Hydrodynamics in the lumen: Triphasic model

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Chapter 3 Results. Biphasic model

We now present results for numerical simulations of the elastohydrodynamics of an EGL-coated microvessel containing a white blood cell. A two-dimensional case is considered and the vessel is represented as a wavy-walled channel. We consider two vessel shapes: symmetric (varicose) and asymmetric (sinuous). Simulations are performed for di erent cell positions within the lumen as well as for cell tracking when the cell starting position is at the inlet.
We examine the impact of the vessel geometry upon the creation of vortices within the EGL region, uid ux into the EGL and motility of the cell. Simulations provide us with uid and elastic shear stress values exerted upon the vessel wall. This information represents a particular interest since it may explain how the EGL transmits mechanical signals to the underlying endothelial cells.
Inlet and outlet regions have a straight wall geometry. This is done to apply the ow bound-ary condition correctly, since an analytical solution is known only for straight wall geometry. As a result, we specify the inlet/outlet velocity boundary condition having the form corresponding to the solution for a straight wall channel (see Appendix A.2).
Parameter values were chosen to be broadly representative of the movement of a cell in a capillary. Hence, the vessel radius was chosen to be H = 5 m, representative of a capillary. We consider a spherical particle having radius R = 2:5 m, which is characteristic of a small lymphocyte. The uid viscosity is assumed to be that of water f = 10 3 Pa s. The EGL thickness  » varies from 0:2 0:4 m up to 1 m (Weinbaum et al., 2007). Although there are currently no direct measurements of hydraulic resistivity within the EGL, estimates range from K = 1010 1011 N s m 4 (Secomb et al., 1998). The shear modulus of the EGL is calculated (Yao, 2007) to be s s = 3:5 10 Pa, and it is generally assumed (Damiano et al., 1996) that the EGL has a small solid fraction s = 0:01. Its Poisson’s ratio is assumed to be = 0:3 (Wei et al., 2003). The mean blood velocity in capillaries is V = 0:8 1 mm s 1(Damiano et al., 2004). An endothelial cell has the length of approximately e = 20 50 m and height of a = 1 2 m (Wei et al., 2003). Table 3.1 summarises these parameters, including values for the non-dimensional quantitites which de ne the dynamics, namely ,  » =  » =H , = H = e, a = a =H and R = R =H .
Table 3.1: Typical non-dimensional parameter values for a small lymphocyte negotiating an EGL-lined capillary. Here R is the vessel radius, is the hydraulic resistivity within the EGL, a is an endothelial cell height, s is the solid phase volume fraction,  » is the EGL thickness, is the aspect ratio and is Poisson ratio.
the channel walls and c = f(x) on the cell, corresponding to a varicose geometry, = 0, and sinuous geometry, = =2. In addition, we present predictions for the elastic displacements in the EGL and associated elastic shear stresses on the wall, s = ( s (h(x) pI n) = ) (here the direction of the tangential vector always coincides with the positive x1 direction). The total shear stress is = f + s.
Figure 3.2: Displacement on the interface (top) and shear stress distribution on the solid wall (bottom) corresponding to Case II and calculated with di erent resolution. Here d = 6:45 10 3 is the reference length of segment. Solutions converge with element re nement.
The simulation with d = 6:45 10 3 requires computational resources of 40Gb and 109CPU hours (73 hours to calculate the matrix coe cients and 48 hours to solve the linear system). It is worth noting that due to implementation of a parallel computational method, the calculation of matrix coe cients (73CPU hours) has been distributed between multiple CPUs. For instance.

