Platelet deposition in stagnation point flow: an analytical and computational simulation

Medical Engineering & Physics 23 (2001) 299–312 www.elsevier.com/locate/medengphy

Platelet deposition in stagnation point flow: an analytical and computational simulation T. David *, S. Thomas, P.G. Walker School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, UK Received 8 December 2000; received in revised form 20 March 2001; accepted 26 April 2001

Abstract A mathematical and numerical model is developed for the adhesion of platelets in stagnation point flow. The model provides for a correct representation of the axi-symmetric flow and explicitly uses shear rate to characterise not only the convective transport but also the simple surface reaction mechanism used to model platelet adhesion at the wall surface. Excellent agreement exists between the analytical solution and that obtained by the numerical integration of the full Navier–Stokes equations and decoupled conservation of species equations. It has been shown that for a constant wall reaction rate modelling platelet adhesion the maximum platelet flux occurs at the stagnation point streamline. This is in direct contrast to that found in experiment where the maximum platelet deposition occurs at some distance downstream of the stagnation point. However, if the wall reaction rate is chosen to be dependent on the wall shear stress then the analysis shows that the maximum platelet flux occurs downstream of the stagnation point, providing a more realistic model of experimental evidence. The analytical formulation is applicable to a large number of two-dimensional and axi-symmetrical surface reaction flows where the wall shear stress is known a priori.  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved.

1. Introduction Due to its serious clinical consequences, thrombus formation, and the subsequent occlusion of arterial vessels, has become one of the most important areas of research in cardio-vascular disease. A thrombus is simply an accumulation or aggregate of platelets, held together by fibrous strands penetrating throughout the interior and exterior of the platelet clump. The cellular platelets and subsequently their aggregate, are crucial in the steps toward stemming bleeding. However the formation of platelet aggregates can be initiated by means other than a simple cut or injury. Most notably the eruption of an atherosclerotic plaque provides a diffusive cloud of chemicals which can initiate the activation of platelets causing them to adhere and aggregate together. These chemicals are subject to both diffusive and convective transport and hence not only cell reaction mechanisms but the flow of blood and the interaction with the

* Corresponding author. Tel.: +44 (0)113-233-2126; fax: +44 (0)233-243-2150. E-mail address: [email protected] (T. David).

arterial wall become important parameters in modelling the initiation of thrombus formation. There have been a number of conflicting hypotheses put forward as to the relationship between these important parameters as set out below. It has been postulated by some that normal flow components are the important parameter for platelet deposition whilst others have put forward the hypothesis that diffusion is the primary transport mechanism. The work set out in this paper aims to introduce a mathematical model which helps to show, through comparison with experimental evidence, the fundamental relationship between reaction, diffusion and convection for platelet adhesion. Although the model is based on stagnation point flow, where the tangential and normal velocity profiles are known, the results can be generalised to any a priori non-negative wall shear stress flow condition where the Peclet number is high. The first stage of thrombogenesis is platelet adhesion on a surface. This may then be followed by platelet aggregation and the formation of mural thrombi [1]. Although the formation of thrombi in stasis is fairly well understood the influence of blood flow characteristics has yet to be fully investigated. Fluid dynamic studies of blood flow, in models of arteries, suggests a set of

1350-4533/01/$20.00  2001 IPEM. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 0 - 4 5 3 3 ( 0 1 ) 0 0 0 4 7 - 9

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T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

fluid dynamic conditions which appear to predispose thrombus formation, principally at arterial bifurcations, T-junctions and curved sections, in particular [2]. Schoepheorster and Dewanjee [3] found that areas of flow stagnation or recirculation were shown to promote thrombus formation. One of the areas where secondary flow occurs is downstream of a stenosis and they have also been observed at the carotid bifurcation, T-junctions and at end-to-side anastomoses, all of which are sites prone to atherosclerosis and thrombogenesis [4]. In all these cases a recirculation zone exists and further downstream, in the eventual transition back to parallel flow, a so-called ‘reattachment point’ is formed [4–6]. Early studies indicated that thrombus formation occurred in areas where the normal flow component was directed toward the wall. In the light of this Kratzer and Kinder investigated streamline patterns in a model of a branching coronary vessel [7]. Here they conducted flow visualisation experiments of flow in T-junctions, obtaining tangential and normal components of flow at some distance from the wall, and compared them to platelet deposition results for the same geometry. Their conclusion that platelet deposition and hence thrombus formation was more likely in areas of normal flow directed toward the wall was based on qualitative evidence. Due to the lack of available flow visualisation technology they could not obtain sufficient detail of the flow in the stagnation point area compared to the thickness of the mass transport layer. A study of their photographic evidence indicates that only a small amount of platelet deposition (if any at all) occurs at the stagnation point streamline, which is directly below the maximum normal velocity component. This evidence seems to contradict the simple hypothesis of platelet deposition correlates with normal flow components directed to the wall. Although an increase in cell flux will clearly be the case for general convection toward an adhesive surface it does not show conclusively that the platelets will adhere or whether any more local phenomena are involved. Petschek, Adamis and Kantrowitz [8] suggested use of a stagnation point flow geometry for the investigation of thrombus growth under controlled conditions for varying thrombogenic surfaces. They assumed a simple fluid model of the stagnation point with the tangential flow component written as a linear function of radial distance. This restricted their domain of investigation. Again, their results seemed to indicate a non-constant deposition of platelet adhesion as a function of distance measured away from the stagnation point. Wurzinger et al. [9] suggested that there is a close correlation between flow conditions and the location of platelet deposits, in particular with stagnation point flow, where there are positive normal components of flow to the wall. Wurzinger et al. [10,11] used a ‘stagnation point flow’ experiment with blood impinging upon a rotating cylinder, to reduce influence of platelet aggregation before adhesion.

