Thermodynamic constraints on leukocyte kinetics in the blood

BioeZectroclretnkby and Bioenergetics3, 1-14 (x976)

Thermodynamic in the Blood

Constraints on Leukocyte Kinetics

by G. B_-ucIL_+TI~,A. CHIABRERA~ 0 Department of Electrical 16145 Genoa, Italy

Engineering,

:

and S. RIDELLAOO

University

00 Electronic Circuit Laboratory, CNR, Via all’opera Manuscript received dctober 10th 1975

of Genoa,

Viale F. Causa 13,

Pia B, 16145

Genoa,

Italy

The OXSAGER principle can be used as a thermodynamic test of the physical .reliability of biological models. Furthermore, an’ electrochemical constraint exists among the constant parameters of a model. These methods are applicable to the peripheral system of the leukocytes The results obtained regarding the blood granulocyte compartments in human beings are extensively discussed_ Introduction The physical modelling of biological systems, which describes the biological processes by means of a set of mathematical relationships is a technique extensively used today. The model relies quite often on rough simplifying hypotheses concernin,= the phenomena involved, while neglecting many physical variables. The purpose is to achieve a number of equations and variables which are easy to handle, and to model those biophysical processes which are scarcely known by means of reasonable phenomenologi’cal parameters. Their values are evaluated, for instance, by fitting the theoretical results into .the available experimental data. The purpose of this paper is to find thermodynamic or ehxtrochem-ical criteria which will allow us to evaluate the physical_ retiabzlity of biological models, Le., the degree of confidence in the adherence of models to the physical world, or to find physical comtmivtts among those parameters which are unknown. Our approach will- be described through its application to a model of the blood granulocyte system in- human beings. .: The Mood leukocyte system White cells are produced by the stem cells of the bone marrow. After attaining their mature state, they reach the external side of some bone marrow capillary vessels and form the reserZre pool. The leukocytes

2

-Barilati,

Chiabrera

and

Ridella

which enter the blood vessels flow through the circulatory system, where they form the circzrdati?zg pool. iMany of them adhere to the internal side of blood vessels, or accumulate in the spleen, the Ijver, the lungs, etc., and form the margisated fiooZ_ The marginated leukocytes can either return to the circulating pool or reach the tissues through the vessel walls. A complex humoral mechanism provides the regulation of the production and reserve pqols with the necessary fe’ed-backs (Fig. I). The circulating and marginated compartments form ‘the peripheral leukocyte system, Tvheie it is easy to perform measurements, especially in human beings, which can be repeated frequently.

-~~~_-~~__~~~_~, Marginated

I

pa01

1pkq L _--------

j -i

Fig. I. Model of the white blood cell regulatory system.

The experimental data, which yield information about the peripheral and the whole systems, can be interpreted correctly provided that an accurate model of the peripheral pools is given. Among the six different types of leukocytes classified, the granulocytes are the most numerous ones so that, from now on, we shall deal kvith them only as the most important ones. The purpose of this paper is to test the consistency of a new model of the granulocyte peripheral system (Fig. 2) with some thermodynamic laws. In particular, we want to achieve the foIIo\ving goals :. to verify whether the proposed model is consistent with the ONSAGER principle ; 2. to find electrochemical. constraints among the phenomenological parameters used in the model. I.

Thermodynamic

Constraints

on Leukocyte

Kinetics

in Blood

3

R ~-----_-_-_-_ l4umaral

Feed-back

-_----Fig.

2.

Model of the peripheral granulocyte system (C ru;d Al pools).

