Effect of Glycocalyx on Red Blood Cell Motion in Capillary Surrounded by Tissue

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Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 9-9466 Vol. 4, Issue (June 009) pp. 4 48 (Previously, Vol. 4, No.

1 Available at Appl. Appl. Math. ISSN: Vol. 4, Issue (June 009) pp (Previously, Vol. 4, No. ) Applications and Applied Mathematics: An International Journal (AAM) Effect of Glycocaly on Red Blood Cell Motion in Capillary Surrounded by Tissue Rekha Bali, Swati Mishra Department of Mathematics Harcourt Butler Technological Institute Kanpur, India P. N. Tandon Res. No. 9-9, Karmcharinagar Bareilly, INDIA Received: June, 008; Accepted: March, 009 Abstract The aim of the paper is to develop a simple model for capillary tissue fluid echange system to study the effect of glycocaly layer on the single file flow of red cells. We have considered the channel version of an idealized Krogh capillary-tissue echange system. The glycocaly and the tissue are represented as porous layers with different property parametric values. Hydrodynamic Lubrication theory is used to compute the squeezing flow of plasma within the small gap between the cell and the glycocaly layer symmetrically surrounded by the tissue. The system of non linear partial differential equations has been solved using analytical techniques. The model predicts that decrease in glycocaly thickness reduces the aial velocity of plasma and the resistance to flow increases in presence of glycocaly. Key words: Glycocaly; Capillary Blood low; Pressure; Resistance to low; Velocity Profile MSC (000) No.: 76Z0, 740 4

2 AAM: Intern. J., Vol. 4, Issue (June 009) [Previously, Vol. 4, No. ]. INTRODUCTION Microscopic observations identified a thin, negatively charged macromolecular layer adjacent the luminal surface of vascular endothelial surface of the capillary. This layer was named as glycocaly and was hypothesized to affect the transport properties of the capillary wall. Glycocaly, a layer of macromolecules bounded or adsorbed to the endothelial surface, may retard plasma motion in a zone adjacent to the capillary wall [Sugihara-Seki & Bingmei (00)]. Regulation of the eclusion of blood from this relatively thick endothelial region could contribute, not only to control of capillary red blood cell filling the space and oygen supply to tissue cells, but also to the controlled modulation of transcapillary solute echange and tissue hydration. The state of understanding of the single file motion of red blood cells through cylindrical tubes is relatively mature, beginning with the seminal works of Lighthill (968), itzgerald (969 a, b) and Bernard et al. (968) and cumulating in the models of Zerda et al. (977) and Secomb et al. (986), which are faithful to the constitutive relationships and well characterized the red cell membrane. However, eperimental evidence mounting over the past 0 years has begun to cast doubt on the applicability of these models to capillary blood flow in vivo. Several eperimental studies during the period suggest that the flow resistance measured in vivo was about twice that from estimates based on measurements in glass tubes [Lipowsky et al. (978, 980), Pries et al. (994)]. Although, several mechanisms were considered, the most likely eplanation, as demonstrated convincingly by recent eperiments of Pries and Secomb (997) that the glycocaly is, primarily responsible for the difference [Klitzman & Duling (979) and Desjardins and Duling (990)]. Several theoretical models have recently been presented that are generally consisting with this new concept of microvascular resistance and reduction of capillary tube hematocrit [Damiano (998), Damiano et al. (996, 004), Secomb et al. (998, 00) and Wang and Parker (99), Srivastava (007)]. These authors assumed binary miture theory and account for deformability of the red cells as they travel in single file through capillaries of roughly 6m diameter. They further assume the eistence of a thin lubricating layer adjacent to the capillary wall. These models have not discussed the effect of glycocaly on flow characteristics of single file flow of red cell in capillaries surrounded by tissue and the fluid movement into and out of the tissue through the glycocaly layer. Therefore, our aim is to study the effect of glycocaly on blood flow in very narrow capillary lined with uniform thickness of porous layer (Glycocaly) which is surrounded by tissue. In this paper, we have considered the glycocaly as a porous layer. The tissue is also considered as a porous matri. Darcy s law of fluid flow is assumed to govern the flow in tissue as well as in glycocaly. The shape of red blood cell is assumed to be aisymmetric. Lubrication theory is used to compute the flow of plasma around the cell. Single file flow of red blood cell is considered and cell to cell interactions are neglected. We have obtained the results for resistance to plasma flow, pressure, normal and aial component of velocity in very narrow capillary.

