Coagulation in cell suspensions: Extensions of the von Smoluchowski model

£ theon BioL (1986) 120, 181-189

Coagulation in Cell Suspensions: Extensions o f the von Smoluchowski M o d e l C. D. HARDYt AND J. S. BECK*

t Department of Physics and :~Faculty of Medicine, University of Calgary, Calgary, Alberta, Canada T 2 N 1N4 (Received 8 July 1985, and in final form 20 January 1986) Recently, a long-range interactive force between erythrocytes has been proposed (Rowlands et al., 1981, 1982a, b) based on an apparent increase in the rate of aggregation of erythrocytes in an aqueous suspension over that predicted by one model of Brownian aggregation. Here, we examine the assumptions underlying this model and propose modifications compatible with the biological constraints on the model. The refined model is represented as a series of coupled differential equations representing the change in particle number density as a function of time. Numerical solution of these equations is consistent with the absence of an intercellular force.

1. Introduction Recently, much work has been directed to the demonstration and identification of a long-range attractive force between erythrocytes in aqueous suspension. The basic premise o f these studies is that erythrocytes in suspension will coagulate more rapidly than predicted by a theory of Brownian coagulation if such a force exists. The refe1"ence model used is the coagulation model developed by von Smoluchowski (1917), with refinements by Swift & Friedlander (1964) and Rowlands et al. (1981). As the von Smoluchowski model was constructed to describe the Brownian coagulation of a suspension containing only small, non-interacting, incompressible spheres (particles unlike erythrocytes) it is particularly appropriate to analyze the assumptions underlying the coagulation model, and to make refinements as necessary. Several authors have proposed analytic modifications to the model (Rosen, 1984; Okyama et al., 1984; Williams, 1984; Hendriks et al., 1983; Klett, 1975), but the results invariably require imposing several restrictions or properties o f the model which are inappropriate to cell suspensions. We present here an analytic generalization of the model which is not inconsistent with the biological constraints on the mathematical model, and discuss the implications of these modifications in light of solution of the modified mathematical model.

2. The yon Smoluchowski Model von Smoluchowski (1917) considered the coagulation of a sol to be the consequence of the contact and irreversible adhesion of small particles. This is useful where the fission o f larger particles to form smaller ones is negligible and where the probability of adhesion following contact is close to unity. One might add that, 181 0022-5193/86/100181 +09 $03.00/0

© 1986 Academic Press Inc. (London) Ltd

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assuming independence of events, the probability per unit time of adhesion, P~, is the product of the probability per unit time of contact, Pc, and the conditional probability of adhesion following contact, Poe. Experimental observations on inorganic sols (Lachs & Goldberg, 1922) and erythrocyte suspensions (Rowlands et al., 1981) suggest that these assumptions are not unreasonable in some real systems. Using this model, von Smoluchowski derives the following expression for the time derivative of the particle number density, vk, of complexes with k sub-units (elements of the kth class) dVk

d--/=½ ~. 4~rDoRo.v,vj i+j=k

i=1

4wD,kR,kVlVk.

(1)

Here, D o is the relative diffusion coefficient between pairs of particles, and R o is the geometric distance separating the centers of an interacting pair of particles. Assuming that the particles undergo Brownian motion prior to their collision and irreversible adhesion, von Smoluchowski applied Einstein's (1926) theory of Brownian motion to expand the diffusion coefficients to yield

dt

i+j=k

\%

rjl

co

1

i=1

\%

1

rkl

where D, is the diffusion coefficient of a particle of the nth class and r, is the Stokes radius of a particle of the nth class. To simplify this expression, von Smoluchowski adds the (unrealistic) assumption that the radii of all particles are equal for all time. That is

r~= ~ =

rk,

for all i,£ and k.

(3)

Using this assumption, equation (2) is reduced to

dt =16~rDlrl ½ ~. v, vj-i+]=k

vivk .

