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Mathematical Biosciences 167 (2000) 19±30
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Mathematical modeling of mortality dynamics of mammalian
populations exposed to radiation
O.A. Smirnova *
Research Center of Spacecraft Radiation Safety, Shchukinskaya Str. 40, 123182 Moscow, Russian Federation
Received 1 February 1999; received in revised form 22 June 1999; accepted 3 September 1999

Abstract
A mathematical model is developed which describes the dynamics of radiation-induced mortality of a
non-homogeneous (in radiosensitivity) mammalian population. It relates statistical biometric functions
with statistical and dynamic characteristics of a critical system in organism of specimens composing this
population. The model involves two types of distributions, the normal and the log-normal, of population
specimens with respect to the radiosensitivity of the critical system cells. This approach suggests a new
pathway in developing the methods of radiation risk assessment. Ó 2000 Elsevier Science Inc. All rights
reserved.
Keywords: Individual-based population dynamics; Low-level irradiation; Model

1. Introduction
Accidents in atomic power stations, nuclear weapon tests, and functioning a series of atomic

weapon complexes led to an unfavorable ecological circumstances in some regions of our planet.
The population in these areas reside at the conditions of elevated radiation background. Besides,
workers of some professions (nuclear power plant employees, radiologists, nuclear physicists and
technicians, and others) are also subjected to low-level irradiation. Therefore one of the urgent
ecological problems is ensuring the radiation safety of large groups of population exposed to
small dose rate chronic radiation. To resolve this problem, it is necessary, ®rst of all, to develop
new approaches to radiation risk assessment. Traditional methods, as noted in [1], are not always
applicable in the case of low-level irradiation because of ambiguity of radiobiological e€ects of
such exposures that were observed in a number of experiments.

*

Tel.: +7-095 190 5131; fax: +7-095 193 8060.

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 2 2 - 5

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O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

In [2,3], a new feasible approach for the risk estimation of low dose rate chronic radiation has
been proposed. It is based on the experimental studies suggesting that populations of various
mammalian species, including humans, contain a small proportion of individuals (from 5% to
12%) that show hyperradiosensitivity [4]. The approach implies the importance to take into account the variability of the individual radiosensitivity of specimens when estimating the radiation
risk of low-level exposures.
Implementation of this approach calls for development and investigation of a mathematical
model of radiation-induced mortality dynamics for non-homogeneous (in radiosensitivity)
mammalian populations. It is this objective that the present paper is devoted to. The starting
point is our model of mortality for homogeneous mammalian populations [3,5±7].

2. Mathematical model
The model of radiation-induced mortality of a non-homogeneous mammalian population rests
on the agreed-upon radiobiological concept of a critical system [8]. According to this concept, for
de®nite intervals of doses and dose rates of acute and chronic irradiation, one can pick out a
speci®c critical system in mammalian organism the radiation damage of which will play a key role
in development of radiation sickness and ultimately in the death of mammals. In turn, the damage
of the critical system manifests itself in reducing the number of its functional cells below the level
required for survival.
In radiobiology it is assumed that the inhomogeneity of a mammalian population with respect

to radiation exposure is mainly due to the dissimilarity in radiosensitivity index of critical system
cells among the specimens constituting the population [3,8]. The index of cell radiosensitivity is
the conventional radiobiological dose D0 : after exposure to this dose the number of cells left
undamaged is 2:718 . . . times smaller than their initial number [8]. The radiosensitive cells in two
major critical systems, hematopoiesis and small intestine epithelium, are division-capable precursor cells in bonemarrow and on crypts, respectively. Thus, the parameter D0 characterizing the
radiosensitivity of division-capable precursor cells of the relevent critical system can be used in the
model as the index of inhomogeneity (in radiosensitivity) of mammalian populations.
Taking this into account, let the specimen distribution in the radiosensitivity index D0 in a nonhomogeneous mammalian population be described by a continuous function u…D0 †. Then, for
simpli®cation of the problem, let us go over from the continuous distribution of the random
variable D0 to a discrete one. For this end, we break the range of the continuous random variable
D0 into a ®nite number of intervals, I. Accordingly, we have I groups of specimens whose critical
cell radiosensitivity index D0 belongs to the respective ith interval …D00i ; D000i † …i ˆ 1; . . . ; I†. The
fraction of animals constituting the ith group is expressed through the probability density function u…D0 † as follows [9,10]:
ni ˆ

