Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol167.Issue1.Sept2000:

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.

www.elsevier.com/locate/mbs

*

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

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 ac-count 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 pre-cursor cells in bonemarrow and on crypts, respectively. Thus, the parameterD0characterizing 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 indexD0 in a non-homogeneous 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 variableD0 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 haveI groups of specimens whose critical cell radiosensitivity index D0 belongs to the respective ith interval …D00i;D000i† …iˆ1;. . .;I†. The

fraction of animals constituting the ith group is expressed through the probability density func-tion u…D0† as follows [9,10]:

niˆ

Z D00_0i

D0 0i

u…D0†dD0: …2:1†

We approximate the initial continuous distribution u…D0† by a discrete distribution in the following way. Assume that the random variable D0 takes on discrete values D0i …iˆ1;. . .;I†

(Eq. (2.2)) with the probability

P_{D0 ˆD0}

i† ˆni …iˆ1;. . .;I†; …2:3†

whereni 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 theith group of the initial population whoseD0belongs to theith interval. The parameterD0ithat shows the radiosensitivity

of the critical system cells for individuals of theith homogeneous subpopulation is equal to the mean value ofD0 for the individuals of theith group of the initial population. It should be noted that with any method of subdivision of the range of continuous random variableD0into 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 parametersniandD0i …iˆ1;. . .;I†are equal. This follows from the equality of the expected value

of the random variable D0, described by the continuous distribution u…D0†, and of the expected value of the random variableD0 described by the above derived discrete distribution (2.3).

Two distribution types most popular in biology (normal and log-normal) [11] are used to de-scribe the specimen distribution in the radiosensitivity index D0 in a non-homogeneous mam-malian population. The explicit formulas for calculating the values of the model parametersniand

D0i …iˆ1;. . .;I† are obtained in a straightforward way (see Ref. [3, Chapter IV]).

For description of the dynamics of population mortality, two statistical biometric functions will be considered. The ®rst is the life span probability densityw…t†, which is the ratio of the fraction of animals that die at the timet to their initial number:w…t† ˆ ÿ…dh=dt†=h0. The second biometric function is the mortality rateq…t†, which is the ratio of the fraction of animals that die at the timet

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† ˆh…t†=h0:

Proceeding from these de®nitions, it is easy to express the biometric functionsq_R…t†andwR…t†, characterizing the mortality dynamics of the population as a whole through the biometric func-tions q_i…t†, vi…t†, andwi…t†, which describe the mortality dynamics of the homogeneous

Using the functionwR…t†, the basic demographic parameters can be calculated: the average life

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

aˆ

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 ki-netic 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 ex-ceeds the critical valueL, 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 timetaround the mean valuez…t†

with the variance S2_;

2. natural aging process in the result of which the mean value z…t† of the functional cell concen-tration 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 tˆtr) by zr…t†.

After introducing dimensionless parametersQˆS=Landqˆz0=L, the coecientkˆK=Lwith dimensionTÿ1, and a dimensionless variable_~zr…t† ˆzr…t†=z0, the statistical biometric functions are de®ned by the formulas (see detailed derivation in Ref. [3, Chapter III])

q…t† ˆq_rRexp‰…1ÿR2†=2Q2Š; …2:8†

U…U† ˆ …2p†ÿ1=2

Z U

ÿ1

eÿu2₌₂

du: …2:13†

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 in-testine epithelium) in irradiated mammals [3,13,14] (see Appendix A). The functionR…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 timet. The functionR…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 population is considered as the random variableD0. The parameterD0 is equivalent to the mean value of the radiosensitivity index of the above-mentioned cells for specimens of this non-homogeneous population. The dimensionless variable _~zr…t† in (2.11) is equivalent to the dimen-sionless 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 andSare available from the radiobiological literature. The coecientskandq₀are determined by ®tting experimental data on mortality rateq…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 cor-relation between mortality and degree of inhomogeneity of the population, i.e., the spread of values of the random variable D0 about a ®xed value of mean D0 of the probability density functionu…D0†. As a measure of inhomogeneity, we chose a dimensionless parameterj, which is equal to the ratio of the square root of the varianceV…D0† to the mean D0

jˆ V…D0†

p

=D0: …3:1†

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 homo-geneous 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 precursor cells for specimens from the homogeneous population is equal to the expected valueD0 of the random variable D0 which describes this index for specimens of the initial non-homoge-neous 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 popu-lations 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. Compari-son 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 functionsq…t†andqR…t†describing the mortality rate of homogeneous and non-homogeneous

(normal distribution,jˆ0:3) populations of mice not exposed (curves 1 and 10_{) and exposed to chronic irradiation at}

dose ratesNˆ0:022 Gy/day (curves 2 and 20_{),N ˆ0:044 Gy/day (curves 3 and 3}0_{),Nˆ0:088 Gy/day (curves 4 and 4}0_).