(4)

i=l

Summing over all k gives

d{,

, Vk] = d { N ( t ) } = - 8 ~ r D , r , { ~ 1 l~k]2

(5)

which is easily integrated to (Swift & Friedlander, 1964) 1

1

N( t-----S= 8"rrD1 rl t + N(0"----)"

(6)

The linear relation between the inverse of the total particle number density and time predicted from equation (6) has been confirmed experimentally for inorganic suspensions and aerosols in which the Stokes radius of the fundamental particle is no greater than 0-1 vLm (Lachs & Goldberg, 1922; Kruyt & van Arkel, 1923; Hidy & Brock, 1972).

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Conceptually, this model is lacking as a description of the flocculation of nonspherical deformable bodies roughly 10 ~m across, such as the human erythrocyte. Given that an observed rate of coagulation in agreement with von Smoluchowski's model is consistent with purely random (Brownian) collisions, then any force altering the collision frequency will cause the observations to differ from the predictions of the model. Rowlands et al. (1981, 1982a, b) have introduced a dimensionless factor (the ratio of a particle's interaction radius to its Stokes radius) which, in principle, will provide not only an indication of whether or not a force exists, but also its effectiveness defined as a change in the coagulation rate relative to that predicted for Brownian coagulation. Its magnitude has loosely been interpreted as a measure of the range of the force (Rowlands et al., 1981, 1982a, b), but the force law is not known, and the range of a force has not been clearly defined. We find four points on which the use of the von Smoluchowski model to represent coagulation of erythrocytes in aqueous suspension is particularly inappropriate. First, it is assumed that only particle fusion can occur. This requires the equilibrium state of the suspension to be a single large particle. Since fission does occur, the true equilibrium is a spectrum of particle sizes, with the most populated state being the mode of a distribution with the lowest free energy, which is determined by the thermodynamic state of the suspension. Short of equilibrium, the appropriateness of this fusion-only assumption depends on the time of observation and the relative rates of fission and fusion within each class: Until relatively large complexes have formed in appreciable numbers, it is a reasonable assumption. Initially it must be correct, as fission of the fundamental particles cannot occur. Second, it is assumed that the probability of adhesion following contact, P~c, is unity. While this is a difficult assumption to test, it is not an unreasonable one to use as a starting point. It must be remembered that a fusion rate below that predicted by the von Smoluchowski model need not indicate a repulsive force; it may simply result from imperfect sticking. Third, it is assumed that the Stokes radius of each particle is equal to its geometric radius. While this is quite reasonable for small inorganic aerosol particles, it is not clear that it is acceptable for the erythrocytes and rouleaux observed during aggregation. Fourth, it is assumed that the Stokes radii of all particles are equal at all times. von Smoluchowski recognized the influence of varied particle radii, but as he analyzed situations in which the dependence was weak he accepted as a good approximation the premise that the aggregate radii did not alter during coagulation in order to gain an analytic solution. For our biological system, this is the most unrealistic of all assumptions imposed by using yon Smoluchowski's model. As stated in the introduction, previous efforts to refine the model in this respect imposed other restrictions incompatible with the biological system. In the following extension to von Smoluchowski's model, in addition to the interaction factor, E, proposed by Rowlands and co-workers (1981, 1982a, b), corrections for the size and shape of the erythrocytes and rouleaux will be incorporated into equation (2). While the emphasis here is on erythrocyte aggregation, the general form of the correction applies to any diffusion-limited Brownian aggregation.

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3. Generalizations of von Smoluchowski's M o d e l

Using the same conceptual model of coagulation as von Smoluchowski, we arrive at the set of k coupled differential equations given in equation (1). Solution requires evaluation of

DijR#=Dir,(l+l~(ri+rj). \r~ rj/

(7)

The modifications of von Smoluchowski's model which we propose focus on particle size and shape (which affects the diffusion coefficient), and on the interaction radius of the particle. (A) PARTICLE SIZE

Rather than use the restrictive equation (3) we write equation (7) as

Dirl( l+ll(r, \ r~ rj/

+ rj) = Dlr~(2+ r_~+~\

rj ri)"

(8)

This is justified as the product Diri is independent of i (see equation (15)). If one knows the radii for all particle classes, it becomes straightforward to substitute equation (8) into equation (2) and to solve them numerically. Our tactic is to relate the radius (i.e. the radius of the projection of the erythrocyte disc) of each class of particle to the radius of the fundamental particle. Knowing the radius of the fundamental particle (a single erythrocyte), we achieve this by imposing either area or volume conservation on an aggregate. It is easily demonstrated that, if either volume or area is considered as the particles aggregate, then the radius of the ith class is approximated by

ri = i'°r~,

f½ for area conservation where to = ~ [½ for volume conservation.

