Transient Processes in Cell Proliferation Kinetics

223 Pages·1989·9.63 MB·English

by  Andrej Yu. Yakovlev, Nikolaj M. Yanev (auth.)

Lectu re Notes in
Biomathematics
Managing Editor: S. Levin
82
Andrej Yu. Yakovlev
Nikolaj M. Yanev
Transient Processes
in Cell Proliferation Kinetics
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo Hong Kong
editorial Board
M. Arbib J. D. Cowan Ch. Delisi M. Feldman J. B. Keller M. Kimura
B. Kosko S. LEwin (Managing Editor) R. May J. Murray G. F. Oster
A. S. Perelson T. Poggio L. A. Segel
Authors
Andrej Yu. Yakovlev
Leningrad Polytechnicallnstitute
Polytechnicheskaya ul., 29
Leningrad 195 251, USSR
Nikolaj M. Yaney
Institute of Mathematics
Bulgarian Academy of Sciences
8 Acad. G. Bonchev str.
1113 Sofia, Bulgaria
Translator
B.I. Grudinko
1s t Leningrad
Medical Institute
Mathematics Subject Classification (1980): 60J85, 68Jl0, 62Pl0
ISBN-13: 978-3-540-51831-0 e-ISBN-13: 978-3-642-48702-6
001: 10.1007/978-3-642-48702-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction 01'\ microfilms or in other ways, and storage in data banks. Duplication
of this publication or parts thereof is only permitted under the provisions of the German Copyright
Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be
paid. Violations fall under the prosecution act of the German Copyright Law.
C Springer-Verlag Berlin Heidelberg 1989
2146/3140-543210 - Printed on acid-free paper
In memory of our fathers
Vurij VAKOVLEV and Hichail VANEV
TABLE OF CONTENTS
INTRODUCTION 1
References 8
CHAPTER I SOME POINTS OF THE THEORY
OF BRANCHINS STOCHASTIC PROCESSES 13
1.1. Introduction 13
1.2. The Salton-Watson Process 13
1.3. The Bellman-Harris Process 17
1.4. Asymptotic Behaviour of the
Bellman-Harris Process Characteristics 24
1.5. The Multitype Age-Dependent
Branching Processes 32
References 36
CHAPTER II INDUCED CELL PROLIFERATION KINETICS
WITHIN THE FRAMEWORK OF A BRANCH INS
PROCESS MODEL 37
2.1. Introduction 37
2.2. The Subsequent Senerations of Cells
Induced to Proliferate 38
2.3. Age Distributions in Successive
Senerations 50
2.4. A Multitype Branching Process
Model and Induced Cell Proliferation
Kinetics 52
2.5. Srain Count Distribution and
Branching Stochastic Processes 62
References 82
CHAPTER III SEMISTOCHASTIC MODELS OF CELL
POPULATION KINETICS 84
3.1. Introduction 84
3.2. Integral Equations of Steady-State
Dynamics of a Transitive Cell
Population 87
3.3. Investigation of Periodic Processes
in Cell Kinetics 94
3.4. Basic· Integral Equations for
Unsteady State Cell Kinetics 104
VI
3.5. Construction of the q-index
of the S-phase in a Special Case 113
3.6. Examples of Constructing Transient
Processes for Particular States
of Cell Kinetics 120
3.7. Analysis of the Process of Cell
Blocking in the Mitotic Cycle 125
References 134
CHAPTER IV THE FRACTION LABELLED MITOSES CURVE
IN DIFFERENT STATES OF CELL
PROLIFERATION KINETICS 137
4.1. Introduction 137
4.2. "Flux-expectations" Concept and the
Fraction Labelled Mitoses Curve 138
4.3. Mathematical Model Based on
Transient Phenomena in Cell Kinetics 149
4.4. Investigation of Labelled Mitoses
Curve Behaviour under Unsteady
State Cell Kinetics Conditions 161
4.5. Labelled Mitoses Curve under the
Conditions of the Diurnal Rhythm
of Cell Proliferation Processes 168
References 173
CHAPTER V APPLICATIONS OF KINETIC ANALYSIS.
RAT LIVER REGENERATION 176
5.1. Introduction 176
5.2. Kinetic Analysis.of Induced Hepatocyte
Proliferation in Regenerating Rat Liver 177
5.3. Dynamic Replacement of Hepatocytes,
a Mechanism Maintaining Specialized
Functions of the Regenerating Liver 194
5.4. A Simple Mathematical Model of
Liver Response to Partial Hepatectomy
of Different Extent 201
References 204
CDNCLUSION 208
SUBJECT INDEX 213
INTRODUCTION
A mathematician who has taken the romantic decision to devote
himself to biology will doubtlessly look upon cell kinetics as the
most simple and natural field of application for his knowledge and
skills. Indeed, the thesaurus he is to master is not so
complicated as, say, in molecular biology, the structural elements
of the system, i.e. ceils, have been segregated by Nature itself,
simple considerations of balance may be used for deducing basic
equations, and numerous analogies in other areas of science also
add to one"s confidence. Generally speaking, this superficial
impression is correct, as evidenced by the very great number of
theoretical studies on population kinetics, unmatched in other
branches of mathematical biology. This, however, does not mean
that mathematical theory of cell systems has traversed in its
development a pathway free of difficulties or errors. The seeming
ease of formalizing the phenomena of cell kinetics not
infrequently led to the appearance of mathematical models lacking
in adequacy or effectiveness from the viewpoint of applications.