The point of view which will be adopted in the following sections is worth describing. Both the influx of granulocytes into the blood or into the tissues and the margination process depend on electrochemical forces which are not yet completely known Therefore, the constitutive

relationships

of the peripheral granulocyte

system cannot be computed

a rigorous physical basis, and a few phenomenological parameters must be introduced into the model. Nevertheless, the large number of particles involved allow us to perform time averages, in which the microscopic fluctuations have been smoothed out, so that the thermodyuamits of irreversible processes can be applied. The thermodynamic analysis of the peripheral pools is performed for non-equilibrium states, and is restricted to first-order deviations from equilibrium. Such a simplified approach does not avoid some difficulties which arise from the need of defining a thermodynamic equilibrium for biological systems. The usual state of normal subjects is the steady state, which exists when the time .derivatives of forces and fluxes are about zero. Under steady-state conditions, fluses are about time-independent, but are different from zero, so that one has to deal with systems which may be very far from equilibrium. Therefore, it is not possible to obtam a thermodynamic equilibrium of the granulocyte peripheral system Gz viva, so that the meaning of small deviations from it is not clear. These difficulties are overcome if one agrees with the following statements. First of all, we treat only phenomena lasting a limited ‘length of time, and we do not pay any attention to the time evolution of the individual.. We suppose to freeze all biological processes inside each granulocyte, for a lapse of time which must be long enough to allow us to perform the microscopic averages and macroscopic measurements on the dynamic behaviour of the granulocytes in each pool, but short enough with respect to their life. Thus, the proposed model can be used only for the analysis of those experiments which involve short periods compared with the granulocyte lifetime, and which do not require any variUnder conditions advanced in the ations of the granulocyte activity. above hypothesis, the granulocytes behave like inert particles, so that the on

Barilati,

4

Chiabrera

and

Ridella

thermodynamic equilibrium is defined as the ideal state in which all fiuxes of granulocyte particles are zero. Such a state, which cannot be obtained in Go, as well as small deviations from it, can easily be simulated on the model, as will be discussed in Section 4. The thermodynamic and electrochemical analysis of the model will be performed in Sections 5 and 6, where its physical reliability will be checked and a constraint among its unknown parameters will be found. We conclude this section %ith the follow&g general remarks : conditions, or dur+g transients, the granI. under normal steady-state ulocyte peripheral pools are open systems, 2. the pools are at the same constant temperature T, and also their volume is constant, 3- during transients, the net flow of granulocytes into a pool changes, so that the number and concentration of granulocytes stored in the pool also changes at the same rate.

Model

of the

grandocyte

peripheral

system

The features of the model of the granulocyte peripheral system showu in Fig. 2 are summarized in this section. Other models have been described in literature. Those used by ATHENS, 1 MORLEY 2 and RUBIxow 3 are among the most interesting ones. As far as the peripheral system is concerned, all of them are rather oversimplified, if compared with our model. The biological foundations of our model, which has been successfully used to explain many experimental data on granulocyte kinetics both in normal and pathological subjects, are described elsewhere.* We let : the number of granulocytes in the circulating pool ; c _M the number of granulocytes in the marginated pool ; the number of granulocytes in the reserve pool ; +-he flux of granulocytes from the reserve pool into the c&.latiug 21 one ; the flux of granulocytes from the marginated pool into the tissues ; 3 the net flux of granulocytes from the circulating pool into the marginated one ; Q the value of a fuzzy factor, which takes into account all the pathogenic agents in the tissues. The continuity

equations

dC -= dt

are :

Kr-Q

dM

and dt

= @,-I&

(1)

The functional dependence of the fluxes among the pools are : 1.2~3 K~

=

&

‘(I?, C),

K2 =

K2 (M,Q)

and

4 = 4 (CM)

The flux K, of the granulocytes coming from the reserve pool is the Therefore, its analytical expresinput .signal for the peripheral system.

Thermodynamic

Coristraints on Leukocyte-- K&et& _. _.-

&- Blobd :_:+;---:_. ‘3.; -:

’ -.

_‘

-.,

.

