A NEW METHOD FOR STUDYING THE RED BLOOD SYSTEM
G.Ormotsadze, K.Nadareishvili
Abstract
Adopted Designations
Introduction
Methodological Points Of Rbs Study
Test On Reversible Osmotic Spherulation
Preparation Methods And Equipment Base
Principle Of Interpretation Of Population
Spectra
Conclusion
Reference
In the framework of systems analysis we have developed a new methodological
approach for studying the functional state of the red blood system (RBS),
based on the analysis of population spectra (PS) of the peripheral blood
erythrocytes (PBE) - distribution of PBE according to their physiological
state. The physiological state of erythrocytes with the use of this methodological
approach is estimated by two parameters: the cell spherulation degree and
their volumes.
By a comparative analysis of PS oblained in humans under normal
physiological conditions and in different pathologies, as well as in rats
under radiation pathologies, consistent changes were revealed in the surface
area and volume of PBE in the course of their survival cycle. Based on
them, a number of criteria for the estimation of the RBS functional state
were established. Within stationary approximations, on the strength of
a single PS analysis, it is supposed to be possible: 1) determination of
overall PBE concentration and hematocrit; 2) detection of a variation in
the intensity of destruction (rate of aging) and consequently, also the
PBE survival and intensity of stochastic perish of cells in respect to
physiological norm; 3) estimation of the degree of erythropoiesis by way
of distribution analysis of volumes and surface area of young fractions
of PBE; 4) evaluation of changes occurring in rheological properties of
the circulating blood. In nonstationary (transient) conditions of RBS functioning,
dynamics of PS variations allows for practically complete visualization
of the process of PBE aging.
Key words: red blood system, method, erythrocyte, erythropoiesis,
erythrocyte aging and elimination.
NaHb(t)
- required oxygen capacity of circulating blood.
NrHb(t) - real
oxygen capacity of circulating blood.
D NHb(t)
- difference between the required and real oxygen capacity
of circulating blood.
ef - efficiency of RBS functioning.
D NcHb
- end effect of RBS response to a single disturbance.
x -
time needed for a full relaxation of RBS.
m - Energy and plastic material
requirements of the system for a minimal D
N Hb during a relaxation period.
Y1 ,Y2 and
Y3 reflect the essence of
RBS, as subsystems of a broader system and namely of the body as a whole.
D
` N HB
- mean value of blood oxygen capacity for a certain
interval of time.
` m -
mean value of energy and plastic material requirements for a certain interval
of time.
nHb - amount of the
functionally active hemoglobin.
W -
biological age of erythrocytes.
t -
chronological age of erythrocytes.
T - maximal life span of PBE.
g - a parameter characterizing
the difference in the cells properties due to a set of all factors at various
stages of hemopoiesis.
p=p(g,W ,t) PS
of PBE.
I(g,t) - intensity of erythropoiesis.
v=v(W ,g) -rate
of cell aging
D(W ,g) - cell
flow between the depot and circulating blood
E(W ,g) -
probability of cell elimination.
q - degree of PBE spherulation.
V - PBE volume.
S - PBE surface area.
pH i,pH o -
intracellular and extracellular pH.
CCli,CClo
. - intracellular and extracellular chlorine.
Cict - concentration
of intracellular cations.
Ci and Co
- overall concentrations of cations and anions of intracellular
and suspending medium.
CB - glycerine concentration.
NE and NB
- amount of intracellular solutes and glycerine, respectively.
VE - volume of
liquid phase of erythrocyte cytoplasm.
Vmax - maximal amplitude of volume
change of erythrocytes, preliminarily loaded with glycerine, during suspending
them in isotonic medium.
c - maximal
value of relative volume changes in erythrocytes loaded with glycerine
during their suspension in isotonic medium.
c CR.
- critical for the cell fraction value of relative change of
volumes.
zHb - hemoglobin charge.
QHb - overall amount
of hemoglobin in erythrocytes.
pI - hemoglobin isoelectric point.
pn(S,q) and pn(V,q)
- distributions reflecting dependence of S
and V, corresponding to maximal amplitudes
of distribution function, on the degree of spherulation.
Vm(q) and Sm(q)
- mean, for euch separate q, values of volume and PBE surface area.
t n
- normalized chronologic age of erythrocytes for their maximal
survival.
nv=dV/dt
n and ns=dS/dt
n - velocities of change in PBE volume and surface area.
Taking insight into the laws governing the generation, development and
formation of a pathologic process, demanded the formation of a new integrative
biology and medicine based on systems approach to considering the principles
of structural and functional organization of living organisms [1-4]. In
the approach in question attention is accentuated on regularities of coordination
of separate links of biological objects, revealing new integrative properties
of the object as an entity. Such a consideration necessitated recomprehension
of all methodologies for the estimation of physiological functions of the
body and it gives impetus to the creation of a new research base for its
realization in practice.