Z

D000i
D00i

u…D0 † dD0 :

The mean value of D0 for animals of the ith group is

…2:1†

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

 0i ˆ 1
D
ni

Z

D000i

D0 u…D0 † dD0 :

21

…2:2†

D00i

We approximate the initial continuous distribution u…D0 † by a discrete distribution in the
 0i …i ˆ 1; . . . ; I†
following way. Assume that the random variable D0 takes on discrete values D
(Eq. (2.2)) with the probability
 0i † ˆ ni …i ˆ 1; . . . ; I†;
P…D0 ˆ D
…2:3†
where ni are determined by Eq. (2.1).
This operation is equivalent to the representation of the initial non-homogeneous population as
a set of ®nite number of homogeneous subpopulations. The number of specimens in the ith
homogenous subpopulation coincides with the number of specimens in the ith group of the initial
 0i that shows the radiosensitivity
population whose D0 belongs to the ith interval. The parameter D
of the critical system cells for individuals of the ith homogeneous subpopulation is equal to the
mean value of D0 for the individuals of the ith group of the initial population. It should be noted
that with any method of subdivision of the range of continuous random variable D0 into intervals

and irrespective of the number of the latter, the mean values of the radiosensitivity index of the
critical system cells for individuals of the initial non-homogeneous population and for individuals
of the non-homogeneous population that is a set of the homogeneous subpopulations (with the
 0i …i ˆ 1; . . . ; I† are equal. This follows from the equality of the expected value
parameters ni and D
of the random variable D0 , described by the continuous distribution u…D0 †, and of the expected
value of the random variable D0 described by the above derived discrete distribution (2.3).
Two distribution types most popular in biology (normal and log-normal) [11] are used to describe the specimen distribution in the radiosensitivity index D0 in a non-homogeneous mammalian population. The explicit formulas for calculating the values of the model parameters ni and
 0i …i ˆ 1; . . . ; I† are obtained in a straightforward way (see Ref. [3, Chapter IV]).
D
For description of the dynamics of population mortality, two statistical biometric functions will
be considered. The ®rst is the life span probability density w…t†, which is the ratio of the fraction of
animals that die at the time t to their initial number: w…t† ˆ ÿ…dh=dt†=h0 . The second biometric
function is the mortality rate q…t†, which is the ratio of the fraction of animals that die at the time t
to the number of animals that have survived to the time t: q…t† ˆ ÿ…dh=dt†=h…t†. The third
function is the life span probability:
v…t† ˆ h…t†=h0 :
Proceeding from these de®nitions, it is easy to express the biometric functions qR …t† and wR …t†,
characterizing the mortality dynamics of the population as a whole through the biometric functions qi …t†, vi …t†, and wi …t†, which describe the mortality dynamics of the homogeneous subpopulations
wR …t† ˆ

I
X

wi …t†ni ;

…2:4†

iˆ1

qR …t† ˆ

I
X
iˆ1

qi …t†vi …t†ni

,

I
X
iˆ1

vi …t†ni :

…2:5†

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O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

Using the function wR …t†, the basic demographic parameters can be calculated: the average life
span A [12]
Z 1
twR …t† dt
…2:6†
Aˆ
0

and the average life span shortening a, which is the di€erence between values of A for two
identical populations unexposed and exposed to radiation
Z 1
Z 1
aˆ
twRC …t† dt ÿ
twRR …t† dt:
…2:7†
0

0

In Eq. (2.7), wRC …t† and wRR …t† are the life span probability density functions which describe the
mortality dynamics of the control (unexposed) population and the exposed population.
To calculate the biometrical functions characterizing the dynamics of the subpopulations, use is
made of the mathematical model of radiation-induced mortality for a homogeneous mammalian
population [3,5±7]. This model relates the statistical biometric functions with statistical and kinetic characteristics of a critical system. These characteristics are considered to be identical for all
the members of the population. It is assumed that the situation, when the deviation of the
functional cell concentration from the normal level z0 in the critical system of an individual exceeds the critical value L, can be treated as a death analog. The reasons of such deviation in our
model are the following:

1. random ¯uctuation of the functional cell concentration at the time t around the mean value z…t†
with the variance S 2 ;
2. natural aging process in the result of which the mean value z…t† of the functional cell concentration decreases linearly with time at low rate K;
3. radiation-induced change of the mean value of the functional cell concentration which
describes after the onset of irradiation (at time t ˆ tr ) by zr …t†.
After introducing dimensionless parameters Q ˆ S=L and q ˆ z0 =L, the coecient k ˆ K=L with
dimension T ÿ1 , and a dimensionless variable ~zr …t† ˆ zr …t†=z0 , the statistical biometric functions are
de®ned by the formulas (see detailed derivation in Ref. [3, Chapter III])
q…t† ˆ qr R exp‰…1 ÿ R2 †=2Q2 Š;
w…t† ˆ q…t† exp



ÿ

Z

t
0

…2:8†


q…s† ds ;

…2:9†

where
R ˆ R…t† ˆ 1 ÿ q‰1 ÿ f…t†Š ÿ kt;
f…t† ˆ



1
~zr …t†

…2:10†

for t < tr ;
for t P tr ;

…2:11†

qr ˆ q0 U…1=Q†=U…R=Q†:
In the last equation, q0 is a constant parameter with dimension T
normal distribution of a standardized random variable u [10]

…2:12†
ÿ1

and U…U † is the standard

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

U…U† ˆ …2p†ÿ1=2

Z

U

2

eÿu =2 du:

23

…2:13†

ÿ1

The dimensionless variable ~zr …t† in (2.11) is calculated in the framework of the mathematical
model which describes the dynamics of the relevent critical system (hematopoiesis or small intestine epithelium) in irradiated mammals [3,13,14] (see Appendix A). The function R…t† in (2.10)
is proportional to the di€erence between the critical deviation of the functional cells concentration
from its normal level and the mean value of this deviation at the time t. The function R…t† is always
positive because of the assumptions adopted in deriving the model.
Thus, we have built, in the general form, the mathematical model that enables one to calculate
the statistical biometric functions which describe the mortality dynamics for a non-homogeneous
(in radiosensitivity) population of mammals exposed to radiation.

3. Results of mathematical modeling
The model has been used to simulate the mortality of non-homogeneous populations of mice
subjected to chronic low dose rate irradiation, duration of which is comparable to the maximum
age of unexposed animals (the duration of the `model experiment' and the age of mice at the onset
of irradiation are chosen to be 1000 and 100 days, respectively). In this case, the critical system is
hematopoiesis (namely, thrombocytopoiesis) [15]. Therefore, the radiosensitivity index of the
division-capable precursor cells in thrombocytopoiesis system of mice in non-homogeneous
 0 is equivalent to the mean
population is considered as the random variable D0 . The parameter D
value of the radiosensitivity index of the above-mentioned cells for specimens of this nonhomogeneous population. The dimensionless variable ~zr …t† in (2.11) is equivalent to the dimensionless averaged concentration of the functional cells (thrombocytes) in thrombocytopoiesis
system of mice. The values of this dimensionless variable are calculated in the framework of
mathematical model of thrombocytopoiesis [3,13] (see Appendix A). In turn, the values of z0 ; L
and S are available from the radiobiological literature. The coecients k and q0 are determined by
®tting experimental data on mortality rate q…t† in non-irradiated mammalian population. So, we
need not know beforehand the mortality dynamics of irradiated mammals for identi®cation of our
radiation-induced mortality model.
In the framework of the model, we investigate the relationship between radiation-induced
mortality dynamics and the type of probability density function u…D0 †. We also study the correlation between mortality and degree of inhomogeneity of the population, i.e., the spread of
 0 of the probability density
values of the random variable D0 about a ®xed value of mean D
function u…D0 †. As a measure of inhomogeneity, we chose a dimensionless parameter j, which is
0
equal to the ratio of the square root of the variance V …D0 † to the mean D
p
0:
j ˆ V …D0 †=D
…3:1†
Let us ®rst examine the results of modeling the mortality dynamics for the non-homogeneous
population of irradiated mice in the case of normal distribution of specimens in the radiosensitivity index of the thrombocytopoiesis system precursor cells. Fig. 1 presents the biometric
functions qR …t†. They describe the mortality rate of the non-homogeneous population of mice in