(9)

These values of to are exact for spheres. Substitution into equation (8) yields

~, / j \ ~ ]

J.

,lO,

Although this provides a reasonable model of an increasingly polydisperse sol (formed of aggregates of various numbers of a unique fundamental particle), it is not satisfactory for modeling aggregation in a fundamentally polydisperse sol (formed of various aggregates of various fundamental particles). (B) PARTICLE SHAPE

In general, the particles of any particular class will not be spheres. So, one must incorporate some factor relating the Stokes radius of each aggregate particle to that of the fundamental particle. In turn, the Stokes radius of the fundamental particle must be related to a characteristic geometric radius. As the Stokes radius can be

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computed or measured for all bodies, this presents no insurmountable difficulty. In particular, we will take the geometric radius of a single erythrocyte to be the radius of the projected disc, roughly 4-2 Ixm (Sewchand & Canham, 1976). As we are primarily concerned with biological systems, we will confine our discussion to the spherical approximation and assume that the Stokes radii o f all complexes are equal to their physical radii. In particular, this has been confirmed reasonablywell for erythrocytes and rouleaux of up to ten cells (Groom & Anderson, 1972; Skalak et al., 1981). In the general case, one has (Lamb, 1932)

r=sp

(11)

where p is the characteristic geometric radius of a body, s is Perrin's correction, and r is the Stokes radius of the body. Following Rowlands et al. (1981), we incorporate the interaction coefficient, E, into equation (2). E is defined by Ri "~ = - (12) r~ where R~ is the effective interaction radius of members of the ith class and r~ is the geometric radius of members of the ith class. This interaction coefficient has been interpreted as a test parameter for Brownian coagulation. Specifically E < 1 -> a potential barrier between particles a n d / o r a failure to stick after collision E = 1 -~ a Brownian (random) motion and perfect sticking E > 1 -~ an attractive interparticle force with sufficiently perfect sticking. Though the last case above violates the premises o f yon Smoluchowski's model, the anticipated behaviour would be the same as that with von Smoluchowski's model, but with particles o f radius E r , Only the case E = 1 is consistent with the model as presented by von Smoluchowski. Further complications arise if one is modeling a process occurring in a space which is not truly three-dimensional. Using Einstein's (1926) expression for the diffusion coefficient of a particle implicitly restricts the analysis to a threedimensional space. Indeed, trying to solve the problem in a two-dimensional (or, more generally, an even-dimensional) space leads to Stokes' paradox (Krakowski & Charnes, 1951; Happel & Brenner, 1973). Again, we restrict ourselves to the three-dimensional case for further discussion. The generalizations discussed above are now easily incorporated into equation (2) to yield i+j~k

i=1

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In equation (13), f(i,j) is a general representation defining a relation between the radii of particles from the ith and jth classes. For all subsequent numerical analysis, we have taken

f(i,j)= 2 + ( j )

+~-J.

(14)

Note that E may be incorporated into the coefficient common to the differential equation (13). If it were a constant, it is nothing more than a temporal scaling factor. However, we choose to express it explicitly since it may (more reasonably) be considered a function of i and j. Furthermore, its value represents the physical property which is at issue. The results below were obtained by taking E as constant for the numerical analysis. In a limited sense, the von Smoluchowski model has now been generalized to include the effects of cell size and shape. We will refer to the model represented by equation (13) as the generalized model of thermal coagulation. We have chosen to solve the equations numerically for some representative examples. Analytic solution of equation (13) with the kernel of equation (14) is, at best, challenging. However, considerable insight can be gained from this relation without solving it analytically. 4. Results

Consider first, the dependence on particle size in equations (13) and (14). Clearly, the coagulation rate is independent of the absolute particle size. The only part of the expression containing the particle sizes as anything other than a ratio of first powers is the diffusion term. But, from hydrodynamics we know that