Their results indicated that platelet adhesion on foreign surfaces was mediated by flow components normal to the wall in addition to biochemical stimulation from high wall shear stresses. In these early experiments little emphasis was placed on the actual mathematical definition of the flow. Tippe et al. [12] presented experiments in a stagnation point flow chamber whose flow was steady state. Importantly their initial experiments showed that the number of platelets deposited at the lower wall surface was proportional to the number of platelets contacting the wall. In addition they stated that the constant of proportionality depended crucially on the functional state of the platelets. They subsequently used this premise to investigate whether the stagnation point chamber could be used to detect blood disorders in patients. In defining their experimental method it was shown that in the neighbourhood of the stagnation point the density of platelets deposited was negligible but increased as the radial distance increased. Reininger, Reininger and Wurzinger [13] used the same methodology to investigate the influence of flow on platelet adhesion to intact endothelium. Study of the published photographs indicated that there existed an area in the neighbourhood of the stagnation point which remained ‘platelet free’, even though this area has the highest value for the normal component of velocity, similar to that found by Petschek, Adamis and Kantrowitz [8]. However the majority of platelets adhered in areas where the normal velocity component was negligible. They continued to surmise that the platelet adhesion was, at least partially, mediated by the normal component of the convective particle transport of the stagnation point flow model even though the platelet adhesion seemed to be at a maximal density in a domain removed from the neighbourhood of the stagnation point, where flow convection towards the adhesion surface was, in contrast to the stagnation streamline, negligibly small. Further work in a similar vein can be found in [14]. In contrast Leonard, Grabowski and Turitto [15] stated that diffusion normal to the adhering surface was the determining factor in how blood components made contact with the adhering surface although convection was certainly important in bringing fresh reactants to the adhesion site. In particular they sited two important parameters which characterised the convection/diffusion equations. These were the Peclet number, Pe, and the surface adhesion reaction rate, k. Turitto and Baumgartner [16] assumed that steady state deposition of platelets onto cylindrical tube surfaces was controlled by a diffusion process normal to the surface whose values changed as a function of shear rate. Their mathematical analysis made use of a transformation which included the assumption of a spatially independent linear velocity profile within the mass transfer boundary layer. They were able, by comparing with

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

experimental data, to provide a power law relating diffusion coefficients to shear rate, however this was for whole blood where the presence of red blood cells is known to enhance the diffusivity of platelets. Affeld et al. [17], using a stagnation point flow cell and platelet rich plasma (PRP), were the first to quantitatively measure platelet deposition as a function of radial position for stagnation point flow. Their results showed that in a small neighbourhood of the stagnation point streamline little deposition took place. They proposed that platelet deposition was maximal at the position of a ‘critical’ shear rate. They also incorporated a computational model which provided steady-state flow profiles within the flow cell. On the basis of a simple derivation of the ‘boundary layer’ thickness, whose width was assumed to be that of a platelet diameter they showed how platelet fluxes could be evaluated. In addition, since in comparison with their experiments, platelets were deposited at sites close to the position of maximum shear rate, they hypothesised that the diffusion ‘collision theory’ of platelet deposition was not a plausible one and that it seemed more likely that a ‘critical shear rate’ was required to fully model the process where thrombin was convected downstream to activate further platelets. Pre-activated platelets were used in Affeld’s experiment and hence the flux of thrombin emanating from the adhered platelets would seem to have little effect on already activated platelets. Other effects apart from thrombin convection may help to provide the full explanation for the spatial variation in platelet deposition. Weiss [18] proposed that fluid shear rate evaluated at or very near the surface may have an effect on reaction rate. All of the above work has been carried out in the steady state condition, apart from that of Schmid-Schonbein and Wurzinger [5] where a qualitative description of pulsatile phenomena was given. Wurzinger et al. [11] have shown that the ‘dynamic’ stagnation point flow may very well become important as a consequence of the motion of stagnation points due to pulsatile flow. Specifically where the vessel surface is moving out of phase with the outer flow profile. Detailed analysis for pulsatile wall shear stress has already been attempted in this area by Hazel and Pedley [19]. They showed that significant differences in the time-averaged wall shear stress is obtained when motion of the stagnation point is taken into account. However it is thought that an understanding of the important parameters governing the role of platelet adhesion in flowing blood can be obtained from steady-state models and the presented work leaves the time-dependent model for future publication. Despite the considerable amount of published work on platelet adhesion there are still a number of important unanswered questions regarding the modelling of platelet adhesion in flowing blood. Firstly, what is the mechanism governing spatially dependent platelet adhesion?