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O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

the absence of irradiation and under chronic irradiation with three di€erent dose rates N. The
parameter k in this calculation is close to the maximum admissable for the distribution type
considered. Fig. 1 exhibits the functions q…t†, too. They de®ne the mortality rate in the homogeneous population of mice both unexposed and exposed to chronic irradiation at the same dose
rates as mice of the non-homogeneous population. The radiosensitivity index of the thrombocyte
0
precursor cells for specimens from the homogeneous population is equal to the expected value D
of the random variable D0 which describes this index for specimens of the initial non-homogeneous population. Fig. 1 also demonstrates experimental data [16] on the mortality rate dynamics
for non-irradiated and irradiated LAF1 mice. It is important to note that these experimental data
practically coincide with corresponding values of the mortality rate q…t† for the homogeneous
population of mice calculated in the model.
Fig. 1 shows that the mortality dynamics in the homogeneous and non-homogeneous populations in the absence of irradiation is the same. This could not be otherwise because, according to
the model construction, any di€erences between specimens belonging to the homogeneous and
non-homogeneous populations mainfest themselves only when irradiation is invoved. Comparison of the biometric functions qR …t† and q…t†, calculated at non-zero dose rates N, allows us to
reveal the following. At the same dose rates N, the mortality model for the non-homogeneous
population for most of the `model experiment' duration predicts higher mortality rates than the

Fig. 1. The biometric functions q…t† and qR …t† describing the mortality rate of homogeneous and non-homogeneous
(normal distribution, j ˆ 0:3) populations of mice not exposed (curves 1 and 10 ) and exposed to chronic irradiation at
dose rates N ˆ 0:022 Gy/day (curves 2 and 20 ), N ˆ 0:044 Gy/day (curves 3 and 30 ), N ˆ 0:088 Gy/day (curves 4 and 40 ).
The experimental data [16] on the mortality rate of LAF1 mice not exposed (+) and exposed to chronic irradiation at
dose rates N ˆ 0:022 Gy/day …†; N ˆ 0:044 Gy/day (), N ˆ 0.088 Gy/day (). The abscissa: the age of animals in
days; the ordinate: the functions q…t† and qR …t† in dayÿ1 .

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

25

mortality model for the homogeneous population: qR …t† > q…t†. At the ®nal stage, di€erences
between the functions qR …t† and q…t†, depending on N, either nearly disappear or are reversed:
qR …t† < q…t†. However, in the last case the di€erence between qR …t† and q…t† are so small in absolute value that the change of its sign does not a€ect the ratio between the average life spans A of
specimens in the non-homogeneous and homogeneous populations: the former have a shorter
average life span than the latter at each dose rates N use. Qualitatively similar data are obtained
when modeling the non-homogeneous population mortality at other values of the parameter
j …0 < j < 1=3† and at other (but of the same order of magnitude) values of N.
Let us now address the modeling results for the case of the log-normal distribution of specimens
in the radiosensitivity index of thrombocytopoietic system precursor cells. Fig. 2 represents the
biometric functions qR …t† and q…t† describing the mortality rate for mice of the non-homogeneous
and homogeneous populations in the absence of radiation and under continuous radiation at
three does rates N. Fig. 2 also exhibits the experimental data on mortality rate of non-irradiated
and irradiated LAF1 mice [16], which practically coincide with the corresponding values of the
function q…t†. The parameter j in this calculation is high. At the same dose rates N, the mortality
model for the non-homogeneous population for most of the `model experiment' duration predicts
higher mortality rates than the mortality model for the homogeneous population: qR …t† > q…t†.
Di€erence between the functions qR …t† and q…t† are particularly large within the ®rst year after the
onset of irradiation, and nearly disappear toward the end of the `model experiment'. Therefore,

Fig. 2. The biometric functions q…t† and qR …t† describing the mortality rate of homogeneous and non-homogeneous
(log-normal distribution, j ˆ 1:0) population of mice not exposed (curves 1 and 10 ) and exposed to chronic irradiation
at dose rates N ˆ 0.011 Gy/day (curves 2 and 20 ), N ˆ 0.022 Gy/day (curves 3 and 30 ), N ˆ 0.033 Gy/day (curves 4 and
40 ). The experimental data [16] on the mortality rate of LAF1 mice not exposed (+) and exposed to chronic irradiation at
dose rates N ˆ 0.022 Gy/day …†. The axes are the same as in Fig. 1.