As in any other domain of science, mathematical theory of cell
systems has its own intrinsic logic of development which, however,
depends in large measure on the progress in experimental biology.
Thus, during a fairly long period running into decades activities
in that sphere were centered on devising its own specific
approaches necessitated by new objectives in the experimental in
vivo and in vitro investigation of cell population kinetics in
different tissues.
There are at present quite a large variety of tools for
experimental research in cell kinetics. The method that has
received the widest acceptance is radioautography [2,6,9-13, 59]
and its combinations with quantitative cytophotometry and
time-lapse cinemicrography [11, 21 28, SO, 59], the latter being
the only direct technique for measuring the duration of the
mitotic cycle. By means of that technique information of primary
importance for the theory of cell systems has been obtained on the
2
speci-fic distribution o-f gan. .a tion time [32, 33, 36, 49, 50, 52]
and duration o-f aitosis [53J -for di-f-f ..a nt types o-f cells in
vitro. A ...i -autoaatic .y.tea o-f processing -fil.. using a
coaput. ... ha. bean proposed [SElJ which axtands considerably the
potentialities o-f time-lapse cin. .i crography as a method o-f
studying individual behaviour o-f cells in a culture.
As a kind o-f alternative to radioautographic approach to cell
kinetics studies a aethod Nas proposed na.ed the BISACK system
[56J, based on introducing broadaoMyuridine (BUdR) into DNA o-f
actively proli-ferating cells. In the BISACK system doses o-f BUdR
are used which induce no inhibition of DNA replication, and by
..a ns of di-f-ferential fluorescent staining o-f chroaosoaes cells
are revealed which appear in the metaphase -for the -first, second
and third tiae during the period of observation. The method also
enables deteraination o-f the total -fraction of cells replicating
DNA in the presence o-f BUdR The BISACK system has been
success-fully employed in studying regularities in the kinetics of
huaan skin -fibroblasts in vitro [48J, peripheral huaan blood
lymphocytes sti.ulated Nith phytohemagglutinin (PHA) [54 - 56J and
bone aarrON cells o-f the rat [47,48]. It is North Nhile to coapare
results obtained by means o-f the BlSACK system Nith the findings
of radioautographic experiaent. The principal difficulty involved
in such a coa!parison lies in the fact that special methods must be
used in analyzing radioautographic data Nhich enable evaluation of
cell kinetics not in the traditional terms of mitotic cycle phase
durations but by determining fluxes of cells entering the cycle
pha&eS under study during a specified interval. In the present
monograph Me revieN such a method based on the introduction of the
so-called q-index which characterizes the integral -flux o-f cells
into a given phase of the cell cycle. We have used this aethod, in
combination Nith the algorithm of labelled mitoses curve analysis,
also described in the aonograph (see Chapter IV), in investigating
the kinetics of PHA-induced proliferation o-f huaan blood
lyaphocytes, and our results are not at variance Nith those
obtained by ._ans of the BlSACK system as reported in refer.,ce
[64J.
A particularly effective tool for studying cell kinetics is the
_thod of flON cytofluoroaetry which .,ables a very high-speed
3
(dozens of thousands of cells per minute) reproduction
of histograms of DNA content of cells [27, 39, 46, 60 ,62]. Its
limitation is the applicability only to cell suspensions, however,
procedures for isolating cells from different tissues~ solid
tumors included, have attained a high degree of efficiency. No
doubt, flow cytofluorometry holds much promise for the future,
as regards not only scientific research but also practical
medicine [23,42,57]. Bray [20] developed a method for analyzing
distributions of DNA content of cells on the basis of the model
of multiphase birth-death process. Using his method, Bray
demonstrated a good agreement between the results obtained by the
techniques of flow microfluorometry and radioautography. However,
comparison of the techniques undertaken by other authors [37] has
shown that radioautography may yield underestimated values for the
fract,on of cells in the S-phase of the mitotic cycle, presumably,
due to the low radioactive label content in the slowly
DNA-synthesizing cells of the population. Comparison of the two
methods will also be found in reference [60]. The authors have
created a computer- ~odel of rat spermatogenesis based on
autoradiographic studies of cell cycle phase durations for each
germ-cell type. The data calculated by means of the model and
experimental flow cytometry findings have shown satisfactory
agreement. Unfortunately, so far very few works have come out
demonstrating combined application of all present-day methods for
investigation of cell population kinetics.