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Secondly, what are the critical non-dimensional parameters defining platelet adhesion? And thirdly is there a relationship between initial platelet adhesion surface reaction and wall shear stress? In the light of this a novel model is developed, using both analytical and numerical techniques, of the platelet adhesion in stagnation point flow is developed below, which incorporates convection, diffusion and surface reaction in a non-constant shear rate fluid flow. The objective is the investigation of the adhesion process of platelets in a spatially varying fluid flow. Particular emphasis is made on the coupling of the platelet adhesion surface reaction rate and the flow parameters. Comparison of the results will be made with the experimental data of Affeld et al. [17]. It is important to try and differentiate between effects emanating from shear enhanced diffusion by the presence of red blood cells and shear enhanced adhesion reaction due to the physical action of shear on the surface matrix mediating bonding between platelet phospholipid and endothelium. With this in mind it was felt important to use the experiment of Affeld et al. since no red blood cells were used and the model could correctly neglect shear enhanced diffusion. It is shown that the model can describe platelet adhesion for a variety of different flow conditions as long as the wall shear stress is a known function of the axial (streamwise) co-ordinate. Section 2 describes both the analytic and numerical models based on the conservation equations for mass, momentum and chemical species. The analysis includes the important non-dimensional parameters governing diffusion, convection and surface reaction rate. In order to compare favourably with experiment the model includes a surface reaction rate which is a linear function of wall shear stress. Section 3 presents results of both the analytical and numerical models for platelet concentration and flux at the surface. There is excellent agreement between the numerical and analytical models for a wide variation in reaction rate. Section 4 discusses the results and puts forward the hypothesis that the reaction rate at the surface, modelling platelet deposition, by comparison with experimental evidence must be a non-constant function of wall shear stress. The analytical model is then used to show the positions of maxima for the platelet flux as a function of radial distance. The analysis also shows that the surface reaction rate cannot be of too large a value although a diffusional controlled mechanism is still considered to be a possibly correct model. Section 5 presents conclusions drawn from the two models and puts forward areas for future work. 2. Mathematical model In the case of stagnation point flow experiments the flow chambers have a lower surface upon which

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adhesion takes place. The blood is pumped into the chamber in a jet whose direction is perpendicular to the adhesion surface. The upper surface of the flow chamber is sufficiently far enough away from the lower surface (in terms of reaction and diffusion lengths) such that its influence can be neglected. We begin the modelling by assuming a steady-state stagnation point flow environment where the basic geometry and axes are shown in Fig. 1. A steady-state axi-symmetric jet is centred at the origin O, with z denoting the co-ordinate normal to the surface and R the coordinate in the radial direction.

fluid reaction rate for the ith species and Di is the ith species diffusion coefficient. We assume a Fickian diffusion model such that the coefficient is constant and that activated platelets are the only species to be modelled. The species equation is written in the simpler form of (u·ⵜ)fi⫽Diⵜ2fi Since density is constant then the momentum and species conservation equations are effectively de-coupled. The diffusion coefficient for platelets is determined by the use of the Stokes–Einstein equation,

2.1. Basic theory We assume that due to the relatively high shear rate in the domains of interest then blood can be modelled as a Newtonian fluid. In addition we assume that constant temperature prevails and the system is of constant density. Finally, although physiological blood flows are pulsatile in nature we begin with a steady-state system. This is reasonable since initial platelet adhesion can be modelled in a pseudo-steady formulation as shown by Strong et al. [20]. The steady-state conservation of mass and momentum in vector are given as ⵜ·u⫽0 and r(u·ⵜ)u⫽⫺ⵜp⫹mⵜ2u where u is the velocity vector in cylindrical co-ordinates, r is the density, m the dynamic viscosity and p a pressure which varies due to dynamic variations in fluid velocity alone. A general conservation equation for the ith species can be written, again in vector form, as (u·ⵜ)fi⫽ⵜ(Diⵜ·fi)⫹Ri i⫽1,…,N ri fi is the ith mass fraction defined as fi= , Ri is the bulk r

BT Dpl⫽ 6pmrpl

Here B is the Boltzmann constant, T the absolute temperature, m the coefficient of viscosity of plasma and rpl the radius of the platelet (苲1.1 µm). Using the above and an assumption of core body temperature of 37° for T, Dpl, the platelet diffusion coefficient is calculated as 1.7×10−13 m2 s⫺1, giving a Peclet number of approximately Pe=6×107. This compares favourably with values calculated by many other workers. If RBC augmentation is modelled then this diffusion coefficient value can increase by as much as two orders of magnitude. For the case presented here it is assumed that no aggregation occurs in the bulk fluid (plasma) and that the phenomenon of platelet adhesion to the wall is represented by a simple reaction boundary condition, where platelets are either ‘free’ or permanently adhered. Thus, the reaction mechanism of platelet adhesion to the wall is given by ⬘free activated platelets⬘

(2)

k

⫹wall → adhered platelets⫹wall with k, a surface forward reaction rate. As a first approximation we assume that the platelets become adhered at the surface at a rate proportional to their concentration at the wall. At this stage we consider only initial platelet adhesion and therefore the adhesion sites available will be considered to be constant over time. Adhered platelets are modelled as being part of the surface and, once adhered, stay in that state. We can thus equate mass diffusion to the wall with reaction at the adhesion surface and model the boundary phenomenon as rDpl

Fig. 1. Basic geometry of the axi-symmetric jet impinging on the flow cell lower wall.

(1)

∂fpl ⫽kfpl|wall ∂n

(3)

here n is a co-ordinate normal to the reacting surface. The modelling procedure has been split into two distinct areas, that of an analytical nature and a numerical procedure. The analytical model is described first.

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

2.2. Analytical model As a first step in the solution process we assume that axi-symmetric conditions exist and mass diffusion in the streamwise radial (R) direction is negligible to that in the axial (z) direction. Finally that the diffusion coefficient is spatially independent, thus invoking a classical boundary type analysis. Although in physiological conditions the red blood cell concentration does provide a mechanism for shear dependent diffusion of platelets the experiment of Affeld et al. [17] used platelet rich plasma containing no red blood cells. Since the platelets were pre-activated in Affeld’s experiment we assume the bulk fluid reaction of platelet activation to be effectively zero. With these assumptions, the constant density conservation equation for the mass fraction of platelets fpl can be written in the boundary layer type form [21] as ∂fpl ∂fpl ∂2fpl ⫽Dpl 2 u˜ ¯ ⫹v˜ ∂R ∂z˜ ∂z˜

(4)