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O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

the ratio between the average life spans A of specimens in the non-homogeneous and homogeneous populations is the same, as in the earlier-considered case: the former have a shorter average
life span than the latter at each dose rates N used. Qualitatively similar results are obtained when
modeling the mortality of the non-homogeneous population at other values of the parameter j
and at other (but of the same order of magnitude) values of N.
The modeling results enable us to reveal the following. The distinctions in the mortality dynamics betwen the homogeneous population and the non-homogeneous population (with both the
normal and log-normal distributions of radiosensitive index D0 ) are more pronounced with larger
j. In accord with this, as j increases, the di€erences in the average life span shortening a between
the homogeneous and non-homogeneous populations also grow at the same irradiation conditions. For example, at a dose rate of 0.022 Gy/day, the average life span shortening a for mice of a
homogeneous population is 53 days, for mice of non-homogeneous populations with the normal
distributions of D0 is 54 days (j ˆ 0:15) and 61 days (j ˆ 0:3), and for mice of non-homogeneous
populations with log-normal distributions of D0 is 59 days (j ˆ 0:3), and 113 days (j ˆ 1:0),
respectively.
It is also found that the di€erences in the mortality dynamics of the subpopulations consitituting non-homogeneous populations with both the normal and log-normal distributions of radiosensitive index D0 are more pronounced with higher j. Therefore, as j increases, the di€erence
between values of the average life span shortening for these subpopulations grow. For instance,
with N ˆ 0:022 Gy/day the average life span shortening for the most radiosensitive and least
radiosensitive subpopulations of non-homogeneous populations with the normal distribution are
86 and 48 days (j ˆ 0:15), 203 and 30 days (j ˆ 0:3). At the same dose rate N, the average life
shortening for the most radiosensitive and least radiosensitive subpopulations of non-homogeneous populations with the log-normal distribution are 122 are 26 days (j ˆ 0:3), 494 and 9 days
(j ˆ 1:0), and 611 and 6 days (j ˆ 1:5). These data demonstrate that continuous irradiation at
relatively low dose rates can be very harmful for specimens whose thrombocyte precursor cells are
hyperradiosensitive.
The model results obtained suggest an important conclusion. The greater the scatter in the
values of the individual radiosensitivity index of the thrombocytopoietic system precursor cells in
the non-homogeneous population, the lower is the level of prolonged irradiation dose rate that is
dangerous for this population. For instance, in chronic irradiation at a dose rate of 0.022 Gy/day,
the average life shortening for specimens constituting the non-homogeneous population (lognormal distribution, j ˆ 1:5) is 160 days, or 21% of the average life span for intact animals. The
same level of exposure is less dangerous for a non-homogeneous population with smaller scatter in
the values of the individual radiosensitivity index (j ˆ 0:15). The average life shortening for such
a population is 54 days or 7%.

4. Conclusion
The model of radiation-induced mortality for a non-homogeneous mammalian population is
developed. It includes, as constituent element, the mathematical model of mortality dynamics for
the homogeneous population and the mathematical model of the relevant critical system. This
structure of the model re¯ects the actually existing levels of manifestation of adverse radiation