The experimental data [16] on the mortality rate ofLAF1 mice not exposed (+) and exposed to chronic irradiation at

dose ratesNˆ0:022 Gy/day…†;Nˆ0:044 Gy/day (),N ˆ0.088 Gy/day (). The abscissa: the age of animals in days; the ordinate: the functionsq…t†andqR…t†in dayÿ

1

mortality model for the homogeneous population: q_R…t†>q…t†. At the ®nal stage, di€erences between the functions q_R…t† and q…t†, depending on N, either nearly disappear or are reversed:

q_R…t†<q…t†. However, in the last case the di€erence between q_R…t† and q…t† are so small in ab-solute value that the change of its sign does not a€ect the ratio between the average life spansAof specimens in the non-homogeneous and homogeneous populations: the former have a shorter average life span than the latter at each dose ratesNuse. 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 functionsq_R…t†andq…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 ratesN. 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 functionq…t†. The parameterjin this calculation is high. At the same dose ratesN, 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 functionsq<sub>R</sub>…t†andq…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 functionsq…t†andq_R…t†describing the mortality rate of homogeneous and non-homogeneous (log-normal distribution,jˆ1:0) population of mice not exposed (curves 1 and 10_{) and exposed to chronic irradiation}

at dose ratesNˆ0.011 Gy/day (curves 2 and 20_{),Nˆ0.022 Gy/day (curves 3 and 3}0_{),Nˆ0.033 Gy/day (curves 4 and}

40_{). The experimental data [16] on the mortality rate ofLAF}

1mice not exposed (+) and exposed to chronic irradiation at

the ratio between the average life spans A of specimens in the non-homogeneous and homoge-neous populations is the same, as in the earlier-considered case: the former have a shorter average life span than the latter at each dose ratesNused. Qualitatively similar results are obtained when modeling the mortality of the non-homogeneous population at other values of the parameterj

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 dy-namics betwen the homogeneous population and the non-homogeneous population (with both the normal and log-normal distributions of radiosensitive indexD0) are more pronounced with larger

j. In accord with this, asjincreases, the di€erences in the average life span shorteningabetween the homogeneous and non-homogeneous populations also grow at the same irradiation condi-tions. For example, at a dose rate of 0.022 Gy/day, the average life span shorteningafor mice of a homogeneous population is 53 days, for mice of non-homogeneous populations with the normal distributions ofD0is 54 days (jˆ0:15) and 61 days (jˆ0:3), and for mice of non-homogeneous populations with log-normal distributions of D0 is 59 days (jˆ0:3), and 113 days (jˆ1:0), respectively.

It is also found that the di€erences in the mortality dynamics of the subpopulations consiti-tuting non-homogeneous populations with both the normal and log-normal distributions of ra-diosensitive indexD0 are more pronounced with higherj. Therefore, asjincreases, the di€erence between values of the average life span shortening for these subpopulations grow. For instance, with N ˆ0:022 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 (jˆ0:15), 203 and 30 days (jˆ0:3). At the same dose rate N, the average life shortening for the most radiosensitive and least radiosensitive subpopulations of non-homoge-neous populations with the log-normal distribution are 122 are 26 days (jˆ0:3), 494 and 9 days (jˆ1:0), and 611 and 6 days (jˆ1:5). 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 (log-normal distribution,jˆ1:5) 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 (jˆ0:15). The average life shortening for such a population is 54 days or 7%.

4. Conclusion

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 or-ganism. 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 radio-sensitivity 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 re-spective 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 spec-imens 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 sub-stantial 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 hy-persensitive 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 catas-trophe 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

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 thrombo-cytes 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 av-erage, 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 megak-aryocytoblasts and their precursors is provided by a chalone ± thrombocytopenin. As regards the e€ect of ionizing radiation on the thrombocytopoiesis, experiments have shown that thrombo-cytes 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 mech-anisms of its functioning. For this purpose, we divide all the cells of the system under consider-ation into the following ®ve groups according to the degree of maturity and di€erenticonsider-ation and to their radiation response:

· _{GroupX, the dividing-capable precursor cells in bonemarrow (from stem cells in the respective}

microenvironment to megakaryocytoblasts) that are not damaged by radiation.

