Download Distribution of the Number of Clonogenic Tumor Cells Surviving

Distribution of the Number of Clonogenic Tumor Cells Surviving
Fractionated Irradiation
LEONID G. HANIN
Department of Mathematics
Idaho State University
Pocatello, ID 83209-8085
USA
and
Department of Biostatistics and Computational Biology
University of Rochester
601 Elmwood Avenue
Box 630, Rochester, NY 14642
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 longstanding 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
D1 , D2 , , Dn administered at times  1 , 2 ,..., n ,
where 0  1   2  ...   n . It is assumed that
every clonogen survives exposure to the dose
Dk with the same probability sk , 1  k  n ,
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 timedependent birth rate b  t  and spontaneous
death rate d  t  . 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 Dk is subject
to random killing with the survival probability
sk , 1  k  n . 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 D thus making
a single survival probability s  s  D  accountable for the resultant effect of damage repair processes.
In the particular case where doses of radiation and inter-dose intervals are equal
( D1  D2  ...  Dn  D
and
 k   k 1 ,
1  k  n ), 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 N  t  of surviving
clonogenic cells at times t   n   n 1 and
t  n belongs to the family of generalized negative binomial distributions (see Section 3 be-
low), 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, G2 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 i of clonogenic cells in a
tumor is typically very large (one cm 3 of a solid
tumor contains about 109 cells [12]; also, a clinically detectable tumor is estimated to contain at
least 105 clonogenic cells probably ranging up
to 109 cells or even more [11, 13]). Additionally, the number n of exposures (typically
ranging from 20 to 40) and the doses D1 ,..., Dn
(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
s1s2 ...sn , is very small.
2 Birth and Death Markov Process
and Generalized Geometric
Distribution
Consider a cell population that starts at time
t  t0 from a single cell and which kinetics is
governed by a birth and death Markov process
with birth and death rates b  t  and d  t  ,
respectively. Let  : b  d be the net birth rate
and   t      u  du be the corresponding
t
t0
cumulative rate. It was shown by Kendall [14]
that probability generating function (p.g.f.) of
the size N  t  of the population at time t  t0
equals
 1  z 
  z   1
,
(1)
1   1  z 
where  and  are functions of t given by
p.g.f. of every r.v. X , that has distribution (4)
with parameters r  (0,1] and q  [0,1) , can be
represented in the form (1), where parameters
 and  are given by   r / 1  q  and
  q / 1  q  , and satisfy conditions (3).
The distribution (4) is referred to as
generalized geometric distribution and is
denoted G  r , q  .
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: r  1 represents a plain geometric
distribution G  q  and corresponds to the pure
birth Markov process, while q  0 leads to a
Bernoulli r.v. X that takes values 0 and 1 with
probabilities 1  r and r , respectively, and
corresponds to a pure death Markov process.
Note also that distribution G  r , q  can be
viewed as a mixture of  0 , the degenerate
distribution
at
0,
and
G q :
G  r, q   1  r   0  rG  q  .
  t   e  t  ,
  t   et   b  u  e u  du, t  t0 . (2)
t
t0
Observe that
  0 and 0    1   .
(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.) X such that
Pr  X  0  1  r ,
Pr  X  k   r (1  q)qk 1,
k  1,
(4)
r   / 1    and q   / 1    .
Clearly, 0  r  1 and 0  q  1 . Conversely,
where
3 Generalized Negative Binomial Distribution
If the birth and death Markov process described
in Section 2 starts from a non-random number i
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 N  t  at time t is the sum of i
i.i.d. r.v.’s with generalized geometric
distribution.
For the plain geometric
distribution G  q  , such sum follows negative
binomial distribution NB  i, q  . This motivates
introducing the following class of probability
distributions.
Definition. Generalized negative binomial
distribution NB  i; r, q  is the distribution of the
sum of i i.i.d. r.v.’s having generalized
geometric distribution G  r , q  .
If p.g.f. of the underlying generalized
geometric distribution is represented in the form
(1) then for p.g.f. of the sum N of its i i.i.d.
copies we have

 1  z  
  z   1 
.
 1   1  z  
0  k  n 1.
for  k  t   k 1 ,
Setting
 k : bk  d k and  k :  K 1   k , 0  k  n  1,
we find that for t   n ,
  s1s2  ...  sne
Recall that in the case of birth and death
Markov process N  t  , t  t0 , with N  t0   i,
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 N  t  of the iterated birth and death
Markov process at any time t   n . Let t be
p.g.f. of r.v. N  t  .
Theorem 1.

 1  z  
t  z   1 
 ,
 1   1  z  
i
where
  s1s2  ...  snet  ,
1
(6)
2
  et  [ 1  f  u  du   2  f  u  du  ...
1
0
 n 
n
 n1
f  u  du   f  u  du ] ,
t
n
(7)
 u
  t      u  du, f  u   b  u  e   , and
0
t
 j : s j s j 1  ...  sn , 1  j  n.
Theorem 1 implies that the distribution of the
r.v. N  t  is generalized negative binomial.
Suppose that birth and death rates do not
vary between exposures: b  t   bk , d  t   dk
n1
and
 b

   k 1 k 1  e   e
k
k 0
n 1
k
i
(5)
 k 0 k  k
k
n1
j k
 j  j
.
Let, in particular, birth and death rates be
d t   d , 0  t   n .
constant: b  t   b
and
Then
  s1s2  ...  sn
and
n 1
b


  e n   k 1 e  k  e  k 1 ,

k 0
where   b  d. Furthermore, in the case of


equal doses  s1  s2  ...  sn  s  we obtain
  s n e n
and
b  n 1


  e  n  s n  k 1 e  k  e  k 1 .