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

27

e€ects in mammals. The ®rst level is that of a critical system, whose radiation injury is largely
determined by the radiosensitivity of the constituent cells. Second is the level of the whole organism. Here the probable outcome of irradiation mainly depends on the extent of radiation
injury of the respective critical system, i.e., on individual cell radiosensitivity of this system. The
third level is that of the population which includes animals with di€ering individual radiosensitivity of the critical system cells. Thus, the elaborated model of mortality is actually a
mathematical description of cause±e€ect relationship set up in the course of radiation injury of a
non-homogeneous mammalian population.
The modeling results revealed a direct correlation between the variability of specimen survival
in the non-homogeneous population and the variability of individual radiosensitivity of the respective critical system precursor cells. Under low-level chronic irradiation, the probability to
survive to certain age for a specimen increases with decreasing radiosensitivity of the bonemarrow
precursor cells of the thrombocytopoietic system.
It is shown in the model that allowance for the normal and log-normal distributions of specimens of the non-homogeneous population in the index of the radiosensitivity of the critical
system precursor cells leads to higher mortality rates and lower survival than it could have been
predicted proceeding from the averaged radiosensitivity index alone. The di€erences in predictions
increase as the variance of the individual radiosensitivity index for critical system precursor cells
in the non-homogeneous population increases. The di€erences are particularly pronounced when
the log-normal distribution with a high variance value is used. This modeling result is of substantial practical signi®cance in view of the experimental data [17] with respect to which a human
population is probably characterized by a log-normal distribution with fairly large value of the
variance. Therefore, account of variability of individual radiosensitivity is of great importance in
estimation of radiation risk for human populations.
The model studies also suggest practical conclusions. The greater the spread of values of the
individual radiosensitivity index for the critical system precursor cells in these populations, that is,
the greater the variance of corresponding distributions the lower is the level of dose rates of
chronic exposures that present a certain danger for non-homogeneous mammalian populations.
For specimens having hyperradiosensitive precursor cells even low-level irradiation can have fatal
consequences. These conclusions are especially signi®cant in development of recommendations
for the radiation protection of human populations. This is due to rather high percentage of hypersensitive persons in human populations (from 5% to 25%). Such estimations were obtained by
analyzing clinical data on persons directly involved in the elimination of the Chernobyl catastrophe after-e€ects [3].
Thus, the model developed can be used for prediction, on quantitative level, of the mortality
dynamics in non-homogeneous populations of small laboratory animals (mice) exposed to low
level chronic radiation (for example, during long-term space ¯ights, such as a voyage to Mars),
and for radiation risk assessment for human populations residing in areas with elevated radiation
background. Certainly, in the last case the model must be identi®ed for man.

Acknowledgements
The work was funded in part by DSWA/AFRRI (USA).

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O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

Appendix A. Mathematical model of the thrombocytopoiesis dynamics under chronic irradiation
Thrombocytopoiesis is one of the major lines of bonemarrow hematopoiesis [8]. The generic
cells of the functional cells of this system (thrombocytes) are megakaryocytes, the giant cells of the
bonemarrow. The youngest morphologically identi®able precursor cell of the thrombocytic line is
the dividing megakaryocytoblasts. At the next di€erentiation stage (promegakaryocyte), the cells
do not divide but grow in size by increasing their ploidy. A megakaryocyte can have 4, 8, 16, 32 or
64 nuclei. When the number of nuclei reaches 8, the megakaryocyte starts producing thrombocytes which subsequently leave the bonemarrow and pass into the blood. The number of
thrombocytes produced by one megakaryocyte is proportional to the volume of its cytoplasm,
which in turn is proportional to the number of nuclei of the mature megakaryocyte. On an average, a megakaryocyte produces some 3000±4000 thrombocytes and then dies. The thrombocytes
also undergo a natural process of dying. The control of the reproduction rate in the megakaryocytoblasts and their precursors is provided by a chalone ± thrombocytopenin. As regards the
e€ect of ionizing radiation on the thrombocytopoiesis, experiments have shown that thrombocytes and all cells of the megakaryocyte line beginning with promegakaryocyte are radioresistant.
The megakaryocytoblasts and their precursors are radiosensitive.
When constructing the elementary model of thrombocytopoiesis [3,13], we take into account
only the principal stages of development of hematopoietic cells and the main regulatory mechanisms of its functioning. For this purpose, we divide all the cells of the system under consideration into the following ®ve groups according to the degree of maturity and di€erentiation and to
their radiation response:
· Group X, the dividing-capable precursor cells in bonemarrow (from stem cells in the respective
microenvironment to megakaryocytoblasts) that are not damaged by radiation.
· Group Xd , the dividing-capable precursor cells that are damaged by radiation and die within 1±
2 days (mitotic death).
· Group Xhd , the dividing-capable precursor cells that are heavily damaged and die within the
®rst 4±7 h (interphase death).
· Group Y, the non-dividing maturing bonemarrow cells (from promegakaryocytes to mature
megakaryocytes).
· Group Zr , the mature blood cells (thrombocytes).
When developing the model, we employ the one-target-one-hit theory of cell damage, according
to which the speci®c damage rate is proportional to the radiation dose rate N. As a result, the
model of thrombocytopoiesis dynamics in irradiated mammals comprises the following ®ve differential equations describing the concentrations of undamaged X, damaged Xd , and heavily
damaged Xhd cells, and also the concentrations of radioresistant Y and Zr cells (x; xd ; xhd ; y; zr ,
respectively):
dx
N
ˆ Bx ÿ cx ÿ x;
dt
D0
dy
ˆ /cx ÿ dy;
dt
dzr
ˆ rdy ÿ wzr ;
dt

