T. David

A THREE-DIMENSIONAL, TIME-DEPENDENT ANALYSIS OF FLOW THROUGH A BILEAFLET MECHANICAL HEART VALVE : COMPARISON OF EXPERIMENTAL AND NUMERICAL RESULTS

The flow through a bileaflet mechanical heart valve during the first half of systole was predicted using computational fluid dynamics (CFD). A three-dimensional model of the geometry of the ventricle, valve, sinus and aorta was developed. Flow through the valve was assumed to be Newtonian and laminar. The peak systolic Reynolds number was 1500 based on the aortic radius and the mean aortic velocity. Flow visualisation and laser Doppler anemometry (LDA) experiments were performed and the results were compared to the CFD model. Good agreement between the LDA measurements and CFD predictions was found in the jets through the major orifices of the valve. The global flow fields predicted by the CFD showed reasonable agreement with the flow visualisation. A starting vortex was shed from the valve leaflets of the CarboMedics valve and the prototype valve. As systole progressed the two major orifice jets were directed towards the aortic wall and a weaker central jet was seen in both the experimental and CFD models. Large vortices were present on either side of the central orifice jet in the sinus area of both models. The three-dimensional time-dependent CFD model was considered to give a reasonable indication of the dominant flow patterns downstream of the bileaflet heart valve and has the potential to be an extremely useful tool to analyse the different designs of existing and future bileaflet valves.

A model for the fluid motion of vitreous humour of the human eye during saccadic movement

During saccadic motion the eyewall moves in a manner similar to a sinusoid or at least can be represented by a sine Fourier series. Motion of the vitreous is induced by the saccade and the vitreo-retinal interface is subjected to a time-dependent shear. This force may be a significant factor for retinal tearing in the neighbourhood of small retinal holes or tears. An analytical viscoelastic model and a numerical, Newtonian model of the motion of the vitreous are presented and compared. Under sinusoidal boundary motion the analytical model shows that a viscous wave propagates inward toward the axis of rotation and the characteristic length of this wave is a function of the Womersley number. The numerical solution indicates that the vitreous moves similarly to the analytical result with small secondary motion; however, this motion allows complete recirculation of the vitreous over large timescales. Excellent agreement is found between the analytical and numerical models. The time-dependent fluid shear is evaluated and from the analytical solution the maximum value of this is found to be proportional to R0 square root of v(omega)3, where R0 is the eye radius, v the modified complex viscosity and omega the sinusoidal frequency. This indicates that myopes have a larger shear force exerted on them by virtue of the larger eye size. Further work is directed toward a model which links the stress found in the sclera to that exerted on the vitreo-retinal interface by the vitreous fluid motion.

Numerical Models of Auto-regulation and Blood Flow in the Cerebral Circulation

A two-dimensional time-dependent computational fluid dynamics model of the Circle of Willis has been developed. To simulate, not only the peripheral resistance of the cerebrovascular tree but also its auto-regulation function, a new active boundary condition has been defined and developed using control theory to provide a model of the feedback mechanism. The model was then used to simulate different common abnormalities of the Circle of Willis while a pressure drop, simulating a rapid compression of the right internal carotid artery, was imposed. Test results using a simple tube compared excellently with experiment. The total time-dependent flux for each efferent artery was tabulated and showed the important relationship between geometrical variations in the Circle of Willis and the auto-regulation of blood flow by vascular vaso-dilation and contraction. From this study, it was found that the worst case seemed to be that of a missing or dysfunctional right A1 segment of the anterior cerebral artery. The use of valid physiological models of the peripheral resistance allows for more realistic models of the blood flow in the Circle whilst allowing an easy extension to 3D patient specific simulations.

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

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.

Three-dimensional study of the effect of two leaflet opening angles on the time-dependent flow through a bileaflet mechanical heart valve

A three-dimensional (3-D), time-dependent computational fluid dynamics (CFD) model was used to investigate the effect of leaflet opening angle on the flow through a fully open bileaflet heart valve up to peak systole. A laminar flow model of a Newtonian fluid was used, and the peak systolic. Reynolds number was 1500, based on the aortic radius and the average velocity at peak systole. This resulted in a Reynolds number of 5800, based on the aortic radius and the local maximum velocity. The flow fields through and downstream of the bileaflet valves were complex, with strong time-dependent 3-D vortices being found in planes parallel and perpendicular to the leaflets. The parametric study of the effect of leaflet opening angle showed that, as the leaflet opening angle increased from 78 degrees to 85 degrees, the flow downstream of the valve leaflets became more centralized, and the wake downstream of the leaflet decreased in size. However, as the opening angle increased from 78 degrees to 85 degrees, the maximum shear rate and the maximum velocity increased, suggesting that the design of the central orifice geometry was also an important consideration.

Computational models of blood flow in the circle of Willis.