D~ri=

kT 6~r~

(15)

where k is Boltzmann's constant, T is the absolute temperature of the system, and is the absolute viscosity of the suspending medium. The product, Dir~ is constant for a given system. (Constancy holds so long as the sol is disperse enough to ensure that viscosity is not a function of the particle number density and size, true to a good approximation for dilute sols.) However, the equations do include relative particle size. In fact, differentiation of equation (14) with respect to (i/j) shows that only one extremum exists and that it is an absolute minimum at i = j (or, alternatively, when ri = rj). One concludes that the yon Smoluchowski model implies a coagulation rate which for all t-> 0 is below that implied by the generalized model. It seems at first that there is no limit to the coagulation rate predicted by the generalized model; in contrast, the coagulation rate is constant in the yon Smoluchowski model. The coagulation rate given by equation (13), however, is not a monotonically increasing function of time as the particle number densities of classes with small k (i.e. k = 1, 2, 3) fall to zero as t approaches infinity. We can get some indication of the difference in the predicted coagulation rates for early

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times by rewriting equation (13), summing over all k and rearranging to give dt = -4"~D1 rl~s -1 ~

j

1

f(i,j)~,,v i .

(16)

Equation (16) would be formally identical to equation (5) if corrections for particle shape were incorporated into equation (5) and if i =j. However, for all terms where i ~ j , f ( i , j ) (equation (14)) is greater than 4. Therefore, as the particle number densities predicted by either model are approximately equal (at least initially), equation (16) implies a higher rate of coagulation than equation (5), at least for an initial period, the length of which is determir/ed by the instantaneous coagulation rate at t = 0. As a consequence of the enhanced coagulation rate, equation (16) implies a more heterogenous sol (with respect to particle size) at a given time prior to equilibrium than does equation (5). The increasing polydispersity further helps to increase the coagulation rate. So, we arrive at the same conclusion implied above: equation (13) predicts a value of N ( t ) which is always less than that predicted by equation (2) given an identical state at some earlier time t. We now focus our attention on specific comparisons of the predictions of equation (13) (obtained by numerical solution) and those of equation (6). Programs for all computational and plotting work were written in the programming language C on a VAX 11/750 (with a 32 bit word). Equation (13) was solved using a fourth order Runge-Kutta analysis (Burden et al., 1981) for coupled differential equations. To solve equation (13) numerically, we required an upper limit on k. A value of 10 was chosen, as our primary interest is aggregation of erythrocytes and few rouleaux of ten or more cells were observed at the end of one hour (the observation period generally used by Rowlands et al., 1981, 1982a, b). Also, we used conditions such as temperature (T = 300 K), and absolute viscosity (77 = 0.01 Pas. s) to match the experimental circumstances. Furthermore, the simulations presented here were done assuming area conservation (o~ =½). The user interested in other fields may simply treat the parameters as forming a single constant defined in equation (13). Figure 1 is a composite plot of the inverse normalized particle number density, defined as N ( O ) / N ( t ) , as a function of time obtained from both the generalized and the von Smoluchowski models with the physical conditions representative of the erythrocyte aggregation studies done by Rowlands. For Fig. l(a), E was 1, whereas for Fig. l(b), E was 10. In comparing the simulations with data, two questions are at issue: (1) Is there a useful distinction between the yon Smoluchowski model and the generalized model? and (2) Is there evidence for an interaction between separated red cells? On the first question, regardless of the experimental data available for comparison, there is no data-dependent reason for choosing one model over the other if the simulations are no different relative to the precision of possible experimental data. So, a priori, no choice can be based on data for times sufficiently early in the process beginning from an initial state v~(0)= N(0) and z,~(0)=0, i > 1. (17)

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/

2"00

/

/

1"00~

"

7200

Time Is)

FIG. I. Normalized inverse particle number as a function o6 time. As predicted with the modified yon Smoluchowski model (lower curves) and the generalized model (upper curves} with "~ = I (a) and