The red blood system (RBS) (the aggregate of structures regulating
oxygen volume in peripheral blood) is a constituent link of the cardio-vascular
system subserving the utmost important function of the body i.e. oxygen
supply to metabolic processes. As displayed by our investigations [5-8]
the state of this system, apart from diagnostics and prognosis of hematological
diseases, may serve also as an effective index for adaptational possibilities
of the organism as a whole. Resolution of this problem is associated also
with the early detection of potentially dangerous homeostatic deviations.
This is interlaced with the resolution of the problem of general resistance
of the organism to unfavourable influence of environment, including social
welfare, as well as the problem of estimation of tension of ecological
situation, quality of life, etc [9].
Methodological Points Of Rbs Study
The way taken from general conceptual views to concrete means of research
may be divided into three stages: 1) Detection of optimal, from the point
of view of practical realization, criteria for the estimation of RBS functional
state, 2) Detection of the characteristics most adequately characterizing
RBS, 3) Resolution of methodological questions of laboratory determination
of the chosen characteristics.
As a first step let us fancy the system as a black box (Fig.1) whose
target function is to render minimal the difference between required and
really available oxygen capacity in circulating blood- D
NHb(t)=NaHb(t)-
NrHb(t) . (1),
 |
As a criterion for estimating the state of a given or any other system
serves the efficiency of realization of its main target function [10,11].
In the most general form, efficiency is the function of the final result
of realization of target functions, expenditure of material resources and
time necessary for its realization. Conformably to RBS this may be expressed
in the following way:
 |
In this equation D NcHb
is the final effect of RBS response to a single disturbance,
x is the time required
for a full relaxation of RBS, m is
the outlay of energy and plastic materials spent by the system for the
minimization of . D NcHb..
in the relaxation period. It should be noted that at RBS perturbances,
when minimization occurs at the expense of mobilization of the deposited
blood, x is more than the time of minimization
D NcHb,
while in anemias it may tend to infinity. The forms of functional dependences
of Y1 ,Y2 and
Y3 are determined by the
fact as to which of the three factors are of priority in this particular
case, i.e. reflects the essence of RBS, as a subsystem of a larger system,
namely of the body as a whole [2].
Such a comprehension of efficiency appears to be most adequate. However,
for its practical application it is necessary to create special conditions
that are realizable only under conditions of an experiment. More expedient
from the point of view of practical application, and at the same time no
less adequate seems to be the expression of efficiency via mean for a certain
interval of time values of deficit in the blood oxygen capacity, expenditure
of energy and plastic material.
 |
The higher efficiency of RBS operation, at similar background loadings, the less is the duration of relaxation period. Although the peak expenditure of energy and plastic material at an effective functioning of the system may be also high, ` D NHb and ` m will be low because of the shortness of the entire length of relaxation processes. From the position of the given criterion, RBS functioning condition can be classified minimum as three conditions:
1) Physiological norm, when ` D
NHb=0 and
average expenditure of plastic material is within norm.
2) Heavy duty, when ` D
NHb = 0
, but ` m
is over norm.
3) Pathologic condition, when ` D
NHb > 0
and ` m
exceeds norm.
From the position of the aforesaid experimentally defined RBS characteristics
must meet the following conditions: 1) Allow to qualitatively estimate
the mean deficit of blood oxygen capacity and average expenditure of plastic
material, 2) Allow to identify as far as possible the reasons for the alteration
of the character of RBS functioning. In order to determine them, let us
decompose the system into its constituent elements and reveal the most
important relations between them. Of the values characterizing these relations
one can choose the optimal ones for the given criterion of RBS characteristics.
The blood oxygen capacity is known to be determined by overall amount
of hemoglobin in the vascular bed. Hemoglobin enters into circulation in
the form of micropacks - PBE. In the process of circulation the amount
of functionally active hemoglobin in the cells gets reduced nHb=
nHb(t ) [12-14]. Impaired
is also the quality of packs - W
=W (t ) [15-22]
due to which probability of their elimination increases E=E[W
(t )] (cell elimination is assumed
to be, on an average, a determined process). From this position, W
can be considered as a certain property or a group of properties of erythrocytes
alteration of which during survival period can cause their death, i.e.
is the biological age of erythrocytes. In its turn, as a functional form
of dependences nHb= nHb(t
) and W =W
(t ), as well as their boundary values
may, to a considerable extent, alter at the variation of the character
of erythropoiesis [23,24]. A definite dispersion in these characteristics
is observed at physiological conditions of functioning of the hemopoietic
apparatus. Consequently it is necessary to introduce the parameter that
characterizes the difference in properties of cells conditioned by cumulation
of all factors at different stages of hemopoiesis (g),
i.e. nHb=nHb(g,t
) and W =W
(g,t ). These functions describe
causally interdependent processes and there must be, for rare exception,
a strong correlation between them. Taking into account that t
for a concrete erythrocyte, in the absence of external disturbances, is
unequivocally determined through g
and W , it can be assumed in the first approximation
that function nHb=nHb(g,W
) is not time dependent and is determined by the constitution
of a concrete organism. From the position of the foreging total amount
of circulating hemoglobin may be expressed in the following relation:
 |
where function p(g,W
,t) reflects the ratio between erythrocyte amount and various
values of W and g
in circulating blood (PBE population spectrum) and is determined by the
relation of processes production-destruction-elimination, i.e. characterizes
the flow of plastic material through the RBS. Substituting (4) in (1) and
deriving averaging in time we get:
 |
Time averaged population spectrum enters in the under integral expression.