· _{GroupXd, 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 dif-ferential 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

dt ˆBxÿcxÿ N D0

x; …A:1†

dy

dt ˆ/cxÿdy; …A:2†

dzr

dxd

Herecanddare the speci®c rates of transfer of cells from the groupXto the groupYand from

YtoZr, w is the speci®c decay rate of cells. The coecientsm and lrepresent the speci®c death rates of damaged and heavily damaged cells, respectively. The multiplierN=D0 is the speci®c rate of transition of cellsXfrom 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 damagedXdcells. Experimental data show that.depends on the above-de®ned parameterD0and on the parameterD00. 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=D0ÿ1†ÿ 1

: …A:6†

The parameter B is the reproduction rate of Xcells. With respect of the chalone theory it is described by the following equation:

Bˆaf1‡b‰x‡Uxd‡Cxhd‡Hy‡XzrŠgÿ

1

; …A:7†

where the dimensionless multipliers U, C, Hand X represent the dissimilar contributions of un-damaged X, damaged Xd and heavily damaged Xhd cells, Y cells, and Zr cells to chalone pro-duction.

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 ploidyP…zr†increases:P…zr†>P…z0†. The ratio ofP…zr†toP…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†

In solving Eqs. (A.1)±(A.5), the initial concentration of undamaged X cells, the initial con-centrations of Y and Zr cells, are equal to their stationary values and the concentrations of damagedXd and heavily damagedXhd cells are 0.

References

[1] Radiation protection, ICRP Publication 60, 1990 Recommendations of the International Commission on Radiological Protection, Pergamon, Oxford, 1990.

[2] E.E. Kovalev, O.A. Smirnova, Estimation of radiation risk based on the concept of individual variability of radiosensitivity. in: COSPAR colloquium No. 6, International round table on radiation risk in humans on exploratory missions in space, 11±14 May 1993, Center of the German Physical Society `Physikzentrum', Bad Honnef, Germany, 1993.

[3] E.E. Kovalev, O.A. Smirnova, Estimation of radiation risk based on the concept of the individual variablility of radiosensitivity, AFRRI Contract Report 96-1, Defense Nuclear Agency Contract DNA 001-03-C-0152, 1996. [4] C.F. Arlett, J. Cole, M.H.L. Green, Radiosensitive individuals in the population, in: K.F. Baverstock, J.W. Stather

(Eds.), Low Dose Radiation, Biological Bases of Risk Assessment, Taylor and Francis, London, New York, Philadelphia, 1989, p. 240.

[5] O.A. Smirnova, Dynamics of irradiated mammals lethality in the framework of the mathematical model of haemopoiesis, Radiobiologiya 27 (1987) 913 (Russian).

[6] O.A. Smirnova, Mathematical modeling of the death rate dynamics in mammals with intestinal form of radiation sickness, Radiobiologiya 30 (1990) 814 (Russian).

[7] E.E. Kovalev, O.A. Smirnova, Life-span of irradiated mammals: mathematical modelling, Acta Astron. 32 (1944) 649.

[8] V.P. Bond, T.M. Fliedner, J.O. Archambeau, Mammalian Radiation Lethality, Academic Press, New York, 1965. [9] W.T. Eadie, D. Drijard, F.E. James, M. Roos, B. Sadoulet, Statistical Methods in Experimental Physics,

North-Holland, Amsterdam, London, 1971.

[10] G.A. Korn, T.M. Korn, Mathematical Handbook, McGraw-Hill, New York, 1968.

[11] Report of the Task Group on Reference Man ICRP Publication 23, Pergamon, New York, 1975. [12] L. Boleslawski, Cohort Tables of Life Span, Statistika, Moscow, 1977 (Russian).

[13] O.A. Smirnova, Mathematical modeling the thrombocytopoiesis dynamics in mammals exposed to radiation, Radiobiologiya 25 (1985) 571(Russian).

[14] O.A. Smirnova, Mathematical modeling of the death rate dynamics in mammals with intestinal form of radiation sickness, Radiobiologiya 30 (1990) 814 (Russian).

[15] I. Kalina, M. Praslichka, Changes in haemopoiesis and survival of continuously irradiated mice, Radiobiologiya 17 (1977) 849 (Russian).

[16] G.A. Sacher, On the statistical nature of mortality with a special reference to chronic radiation mortality, Radiology 67 (1955) 250.