…A:1†
…A:2†
…A:3†

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

dxd
N 1
x ÿ mxd ;
ˆ
D0 1 ‡ .
dt
dxhd
N .
x ÿ lxhd :
ˆ
D0 1 ‡ .
dt

29

…A:4†
…A:5†

Here c and d are the speci®c rates of transfer of cells from the group X to the group Y and from
Y to Zr , w is the speci®c decay rate of cells. The coecients m and l represent the speci®c death
rates of damaged and heavily damaged cells, respectively. The multiplier N =D0 is the speci®c rate
of transition of cells X from the undamaged state to the damaged and heavily damaged states. The
coecient . represents the ratio of the number of heavily damaged Xhd cells to the number of
damaged Xd cells. Experimental data show that . depends on the above-de®ned parameter D0 and
on the parameter D00 . The latter is the dose after exposure to which the number of cells that have
not died in the interphase is 2:718 . . . times smaller than the initial number of cells. The form of
this dependence is the following:
. ˆ …D00 =D0 ÿ 1†ÿ1 :

…A:6†

The parameter B is the reproduction rate of X cells. With respect of the chalone theory it is
described by the following equation:
B ˆ af1 ‡ b‰x ‡ Uxd ‡ Cxhd ‡ Hy ‡ Xzr Šgÿ1 ;

…A:7†

where the dimensionless multipliers U, C, H and X represent the dissimilar contributions of undamaged X, damaged Xd and heavily damaged Xhd cells, Y cells, and Zr cells to chalone production.
The description of the complicated process of nucleus duplication in megakaryocytes, which
eventually determines the megakaryocyte ploidy and the number of thrombocytes produced, is
replaced in the model by a new integral quantity: coecient of megakaryocyte ploidy /. It is
known from experiments that in healthy mammals, the thrombocyte concentration in the blood,
z0 the average ploidy of bonemarrow megakaryocytes, P …z0 † and the thrombocyte yield per
megakaryocyte, r, are stable quantities. When the number of thrombocytes is reduced (zr < z0 ),
the average ploidy P …zr † increases: P …zr † > P …z0 †. The ratio of P …zr † to P …z0 † is the ploidy coecient
/. In accordance with experimental data, the coecient / is represented as decreasing function of
thrombocyte concentration
/ ˆ …x ‡ kzr †ÿ1 :

…A:8†

When deriving Eq. (A.3), it is assumed that all Y cells have the same ploidy and after maturation
produce the same number r of thrombocytes. A change in the average ploidy is described in the
model by an equivalent change of the concentration of Y cells. The concentration of X cells that
have just passed into phase Y is multiplied by the coecient /. This way of introducing the
coecient / into Eq. (A.2) enables one to take account in a dynamic form of the delay e€ect
between the control signal (deviation of thrombocyte concentration from the normal level) and
the response (change of average megakaryocyte ploidy). Thus, the variable y accounts for the
dynamics of the total megakaryocyte mass in the bonemarrow.

30

O.A. Smirnova / Mathematical Biosciences 167 (2000) 19±30

In solving Eqs. (A.1)±(A.5), the initial concentration of undamaged X cells, the initial concentrations of Y and Zr cells, are equal to their stationary values and the concentrations of
damaged Xd and heavily damaged Xhd cells are 0.

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