Consequently condition:
 |
may be realized by various functions ` p(g,W
). Bearing in mind that the processes of production-destruction-elimination
are interdependent and are optimized by the system taking into account
the demands and possibilities of the organism, one may expect that under
normal physiological conditions and in the absence of functional loadings
the averaged population composition of PBE must represent quite a definite
functional dependence. From this it directly follows that deviation of`
p(g,W ) from
the physiological norm even when condition (5) is observed would testify
to the tension of RBS functioning. Summing up the above stated, `
p(g,W )
may be considered, with good reason, as an integral characteristic of the
RBS functional state. The structures that are directly implicated in the
regulation of PBE population composition belong to the following level
of RBS organization ierarchy. Pursuant to the presently available common
views, the dynamics of PBE population composition is described as the following
equation:
 |
The given expression is a modified equation of continuity, where v=v(W
,g) is the velocity of aging of the cells biological age W
and with initial properties of g, D(W
,g) - characterize the flow of cells between the depot and circulation,
E(W ,g) - is
the probability of elimination of cells of biological age W
(g,t ),
while I(g,t) is the intensity of erythropoiesis.
Consequently, subelements of the RBS at the second level of decomposition
are the hemopoietic system, elimination mechanisms, the system regulating
the cell flows between the blood depot and circulation, as well as whole
cumulation of factors of internal medium determining the intensity of cell
aging.
Considering the processes of RBS regulation, on an average as stationary,
averaging the equation (6) in respect to t and accepting that the mean
value of cell flow between the blood depot and circulation under conditions
of physiological norm and rest is near to zero (the given condition is
introduced for the simplification of further statement and is not an obligatory
condition) we obtain:
 |
solution of the given equation in relation to `
p(g,W ) looks
like:
 |
Determination of values I(g) and
` E(g,W
) via ` p(g,W
) in a general case is an unsoluble task. However, taking into
account that these functions make a varying contribution to different areas
of determination of function ` p(g,W
), in the framework of certain model representations and reasonable
simplifications, analysis of population spectra offers potential possibilities
of identifying with a sufficient degree of probability the reasons for
any changes in PBE population composition. From the position of the aforestated
the averaged population spectra of PBE can be considered with certainty
as an optimal characteristic for efficiency of RBS functioning.
Error in estimating the RBS functional state with this approach is
determined by the degree of variability of PBE population spectra [`
p(g,W )-p(g,W ,t)
] in normal and pathological conditions of PBE functioning.
Therefore finding of criteria for the applicability of this approximation
is in need of special investigations.
Another no less important problem is a correct determination of population
composition, that is primarily associated with resolving of the question
of determination of minimally sufficient set of parameters, which characterize
adequately the PBE physiological state. The present conventional methods
for PBE analysis of variance (methods of osmotic, acidic, immune, quantitative,
etc erythrograms) differ by the choice of properties considered as the
optimal indices of the physiological state of erythrocytes. In the approach
proposed by us as a set of parameters characterizing PBE are applied: cell
volume (V) and their spherulation degree
(q), determined as the relation of
cell volume to the volume of sphere with the same surface area.
 |
The first of these parameters characterizes the initial dispersion in
the erythrocyte properties during their entry into the circulation bed,
the other is the deformability of cells and is considered as the biological
age.
Aging of erythrocytes is a compound process and as a matter of fact
it involves all the structural and functional aspects of vital activity
of cells. The most firmly established manifestation of aging process is
the increase of cell density. It is namely this fact that is used in the
majority of cases for the fractionation of erythrocytes according to their
age. The most likely explanation of this event is the revealed fact of
cell dehydration in the process of aging. In parallel a decrease is observed
also in the cell surface area (fragmentation), however the relation of
the cell volume to its surface area is increased, i.e. the erythrocytes
are spherulated. Consequently the degree of cell spherulation, or more
correctly the amount of liquid per unit of the cell surface area can be
considered with certainty in the capacity of their biological age.
It has long been known that the sizes of cells are formed at various
stages of hemopoiesis. It has been established that there exists a strong
correlation between the intensity of erythropoiesis and the sizes of cells
entering into the bed [23-25]. Dependence of the type of erythrocyte distribution
according to their sizes (the Praise - Johns curves) on the character of
hemopoiesis are dealt with in a huge body of literature and they are cited
in hematological hand-books. Therefore further argumentation seems to be
extra. We shall only note here that in clinical practice mean diameter
of erythrocytes alongside with colour index of blood has been widely used
as a classification criterion of various pathologies of blood system. From
this position the spherulation degree and volume of cells can be considered
with certainty as a minimally sufficient set of data required for an adequate
characteristics of erythrocyte quality.