-_-1 =

. ‘_L__ .:ir_._ _ -=I

-number of -available-~~it_~ith~~~~~~be-= occupied by granulocytes in the marginated pool.-! .Thu$ the+&ber . of. empty sites is (&Z+W), M being the- number_.of_.occupied:sites.~~~~~;now_; apply the mass-action law. _The probability’ for a gramrlocyte~to- accom<._ plish the transition from the circulating pool: to the marginated. one iS.. We

let M, Abethe maxim&n

-

proportional

to C and_ to (M&M)

s?_ that-

A:; ’

:_.,_--

: _’ :- _‘-~“. ::1_

in such a way that @‘e has the physical dimension of a flux r2 ,. constant. ____ ..-,-:-_L-: :.-: Using the same kind or &urne&s, we be &&&number _.-: of available sites in the circulating -pool so -that :. .... _.-_;i: :.___--.-y.:. c.-,-;j T- ,.

i,t’cl

the

6

Barilati.

Chiabrera

and

Ridella

Equations (I, 3, 5) give the model of the granulocyte peripheral system. Whereas equations (I, 2) are always true, equation (5) relies on rough assumptions, and the parameters T~;. 7,. C1. M2 are not known. As discussed_in Section I, our purpos e is to test the physical reliability of the constitutive relationship (5) from the thermodynamic point of view, and to find an electrochemical constraint among the parameters Tr, r,, Cl. MC. Steady-state

and thevntodynamic

equi~ibvimn

The model outlined in Section 3 can be used very far from equilibrium, and during transients. Nevertheless, the thermodynamic relations which are going to be applicable in Sections 5 and 6 can be applied correctly only for small deviations from equilibrium. Thus, first of all. the thermodynamic equilibrium state for our model is worth discussing. As a first step, we shall deal with the steady state, labelling the variables by the subscript (0) : dC dt

=

dM -= dt

0 ; K1,O = @,, = Kz,,, = +

>O

(6)

Steady-state is the ideal state of a normal subject in which the number of circulating and marginated granulocytcs is always constant in time. Thus, the number of pathogenic agents in the tissues Q, and the tissues demand of granulocytes are constant in time, also. If the constitutive relationship (5) is introduced in equation (6), we obtain a relationship among C and M in steady-state : W=++Z)[

I+(:;.)(&-*)]-’

(7)

It is now possible to define a fictitious state of thermodvnamic equilibrium for the peripheral system. Thermodynamic equ.ilibGum is a particular case of steady state, labelled by the superscript (tJz),in which the fluxes among the pools become equal to zero : &rh

=

@a =

&tk

=

0

(8)

In this state, the peripheral system as a whole becomes closed, i.e., it does not receive any g-ranulocytes from the reserve pool. nor does it send them to the tissues, K, and K, being unidirectional_ Nevertheless, the peripheral pools continue to eschange granulocytes between them, being T > o, i.e., the two fluxes @lt’l and @P are different from zero and equal so that the net flux @h becomes zero. We recall that K,* = o implies F + 00, being _iMfk # o. The limit 7 --+ 00 means that the tissue demand is zero, i.e., Q = o. It becomes now apparent why thermodynamic equilibrium cannot be observed ,~PZ V~UO. Nevertheless, such a state has been easily simulated using equantions (I), (2) and (5) of the model.

where (-dfL~~w/T) are the forces. at

ihe

fiermodp&c

qfi,,c_;

_- _, Jc

A;

J&:_&

6:

Using the cotitinuity equations (I), the f+es.:dJ~,,~ ~.

dJc=d

- .:: 1 -i

‘-.

_ s -1. --‘;: 7.

:_. . >. I ., -. I :

take ~,thelfoi;in -_..:

(KL-@)

where the superscript (tlr) x&eans~that the partial: fleriv&i~es_.are e’tali:r :I uated- at the- thermo_dynarnic equilibrium. T&e.’ elec~~he+i+lL~~p@ntiak Fc_ti depend on C,M and T.’ If there- ak &I&~ &3lc@ t y. ceperi.-

-8

Barilati. Chiabrera

and

Ridella

pheral pools, and the average granulocyte charges qcni are not negligibIe, @CJ~depends also on the electric potential qc,.kf, which, in turn, could depend on C and M. We point out that, in this paper, we deal with the actual number of granulocytes C and M. Thus, the electrochemical potentials are given as potentials per particle. These potentials, times the Avogadro number N, give the more common electrochemical potentials per mole. The charges QC~\I per particle, times N, give the charge in Faraday units per mole. If the contributions of ‘the chemical potentials ~C,M are brought into evidence, we obtain :