Where u˜ and v˜ are the velocity components for the R and z directions respectively, Di is the Fickian diffusion coefficient of the ith species and N the total number of species participating within the domain. By choosing appropriate length and velocity scales, L and U respectively, such that v˜ u˜ u⫽ ;v⫽ ; U U z˜ R˜ R⫽ ;z⫽ L L

cous boundary layer should of the order of 5–10 times v bigger than the mass transfer boundary layer (⇒ ⱖ D 0.2). We may introduce v˜ a ‘friction’ velocity defined by

再 冎

v˜ ∗⫽

t˜ w(R) r

1/2

(6)

where t˜ w(R), is a wall shear stress function whose values are known a priori, and r the blood density. We can define a similarity variable, h given by {tw(R)}1/2(Pe Re)1/3 ⫽zb(R) h⫽z R

冋冕

册

∂fpl ∂fpl 1 ∂ fpl u ⫹v ⫽ ∂R ∂z Pe ∂z2 2

(5)

where Pe, is the Peclet number defined as UL Pe⫽ D The values of U and L are taken from the experimental work by Affeld et al. [17]. Their experiment consisted of a pair of axi-symmetric discs with a circular inlet pipe fixed to the top disc of diameter 670 µm. The platelet rich plasma flux provided a mean velocity of 16 mm s⫺1 through the inlet. L was taken as the inlet diameter and U as the mean velocity. The diffusion coefficients for cells moving in blood plasma are extremely low and thus the Peclet number, Pe, is correspondingly high. For this case the velocity boundary layer thickness is large compared to the species mass transfer boundary layer and the velocity profile can therefore be assumed to have a linear form. A reasonable estimate would be that the vis-

(7)

1/3

9 {tw(g)}1/2dg 0

where the non-dimensional wall shear stress is given by t˜ w(R) tw(R)⫽ rU2⬁ and the Reynolds number Re is defined as Re⫽

U⬁L v

In the case of Affeld et al.’s experiments Re=9 and we take this value as a constant throughout the following work. The non-dimensional species conservation equation using Eq. (7) becomes for the single species mass fraction of platelets fpl. dfpl d 2fpl ⫹3h2 ⫽0 dh2 dh

the non-dimensionalised form of the species equation is given as

303

(8)

A full derivation of the similarity variable definition is given in Appendix B and an excellent example of this type of transformation is given by Kestin and Persen [22], although in their case the application was of a thermodynamic nature. This type of transformation has been used for the simple case of a constant shear stress by Turitto and Baumgartner [16]. It should be noted that in contrast to Turitto and Baumgartners simple analysis the transformation variable defined by Eq. (7) is general in its form, the only restriction that the wall shear stress remain bounded throughout the domain (this can be thought of as the viscous boundary layer remaining attached to the surface). The resulting ordinary differential equation [Eq. (8)] can therefore be used in a variety of fluid flow cases where tw(R) is known a priori. By using Eq. (3) we can write a boundary condition at the surface of the form ∂fpl ∂fpl∂h ⫽ ⫽␬fplwall ∂z ∂h ∂z

␬⫽

kL D

(9)

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It has been postulated by Weiss [18] that the activation/adhesion of platelets is ‘controlled’ somewhat by the local fluid shear stress. Using this as an extension to the model, the reaction rate at the cell chamber surface can be given the simple linear functional representation of the form

␬(R)⫽␬0⫹at(R)

(10)

where t is non-dimensional shear stress at the wall and aⱖ0. The analysis is not necessarily restricted to this linear representation however it is used simply as an initial model. Finally a boundary condition modelling the species mass fraction at a large distance away from the surface is needed and we assume that far away from the stagnation point the concentration of species is a constant. Thus we have the following h→⬁; fpl(h)→fpl⬁

(11)

Heimenz [25] may be used to compare with the numerical simulation and found to be in excellent agreement. A plot of the non-dimensional shear stress is presented in Fig. 3 and from this both b(R) and the similarity variable h can be evaluated using Eq. (7). Clearly at the surface h=0 the normalised platelet concentration is a function of radial distance, so using Eq. (12) we find the concentration to vary in the form given below. fpl(0) ⫽ f¯ pl(0)⫽ fpl(⬁)

3b(R) 1 ␬⌫ ,⬁ +3b(R) 3

冉 冊

(13)

The above equation, using the surface boundary condition given by Eq. (9), provides a solution to the platelet flux onto the surface as a function of radial position and wall shear stress measured from the stagnation point streamline. 2.3. Numerical model

We are now in a position to solve the resulting Eq. (8) subject to the boundary conditions (9) and (11). In fact Eq. (8) is directly integrable and it is easy to show that the solution may be written as

fpl(h) ⫽ f¯ pl(h)⫽ fpl⬁

冕

冉 冊冢 冉 冊

3 1 ⌫ ,⬁ 3

3b(R)

1⫺ 1 ␬⌫ ,⬁ ⫹3b(R) 3

冣

r(u·ⵜ)u⫽⫺ⵜp⫹ (12)

h

3b(R)

3

e−g dg⫹

0

冉 冊

1 ␬⌫ ,⬁ ⫹3b(R) 3

冉 冊 冉 冊冕

In order to not only test the accuracy of the analytical solution but also as a precursor for more complex models the CFD model uses the full steady-state constant density non-dimensional momentum and conservation of species equations, written as

1 Here ⌫ ,⬁ is the Gamma function given by 3 ⬁

1 ⌫ ,⬁ ⫽ e−tt−2/3dt 3 0

Finally, in order to complete the development of the analytical solution, the form of b(R) has to be found. This was done using the fluid flow solution developed from the numerical model (see below). The non-dimensional shear stress at the wall was evaluated as a function of radial distance at a number of discrete points and interpolated using a polynomial form. There is no analytical solution for the stagnation axi-symmetric stagnation point flow for all values of the radial distance. However for small values of R the analytical result of