With the accumulation of experimental material on cell kinetics
in embryonal and definitive tissues, starting from the pioneer
research by Howard and Palc [25] and Quastler and Sherman [41]
there has been a growing need for recruiting adequate mathematical
apparatus for analysis of available results.
At present various approaches have satisfactorily been
developed to the analysis of the following experimental data: (1)
indices of labelled cells, using pulse [1,31] or continuous
[1,26,43,44,64,66] labelling with 3H-thymidine; (2) labelled
mitoses curves with pulse [22, 24, 26,34,43,64] and continuous
[44] labelling; (3) experimental evidence with double labelling
[45,51]; (4) kinetic indices variation curves with the blocking of
cell cycle processes [20, 26, 63]; (5) DNA synthesis and mitotic
4
activity diurnal rhyth. curves [22,30,35,651; (6) di.tributions
~
grain count. in radioautograph. and lab.l dilution dyna.ics
[14-16,51,691; (7) flON data [3, 4, 7,
.icr~luora.etry
17-20,29,68,701; (8) cell d.ath characteristic. [261.
ftany the _thods of applied kinetic analysis are based upon
~
probabilistic .ad.ls of biologic papulation dyna.ics Mhich, in
turn, repr....,t .adification. ~ cOMPrehen.ively studied ~.ls
~ .tocha.tic proc• •••• of certain .tructur.s. various types of
ttarkovian proc. ...., ag_dependent branching proc. .... and ran. ..l
proc. ..... SoIMt of such .odels ... i11 be d.alt ... ith in this book.
Atte.pts have been .ade to foraaliz. cell kinetics on the basis of
the theory of .tochastic int.gral equations [401. The fruitfulne• •
the .tochastic approach has been vividly demonstrated in a
~
recent Mark dealing ... ith plant cell papulation grOMth [81. Ther.
is, indeed, a vast literature on deterministic models ~ cell
kinetics.
It is particularly i.ortant to single out interRllKliate type
.adel. Mhich ..y be arbitrarily called seai-stochastic. SUch
.adels contain random variables side by side ... ith deterministic
para_ters or functions. Besides, the follOMing situation is of
special inter. .t . on the one hand, some para_ters of a lIIOdel (for
in.tance, durations of cell cycle phases) are assu. .d to be randoe
and, on the other, it is only the behaviour the math. .a tical
~
expectations of the principal variables (e.g., nu.ber of cells in
a papulation or age distribution) that is investigated. We shall
call such .adels se.i-stochastic as ....1 1 though a certain
artificiality the classification is evident. The notion
~ ~
_i-stochastic ~el is .i.ilar to Nlsel"s "hybrid model"
introduced in his book [381.
Semi-stochastic models, given in this book quite an ample
.pace, have a fairly ... ide range of applicability as regards
description cell kinetics phana.ena and, at the sa. . time, they
~
ar. si.l. enough to serve as the basis for dev.loping applied
..t hods for analyzing experi.antal findings. In the pa.t f .... years
research associat.d ... ith the develap.ant of stochastic si.ulation
models of c.ll proliferation kinetics has been gaining in
hlportance both in theoretical and applied respects [5,661.
The foregoing shews that, as regards present-day ..t heeatical
5
biology,there is no lack of models proposed for description and
analysis of cell population kinetics."uch progress has also bean
made in developing experimental methods for exploring the
structure of the cell cycle and their related software. HDNavar,
in our opinion it is the abundance and diversity of the proposed
mathematical methods that hinder their introduction into
experimental practice. Some of the models are unjustifiably
intricate for practical realization while others, on the contrary,
contain biologically unacceptable assumptions. The user needs a
single working apparatus embodying a reasonable compromise between
achievements of the mathematical theory of cell systems and actual
requirements of biologic experimental methodology. It is from this
standpoint that problems involved in the description of transient
in cell kinetics are considered in this book.
proces~es
The investigator comes across the unsteady nature of cell
kinetics in the majority of practically interesting situations.
The latter, first and foremost , are associated with systems with
induced or stimulated cell proliferation. In the meantime, many
experts on simulation of population dynamics seek to obtain
asymptotic results or concentrate on interrelations between
different characteristics of an already steady state.
The purpose of the present book is to fill the gap caused in
the present-day literature by the lack of the attention to
transient processes in cell kinetics. We use the term "transient
process" in the sense adopted in the theory of dynamic systems,
i.e. applying it to time variations in the characteristics of a
system due to initial conditions. It should be emphasized that the
term, as used in the theory of branching stochastic processes, has
an altogether different meaning.
Consideration for the effect of transient processes calls for
modification of the existing methods for applied kinetic analysis.
We have attempted to substantiate certain ways of such
modification and to demonstrate their effectiveness on real
biologic material, i.e. in studying regular features of the
process of the rat liver regeneration following partial
hepatectomy. Wherever possible, we endeavoured to provide a full
probabilistic description of cell kinetics, giving up it only when
the difficulties seemed insurmountable. InCidentally, in some