3 6 Bali el al.. ORMULATION O THE PROBLEM We have considered the channel version (igure) of an idealized Krogh capillary tissue cylinder as the geometrical representation of the capillary beds. The interior surface of capillary is lined with a glycocaly layer, which is assumed as a porous matri. Red blood cell is assumed aisymmetric. Single file flow of red blood cell is considered. Hydrodynamic lubrication theory is used to describe the motion of plasma around the cell. Gap between the cell and capillary wall is given by h h 0, () where is the shape parameter. The region is divided into three sub regions (i) luid ilm Region, d r h. (ii) Glycocaly Layer, 0 r d. (iii) Tissue region, H r 0. We introduce the following non dimensional scheme:,, y y, U 0 u u, U 0 v v, U 0 U 0 Re,,, d d,. To write the governing equations for flow in three regions as given below: (A) luid ilm Region In between the red cell and the glycocaly surface there is a thin lubricating layer of plasma. Therefore, introducing lubrication theory, the governing equations of motion and continuity for two dimensional flow of plasma (considered as Newtonian fluid) may be written as follows: u Re, () y y 0, () u v 0, y (4)

4 AAM: Intern. J., Vol. 4, Issue (June 009) [Previously, Vol. 4, No. ] 7 where u and v are the velocity component along aial and transverse directions and is the viscosity of plasma in the capillary. P is the pressure in fluid film region. (B) In Glycocaly and Tissue Region The flow of viscous fluid in porous matrices is governed by Darcy s Law. Therefore aial and normal component of velocity are given as: ui i ki Re and vi i ki Re, () y ki where ki, i = stands for the glycocaly layer and i = stands for tissue region. u i and v are the aial and normal velocities of the fluid in the porous matri of glycocaly and tissue. i Pressures in the two porous regions satisfy the Laplace equation. Thus,, the pressure in the glycocaly layer of thickness d and, is the pressure in the tissue region of thickness H satisfies the Laplace equations: 0. 6) i Boundary and Matching Conditions u at y h (7a) u u y at y d (7b) v 0 at y h (7c) 0 y at y H (7d) 0 at (7e) at y d (7f) k k y y at y 0 (7g) at y 0, (7h) where Uo is the cell velocity, is the slip parameter, 0 is the reference pressure, k and k are the permeability of glycocaly layer and tissue. and are the partition coefficients, is the minimum gap width

5 8 Bali el al. ; U 0 Re ; y y ; ; U 0 ; ; u u ; U 0 d d ; v v ; U 0. (8) SOLUTION O THE PROBLEM Capillary Region Solving equation of motion and equation of continuity with the help of boundary condition 7(a) and 7(b) we get the solution for velocity distribution in the capillary region as given by where y u Re Ay B, (9) Re d d A, h d B d h d h d h Re h d hd d h d. Porous Region Pressure in porous tissue and glycocaly are governed by the Laplace equation 0. (0) i Thus,, the pressure in the glycocaly layer of thickness d, satisfies the equation y 0. () Integrating equation () with respect to y over the layer thickness d and using boundary condition (7g) y yd d k dy k 0 y. () y0

6 AAM: Intern. J., Vol. 4, Issue (June 009) [Previously, Vol. 4, No. ] 9 Similarly, we integrate the Laplace equation in tissue layer of thickness H and using condition (7d) dy y 0 H 0 y. () rom () and () we get dy k k dy y 0 H d 0 d y. (4) If the layer thickness d and H are assumed to be small, equation (4) reduces to d y H k k d y. () Introducing aial velocity in equation of continuity and using the condition at the interface (7f) pressure distribution in capillary region is obtained as C 4 C 8, (6) where C L L L C 8 L L L 4 8 7

7 40 Bali el al. k Re d 4 H d k d d 4 d d Re Re d 8 4d d 4 Re 7 d Re 8 d Red 6 Red 6(d ) 7 d d 4 8 d æ+s4- ö 9 = ç çè d ø d d 6 d d Resistance to blood flow is given as * 0 L L L R, (7) Q where Q is the volumetric flow rate [Guyton and Hall (996)].