1 2 ⵜu Re

(14)

Here r is the density of blood, p the pressure and Re, the Reynolds number as defined in the analytical model. For the conservation of platelet species (again with constant density and the assumption of Fickian diffusion) we can write (u·ⵜ)fpl⫽

1 2 ⵜ fpl Pe

(15)

As in the analytical case we use a single species, that of activated platelets. The Reynolds number was evaluated using the conditions of Affeld et al.’s [17] experiment, giving Re=9, the Peclet number remains the same as in the analytical model. The geometry of the fluid domain is the same as the analytical case, an axi-symmetric stagnation point flow. The symmetry stagnation point streamline boundary requires that both the radial velocity, u, and the radial derivative of the axial velo∂v city, , are zero, whereas the upper and lower surfaces ∂R require no-slip conditions. The outlet boundary condition was set as a zero-stress condition. The boundary conditions for the velocity solution modelled the experimental conditions of Affeld et al. [17]. Fig. 2 shows the computational domain and the appropriate boundary

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

Fig. 2. Geometry of the computational domain and boundary conditions.

305

Fig. 3. Non-dimensional wall shear rate as a function of non-dimensional radial distance.

3.1. Constant surface reaction rate conditions used in the numerical solution for the momentum and mass conservation equations. For the species equation the variation of platelet concentration occurs across the mass transfer boundary which, due to the high Peclet number, is very small. Thus in the numerical solution, to avoid excessive numbers of mesh elements, the upper boundary is much less than the upper surface of the perfusion cell used in the numerical algorithm, giving a reduced domain for the conservation of species. A simple reactive boundary condition similar to that given by Eq. (9) is used at the surface. Since the momentum and species equations are decoupled the momentum equations were solved first and the velocity components were then used in the solution of the platelet species concentration. Both momentum and species conservation equations were solved using the finite element commercial software code FIDAP [24], using the Galerkin weighted residual method. The domain was discretised into simple 9-noded quadralaterals, where the velocity was of a quadratic formulation and the pressure determined through a piecewise continuous approximation.

Fig. 4 shows the normalised platelet concentration at the wall of the flow cell as a function of radial distance for both numerical and analytical models where ␬, a constant reaction rate (a=0), takes various values and the Peclet number Pe=107. The agreement between these two models is excellent. The platelet concentration is maximum at the stagnation point and thence decreases out to R=2.0. As ␬ the surface reaction rate is increased the maximum value of the platelet concentration decreases. Fig. 5 shows the corresponding platelet flux, q(R), to the surface for two constant reaction rates. For all cases considered the flux to the wall is given by Eq. (10). Here again the agreement between numerical and analytical solutions is very good. The flux maximum lies, as in the platelet concentration case, at the stagnation point, R=0. The flux as a function of radial co-ordinate is monotonic decreasing and the reaction rate has to change considerably to make an appreciable difference to the flux. As expected the higher the reaction rate the higher the flux of platelets to the wall. 3.2. Non-constant surface reaction rate

3. Results Fig. 3 shows the variation in wall shear stress as a function of radial distance evaluated from the numerical model This compares very well with the analytical solution given by Glauert [23]. As can be seen, in the neighbourhood of the stagnation point the function is linear and as the radial co-ordinate increases the shear stress goes through a maximum value. It then decreases monotonically tending asymptotically to zero as R tends to infinity. However, more importantly, for the case presented here the wall shear stress has a stationary point or maximum at R=0.4.

In terms of platelet adhesion the flux is of crucial importance, since for initial adhesion conditions the number of platelets adhering is simply the product of the initial platelet flux and time. Fig. 6 shows non-dimensional platelet flux, derived from the analytical solution, as a function of radial distance for various values of a with ␬0=10.3. As the value of a increases the maximum platelet flux approaches the origin. Fig. 7 compares the analytically derived platelet flux with the experimental data of Affeld et al. [17]. In this case the platelet flux has been evaluated with a non-constant surface reaction rate given by Eq. (11) with parameters ␬0=0 and a=100. Geometrical variations will clearly give rise to variations in the wall shear stress as a function of streamwise

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Fig. 4. Normalised platelet concentration as a function of radial distance for constant reaction rate at the adhesion surface. Symbols are the analytic solution, lines are the numerical solution (䉫) ␬0=323; (+) ␬0=188; (⫺) ␬0=67.6; (䊐) ␬0=20.9; (×) ␬0=10.3).

Fig. 5. Non-dimensional platelet flux, q, as a function of radial distance at the flow chamber surface for two constant reaction rates. Symbols are the analytic solution, lines are the numerical solution (+) ␬0=67.6; (䉫) ␬0=10.3).

Fig. 7. Platelet flux (␬0=0 and a=100) compared with Affeld et al. data [17] and wall shear stress t(R) indicating reasonable agreement for R⬍0.4 but less so for R⬎0.4.

Fig. 6. Non-dimensional platelet flux, q(R), as a function of nondimensional radial distance for various values of a (0, 10, 50, 100, 200, 500, 1000), with ␬0=10.3.

co-ordinate. Using the Faulkner–Skan solutions for viscous boundary-layer flow past a wedge [25] the wall shear stress may be evaluated for various wedge angles. These angles may be parametrised in terms of a single value m, as shown in Fig. 8.

Fig. 8. Graphical relationship between the parameter m and the bifurcation angle.