As been mentioned above, error in estimating the RBS functional state
in the given approach is determined by the rate of variability of PBE population
spectra. At the same time it is known that the cell volume and consequently
their degree of spherulation as well, under the changeable conditions of
internal medium of the body, undergo cyclic fluctuations which are superimposed
on the irreversible, chronological age depending changes of spherulation
degree. Therefore it seems reasonable to consider the factors on which
the cell volume depends in the fixed external conditions, i.e. when they
are suspended in a standardized medium under experimental conditions.
As is known, the erythrocyte may be considered as the Donnan system.
In the state of equilibrium such a system is described by a system of equations
[26]:
 |
The first of these equations reflects the fact of osmotic equilibrium
between the intra- and extra-cellular medium, the second, electroneutrality
of the intracellular medium, the third, electrochemical balance of permeant
anions, the fourth, dependence of common charge of impermeant anion on
pH of internal medium. Solving of the
equation system in respct to V is the
function of total concentration of cations and anions of suspending medium
(Co) and its pHo,
of the intracellular concentration of cations (Cict),
amount of hemoglobin (QHb ) and
its electric properties (a, pI).
 |
(the formula in its explicit mode is not presented because of its bulkiness).
At fixed values of Co.
and pHo, created under experimental
conditions, the cell volume is determined by the concentration of intracellular
cations, as well as by amount and electric properties of hemoglobin. Consequently,
the possible variation of population spectra may be associated with the
variation of namely these properties of PBE.
Test On Reversible Osmotic Spherulation Of Erythrocytes
During suspension of erythrocytes, preliminarily loaded with any neutral,
readily permeable through the membrane substance B
(e.g. glycerine), in isotonic medium, dynamics of changes in their volumes
will be determined mainly by the values of water and glycerine fluxes through
the cell membrane. Since the erythrocyte membrane does not practically
pose an obstacle for water, cells would first swell and then later on,
with the outflux of glycerine, would start regaining their original volume.
Maximal amplitude of volume change of erythrocytes would be dependent upon
the initial concentration diffirence between intra- (Ci
) and extracellular medium (Co
) and upon the ratio of permeability coefficients of erythrocyte
membranes for water and glycerine
 |
where
 |
NE and NB.
are respectively amounts of intracellular solutes and glycerine, while
VE is the volume
of liquid phase of cytoplasm, practically equal to that of erythrocyte.
Let us assume that PB<
< PH2O. Then the
initial equilibrium of osmotic pressure between the cell and suspending
medium ( D C=0 ) would
be imposed only by the water influx:
 |
Considering that NE /VE
= CIZ, from (12) after simple transformations
we get:
 |
i.e. maximal value of a relative change in the volumes of erythrocytes
loaded with glycerine, during their suspension in isotonic medium, for
all population of PBE constant value and is determined by the concentration
of loading (CB). In case
for the definite fraction of erythrocytes it exceeds the critical value
(c CR.),
the given fraction is hemolysed. The other cells gradually,
as glycerine is released from the cells, regain their initial volume. Bearing
in mind that the membrane of erythrocytes is instable to stretch deformation,
and that for the erythrocyte with the volume VE
and surface area SE, the
maximally admissible value of volume is equal to that of a sphere with
surface area equal to the area of erythrocyte, then for c
CR we obtain:
 |
Consequently: for the erythrocyte with the volume VE
and the surface area SE the
value c CR
is a fixed one and shows as to how many times the cell should increase
to turn into a sphere. It can be readily demonstrated that between c
CR and the cell spherulation
degree there is a mutually unequivocal conformity:
 |
If PI(V) and PI+1(V)
distribution according to the volumes of nonhemolysed fractions
of PBE, loaded with glycerine respectively in concentrations CIB
and CI+1B
(CIB >CI+1B),
when D PI(V)
= PI+1(V) - PI(V) according to (13)
and (14) will be distribution by volumes of fractions of PBE possessing
values c CR
in the interval:
 |
Choosing the value {CIB}
and experimentally determining the suitable ones {PI(V)},
by means of simple subtractions, it is possible to determine
histograms of PBE distributions according to their volumes and critical
values of relative changes of volume - P=P(c
CR,V). From these distributions by means of substituting
the system of coordinates (c
CR,V)Þ (q,V) one
can obtain the PBE distribution according to volumes and degree of spherulation.
 |
Really, if PB. and PH20 values are of one order, c may depend also on the ratio PB /PH2O. As is known in erythrocytes water transport is strongly coupled with anion transport, consequently value c , through the Jacobs - Stewart system [26] may be influenced also by the ratio of concentrations of intra- and extracellular permeant anions and pH. However, these parameters would have considerable significance, if PBE possessed considerable dispersion in relation to them. Otherwise they would not affect the character of PBE distribution. Estimation of the effect of these factors on c will be made below.