The chemical potentials are computed as those of dilute solutions, near the thermodynamic equilibrium : s*lo

pc=kTIn

c

111

C1_Cand~L~~=RTln

1wz-

111

(13)

where the BOLTZJIANX constant k appears, because we deal with potentials per particle. If we apply zk vitro a known electric field to a physiological solution of granulocytes, the esperimental results give very low values of their mobility, i.e., of their electric charges. 11pL2 Accordingly, it seems reasonable to neglect the electric effects, i.e., the dependence of ‘po+tfand p*c,~ on C and M :

dC+

H

dIL.\f

rh

a1M

V,T.C

d&i

(14)

,

Equation (13) give :

(15) Introducing equations (14) and (15) in equations dJc=--cc

(IO),

we obtain :

-Lc-sf

(16)

Thermodynamic

Constraints on Leukocyte

The comparison between the off-diagonal and (16) gives:

Kinetics in Blood

terms in equations

9

w

I

=-

T

(17) Computing the partial derivatives of the chemical potentials from equations (13), the ratio among both sides of the above equations gives :

The relationship (18) is the formulation of the OSSAGER principle

for the granulocyte

peripheral system, and it is a necessary condition

which must be fulfilled by the model. The reader can easily verify that if the partial derivatives &D/X and S@//s_M are evaluated from equations and (9) the OSSAGER constraint (18) is satisfied, regardless. of the 7 Etual value of Cfh. This finding is a very important test for our model, and supports the consistency of the constitutive relationship (5) with the physical world. Of course, this result is not sufficient for claiming that the proposed expression of @ is true. If the proposed 6, has to be changed in the future, because it cannot -explain new e.xperimental data, the new constitutive relationship will still have to satisfy the OXSAGER constraints (IS). which is a general thermodynamic test of its physical reliability. .

Electrochemical

constraint

orz the cortstitd&-

relatioushifi

In the previous Section, we have pointed out that the electrockmical potential may be obtained by adding three terms. The first one is the chemical potential ; the second one is the contribution of the internal thermodynamic energy $c,,jf and the third one is the contribution of the electric energy : ~c=liTln

C Cl--c

+EricpJ

tqcgc

At the thermodynamic equilibrium, all the gradients of the intensive

variables must be zero throughout the whole system.

Thus, the elec-

Barilati.

IO

Chiabrera

and

Ridella

trochemical potential must be uniform in the whole system, s T being uniform, too : (&)‘”

=

(@_,f)‘h

Equating the relationships (zg) at the we obtain : (Qc?c

-

QM 9.\fP

+

(tz’c -

pi_If)th =

thermodynamic kT In -

&P

(C,

-

Clh(A&z-

equilibrium, 0”)

AP)

(20)

The right hand side of equation (20) can be evaluated from equation (g), so that equation (20) b ecomes : (qcqQ?--qQaf’9_\f)th

+ kT

(f&ic-p!tf)tk

Pr) 1

The relationship (21) is very important because it gives an electrochemical constraint on the unknown parameters T,, TV, C1, M2. Until better quantified information regarding the left-hand side of equation (20) is forthcoming, we suggest, to neglect it with respect to the right-hand side, for modelling purposes. Such a hypothesis has already been partially discussed in Section 5_ We accept .it because it does not appear inconsistent with the experimental data 4 which have been used until now to test the validity of the model. Accordingly, equation (21) gives the following electrochemical constraint ; (22)

If new experimental data are found which are inconsistent with the previous simplifying hypothesis, equation (22) will have to be dropped, and the electrochemical constraint among the phenomenological parameters +rl, 7”, C,, AZ, will have to be computed directly from equation (zr), which remains valid.