A two-dimensional, steady state model of the circle of Willis has been developed. To simulate the peripheral resistance of the cerebrovascular tree, blocks of porous media were used. Their effective resistance was kept constant, disregarding the effects of arterial auto-regulation. The model was then used to simulate different common abnormalities of the circle of Willis while a range of varying boundary conditions was imposed to the right internal carotid artery (ICA). The total flux was tabulated and compared favourably with both clinical measurements and other models of the circle of Willis. Relevant fluid dynamics effects were also observed and analysed. The present model demonstrates that the use of CFD can produce physiological results if the appropriate boundary conditions are used. We can provide clinicians with a priority list of the severity of the flux reduction for the considered abnormalities for different degrees of stenosis of the right ICA. From this study it is apparent that the redistribution of blood via the circle of Willis is mainly driven by changes in the vascular resistance of the brain rather than in the local arterial geometry. The use of valid peripheral resistances allows for a more realistic model of the circle of Willis but also highlights the need for more accurate means to estimate the vascular resistance of a patient.

Modeling perfusion in the cerebral vasculature

The constant perfusion of a human organ with nutrients and oxygen demands a robust regulatory mechanisms in the face of normal day-to-day pressure variations in the vasculature. The brain, in a similar manner to the heart requires this mechanism to be extremely quick acting, relative to other ways of altering perfusion such as varying systemic blood pressure, since oxygen depravation in the tissues of the brain can be tolerated for only of the order of tens of seconds before significant damage can be done. In recent years computational models, and it must be noted computer architecture have evolved to an extent where mathematicians and engineers can play a large part in discovering how the brain functions physiologically as well as investigating pathological conditions. This review will look at a number of increasingly complex computational models of blood flow to the brain and how variations in arterial geometry can influence the perfusion in the cerebral vasculature. Although these models have provided an insight into complex mechanisms the research area is densely populated with important questions that perhaps only computer models can answer. The review will indicate possible areas of investigation.

Time-dependent stress and displacement of the eye wall tissue of the human eye

Myopia or short sightedness, is the most important predisposing factor to retinal detachment. The relative risk of detachment rises with increasing myopia. The model characterizes that because the severity of myopia increases with the axial length (antero-posterior diameter) of the eyeball, the relative risk of retinal detachment rises with increasing eye size. We present a mathematical model of the time-dependent shear stress force that occurs in the thin eye wall shell supporting the vitreous humour inside the eye globe during the acceleration and deceleration phases of saccadic eye movement. Results show that the shear force increases as the thickness of the eye wall decreases. It is common for myopes to have thinner eye wall tissue than emmetropes. In addition, if account is taken of the increased force required to provide normal saccadic movement of myopic (larger) eyes, then the shear force is up to seven times greater than that experienced for emmetropes.

Disturbed flow promotes deposition of leucocytes from flowing whole blood in a model of a damaged vessel wall

Departure from simple laminar flow in arteries may promote the local attachment of leucocytes either to intact endothelium or platelet thrombi. We perfused blood through a chamber with a backward facing step, to observe whether adhesion from whole blood to P‐selectin was indeed localized to a region of recirculating flow, and whether platelets binding to collagen in such a region could capture leucocytes. Blood flowing over the step established a stable vortex, a reattachment point where forward and backward flow separated, and a simple laminar flow with wall shear rate c. 400/s further downstream. Fluorescently labelled leucocytes were observed to attach to P‐selectin immediately upstream or downstream of the reattachment point, and to roll back towards the step or away from it, respectively. There was negligible adhesion further downstream. When a P‐selectin‐Fc chimaera was used to coat the chamber, stable attachment occurred, again preferentially in the disturbed flow region. Numerous platelets adhered to a collagen coating throughout the chamber, although there were local maxima either side of the reattachment point. The adherent platelets captured flowing leucocytes in these regions alone. Leucocytes may adhere from flowing blood in vessels with high shear rate if the flow is disturbed. While platelets can adhere over a wider range of shear rates, their ability to capture leucocytes may be restricted to regions of disturbed flow.

A mathematical model of post-canalization thrombolysis

During the initial phase of lysis of an occlusive thrombus using lytic agents such as tissue plasminogen activator, blood flow through the centre of the clot is established (the process of recanalization). Following canalization, the clot remains on the vessel wall and further lysis is required. This paper develops a multi-species mathematical model to describe the bulk chemical reactions in the bloodstream and the convective and diffusive transport of chemical species to and from the clot surface in conditions following canalization. For the steady state case, the model indicates that the process of clot lysis following initial recanalization is dominated by surface chemical reactions and the bulk reactions play little role in the lytic process. Lytic rate is dependent on the clot geometry and flow conditions. The rate of clot dissolution is greatest at the upstream end of the clot and decreases steadily downstream due to lytic agent being removed from the flowing blood as it binds to the clot surface. This model may be further developed and used to simulate and compare different lytic regimes.