Preparation Methods And Equipment Base
Blood drawn from a finger in the amount 15 m
l is diluted in 1ml
solution of the following content: NaCl - 0,150 mM/
l, EDTA-Na2 - 0.03 mM/L,
5% formalin - 5 ml/l, glycerine
- 1,8 M/l, "HEPES" - 10 mM/l,
pH=7. Fractionated hemolysis of erythrocytes is performed after 15-20 min,
by means of introduction of 50m l.
glycerine loaded erythrocyte suspension into 1 ml samples 1,3; 1,5; ...
1,9 M/l solution of NaCl, ethanol -
4%, "HEPES" - 10 mM//l, pH=7,3. After
3-5 min 100m l
suspension from each separate sample is diluted in 5 ml
basic solution.
Measurement of distribution according to the volumes of suspended particles
(erythrocytes and erythrocyte ghosts) in various samples is made 10-15
min after full restoration of the volume in nonhemolysed erythrocytes.
Primary information is obtained in the form of histograms of distribution
of concentration of erythrocytes and ghosts of erythrocytes (Fig.2).
 |
Fig.2 Distribution of erythrocytes and erythrocyte ghosts according
to volumes, corresponding to different concentration of glycerine loading
prior to (dots) and after primary treatment (solid line).
|
Primary treatment involves filtration of areas of noised erythrocyte
ghosts, curve smoothing and transformation of coordinates
 |
Experiments are carried out on a systems engineering complex of original design [6] based on the conductometric method for the measurement of sizes of particles dispersed in the electrolyte. The system is realized on the base of conductometric counter "PICOSKEL"-4 (Medicor, Hungary) and DEC-compatible computers of Russian production DBK-4. Automatic measurement of amplitudes of output impulses and data input in computer's memory are accomplished by means of specialized A/D converters realized in the form of printed circuit board . More detailed description of hardware complex is given in [6].
Principle Of Interpretation Of Population Spectra
Using the method of comparative analysis of normal PBE population spectra,
at various conditions of RBS fuctioning, taking into account species specificity,
one can ascertain the principles of interpretation of PBE population spectra,
separate a set of parameters, the most informative from the point of view
of estimation of RBS functional state. Fig. 3.1 - 3.3 presents a typical,
for 25-45 year-old practically healthy men, dynamics of PS in the absence
of functional loadings and (Fig. 4) calculated on their basis averaged
(stationary) for the period of observation population spectra of PBE
 |
 |
 |
|
population spectra (a - PBE distribution according to the volume and degree of spherulation, b - PBE distribution according to surface area and degree of spherulation ) in the absence of functional loadings. Fig.3,1 - the first, 3,2 - ninth, 3,3 - twentieth day of observation. Axis of obscissa: degree of spherulation in relative units. Axis of ordinate: - (a) volume [m 3], (b) surface area [m 2]. Axis of Z - standardized values of PBE concentration. |
.
 |
|
Axis of obsissa: degree of spherulation in relative units. Axis of ordinate: - (a) volume [m 3], (b) surface area [m 2]. Axis of Z - standardized values of PBE concentration. |
Primarily one ought to make sure how much adequately the offered test can determine the degree of erythrocyte spherulation, i.e. ascertain whether c depends only on the amount of intracellular liquid per unit area of cell surface, or PB /PH2O is also determined. At this point let us confine to the solution of this question in an indirect way, determining from the averaged population spectra mean values of volume for PBE populations and degree of erythrocyte spherulation. Through the expression (9) one can compute averaged value of surface area of PBE Sm=140m k 2. Comparison of the result obtained by us with the data reported [27-28], according to which mean surface area of human erythrocytes is within the ranges 134 - 147m k 2 , gives support to the conclusion thet erythrocyte testing by the degree of spherulation in good approximations may be considered as adequate.
Areas of minimal and maximal values of spherulation degree correspond
to the distribution of the least and most deforming erythrocytes and consequently
young and old fractions. One can be convinced in this even by visual observation
on mixing separate PBE fractions with population spectra of rats irradiated
in minimal lethal dose (Fig. 12).
 |
 |
 |
|
|
||
As is known, in physiological conditions daily one percent of erythrocytes
perish
 |
Consequently the contribution of function E(V,q)
to p(V,q) with up to 1% accuracy can
be ignored. In this case expression (8) acquires a simple appearance linking
the population spectrum with the input flow of erythrocytes and their spherulation
rate:
 |
Distribution function (Fig.4) is a convex surface acquiring a maximal
value in area q=0.52-0.62 and V=75-80
m k 3.
From this it emerges that PBE spherulation rate acquires maximal
value in the boundary areas of distribution functions, i.e. it is maximal
in the young and old PBE fractions.
