Thank you, Dr.Sydney Bush for your elaborate and honest answer. It is not certainly obvious to us. Dr.K. Shailendhra.

Dear Dr.Sydney Bush,

Thank you for the pains you have taken to give an elaborate answer. We are working on mathematical modeling of blood flow in large arteries. There a good number of publications and some of them very useful (not all of them) for us to understand the dynamics. We are focused on modeling in such a way that ultimately our results match with reality as much as possible. Blood in large arteries can be considered to be Newtonian (non-Newtonian effects being significant only in tubes of small radius). We are presently focusing on Pulmonary Artery, since mathematical modeling of the pressure gradient matching with the experimental data is readily available (in the form of Fourier Series with nearly 10 terms). So, we are using Navier Stokes Equations to model the blood flow in pulmonary artery. Now, the arterial wall is porous and there is a flow inside the wall. We have to model the flow in the wall and then mathematically there is a necessity to match these two flow at the boundary / interface. One of the methods employed in the literature is Darcy Law along with Beavers-Joseph slip boundary condition. But this is possible only if the wall thickness is more than the radius of the lumen. So we are not happy with this. Some researchers have used Brinkman Equation in the wall with Williams (or Beavers-Joeseph-Rudraiah) slip boundary condition. But, this is possible only if the porosity tends to unity and if the viscous effects are dominating in the wall. Again, we are not happy with this model. However, following these two methods we too have done something, as a preliminary analysis and we are in a position to match our mathematical predictions to some extent with the available realistic data like flux and wall shear stress. However, we would like to further improve the model to go towards reality still further. We are very much aware that mathematical modeling of blood flow is an esoteric area and that such a modeling may not dynamically match with reality. Still, we have to work and keep improving the model and perhaps one day something very useful can be predicted. Further, we are also applying magnetic field and studying its effect on flux, wall shear stress and resistance to flow. Such a study, we believe, will be useful to understand what really happens to blood during MIR scan etc. In our modeling we go up to 8 Tesla though generally in commercial MIR scan instruments the field strength goes up to 1.5 Test (or up to 3 to 5 Tesla in experimental instruments). We are consulting some medicos to give us some inputs. It should be an inter-disciplinary collaborative research work and your honest feed back is very useful to us. Thank you once again.

Yes, Dr.Sydney Bush, if your consider "harmonics" to the required accuracy, modeling of blood flow becomes a bit complicated. Added to this, the porous nature of the wall is posing a problem : the question again remains unanswered; What is the governing law for fluid flow in the porous wall and what are the matching conditions at the interface? We proceed with the available literature and we are able to match velocity profile, Wall Shear Stress (WSS) and other well-known indices like Oscillatory Shear Index, Oscillatory Flow Index etc for normal human beings. The next stage is to model the problem for atherosclerosis or aneurysm. One way or the other, the complexity of the problem poses severe threats to mathematical modeling and hence is quite challenging and interesting. Even the in vivo measurements made using non-invasive biomedical instruments, (phase-contrast MR imaging, Pulsed Doppler sonography, quantitative arteriography (pretty old perhaps)), however sophisticated they are, are subject to a lot of errors. Exhaustive literature survey reveals the fact that studies on WSS should have multifactorial multidisciplinary approach due to the intrinsic complexities involved.

Dear Dr.Sydney Bush,

I am absolutely surprised to see such an elaborate description from your end. Honestly, I understand some portions of the above discussion but not all. I have planned to seek the help of my colleagues in the Medical School to help me understand.

Meanwhile, I would like to present to you the literature on the haemodynamic factors of atherosclerosis. When Robert J Hall wrote an article in 1990 titled "Atherosclerosis Past, Present and Future" in Texas Heart Institute Journal, an instant reply was given by Meyer Texon challenging the conventional wisdom and established concepts regarding the cholesterol-heart disease hypothesis. He mainly pointed out that such studies are statistically flawed and are hence (and due t many other reasons as well) inconclusive. He further expressed his opinion, based on his research works, that mechanical cause of atherosclerosis relating its pathogenesis to a 'reactive biologic response of blood vessels to the forces generated by the flowing blood". This was published in the "Correspondence" section of the said journal subsequently along the editor's (Robert J Hall) reply. Texon has also written a book "Hemodynamic Basis of Atherosclerosis". During the same period (late 80's and early 90's) Stehbens WE also challenged the lipid hypothesis of atherogenesis in his letter to the journal Pathology (1988). Dr.M.Texon and Dr.Stehbens are considered to strong proponents of mechanical causes of atherosclerosis. Following this a lot of attention was shed on the hemodynamic studies, since 1990 - which continues till date, to find out the actual hemodynamic factors that form the basis of atherosclerosis. The main factor identified is the '"Wall Shear Stress (WSS)". WSS is the tangential stress (measured in dynes / square cm) produced by the flowing blood on the endothelial wall. In this connection, it is worth referring to the excellent review article of Peter F Davies in Physiological Reviews in 1995. He has exhaustively documented as to how the endothelial wall behaves as a sensor of WSS and accordingly triggers very many responses by a process called mechanotransduction. The various responses of the endothelial wall, depending on the magnitude of WSS is given in the form of table in this article. Another review article worth reading is the article by Akram M Shaaban and Andre J in American Journal of Radiology in June 2000.

In 2005 Kristopher SC and Avrum IG came out with an article entitled 'The role of shear stress in the pathogenesis of atherosclerosis" in the journal "Laboratory Investigations" explaining how the WSS is mechanotransduced into biochemical signal that results in changes in vascular behavior. Currently, it is believed that vascular endothelium has different behavioral responses to altered flow patterns both at the molecular and cellular levels and these reactions are proposed to promote atherosclerosis in synergy with other well-defined systemic risk factors.

Meanwhile, in 1960 Burton proposed to mathematically model the pressure gradient in the arteries in a simple way (constant term superimposed with a coine wave form). Many started using this and it is also used till date. But, McDonald in the early 1960's proposed that it will be better, to be realistic, if the pulsatile pressure gradient is approximated by a Fourier Series with adequate number of harmonics. Mc Donald's model is used by only by very few researchers since the corresponding mathematical problem is somewhat difficult to address.

In our present work, we have considered the Burton's model and Mc Donald's model and have compared the results to highlight the role of various parameters on WSS. We have considered three arteries - two from the systemic side and one from the Pulmonary side 'Branchial, Femoral and Pulmonary". It is believed that low WSS causes atherosclerosis and high WSS causes aneurysm. Our results indicate that Mc Donald's model gives a better picture on the role of various parameters on WSS compared to Burton's model. We are aware of the limitations of our mathematical model and we are planning to explicitly document them in our article. We shall communicate our work in the near future.

We once again thank you for your kind interest.

Shailendhra Karthikeyan.