Unruled Notebook

By 2020 seventy percent of the heart patients of the World, a study suggests, would be in India. The cause seems genetic. The gene that codes the enzyme called PON1 is defective in Indians and predisposes them to heart ailments and diabetes. Coupled with degenerating lifestyle – eating habits – leads to such a dire prediction. For more details, read the papers in reference [1] and [2]. This note is not on PON1.

Coronary artery diseases are a result of formation of plaque in our arteries that block blood transport. To know more on this one would first need to know more on how movement of blood flow is affected by relative movement of what it contains – red blood cells, for example. This essay is on blood flow and in particular, the shape of the red blood cells.

Blood is a connective tissue made of cells in an extracellular matrix. The extracellular matrix of blood is blood plasma, formed mainly by water (90 percent) and plasma proteins. The main proteins are albumins, globulins and fibrinogen. Fibrins primarily help the blood to clot. Cells and platelets found in blood are called forming elements. Red blood cells (RBCs), also called erythrocytes, occupy approximately 45 percent of the blood volume. These cells are round, about 8 micro-meter in diameter (90 micro-m³ in volume), with concave sides. They have few organelles, no nucleus, or mitochondria. With no nucleus, RBCs cannot divide. They normally live 120 days.

Here is a close-up view of a blood clot from a recent paper in Science. It is taken using a scanning electron micrograph from the coronary artery of someone who had a heart attack.

The brown mesh is fibrin molecules and are strengthened by the purple-grey platelets. The RBCs are shown in red color while the white blood cells are colored green. The fibrin mesh bends and stretches without breaking to catch RBCs and infection-fighting WBCs. Such flexibility is primarily due to how each molecule unfolds when tugged, exposing hidden inner parts of the fibrin string that then actively expel water. In the recent paper in Science [3] (from which the above SEM clot picture is taken), Penn State biologist John Weisel proposes a molecular basis for such extensibility of fibrin gels based on the structural and mechanical properties of clots at several levels. The whole clot volume is reported to decrease about ten-fold with three-fold stretching.

That is about the clot. But what happens when blood flows?

Blood circulation in human body is a closed loop. Within this loop, blood requires to be pumped for flowing. Pumping requires energy, which is provided by the heart. Clots and fats in the blood vessels (veins and arteries) increase the pumping power load of the heart – a potential reason for heart attack.

Blood viscosity affects the power required from the heart. If the viscosity is too high, the required power increases; if the viscosity is too low, the required power again increases to hold the blood from flowing by gravitational effect. Due to heterogeneity in content, blood viscosity behaves differently from the viscosity of simple liquids.

Fluids like air and water have their viscosity function in such a way that the applied shear stress (or force) is linearly proportional to their strain rate (or how velocity changes in space). This is referred to as the Newtonian behavior of fluid viscosity. Although made of 90 percent water, blood, in some circumstances, does not behave as a Newtonian fluid.

The heterogeneous character of blood affects the blood viscosity when the length scale of the blood vessel (artery etc.) is comparable to the length scale of the components of the blood. When flowing through very large vessels, much larger than the size of the components of blood, blood exhibits as a homogeneous fluid (read on homogeneity and length scales). Hence, in those instances, blood viscosity exhibits Newtonian characteristic. Blood viscosity varies with concentration of components (e.g., amount of RBCs), and with diameter of blood vessel.

Small blood vessel diameter has the effect of reducing the apparent blood viscosity because the blood components like RBCs tend to concentrate near the center of the vessel. This induces a small layer of more homogeneous fluid to surround the vessel wall. The cell components then slide more freely as they flow downstream. For example, at a given concentration of RBCs (how many per volume of blood), the apparent blood viscosity reduces more than 50 percent (3 to less than 2 centipoise), when the blood vessel diameter reduces 10 times (say, from around 2 mm to 0.2 mm). Under these circumstances, blood viscosity differs from Newtonian characteristic.

At least that was the understanding until recently. The crowding of the RBCs near the center of the vessel is possible due to the shape changing ability of the RBCs. Many recent research have attempted to simulate the RBC shape change and its effects through computer simulations and model experiments. Check the video on the MIT news item [4] for one such simulation experiment of the squeezing of RBC during ‘blood’ flow.

Now for a recent research [5] that brings additional insight on the shape of RBCs and the resulting viscosity and blood flow.

A normal blood vessel is circular in cross section. One could imagine the blood flow coming out of it to be symmetric with reduced speed near the wall – due to wall shear – and compensatory increased speed near the axis. Such a flow profile is a laminar flow solution for the Navier Stokes equation that govern the momentum exchange for fluid flows. Our household water coming out of pipes exhibits such a characteristic. Read the pipe turbulence essay for more details.

As the picture above shows, RBCs are bi-concave sided discs. They have a cytoskeleton protein network underneath the lipid bilayer membrane, which gives the system a shear elasticity and supports a biconcave shape. When blood flows, this changes. Each of the RBCs flowing along with the blood are separate particulates that also get elongated near their center and bent near their edge due to the viscous forces from the surrounding flow. In effect, from the side, they look like a parachute or a boomerang as the picture below shows.

For blood vessels with diameter compared to the diameter of the RBCs, the situation is different. More shapes are possible. In vivo studies have already demonstrated that RBCs form asymmetric shapes in vessels that are less than 20 mm, which is only a little larger than the cell itself. A RBC can tumble as a rigid body. It can also tank tread. That is, while the RBC orientation is fixed in a flow, the membrane rotates around the cell’s cytoplasm. Like the veering of a spinning top on the floor.