Discussion In Section 3, it was shown that the proposed model is consistent with the OXSAGER principle. Furthermore, the electrochemical constraint

% C1 = IC?M, was found in Section 6 among the model parameters. Consequently, the constitutive relationship results strongly simplified :

so that the flux Q becomes linearly dependent on C and M. This finding coincides apparently with the lvidely used method of appro,ximating an

Thermodynamic

Constraints

on

Leukocyte

Kinetics

in

Blood

II

function by means of a linear dependence on the variables for small deviations from equilibrium. \Ve recall that equation (23) has been evaluated from equations (5) and (32) so that the above relationship is still true even for large deviations of C and M from equilibrium. A deeper insight into equation (33) can be obtained by discussing an experiment currently used in clinical practice, I3 where the introduction of radioactive tracers as cell labels has led to a rapid accumulation of knowledge of the granulocyte kinetic behaviour in the blood and marrow of man. A small amount of granulocytes are withdrawn from a subject and labelled in ‘c:itrowith a radioactive tracer, like di-iso-propylfluorophosphate DF32P. The labelled granuloc_ytes L are then injected intravenously into the donor subject. At the end of the autotransfusion (t = o), the number of labelled circulating granulocytes CR decays with time. The analysis of the radioactive transient gives information about both the peripheral and the whole system which are about in steady state. 3~4~13The amount of available information increases if during the decay of the labelled granulocytes an endotoxin, such as Pyrexal, obtained by the Salrnonelln abortzss eqlsi, is injected into the subject. The granulocytosis followig the administration of bacterial endotoxin is due to the rapid influx of cells from the bone marrow reserve pool into the blood, so that KI increases very rapidly and both C and 1111undergo a transient which modifies the decay of CR_ The behaviour of CR versus time is shown in Fig. 3 for both espcriments performed on the same subject. The values of C measured during the two experiments are plotted in Fig. 4. In the first case (dotted line),

unknown

\

\

End OFthe autotransFusion /

o\\

\

\

Fig. 3. Labelled (DFJ’P) granuiocyte decay versus time in the circulating pool. with (dots) and without (circles) Pylexal. in a normal The subject is the same as in subject. Fig. 4.

Barilati,

12

Chiabrera

and

Ridella

C is about constant, the subject being in his normal steady state. In the second case, the increase of C induced by the endotoxin becomes apl The detailed analysis of the experimental data is beyond the parent. scope of this paper. 4,1P It will suffice it to point out that from Fig. 3 it is apparent that the decay of CR is strongly affected by the transient of C.

Pyrexal

‘injection

End OF the autotransfusion /J

I

t(h)

,

I 10

I 20

Fig. 4. Granuioq-te kinetics versus time in the circulating pool. with (do&.) and without (circles) ~-rcxal. in a normal subject. The subject is the same as in Fig. 3_

This finding can be understood if one writes the equations for the radioactive granulocytes (subscript R) :

dCR dt

=-@R

d-!l!fR ;

dt

R-

’ydt+CR(O) fi!R

under the boundary condition

1iR

=(I,

(24)

-

7

+l+fR(o)

=L,

/ -cl

-Aa) being the beginning of the autotransfusion. The closed-form expression of @R can be easily derived from equation (5). recalling that the margination probability for a labelled granuIoc_yte is proportional to CR times the number of empty sites (M,;-&I) in the marginated pool. On the same basis, the demargination proba(1 =

Thermodynamic Constraints on Leukocyte Kinetics in Blood

13

hility for a labelled granulocyte is proportional -to fifR times the number of -empty sites (C,-C) in the circulating pool :

(25) The set of equations (24) and (3,s) shows actual value of C (t) and. M (t), and gives the known experimental data. 4~14 _ If one starts constitutive relationship, like equation (23), equilibrium, one will obtain :

that CR (i) -d&pen& on~the correct explanation Qf, the directly- from a linearized for small .deviations from

This relationship is incorrect because eiuaticns (24) and (26) ca& not explain the different behaviours,,of CR (t) with and without Pyrexal. In fact, equations (24) and (26) do not depend on C (t), nor on_ M‘(t)_ The reason why it is necessary to remember that equation (23) iS obX&ned by equation (5) thanks to the electrochemical constraint (23) becomes Ijloti evident ; accordingly, the correct expression (25) of_ @R can be found. Our final remark~is that the experiments performed in the clinical practice 3.13*14on the granulocyte system gives a limited amount of information so that the system cannot be completely identified. For in; stance only three relationships are found among 71, G, C,, iWZ so that it was impossible until now to evaluate the four parameters. The new independent relationship (22) permits the evaluation of ali four model parameters, without performing further experiments on-man. .

conclnsion

.