In terms of expression (9) dynamics of change in PBE spherulation degree
is due to the dynamics of changes in volumes and surface area. Therefore
it seems expedient to clarify general laws governing the changes of these
values during survival of cells. Since S
is unequivocaly determined through V
and q, then from p(V,q)
by means of simple transformation of coordinates V,qÞ
Sq one can obtain PBE distribution according to S
and q:
 |
General tendency to the dependence of S
and V on q
is clearly traced in the distributions reflecting q-dependence of
values S and V,
corresponding to maximal amplitudes of distribution function (Fig. 5)
 |
 |
| Fig.5. Explanations see in the text. |
Red colours in Fig. 5 correspond to maximal values of distribution functions
for respective q. From graphs it is
clearly seen that increase of q in
the original distribution area largely depends on the volume increase.
Beginning with q=.53 change of the
degree of spherulation depends also on S.
Let us compute mean value of volume and surface area for each separate
q
 |
Functions Vm(q) and Sm(q)
in stationary approximations may be considered as the functions reflecting
dynamics of erythrocyte volume and surface area change as dependent upon
their biological age (Fig. 6).
 |
Fig.6. Dependence of mean values of surface area Sm=Sm(q) and volume Vm=Vm(q) PBE fraction from their degree of spherulation . |
From these functions one can readily obtain the functions of dependence
of S and V
already upon their chronological age: integrating p(V,q)
into V for a fixed q,
from two-dimensional population spectra let us go over one-dimensional,
reflecting dependence of PBE quantity only on the spherulation degree:
 |
In conformity with (18) and (19)
 |
From (20) dependence between the chronological age and q can be calculated
by the formula:
 |
Determination of the dependence of chronological age on the spherulation
degree and thereby also maximal survival period, on the basis of expression
(21) is possible only with the accuracy up to the constant multiplier I,
which in the framework of the given methodological approach cannot be defined
directly. Consequently, determination of function t
=t (q) in
its absolute values is possible only in physiological conditions, when
it is known that input stream of erythrocytes makes up approximately 1%
of total PBE amount. Therefore, for further analysis more convenient is
the operation of fixed for maximal erythrocyte survival function (t
/T ), which
allows to characterize dynamics of S
and V in the parts of PBE maximal survival:
 |
By substitution of coordinates in functions Vm(q)
and Sm(q), Vm(q)
Þ Vm[q(t n)]
Þ Vm(t n)
and Sm(q) Þ
Sm[q(t n)] Þ
Sm(t n) we obtain function
of volume and surface area dependence upon the fixed chronological age
t n,
while their differentiation by t n
yields the function of V and S
change of velocity: n
v=dV/dt
n , n s=dS/dt
n , (Fig.7).
 |
|
|
As is evident from the graphs (Figs.6-7) in physiological condition
of RBS functioning a young fraction of erythrocytes entering the bed has
a tendency to volume increase - swelling. They are distinguished by their
original high swelling velocity and relatively low velocity of fragmentation.
With t n
increase, the velocity of fragmentation sharply increases, while
that of swelling sharply falls. In this period spherulation rate -
n (t n)
rapidly attenuates
 |
Consequently, spherulation rate in the initial period of life cycle
is determined by velocity of swelling. Beginning with a definite moment
(point A) the velocity of fragmentation although keeps increasing,
occurs more slowly. It is paralleled by a decrease in the swelling velocity
variation and at point C it crosses the coordinate axis. At this
point the change of spherulation rate is determined only by velocity of
fragmentation. Beginning with this point the swelling velocity dV/dt
n alters a sign, the cell volume starts attenuating
and increase in the spherulation degree due to a decrease of S
is compensated for, to a certain extent, by a decrease of V.
Consequently, further attenuation of spherulation rate depends namely on
this factor. From point B, velocity of fragmentation starts increasing
sharply. During a definite period (fragment B-D) there is a parallel
increase in the velocity of volume decrease
dehydration), due to which n q(t
n) is maintained on a low level (compensatory period),
however, later on, velocity of fragmentation starts to dominate considerably
over the velocity of cell dehydration and the spherulation rate starts
to sharply increase - the cells enter the last phase of their existence.
In the light of our findings in physiological conditions the life cyle
of erythrocyte can be divided into four periods: 1- phase of a sharp increase
of fragmentation velocity and an acute attenuation of swelling velocity,
2-a phase of relative stabilization of fragmentation velocity and inversion
of velocity of volume change, 3-a phase of compensation and 4-a preelimination
phase.
Dynamics of fragmentation and volume change under conditions of circulation
are strictly correlated both in time and magnitude. In the periods of erythrocytes
survival, when velocity of fragmentation changes with acceleration, there
is an accelerated change also in dV(t
n)/dt n, i.e.
dynamics of these processes is determined by one and the same generalized
force. Consequently one may affirm that they are different manifestations
of one and the same process.