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

The wedge angle bp is given as bp⫽

2m m+1

(16)

Using Eq. (13) and the Faulkner–Skan definition of the boundary-layer velocity profile we can evaluate the similarity transformation function, as a function of streamwise co-ordinate x, so that b(x)⫽

冋

册

Pe Re A(m+1) 12

1/3

x(m−1)/2

A is a constant taking into account the magnitude of the free-stream velocity and the kinematic viscosity. With this definition and assuming that the adhesion rate is a linear function of wall shear stress as set out by Eq. (10) then the flux at the surface of the wedge can be written in the general form of x(3m−1)/2 q(x,m)⫽ 1+axm Fig. 9 shows the platelet flux as a function of streamwise co-ordinate (measured from the stagnation point) for various values of m.

4. Discussion We can easily show that b(R) is a monotonic decreasing function over the domain of interest and, for a constant reaction rate, the platelet flux to the surface are also decreasing functions. For this case it is easy to show that the maximum platelet flux is at R=0, since by definition the flux is simply a multiple of the platelet concentration at the wall. Hence, from the analysis it can be seen that, for all values of the constant reaction rate ␬ presented, the platelet concentration and, more importantly, the flux is a maximum at the stagnation point, R=0. The species

Fig. 9. Platelet flux q(x) as a function of downstream co-ordinate for various values of bifurcation angle parameter m.

307

diffusion boundary layer is of minimum depth at the stagnation point (and hence the platelet flux is a maximum). In fact this mass transfer layer is of an approximate constant depth, proportional to Pe1/3, throughout the linear portion of the shear stress function t(R), 0ⱕRⱕ0.25. The result of a maximum of platelet flux at the stagnation point or high rate of adhesion at this point is in opposition with the results of platelet deposition experiments using stagnation point flow [9,11,17,26]. This leads to the important conclusion that, on comparing with experimental results of platelet adhesion in stagnation point flow, platelet adhesion at the lower wall of an experimental flow cell cannot be successfully modelled using a constant reaction rate function, even though both convection and diffusion are fully taken into account. However when the wall reaction rate is not constant then both concentration and flux profiles at the wall change considerably in comparison to those for constant surface reaction. Since the analysis is general in formulation the same may be said for constant flow conditions and any other 2D flow profile as long as the viscous boundary layer stays attached to the adherent surface. The platelet flux profile for a shear rate dependent reaction rate is significantly different from the case of constant reaction rate. Fig. 6 presents profiles of platelet flux, using Eqs. (10) and (13), for increasing values of the linear coefficient a ranging from 0 to 1000. This figure shows that the maximum platelet flux now occurs at some non-zero distance from the stagnation point. Additionally as a increases the reaction rate increases and the position of maximum flux tends towards the stagnation point, R=0. The maximum platelet adhesion occurs at some position downstream of the stagnation point but still upstream of the maximum of the wall shear stress. We can easily show (see Appendix A for proof) that, for reaction rate which is a linear function of wall shear rate the value of R where the platelet concentration is a minimum, Rp min, is greater than the radial position of maximum shear Rt max, so that 0⬍Rt max⬍ Rp min. The shear rate maximum occurs for R=0.4 as shown by Fig. 3 (and evaluated by simple numerical root finding techniques). Fig. 6 shows that the platelet flux maximum occurs at a value of R less than the maximum shear rate. We can show from the analytic solution (see Appendix A for proof) that the position of maximum platelet flux, R=Rf max, for a reaction rate that is modelled by a linear function of wall shear rate, is such that 0ⱕRf maxⱕRt max, i.e. at a radial position less than that of maximum shear rate. This is in accordance with the experimental results of Affeld et al. [17], where the maximum platelet deposition also occurred upstream of the maximum shear rate and also in accordance with the evidence laid out by Tippe et al. [12]. The assumption that normal flow components to a wall are required for

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platelet deposition proposed by Wurzinger et al. [9] are shown to be at least too simplistic and at most incorrect since using this hypothesis one would expect that the stagnation point streamline to be the position of maximal platelet flux, whereas experiments show otherwise. Earlier work indicated that the adhesion process may be diffusion controlled meaning that the adhesion rate ␬ is large compared to other phenomena. From the analytical solution we can show that as a, the coefficient in the linear approximation for the surface reaction ␬ given by Eq. (10), grows larger then the position of maximum flux and hence maximum platelet adhesion moves towards the stagnation point. Again this is given in detail in Appendix A. if we look at the limit of the platelet flux as the reaction rate becomes increasingly large then we have using Eqs. (9) and (13) that the platelet flux is limited to a finite value as shown below.

lim q(R)⫽lim

␬→⬁

冦冉

␬→⬁

冧

3b(R) 3␬b(R) ⫽ ⌫ 1 ␬⌫ ,⬁ +3b(R) 3

冊

(17)

Thus for large adhesion rate the maximum platelet flux, for any functional form of ␬, appears at the stagnation point contrary to experimental evidence. This therefore tends to suggest that the surface reaction rate may be smaller than at first imagined. Given that from the experimental evidence the maximum flux occurs away from the stagnation point streamline we surmise that the adhesion rate cannot be substantially large. Again from experimental evidence, the adhesion of platelets is quite small (or even zero) in the neighbourhood of the stagnation point the surface reaction rate, and tends therefore to suggest that ␬0 cannot be large. It is recognised however that ␬ could be large enough such that the adhesion mechanism is, in effect, diffusionally controlled by a suitable choice of the parameter a. Affeld et al. [17] hypothesised that the positioning of maximal platelet deposition was due to thrombin diffusing away from the adhered platelets on the surface. This thrombin diffusion does occur physiologically, however in their experiment pre-activated platelets were used so that any thrombin diffusing from the surface would possibly have only a small influence on whether platelets adhere since they are already activated. The model presented here does not include thrombin diffusion or reaction, however it is accepted that thrombin concentration profiles could provide an additional phenomenon which, along with non-constant surface reaction rates, explains the variation in platelet adhesion as a function of radial position. It is important to note that the non-dimensional surface reaction rate, ␬, is not dependent on any special