In the presence of large viscous forces, it is established by observation that the RBC deforms asymmetrically into what is known as the slipper shape. Slipper as in human foot wear – the shape in the bottom portion of the RHS figure above. This slipper shape is due to RBC getting confined in a small blood vessel.

The curious thing about this shape changing activity is, it happens away from the wall of the vessel through which the blood is flowing. The reasons of why RBCs often show asymmetric shapes in a symmetric tube are a puzzle in blood micro-circulatory research. So, the obvious question is, under what conditions such a shape change from a symmetric parachute to asymmetric slipper happen for RBCs? Answer to this question is attemped through computational fluid dynamics simulations in a recent paper by Kaoui et al. [5]. The above figure is a detail redrawn from their results and reported in Physics viewpoint [6] that discuss this paper.

Assuming Stokes flow (very low Re flow) conditions in 2D and accounting for the RBC membrane forces properly with the internal and surrounding fluid of identical viscosity, Kaoui et al. show that there is a loss in stability of RBC shape under certain conditions. They define a dimensionless vesicle deflation number v, defined as the ratio of the actual RBC area to the area of a circle with a circumference equal to the perimeter of the RBC. If this v is below a certain critical value of 0.7 (v is always less than 1, unless the cell is a circle), the symmetric parachute like shape develops an instability and the cell transforms into an asymmetric slipper like shape. This is shown in the RHS of the figure above.

As the v reduces, the simulations show that slip between the RBC and the surrounding flow increases. That is, the the lag increases with decreasing v and attains a maximum (within numerical uncertainties) at the bifurcation point or the critical v value of 0.7. Here onwards for further decrease in v, a slipper shape persists with accompanied decrease in the lag. It however has the consequence of inducing the tank treading mechanism. To quote a relevant sentence from the paper:

By assuming an asymmetric shape, the vesicle reduces its lag, albeit at the price of inducing membrane tank-treading and internal flow. The reduction of lag occurs as a subtle interplay between the shape adaptation to the flow and a compromise between additional membrane and vesicle internal dissipation. As a consequence, the slipper becomes a favorable shape.

Here is a Video link of the simulation, obtained from the supplemental material provided for the paper at the APS ftp site [7].

The movie shows the notion of parachute instability. The parachute shape is unstable in venules conditions while the slipper shape is stable.

What is interesting is, their simulations confirm that the transition from parachute to slipper shape is not dependent on either the confinement of the surrounding blood vessel walls or membrane shear elasticity. Also, the shape transition causes a decrease in the velocity difference between the cell and the flow. In other words, the slipper shape enhances transport efficiency. Implications of this result is yet to be seen.

The simulations are two dimensional and assume a variety of things including the co-panarity of the RBCs for simplicity. Each of these could be relaxed in more complex simulations, but the insight about the instability of the asymmetric shape under certain flow and geometry conditions should remain as a contribution of this research paper. More details of the work at an introductory level is available in the write-up given in the reference.

More information exist on this topic as it is actively pursued in current research. Few more papers are given in the reference (from [8]) for further reading.

References

1. Paraoxonase 1 gene polymorphisms contribute to coronary artery disease risk among north Indians, Agrawal et al., Indian Journal of Medical Sciences, 63, pp. 335-344, 2009. DOI: 10.4103/0019-5359.55884
2. Paraoxonase gene polymorphism and coronary artery disease in Indian subjects, Nirupma Pati, Uttam Pati, International Journal of Cardiology, 66(2), pp. 165-168, 1998. [DOI: 10.1016/S0167-5273(98)00209-5]
3. Multiscale Mechanics of Fibrin Polymer: Gel Stretching with Protein Unfolding and Loss of Water, A. E. X. Brown, Science, Vol. 325. no. 5941, pp. 741 – 744, 2009 [DOI: 10.1126/science.1172484] – [News Item]
4. MIT shows how blood cells change shape
5. Kaoui, B., Biros, G., & Misbah, C. (2009). Why Do Red Blood Cells Have Asymmetric Shapes Even in a Symmetric Flow? Physical Review Letters, 103 (18) DOI: 10.1103/PhysRevLett.103.188101
6. Slipping through blood flow, H. A. Stone et al., Physics Viewpoint, Physics 2, 89, 2009.
7. See EPAPS Document No. E-PRLTAO-103-004942 for movies. For more information on EPAPS, see [http://www.aip.org/pubservs/epaps.html]
8. Shape transitions of fluid vesicles and red blood cells in capillary flows, Hiroshi Noguchi† and Gerhard Gompper, PNAS, vol. 102 no. 40, 14159-14164, 2005 doi: 10.1073/pnas.0504243102
9. Measuring red blood cell flow dynamics in a glass capillary using Doppler optical coherence tomography and Doppler amplitude optical coherence tomography, J. Moger et al., Journal of Biomedical Optics, Vol. 9, No. 5, pp. 982–994, September 2004, doi:10.1117/1.1781163
10. Rekha Bali, Swati Mishra, Shraddhya Dubey, A mathematical model for red cell motion in narrow capillary surrounded by tissue, Applied Mathematics and Computation, Volume 196, Issue 1, 15 February 2008, Pages 193-199, ISSN 0096-3003, DOI: 10.1016/j.amc.2007.05.065
11. Red Blood Cells: Change in Shape in Capillaries, M. Mason Guest et al., Science, Vol. 142. no. 3597, pp. 1319 – 1321, 1963 DOI: 10.1126/science.142.3597.1319

[Thanks to Lakshmi, for additional references and Dhiman, for spotting a typo.]