An analogous picture is observed in the conditions of RBS functioning
corresponding to various pathological states of the body (Figs.8-11).
 |
Fig.8. PBE distribution according to the
volume and degree of spherulation (8,a), PBE distribution according to
surface area and degree of spherulation (8.b), functions of dependence
of mean value of surface area Sm=Sm(q) and volume Vm=Vm(q) in a PBE fraction
on their degree of spherulation (8,c) and (8,d) function of dependences
of spherulation rate ( ), velocity of surface
area (O) and volume (Ñ ) alteration on
chronological age of erythrocytes, normalized on their maximal survival
time (t /T).
-Patient -a 68-year old woman -Diagnosis - planocellular cancer of thyroid gland, 3 rd stage, anemia. |
|
|
Fig.9. PBE distribution according to the
volume and degree of spherulation (9,a), PBE distribution according to
surface area and degree of spherulation (9.b), functions of dependence
of mean value of surface area Sm=Sm(q) and volume Vm=Vm(q) in a PBE fraction
on their degree of spherulation (9,c) and (9,d) function of dependences
of spherulation rate ( ), velocity of surface
area (O) and volume (Ñ ) alteration on
chronological age of erythrocytes, normalized on their maximal survival
time (t /T).
- Patient - a 25 year old man - Diagnosis - essential hypertension |
|
|
Fig.10. PBE distribution according to the
volume and degree of spherulation (10,a), PBE distribution according to
surface area and degree of spherulation (10.b).
Patient - a 65-year old man Diagnosis - bronchial asthma. |
|
|
Fig.11. PBE distribution according to the
volume and degree of spherulation (11,1), PBE distribution according to
surface area and degree of spherulation (11.2).
- Patient - a 25 year old man - Diagnosis - nasopharyngeal cancer, 3B stage, anemia |
At the same time the higher the original PBE swelling velocity (Fig.9),
the higher is the ultimate value q
with which the preelimination phase begins. From this it is automatically
obtained, that the spherulation degree (deformability) is not the basic
property change of which causally determines the cell elimination, since
between it and the PBE elimination there exists mutually unequivocal correspondence.
Though the degree of cell fragmentability in the majority of cases does
reflect more adequately the PBE functional state, it, however, cannot be
considered as the basic reason for cell elimination. This can be testified
at least by the fact that a dramatic shortening of PBE survival time in
irradiated rats is bound with the rise of spherulation rate on the background
of a decrease in fragmentability degree.
In this context it should be noted that on (Sq)
population spectra in the areas corresponding to macrocytes one can observe
small protrusions practically parallel to the axis q (Figs.3,1-3,3). These
protrusions may be identified with the erythrocyte fractions whose velocity
of volume change, for some reasons, begin to acutely predominate over that
of fragmentation. They get rapidly spherulated and disappear from the vascular
bed. These fractions are observed in all age groups and, as assumed, may
be considered as the cells with decreased vital potential - as stochastically
perishing cells. In support of this assumption is the fact that in rats
distinguished by a high per cent of PBE stochastic death the existence
of protrusions is more pronounced than in man (Fig.12).
Summing up the afore-stated it emerges that the dynamics of evolution
of spherulation rate is determined by: 1) the factor regulating velocity
of fragmentation and PBE volume change, as well as, 2) the factor determining
the original swelling velocity. As far as the PBE survival time is concerned,
it is determined neither by the velocity of fragmentation nor the dynamics
of volume change separately, but by their balance. Any disturbance of the
balance between these processes would lead to the shortening of PBE survivability.
Concrete definition of the intracellular mechanisms maintaining the balance
between these processes, on the strength of presently available factual
material, does not seem plausible. Howerer, on their basis, already at
this point of development of the method, one can obtain a number of criteria
for the evaluation of the RBS functional state by population spectra.
1. Criteria characterizing variation of PBE survival time
Variation of PBE maximal survival time, whether it is induced by an increase in the original swelling rate, or is associated with the acute acceleration of fragmentation rate, or is due to both factors together, should be expressed in characteristic deviations of dq/dt n. This would be reflected in PBE population spectra, too. In particular: representing the velocity of volume change in the form of two components depending and not depending on the fragmentation rate and substitute in (22) we obtain:
 |
f(dS/dt n) an increasing function from dS/dt n. Integrating (23) from 0 up to t n
 |
According to (24) and taking into account that area of an erythrocyte
can only decrease, for one and the same chronological age in various situations,
it is obtained:
1. At the increase of original swelling velocity the value of the second
integral would be increased. Consequently would increase the spherulation
rate too. The value of the volume exceeding, but the value of surface area
being equal or less than the normal one (Fig.9).
2. At increused fragmentation rates although the value of the first
integral increases, however large negative contribution of the second integral
will result in a decrease of the value of spherulation rate. At this time,
the values of volume and surface area would be below the normal (Fig. 8).
3. At the concomitant increase of the original swelling velocity and
fragmentation rates members in the second integral would be mutually compensated
for and the increase in spherulation rate would be determined only by the
contribution of the first integral. At the same time taking into account
that at intensive fragmentation its velocity sharply increases and emerges
from the plateau - one should expect smearing of population spectra in
the area of its definition. Volume deviation from the original value would
be insignificant, while change of surface area would be maximal (Fig.10).