..

In biological modelling, one very often makes up fori+e ?.@orance of the phenomena involved with phenomenological parameters, -which cannot be computed on a physical basis. Then; it is important. that thermodynamic criteria should be available to test the physical coherence of the proposed constitutive relationships, or to find constraints among the unknotparameters. . If a.model of the biologic& system is found, it is possible to simulate its thermodynamic equilibrium state, although the normal state of the _biological system is out of equilibrium.. The thermodynamic consistency of the model with Ahe &A& _ principle can then be tested. Such a test is a necessary condiG% for &e- phy&i& reliability-.&f the proposed model..

Barilati.

=4

Ctiabrera

and

Ridella

Furthermore, keeping in mind that the electrochemical potentials must be uniform at the thermodynamic equilibrium, it is possible to find an electrochemical constraint on the constant parameters of the model. These results turn out to be very useful because they limit the allowed ranges of the parameter values. This approach has been applied to a model of the granulocyte peripheral system, showing that it is consistent with. the ONSAGER principle. and finding an electrochemical constraint among its Darameters which makes line>r the ,dependence of the granulocyte gow bkween the circulating and ‘marginated compartments on their populations.

Acknowledgments This research was supported by the (Italian) National Research Council. The authors are indebted to Prof. C. SACCHETTIand Dr. G. S. MELA of Istituto Scientifico di Medicina Intema for stimulating discussions, and to Prof. G. MILAZZOof Istituto Supexiore di Sanitti for his helpful suggestions.

References 1 2 3 4 5 6 7

8

9 10 11 If 13 14

H.R. WARSER and J.W. ATHESS. Ann. N-Y. Acad. Sci. 113, 523 (x964) E._4. &SC--SMITH and -4. MORLEY, Blood 36, 254 (1970) S-1. RUBIKOW and J.L. LEBOWITZ, J- Math. BioZ. 1, 187 (1975) A. CHIABRERX. G.S. MELX ei al.. Labelled Granztlocyte Kinetics in the Blood: Normal Subjects ; PathoZo,oicaZ Szrbjects. to be published. L. OSSAGER, Phys. Rev. 37, 405 (1931) ; Phys. I&v. 38, 2265 (1931) P. GLAXSDORFF and I. PRIGOGISE, Strucfrcre, Stabilitk et FZuctuations. Masson et C. Editeurs, Paris (1971) p_ 4r A. KATCHXLSKY and P.F. CURRAN, Non-Equilibrizcm Thermodynamics in Mass. (1967) p_ 85 Biophysics, Harvard University Press, Cambridge, K. DEXBIGH, The Principtes of Chemical Equilibrizrm. Cambridge University Press, Cambridge (1971) p_ 86 A.C. SJIITEX, J.F. JXNXK and R-B. ADLER. Electronic Conduction in Solids, McGraw-Hill, New York (1967) pp_ 8, 21, 48 T.L. HILL, Statistical Mechanics, MC Graw-Hill, New York (1956) p_ 402 R. ROBIKEAUX and S. BAZIN. Le Sang 22, 241 (1951) M.& LICHTMAX and R-1. WEED, BZood 35, 12 (1970) _4_M. BLXVER, J_\V_ ATHEXS, H. AWENBRUCKER, G-E. CARTWRIGHT and M.M. \VISTROBE. J_ CZin. Invest. 38, 1481 (1960) G.S. MELA. B. BIANCO, S. -DELLA and C. SACCHETTI. Huematologica 58, 843 (1973)