functional relationship of shear rate. Thus, the platelet flux maximum could in principle, be positioned at the point of maximum shear rate by a judicial choice of a reaction rate which is a function of the shear rate itself, ␬(t(R)). The determination of this functional relationship is left for further investigation. However, as a start in this process Fig. 7 compares Affeld’s data with a specific solution derived from a non-constant surface reaction rate [given by Eq. (10)]. Here the parameter values for a and ␬0 have been chosen such that the platelet flux maximum occurs at a similar position to that found by Affeld. Although the present model can provide good agreement with experiment for adhesion in the neighbourhood of the stagnation point streamline and someway downstream the analysis indicates a larger adhesion than that given in the experiment. The significant number of adhered platelets after some time into the experiment may have affected the local flow dynamics in the small mass transfer layer near the adhesion surface. In addition the adhesion process is essentially time dependent and the adhered platelets may have an influence on the adhesion of other platelets which are very close to the adhesion surface but not yet adhered. In addition even though only platelet rich plasma has been used in the experiment the reaction mechanisms for activated platelets may be relatively complex and this is not included in the present model. We note that the convection of thrombin is certainly important for the case of modelling the activation of platelets and their subsequent adhesion. However, once the platelets are activated and for situations where diffusion and reaction at the adhesion surface are not too dissimilar in magnitude, it seems likely that the model of platelet adhesion reaction becomes the important parameter in determining the flux distribution; especially if it varies as a function of wall shear stress. This also seems to have been hypothesised by Leonard et al. [15] based on experiments done by Monsler, Morton and Weiss [27]. Although no experimental evidence seems to exist which explicitly measures the surface reaction rate for platelet adhesion Turitto and Baumgartner [16] state that since adhesion is dependent on flow parameters then a diffusion controlled mechanism is expected. The result that the reaction rate depends on wall shear stress may have interesting implications for the endothelial cells, which line the arterial walls and are known to respond to wall shear stress. This line of investigation is left for further work. In the adhesion of cells (specifically monocytes) to vascular endothelium, it is observed that these cells roll along the endothelial surface before adhering. When platelets roll, they do so as an increasing function of shear rate, since it is the dynamic fluid pressure (and viscous shear) acting on the upstream cell face that provides the external force. This being the case platelets cannot roll away from the stagnation point streamline

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

(or for that matter within a small neighbourhood of the stagnation point streamline) since the shear stress (and hence shear rate) is vanishingly small. Platelets impinging on the surface in the neighbourhood of the stagnation point streamline will either stay put or roll only a very small amount. However platelets diffusing and impinging on the surface in the neighbourhood of the maximum shear stress will therefore roll downstream before adhering permanently. This is contrary to experimental evidence, where it is found that the maximum platelet flux lies upstream of the maximum shear rate. Bearing these points in mind it is highly unlikely that rolling (although clearly a real phenomenon) is the dominant mechanism in the radial distribution of adhered platelets. When comparing the flux values plotted in Fig. 9 with those for the axi-symmetric flow it should be noted that the velocity profiles, assumed for the outer inviscid flow in the Faulkner–Skan solution, are only valid for small values of the free-stream co-ordinate x (x⬍1.5). However, some interesting phenomena are worth high1 lighting. For values of m⬍ there exists a singularity in 3 the platelet flux at the stagnation point x=0 and the flux 1 is a monotonic decreasing function. For m= the flux 3 1 varies only slowly and for m⬎ the singularity no longer 3 exists but the flux is zero at x=0 and the flux increases indefinitely. This indicates a rich area of investigation for a variety of flows where the convection, diffusion and surface reaction interact together. The group at Leeds is currently investigating a range of geometries modelling physiological conditions. Finally, the similarity transformation given by Eq. (7) is a general formulation of that originally given by Turitto and Baumgartner [16]. It can therefore be used in a whole variety of 2D as well as axi-symmetric flow cases where the wall shear stress is known a priori.

309

maximum platelet deposition occurs at some distance downstream of the stagnation point. However if the wall reaction rate is chosen to be linearly dependent on the wall shear stress then the analysis (and the numerical model) shows that the maximum platelet flux occurs downstream of the stagnation point, correctly modelling experimental evidence. Using the analytic representation of the platelet flux and concentration along the wall proofs are given that show this relationship between wall shear stress and platelet concentration and platelet flux. Such that 1. for constant wall reaction rate the maximum platelet flux and concentration occur at the stagnation point. 2. for a wall reaction rate, linearly dependent on wall shear stress, the platelet flux occurs at a radial position less than the position of maximum wall shear stress but greater than the stagnation point streamline. 3. for a wall reaction rate, linearly dependent on wall shear stress, the platelet concentration occurs at a radial position greater than the position of maximum wall shear stress. Comparison of the presented model analysis and experimental evidence indicates that it is highly unlikely that rolling (although clearly a real phenomenon) is the dominant mechanism in the radial distribution of adhered platelets. The result that the reaction rate depends on wall shear stress may have interesting implications for the endothelial cells, which line the arterial walls and are known to respond to wall shear stress. On the basis of the excellent agreement between numerical and analytical results the approximation of high Peclet number and the associated model is considered to be an excellent framework to investigate further reaction and species models in complex flows especially near reattachment points.