Proceeding from the foregoing it becomes possible by mere visual analysis
to fix with high degree of probability the decrease of PBE survival time
in relation to physiological norm and identify reason for its occurrence.
Rise in the rate of PBE stochastic death, that is observable in some
pathologies, would be reflected in population spectra by an increase of
amount and sizes of protrusions in the area of high S.
2. Criteria characterizing erythropoiesis
A classical method for evaluating the state of erythropoiesis is the
PBE characteristics according to size and colour indices. Population spectra
allows to determine the distributions by volumes and surface area in young
fractions of erythrocytes. It has been indicated above that the cell volume
alongside with the total cation concentration is determined by the concentration
and electric properties of hemoglobin. This makes it possible to estimate
the erythropoietic function of the body. In particular if Vmn
and qmn
are mean values of volume and spherulation degree in a young fraction of
erythrocytes at normal erythropoiesis, then Vm³
Vmn and qm>qmn
testify with a high probability to a decrease in hemoglobin
amount per a single cell, i.e. indicates a hypochromic character of erythropoiesis
(fig.9, fig.11). If atVm>
Vmn and qm
=qmn there is a normochromic macrocytosis
(Fig.8), while at Vm<
Vmn and qm
<qmn
there is hyperchromic microcytosis (Fig. 10).
To the parameters which characterize erythropoiesis belong both the
initial swelling and fragmentation rates. Yet the diagnostic and prognostic
values of these parameters are in need of further studies.
The value being the function of PBE amount, mean values of their sizes
and spherulation degree may serve as characteristics of the blood rheological
properties. Definition of a concrete type of functional dependence is also
in need of further investigation.
The indispensable condition for a correct determination of above-said
characteristics is a steady-state of RBS functioning. In order to clarify
in which approximation one can consider the population spectrum, determined
at an arbitrary moment of time, as a stationary value, let us analyse variation
of mean population value of spherulation rate (qm)
for the observation period in the absence of functional loading in a 45
year-old man. Computed from the population spectra mean value of spherulation
rate is equal to qm =0,57,
while variation coefficient CVq=3%.
Taking into account how many times qm
may change at the change of the whole system's
functioning condition, the population spectrum, determined for the observation
period under normal physiological conditions, can be reasonably considered
as the stationary distribution. This is vividly seen at the comparison
of Fig. 3,1-3,3 with PBE distribution during various pathologies (Figs.8-11).
It should be noted that the presented distributions, although might be
nosologically specific, demonstrate clearly the resolution capacity of
the method. Consequently, the RBS functioning condition in the lack of
physical and psycho-emotional loadings can be considered, in good approximation,
as stationary.
Direct extrapolation of this inference to pathological states may seem
not quite correct. However, lack of characteristic for nonstationary conditions
abrupt change of amplitudes of population spectra, as well as of functions
V(q) and S(q)
(Fig.8) in the basic area of their determination, indicates the validity
of this assumption.
In nonstationary (transient) conditions of the RBS functioning the
analysis of dynamics of variation of population spectra with the use of
equation (6) allows to evaluate the RBS main characteristics. The main
principles of population spectrum analysis in transient conditions of RBS
functioning have been described in [8], we shall only emphasize here that
the presented dynamics of population spectra variations in irradiated rats
demonstrates the possibility of practically full visualization of a PBE
aging process.
Conclusion
Though final evaluation of the degree of adequacy of the proposed criteria
will become possible after performing parallel experiments with the application
of already approbated method for RBS study, high informative value of the
proposed methodical approach to resolving a number of fundamental, as well
as applied tasks of contemporary biology and medicine is evident. It can
be successfully applied for the detection of intracellular mechanisms which
maintain the balance between the dynamics of fragmentation and cell volume
change in the conditions of circulation. In the light of our findings these
mechanisms are of primary importance in regulating the PBE survival duration.
Perspective of evaluating the efficiency of medicamentose correction of
the disturbance also emerges. Another, no less important direction of research
is the detection of regularites in PS evolution during changes in the character
of functioning of its subsystems. In any system, as in RBS alteration of
the character of functioning of one subsystem leads to the alteration of
the character of functioning of other subsystems. Therefore detection of
concurrence between the initial and final (stationary) type of population
spectra change is the necessary condition for a correct identification
of the reasons for bringing about change of RBS functioning condition.
On the other hand, manifestation of the laws governing the PS evolution
at various initial perturbances, will help to deepen our understanding
of delicate processes of RBS regulation, estimate the significance and
predict the final outcome of one or another change in RBS, due to both
external and internal factors.
Bearing in mind the comparative facility of the method, possibility
of virtually full automatization of investigations, negligible material
labour expenditures which permit for carrying out mass examinations, there
are all grounds to suggest that the method can be successfully applied
for resolving quite a number of practical tasks of clinical and sports
medicine, medical ecology, etc.
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