8 AAM: Intern. J., Vol. 4, Issue (June 009) [Previously, Vol. 4, No. ] 4 Normal component of velocity is obtained as Re v 6 A y h B y h y h, (8) where A a a a a B a h a h a h Re a 4 4 a a 6 4 a a h hh a a h h Re d a Re d d a d d a 4 d.. RESULTS AND DISCUSSIONS The role of glycocaly has been described here for blood vessels when red cells flow in a single file. Their effects on pressure distribution, resistance to flow, aial velocity and normal component of velocity have been presented through figure to 9 as discussed below. The presence of the glycocaly reduces the crossection available for flow of red cells. The additional energy may be dissipated due to narrowing of the lubrication layer. igures and depict the variation of pressure distribution and flow resistance for different values of glycocaly layer thickness d. These figures demonstrate that both, after attaining a maimum value at the origin, decreases sideways symmetrically. This is due to the assumption of geometrical symmetry and reduction of the gap between the cell geometry and the capillary wall. igure 4 and igure present the variation of normal and aial velocities for different values of glycocaly thickness. Normal velocity as well as aial velocities both decrease with increasing values of the thickness. Both after attaining maimum value at the origin decrease sideways symmetrically and both the results support each other. Similar results have been observed by Secomb and Hsu (997). This layer slows down the plasma flow due to the movement of the

9 4 Bali el al. fluid from the gap into the layer. The aial velocity profiles of the fluid in the lubricating layer in presence of glycocaly layer at its luminal surface have also been shown through igure. The glycocaly acts as a transport barrier. urther work is needed to eplain the effects of glycocaly on nutritional transport to the cells of the tissue. The present model also studies the effect of various shapes of the red blood cell through the variation of parameter. The effect of red cell shape parameter has been discussed through figures 6 to 9. Aisymmetric shape of red cell is assumed throughout in the model. In general, red blood cell shapes are not aisymmetric but this has little effect on flow behavior. Pressure distribution and the Resistance to flow have been presented in the igure 6 and igure 7. Pressure in fluid film (lubrication layer) increases (igure 6) and resistance to flow also increases (igure 7) with increasing values of. Normal component of the fluid velocity increases at the capillary-tissue interface as increases. One may also observe that as increases, the red cell gets elongated and lubrication layer thickness decreases. Without the glycocaly, the red cell almost fills the gap width. The presence of the glycocaly leads to longer and narrower red blood cell shapes, and the width of Lubricating layer changes with. igures 8 and 9 represent the variation of normal and aial velocity of plasma in capillary. Results support the observation in igures 6 and 7 for different values of. 4. Concluding Remarks Introducing the concept of lubrication and the forming a wedge in between porous glycocaly layer and the assumed shapes of red blood cell, this study presents the effects of glycocaly layer on physiological parameters of the model. The results support the eperimental findings of various researchers [Damiano et.al. (996); Damiano (998); Secomb et.al. (998, 00); Wang and parker (99)] Decreasing of the fluid flu into the tissue simultaneously decreases the nutritional transport and oygen supply to the tissue cells. This would form the basis for further study of coupled diffusion in tissue in presence of glycocaly layer on inner side of the capillary. Acknowledgement The authors gratefully acknowledge the suggestions of the reviewers of original manuscript of the paper for better presentation.