Acknowledgements 5. Conclusions A mathematical model of platelet adhesion, using a high Peclet number approximation in a stagnation point flow chamber, has been presented which incorporates convection, surface reaction and diffusion. Excellent agreement exists between the analytical solution and that obtained by the numerical integration of the full Navier– Stokes equations and decoupled conservation of species equations. It has been shown that for a constant wall reaction rate modelling platelet adhesion the maximum platelet flux occurs at the stagnation point streamline. This is in direct contrast to that found in experiment where the

We would like to acknowledge the kind support of the British Council and The Leslie and Ann Reid University Scholarship awarded to S. Thomas.

Appendix A We show that given ␬(R)=␬0+at˜ w(R), a⬎0 then the position of minimum platelet concentration is 0⬍Rt max⬍Rp min Proof. Since ␬ is a linear function of tw then

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d␬ dt˜ w =a and this derivative cannot be zero other than dR dR at some non-zero distance from the stagnation point streamline by virtue of Eqs. (10) and (13). In addition the following criteria hold true; 0ⱕfⱕ1, ∀ R 苸[0,Rmax]

(A1)

then for a stationary point we have that

冋

b⬎0, ∀ R 苸[0,Rmax] bRⱕ0, ∀ R 苸[0,Rmax] t˜ wR(R)⬍0, R 苸(Rtmax,Rmax] bR␬(1−f)⬍0 and b␬Rf=−bRk(1−f)

⇒␬Rⱕ0

(A2) (A3)

d␬ db ␬R⫽ ; bR⫽ .etc. dR dR Now the condition given by Eq. (A3) can only occur for values of R which satisfy

␬R(R)⫽at˜ Rw(R)ⱕ0⇒Rp min苸(Rt max,Rmax]

(A7)

t˜ wR(R)ⱖ0, R 苸[0,Rt max]

From equation [13] we have that

⫽bR(1⫺f)ⱕ0

(A6)

␬⬎0, ∀ R 苸[0,Rmax]

thus

3

冊

⇒b␬Rf⫹bR␬(1⫺f)⫽0

db ⱕ0, ∀ R 苸[0,Rmax] dR

1 ,⬁ 3

冉

0ⱕfⱕ1, ∀ R 苸[0,Rmax]

where Rmax is the upper bound for the domain of the polynomial approximation to the wall shear rate. We can also show, by graphical means, that b(R) is a monotonic decreasing function of R. Hence

冉 冊

册

3␬ f dq b ⫺ [3b ⫹⌫␬R] ⫽0 ⫽␬ f⫹ dR R 3b⫹k⌫ R 3 R

Now

dt˜ w(R) ⬍0, R苸[Rt max,Rmax] dR

f␬R⌫

(A5)

f ⇒(3b⫹␬⌫)␬Rf⫹3␬ bR⫺ [3bR⫹⌫␬R] ⫽0 3

db ⱕ0, ∀ R 苸[0,Rmax] dR dt˜ w(R) ⱖ0, R苸[0,Rt max] dR

q(R)⫽␬f|wall

(A8)

⇒␬R⬎0 Since the reaction rate is a linear function of the wall shear rate then the derivative of reaction rate can only be positive for values of R where the shear rate gradient is positive, that is where 0ⱕR⬍Rt max hence 0ⱕRq max⬍Rt max. 䊐

(A4)

thus 0⬍Rt max⬍Rp min

Appendix B

䊐 We now look at the platelet flux, where we show below that the maximum platelet flux for a reaction rate that is modelled by a linear function of wall shear rate occurs for R=Rq max such that 0ⱕRq max⬍Rt max Given that the platelet flux to the wall is written as q(R)=␬f|wall then the maximum value of this function qmax occurs for a value of R=Rq max such that 0ⱕ Rq max⬍Rt max Proof. The platelet flux to the wall is defined as

Transformation of the conservation into the o.d.e. by similarity variable. We have the non-dimensionalised conservation equation for the platelet mass fraction. u

∂fpl ∂fpl 1 ∂2fpl ⫹v ⫽ ∂R ∂z Pe ∂z2

(B1)

The velocity profile is assumed to be of a linear form since the ratio of mass transfer boundary layer thickness to the viscous boundary layer thickness is small. A stream function ␫ is thus defined as 1n˜ 2 ˜ (R,z)⫽ ∗z˜2 y 2n

(B2)

T. David et al. / Medical Engineering & Physics 23 (2001) 299–312

References

where v* is the ‘friction’ velocity defined by v˜ ∗⫽

再 冎 t˜ w(R) r

1/2

(B3)

t˜ w(R), a wall shear stress function whose values are known a priori. If t˜ w(R) is non-dimensionalised by rU2⬁ then the non-dimensional velocity components can then be written as u⫽v˜ 2∗z˜

1z˜2d(v˜ 2∗) z˜2d(v˜ 2∗) U⬁L 2 ⫽ ⫽v∗z Re; v⫽⫺ Re v 2 v dR˜ 2 dR

(B4)

We define two new variables thus, x⫽R; ␵⫽v∗zRe

(B5)

and substituting the velocity expressions this into the convective part of the conservation equation we have ∂f Re2 ∂2f ␵ ∂f ∂fi ∂fi u ⫹v ⫽v2∗z Re ⫽ v∗2 2 ⇒ ∂R ∂z ∂x Pe ∂␵ v∗∂x ⫽

(B6)

Re2∂2f Pe ∂␵2

By transforming again through the variables s⫽␵

冉 冊 冕 Pe Re2

1/3

(B7)

x

c⫽ v∗dx x0

and h⫽

s (9c)1/3

(B8)

Upon substituting into Eq. (B6) we have, with a little algebra, d2fpl dfpl ⫹3h2 ⫽0 dh2 dh the similarity variable h can be written now in terms of the initial variables thus, h⫽z

{tw(R)}1/2(RePe)1/3 ⫽zb(R) R

冋冕

9 {tw(g)}1/2dg 0

册

1/3

311

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