-=-- 10 (b). The possibility remains that for a system where sufficiently long coagulation times can be observed one or the other model (or both) may prove inconsistent with the later data. On the second question, we note that, as the interaction coefficient increases, so too will the discrepancy between the predictions of the simulations. When one suspects an attractive non-Brownian interaction, a simple fit of a straight line to the data may result in an unreasonably high estimate of --~. However, any attempt to minimize the g 2 statistic for the generalized model requires a great deal of computing time and increases the risk of introducing other numerical errors. As a consequence of this analysis, it is clear that demonstrating the existence of a long range attractive force with E = 3 (the factor reported by Rowlands et al., 1981, 1982a, b) will require careful statistical analysis of the data. Using the z statistic and assuming normal distributions for all of the measured variables, the probability that E differs from 1 is estimated to be less than 0.01 in the several experiments represented by their published data (Hardy, 1982). We conclude that the published studies do not answer this last question. Any experimental design should include the extension of the von Smoluchowski model which we have suggested here. The experiments published do not conform to the extended model, nor to the three-dimensional nature ofvon Smoluchowski's original model. Furthermore, apart from the existence of any long-range intercellular force, any test of the model extensions themselves must provide much higher precision. We wish to thank Dr J. S. Murphree for allowing us to do all of the computational work on the VAX-11/750 in the Physics department at the University of Calgary. REFERENCES

BURDEN,R. L, FAIRES,J. D. & REYNOLDS,A. C. (1981). NumericalAnalysis, 2nd edn. Boston: Prindle, Webber and Schmidt. EINSTEIN, A. E. (1926). Investigation on the theory of Brownian movement, with notes. New York: Dover Publishing. GROOM, A. C. & ANDERSON, J. C. (1972). J. cell Physiol. 79, 127.

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HAPPEL, J. & BRENNER, H. (1973). Low Reynold's number hydrodynamics. Leyden: Noordhoff International Publishing. HARDY, C. D. (1982). An investigation of Long Range Forces Between Erythrocytes. (University of Calgary, Master's Thesis.) HENDRIKS, E. M., ERNST, M. H. & ZIFF, R. M. (1983). J. Stat. Phys. 31, 519. HIDY, G. M. & BOCK, J. R. (1972). The dynamics ofaerocolloidalsuspensions, VoL 1. New York: Pergamon Press. KLETr, J. (1975). J. Atm. Sci. 32, 380. KRAKOWSKI,M. g/. CHARNES, A. (1953). Stokes paradox and biharmonicflows. (Carnegie Institute of Technology, Department of Mathematics: Technical report 37.) KRUYT, H. R. & VAN ARKEL, A. E. (1923). Kolloid.-Ztschr. Dresd. u. Leipz. 32, 29. LACHS, H. & GOLDBERG, S. (1922). Kolloid.-Ztschr. Dresd. u. Leipz. 31, 116. LAMa, C. E. (1932). Hydrodynamics, 6th edn. Cambridge; Cambridge University Press. OKYAMA, K., KOUSAKA, Y. & HAYASHI,K. (1984). Z Coll. Inter. Sci. 101, 98. ROSEN, J. M. (1984). J. Coll. Inter. Sci. 99, 9. ROWLANDS, S., SEWCHAND, L. S., LOVLIN, R. E., BECK, J. S. & ENNS, E. (1981). Phys. Lett. 82a, 436. ROWLANDS, S., SEWCHAND, L. S. & ENNS, E. (1982a). Phys. Lett. 87A, 256. ROWLANDS, S., SEWCHAND, L. S. & ENNS, E. (1982b). Can. J. Physiol. Pharmacol. 60, 52. SEWCNAND, L. S. • CANHAM, P. n. (1976). Can. J. Physiol. Pharmacol. 54, 437. SKALAK, R.s ZARDA, P. R., JAN, K.-M. & CHIEN, S. (1981). Biophys. J. 35, 77L SWIFT, D. L. gz FRIEDLANDER, S. K. (1964). J. Coll. Inter. Sci. 19, 621. YON SMOLOCHOWSKI, M. (1917). Ztschrft. Phys. Chem. 92, 129. WILLIAMS, M. M. R. (1984). J. Coll. Inter. Sci. 101, 9.