Algorithmic support of a testing unit for screening geroprotectors and ...

Biomedical EngineerinN Vol. 31, No. 4, 1997

ALGORITHMIC

SUPPORT OF A TESTING UNIT FOR SCREENING

GEROPROTECTORS

AND GEROPROMOTORS

P. S. K u d r y a v t s e v , L. Yu. P r o k h o r o v , E. V. K h i r o v a , a n d A. N. K h o k h l o v

According to some modern concepts of aging, the leading role in this process belongs to a proliferation activity of the cells constituting the human or animal body (aging/n vivo) or a cell culture (aging/n v/tro, stationary aging) [6]. Because changes in the proliferation activity affect the cell growth kinetics, screening of geroprotectors and geropromotors can be implemented using corresponding model systems of cell populations. However, specific changes of cell growth kinetics induced by aging both in vivo and in vitro remain obscure. Various methods are presently used for studying these processes (auroradiography, cytofluorimetry, high-speed filming, cell counting in a Goryaev chamber, etc.), and there is a discrepancy, between the results obtained by different methods [7, 11]. This, in turn causes a controversial interpretation of the test results. This controversy can be usually resolved by mathematical simulation of cell population growth. The results of experimental tests are used for probing the adequacy of the model used, and this avoids the uncertainty of the testing method. The model of limited proliferation activity of normal cells proposed by Hayrick [10] is one of such mathematical models of cell population growth. However, this model does not take into account the phases of dormancy of the cell cycle and does not allow the population growth dynamics to be described quantitatively within one passage. A number of mathematical models are based on simulation of periodic biochemical reactions in cell plasma. These models describe cell division under ideal conditions and they fail to describe cell population growth kinetics under changeable environmental conditions [3]. Many mathematical models are based on simulation the membrane mechanism of cell division [2]. These models allow the role of such factors of regulation as antioxidants or lipids to be evaluated, the phases of dormancy R1 and/or R2 to be described. However, these models fail to describe the age-dependent changes in the proliferation activity of cells and effects of some additional factors (e.g., environmental factors). Branching Markov processes are the most comprehensive class of mathematical models of cell development [5]. However, these models are usually used to study asymptotic characteristics of cell development, rather than problems of identification of experimental results and quantitative evaluation of population dynamics within, finite time intervals. We suggested a mathematical model of cell population growth within one passage. This model takes into account the phases of dormancy of cell cycle, age-dependent changes in the proliferation activity of cells, and effects of some environmental factors. A procedure for identification of this model was developed on the basis of integral parameters, proliferation activity index, and index of labelled nuclei (in experiment with radioactively labelled precursors) or individual cell dynamics (experiments with high-speed filming detection). The results of the mathematical simulation provided a basis for the development of software for a testing unit for screening geroprotectors and geropromotors/n vitro. Consider this model in more detail. Assume that a developing population grows under conditions of sufficient supply of nutrients within a sufficiently large space within a given time interval. There is a finite number of cell types m , which belong to subpopulations with differing duration of the cell cycle. The cell cycle duration is regarded as the total time interval from its birth to division, including phases of dormancy. In this case, each cell of type A i, i E (1, m) has a random duration of life r i with the distribution function: ~'l-c~ ~ t} = F~(t), F~(-O) = O.

(1)

School of Biology, Lomonosov Moscow State University. All-Russian Scientific-Research Institute for Medical Instrument Engineering, Russian Academy of Medical Sciences, Moscow. Translated from Meditsinskaya Tekhnika, No. 4, pp. 22-26, July-August, 1997. Original article submitted March 4, 1997. 0006-3398/97/3104-0209S18.00

9

Plenum Publishing Corporation

209

At the end of the life cycle, each cell is converted into two cells of zero age, regardless of the type of the cell and presence of cells of other types. The zero age calls belong to one of the following types A1, A2, ..., Am. Let the conventional i

probability of conversion of a cell of type A i into two cells Aj and A t be qie(v). Let the value of q/~ be not dependent on v and i

let qA = q~/. Therefore, the following equation is valid:

qil = 1,

i, j, l ~ (1, m).

(2)

j, l

Each newborn cell undergoes the same life cycle. In a certain time interval t, the cell population A i with the probability

Pie(t) would have progeny c~ = (czl, cz2..... am), where a i ~ N, c~ ~ N m, N = {0, 1.... }. The state of the population at moment t is characterized by the following vector:

X(O = (x, (t),xa(t),...,x, (t)),

(3)

where xi(t ) is the number of cells of the ith subpopulation at the moment t. A random vector X(t) is a branching Markov process, because the transitional probabilities P~s(t)

P ~ = P{X(t + t,) = 13IX(t1) = a}

(4)

P.~(0 ~ 0, ~ e.~(0 = 1,

(5)

~) = ~ P ~ ( t ) P ~(z),

(6)

meet the following conditions

P.~(t

+

t, ~ > O,

.f

P.~(O =

~

(e~. (0)~' (P,. (0)~.--(P,.(t)) ~'-

(7)

l~t*13z-.- *13.=t3

Equation (7) shows that evolution of any cell of the group a does not depend on the evolution of other cells, and the probability of transition of the population from the state a to the state/3 within the time interval t is a composition t31, t32, ...,/3 of the distribution Pla for the population to be in the state cz [4]. Assume that the s~bpopulations are ordered by the mean cycle duration time:

T~ = fvaF~(v),

i e (1, m),

(8)

B

i.e., T1 < T2 < ...