose is subject to random killing with the survival probability , . The combined stochastic process will be called in what follows iterated birth and death Markov process, compare with [3].
It is well known that tumor response to radiation depends critically on intracellular damage repair processes. They operate on the time scale of minutes to hours while, in the case of fractionated radiation, the inter-dose intervals typically range from one to several days. Such time intervals are long enough for the accomplishment of transient processes of inactivation and recovery of damaged cells, so that survival probability of an irradiated cell depends, conditional on its survival under previous exposures, only on the radiation dose thus making a single survival probability accountable for the resultant effect of damage repair processes.
In the particular case where doses of radiation and inter-dose intervals are equal ( and , ), and birth and death rates b and d are constant, the problem at hand was solved within an approach based on iterated birth and death Markov processes in [3]. It was shown that the distribution of the number of surviving clonogenic cells at times and belongs to the family of generalized negative binomial distributions (see Section 3 below), and an explicit computationally feasible formula for the distributions from this family was obtained.
A survival model based on the findings of [3] was successfully applied to statistical analysis of data on post-treatment recurrence of prostate cancer [5]. For a detailed discussion of a wide range of biomedical and statistical implications of the work [3], the reader is referred to [4]. The points made there promise even a larger impact for a far more general and realistic model studied in the present work. The rationale for the extension of results obtained in [3] to arbitrary schedules of fractionated radiation is two-fold. First, the most commonly used schedules of fractionated dose delivery consist of daily irradiation with equal or variable (usually escalating) doses on business days followed by weekend breaks. Second, the search for optimal schedules of fractionated irradiation presupposes variation of fractional doses and inter-dose intervals. Furthermore, consideration of variable birth and death rates is equally important for the following reasons:
(1) A large fraction of tumors is detected and treated beyond the initial exponential phase of their progression. For such tumors, the birth and death rates depend on tumor size and thereby are functions of time;
(2) It is well known that exposure to ionizing radiation induces blocking of irradiated cells in various (most notably, and M) phases of the mitotic cycle, see e.g. [8-10]. After remaining dormant for some time, such cells either continue to proliferate or disintegrate. This causes prolongation of the life cycle of surviving cells and leads to a complex dependence of birth and spontaneous death rates on time;
(3) During the first weeks of radiation treatment the effective clonogen doubling time is relatively long after which it becomes much shorter, see [11] and references therein.
The initial number of clonogenic cells in a tumor is typically very large (one of a solid tumor contains about cells [12]; also, a clinically detectable tumor is estimated to contain at least clonogenic cells probably ranging up to cells or even more [11, 13]). Additionally, the number of exposures (typically ranging from 20 to 40) and the doses (usually from 1 to 2 Gy) are selected in such a way as to ensure that the overall survival probability of a clonogenic cell, which equals , is very small.
2 Birth and Death Markov Process and Generalized Geometric Distribution
Consider a cell population that starts at time from a single cell and which kinetics is governed by a birth and death Markov process with birth and death rates and , respectively. Let be the net birth rate and be the corresponding cumulative rate. It was shown by Kendall [14] that probability generating function (p.g.f.) of the size of the population at time equals
, (1)
where and are functions of given by
,
. (2)
Observe that
and . (3)
For any numbers and that satisfy conditions (3), formula (1) represents p.g.f. of a non-negative integer-valued random variable (r.v.) such that
,
(4)
where and . Clearly, and . Conversely, p.g.f. of every r.v. , that has distribution (4) with parameters and , can be represented in the form (1), where parameters and are given by and , and satisfy conditions (3).
The distribution (4) is referred to as generalized geometric distribution and is denoted . Thus formula (1) with parameters subject to (3) gives a general form of p.g.f. of generalized geometric distribution. In particular, the state at any given time of a birth and death Markov process starting from a single cell follows a generalized geometric distribution. Two particular cases of the generalized geometric distribution are worth mentioning: represents a plain geometric distribution and corresponds to the pure birth Markov process, while leads to a Bernoulli r.v. that takes values 0 and 1 with probabilities and , respectively, and corresponds to a pure death Markov process. Note also that distribution can be viewed as a mixture of , the degenerate distribution at 0, and .
3 Generalized Negative Binomial Distribution
If the birth and death Markov process described in Section 2 starts from a non-random number of cells then, due to the fact that the sizes of populations emerging from initial cells are independent and identically distributed (i.i.d.), the total size at time is the sum of i.i.d. r.v.’s with generalized geometric distribution. For the plain geometric distribution , such sum follows negative binomial distribution . This motivates introducing the following class of probability distributions.
Definition.Generalized negative binomial distribution is the distribution of the sum of i.i.d. r.v.’s having generalized geometric distribution .
If p.g.f. of the underlying generalized geometric distribution is represented in the form (1) then for p.g.f. of the sum of its i.i.d. copies we have
(5)
Recall that in the case of birth and death Markov process , , with parameters and are specified in (2).
4 Distribution of the State of Iterated Birth and Death Markov Process
In this section we identify the distribution of the state of the iterated birth and death Markov process at any time Let be p.g.f. of r.v.
Theorem 1.
where
, (6)
, (7)
and
Theorem 1 implies that the distribution of the r.v. is generalized negative binomial.
Suppose that birth and death rates do not vary between exposures: for Setting and we find that for
and
Let, in particular, birth and death rates be constant: and Then
and
where Furthermore, in the case of equal doses we obtain
and
The following theorem obtained in [3] provides an explicit computationally feasible formula for the generalized negative binomial distribution. In combination with formulas (6) and (7) it solves the main problem of the present work.
Theorem 2. Let p.g.f. of a generalized negative binomial distribution be given by (6). Then
(8)
where and polynomials forare given by (9)
5 Discussion
It is widely accepted that the survival probabilities , follow the linear-quadratic model, where and are positive constants that can be interpreted in terms of cell radiosensitivity and sublethal damage repair capability, respectively [15]. Thus, in the simplest case of homogeneous iterated birth and death Markov process, the model of fractionated radiation cell survival discussed in this work depends on five unobservable parameters . More importantly, the number of surviving clonogens is unobservable as well. To make use of the computed distribution of the size of the surviving fraction of clonogens, one has to relate it to an observable endpoint, such as the time to tumor recurrence. Specifically, let be the time to tumor recurrence counted from the moment of delivery of the last fraction of radiation.
According to the clonogenic model of post-treatment tumor recurrence [2], a recurrent tumor arises from a single clonogenic cell. Every surviving clonogen can be characterized by a latent progression time during which it could potentially propagate into a detectable tumor. It is assumed additionally that progression times of surviving clonogens are i.i.d. r.v.’s. Suppose that the number of surviving clonogens immediately after the end of treatment is equal to . Then for the observed time of tumor recurrence we have , where is the progression time of the j-th clonogen. Let be p.g.f. of r.v. N. It follows from the assumptions of the clonogenic model that the conditional survival function of r.v. is given by where is the common survival function of the progression times of surviving clonogens. Then
(10)
Significance of formula (10) is two-fold. First, it suggests that knowledge of the entire distribution of the number of surviving clonogens (not only of the tumor control probability ) is critical for developing biologically motivated post-treatment survival models. Second, assuming some parametric form for the function and using for the distribution of r.v. its exact form (6-9), one can estimate the unknown initial number of clonogenic cells and especially the all-important kinetic parameters and from the observed times to tumor recurrence.
Acknowledgements:
Research of the author was supported by NSF grant DMS-0109895.
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