10 AAM: Intern. J., Vol. 4, Issue (June 009) [Previously, Vol. 4, No. ] 4 REERENCES Barnard, A.C., Lopez, L. & Hellums, J.D. (968). Basic theory of blood flow in capillaries. Micro vascular research, vol., pp. -4. Christafakis, A. et al. (009) Modelling of two phase incompressible flows in ducts. AMM. Vol., issue, pp. 0-. Damiano, E.R., Duling, B.R., Ley, K. & Skalak, T.C. (996). Aisymmetric pressure-driven flow of rigid pellets through a cylindrical tube lined with a deformable porous wall layer. J. luid Mech., vol. 4, pp Damiano, E.R. (998). Blood flow in micro vessels lined with a poroelastic wall layer. In Poromechanics (ed. J.. Thimus, Y. Abouseiman, A.H.D. Cheng, O. Coussy & E. Detournay), pp , Balkema, Rottderdam. Damiano, E.R., Long, D.S., El-Khatib,.H. and Stace, T.M. (004). On the motion of a sphere in a Stokes flow parallel to a Brinkman half-space. J. luid Mech., vol. 00, pp Desjardins, C. & Duling, B.P. (990). Microvessel hematocrit: measurement and implications for capillary oygen transport. A.J. Physiol., vol., pp. H494-H0. itzgerald, J.M. (969a). Mechanics of red cell motion through very narrow capillaries. Proc. Roy. Soc. Lond. B. vol. 74, pp itzgerald, J.M. (969b). Implication of a theory of erythrocyte motion in narrow capillaries. J. Appl. Physiol. Vol. 7, pp Klitzman, B. & Duling, B.P. (979). Microvascular hematocrit and red cell flow in resting and contracting striated muscle. Am. J. Physiol., vol. 7(Heart circ. Physiol. 6): pp. H Lighthill, M.J. (968). Pressure forcing of tightly fitting pellets along fluid filled elastic tubes. J. fluid Mech., vol. 4, pp. -4. Lipowsky, H.H., Kovalcheck, S. & Zweifach, B.W. (978). The distribution of blood rheological parameters in the microvasculature of cat mesentery. Circ. Res., vol. 4, pp Lipowsky, H.H., Usami, S. & Chein, S. (980). In vivo measurements of apparent viscosity and microvessels hematocrit in the mesentery of the cat. Microvasc.Res. vol. 9, pp Pries, A.R., Secomb, T.W., Gessner, T., Sperandio, M.B., Gross, J.. & Gaehtgens, P. (994). Resistance to blood flow in microvessels in vivo. Circ. Res., vol. 7, pp Pries, A.R. & Secomb, T.W. (997). Resistance to blood flow in vivo: rom poiseuille to the in vivo viscosity law. Biorheology, vol.4, pp.69-7 Secomb, T.W., Skalak, R. Ozkaya, N. & Gross, J.. (986). low of aisymmetric red blood cells in narrow capillaries. J. luid Mech., vol. 6, pp Secomb, T.W., Hsu, R., 997. Resistance to blood flow in nonuniform capillaries. Microcirculation. 4, Secomb, T.W., Hsu, R. & Pries, A.R. (998). A model for red blood cell motion in glycocaly lined capillaries. Am. J.Physiol. Heart circ. Physiol., vol.74, pp. H 06-H0. Secomb, T.W., Hsu, R. & Pries, A.R. (00). Motion of red blood cell in a capillary with an endothelial surface layer: effect of flow velocity. Am. J. Physiol. Heart circ. Physiol., vol. 8, pp. H69-H66. Srivastava, V.P. (007). A theoretical model for blood flow in small vessels. Application and Applied Mathematics vol., no., pp. -6. Sugihara-Seki, M. and Bingmei, M.u (00). Blood flow and Permeability. luid Dynamic Research, vol.7, pp. 8-.

11 44 Bali el al. Wang, W. & Parker, K.H. (99). The effect of deformable porous surface layers on the motion of a sphere in a narrow cylindrical tube. J. luid Mech., vol. 8, pp Zerda, P.R., Chien, S. & Skalak, R. (977). Interaction of viscous incompressible fluid with an elastic body. Computational methods for fluid solid interaction problems, ed. T. Belystschko and T.L. Geers, New York. American Society of Mechanical Engineers, pp. 6-8.

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