k 0


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
pk 


   

 ,

 Qk 
 1   
 1        
(8)
k  0, where Q0  x   1, and polynomials Qk

 
1 

 1  
i
k
k 1
are
given
 k  1  i  j  1 j
Qk  x    

x .
j
j 1  k  j 

for
by
k
(9)

  pk F k  t    N  F  t   , t  0.
5 Discussion
It is widely accepted that the survival
probabilities sk  s  Dk  , 1  k  n , follow the
linear-quadratic model, s  D   e
  D  vD 2
, where
 and v 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 i, b, d ,  , v . 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 T be the time to
tumor recurrence counted from the moment of
delivery of the last fraction of radiation.
According to the clonogenic model of posttreatment 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 N of surviving clonogens
immediately after the end of treatment is equal
to k . Then for the observed time T of tumor
recurrence we have T  min1 j  k T j , where T j is
the progression time of the j-th clonogen. Let

 N  z    k 0 pk z k be p.g.f. of r.v. N. It
follows from the assumptions of the clonogenic
model that the conditional survival function
FT |N k  t  : Pr T  t | N  k  , t  0, of r.v. T is
given by FT |N k  F k , where F is the common
survival function of the progression times of
surviving clonogens. Then

FT  t    FT | N  k  t  Pr  N  k 
k 0
(10)
k 0
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 p0 ) is
critical for developing biologically motivated
post-treatment survival models. Second, assuming some parametric form for the function
F and using for the distribution of r.v. N its
exact form (6-9), one can estimate the unknown
initial number of clonogenic cells i and
especially the all-important kinetic parameters
b, d ,  and v from the observed times to tumor
recurrence.
Acknowledgements:
Research of the author was supported by NSF
grant DMS-0109895.
References:
[1] S.L. Tucker, H.D. Thames and J.M.G.
Taylor, How well is the probability of tumor
cure after fractionated irradiation described
by Poisson statistics?, Radiation Research,
Vol. 124, 1990, pp. 273-282.
[2] A.Y. Yakovlev, B. Asselain, V.-J. Bardou,
A. Fourquet, T. Hoang, A. Rochefordière
and A.D. Tsodikov, A simple stochastic
model of tumor recurrence and its
application to data on premenopausal breast
cancer. In: B. Asselain, M. Boniface, C.
Duby, C. Lopez, J.-P. Masson, J.
Tranchefort, eds, Biométrie et Analyse de
Données Spatio-Temporelles, Vol. 12,
Rennes, France: Société Française de
Biométrie, ENSA, 1993, pp. 66-82.
[3] L.G. Hanin, Iterated birth and death process
as a model of radiation cell survival, Mathematical Biosciences, Vol. 169, 2001, pp.
89-107.
[4] L.G. Hanin, M. Zaider and A.Y. Yakovlev,
Distribution of the number of clonogens
surviving fractionated radiotherapy: A longstanding problem revisited, International
Journal of Radiation Biology, Vol. 77, No.
2, 2001, pp. 205-213.
[5] M. Zaider, M.J. Zelefsky, L.G. Hanin, A.D.
Tsodikov, A.Y. Yakovlev and S.A. Leibel,
A survival model for fractionated
radiotherapy with an application to prostate
cancer, Physics in Medicine and Biology,
Vol. 46, 2001, pp. 2745-2758.
[6] W.S. Kendal, A closed-form description of
tumour
control
with
fractionated
radiotherapy and repopulation, International
Journal of Radiation Biology, Vol. 73, 1998,
pp. 207-210.
[7] M. Zaider and G.N. Minerbo, Tumor control
probability: A formulation applicable to any
protocol of dose delivery, Physics in
Medicine and Biology, Vol. 45, 2000, pp.
279-293.
[8] S. Okada, Radiation Biochemistry, Vol. 1:
Cells, Academic Press, New York, 1970.
[9] B.G. Weiss, Perturbations in precursor incorporation into DNA of X-irradiated HeLa
S3 cells, Radiation Research, Vol. 48, 1971,
pp. 128-145.
[10] J.B. Mitchell, J.S. Bedford and S.M. Bailey, Observations of the first postirrradiation
division of HeLa cells following continuous
or fractionated exposure to γ rays, Radiation
Research, Vol. 80, 1979, pp. 186-197.
[11] M. Klein and R. Bartoszyński, Estimation
of growth and metastatic rates of primary
breast cancer. In: O. Arino, D. E. Axelrod
and M. Kimmel, eds., Mathematical Population Dynamics, Marcel Dekker, New
York, 1991, pp. 397-412.
[12] S.L. Tucker and J.M.G. Taylor, Improved
models of tumour cure, International
Journal of Radiation Biology, Vol. 70, 1996,
pp. 539-553.
[13] S.L. Tucker, Modeling the probability of
tumor cure after fractionated radiotherapy.
In: M.A. Horn, G. Simonett and G. Webb,
eds., Mathematical Models in Medical and
Health Sciences, Vanderbilt University
Press, Nashville, 1999, pp. 1-15.
[14] D.G. Kendall, On the generalized “birth
and death” process, Annals of Mathematical
Statistics, Vol. 19, 1948, pp. 1-15.
[15] E.J. Hall, Radiobiology for the Radiologist,
4th Ed., Lippincott-Raven, 1994.