Simulation results

No cell

Firstly, in gure 3.4 we compare the stresses on the interface with those on the wall, and nd that the EGL acts to reduce the uid shear stress (as previously reported), but increases the stress in the solid phase. Combining these two contributions, we observe that in general the total shear stress on the wall is in fact greater than that on the interface. The exception is at the widest section of the vessel, although this is only non-negligible for a varicose vessel (see gure 3.5).
In gures 3.6(e), 3.10(e), 3.15(e) and 3.16(e) we show the ow elds, stresses (both uid and elastic), and elastic displacements for both sinuous and varicose microvessels in the absence of the cell. These act as base cases, against which we can compare the cases where a cell is present. We observe that elevated stresses and elastic displacements occur at the geometric constrictions, as expected. Moreover, we note that the magnitude of the shear stress on the wall due to the solid phase dominates over that due to the uid phase for all cases considered. This appears to support ideas around the solid phase being the main transducer of mechanical.

Varicose geometry with the cell

In gure 3.6 we examine the ow elds and uid shear stresses for the varicose geometry in the presence of a cell. When the cell is located in the geometric constriction (Case I), in gure 3.6(a) we observe a local ampli cation of stresses and ow velocities, over that seen when the cell is absent (see gure 3.6(e)). However, immediately above the cell we observe a reduction in the shear stress. This is highlighted further in gure 3.7 which shows that the presence of the cell leads to increased wall stress (as compared to the cell-free vessel) immediately upstream and downstream of the cell, but decreased stress directly above the cell (i.e. x1 = 2). When the particle is located on the centreline of the vessel, and in the widest part of the vessel, gure 3.6(c), we observe that the in uence of the cell upon the uid shear stresses on the wall is fairly minimal.
However, when the cell is positioned in the expanding section of the vessel, Case II as shown in gure 3.6(b), we again see elevated levels of wall shear in the vicinity of the particle, and a local reduction in shear immediately above and below the cell. In fact, in this scenario the cell-induced local stress reduction occurs to such an extent that we see a region of negative uid shear stress on the wall. A similar situation occurs when the cell is placed in the widest section of the channel, but close to the upper wall (Case IV, see gure 3.7, dashed line). Upon closer examination of the associated ow elds for these two cases (see gure 3.8), we see that this is associated with the presence of a vortex. It was shown by Wei et al. (2003) that vortices can appear in a varicose vessel in the absence of a cell, although these were noted to appear in the widest part of the vessel, and for values of 1600 which is greater than those considered here. From a physiological standpoint these ow features are important since they have the capacity to increase the residence time of circulating substances within the EGL. Moreover, the accompanying variations of the shear stress pro le could have important implications for mechanotransduction in the microvasculature, since mechanoreceptors located on the surface of the endothelial cells are liable to experience shear stresses exerted by ow within the EGL ( uid component of the total stress).

1 Introduction 
1.1 Structure and function of the EGL
1.2 Mechanical properties of the EGL
1.3 Clinical implications
1.4 Models of the EGL
1.5 Gaps in the existing models
1.6 Objectives
1.7 Thesis structure
2 Biphasic model
2.1 Geometry
2.2 Biphasic model assumptions
2.3 Hydrodynamics in the lumen
2.4 Elastohydrodynamics in the Poroelastic Layer
2.5 Boundary conditions
2.6 Non-dimensionalisation
2.7 Numerical method
3 Results. Biphasic model
3.1 Parameter values
3.2 Computational verication
3.3 Simulation results
3.4 Discussion
4 Charge eects in the EGL
4.1 Quadphasic model assumptions
4.2 Hydrodynamics in the lumen: Triphasic model
4.3 Elastohydrodynamics in the EGL: Quadphasic model
4.4 Boundary conditions
4.5 Non-dimensionalisation
4.6 Straight-walled vessel
4.7 Arbitrary vessel shape: thick Debye layer
4.8 Arbitrary vessel shape: thin Debye layer
5 Results. Quadphasic model
5.1 Parameter values
5.2 Verication
5.3 Simulation results
5.4 Discussion
6 Conclusions 
6.1 Elastohydrodynamics of porous medium
6.2 EGL dynamics
6.3 Future work
Appendix
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Modelling the Endothelial Glycocalyx Layer

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