# Statistical dynamics of flowing red blood cells by morphological image processing

Original Entry by Michelle Borkin, AP225 Fall 2009

## Contents

## Overview

"Statistical dynamics of flowing red blood cells by morphological image processing."

J. Higgins, D. Eddington, S. Bhatia and L. Mahadevan. PLoS Computational Biology, 5, e1000288, 2009.

## Keywords

Rheological behavior, Microfluidics, hemodynamics

## Summary

This paper investigates the complex random motions of individual red blood cells to better understand the role of individual cell movements in nutrient transport, gas transport, clotting, and hematological diseases. With this microscopic view, versus studying just the bulk flow, they were able to see the importance of these random motions. For example, patients with sickle cell disease who have irregularly shaped cells, have decreased random cellular motions suggesting an increased risk of vessel occlusion. The experiments were conducted by passing blood through microfluidic devices with a cross-sectional area of 250 μm x 12 μm (red blood cells have a radius of ~4 μm and thickness of ~1-2 μm) thus confining the motion of the cells to one direction. Also, cells only in the middle fifth of the flow were studied since the shear rate (~10/sec) is in the human physiological range for microcirculation. This "quasi-2D" set-up allowed for easy video imaging of the cells and subsequent image analysis to determine the random motions.

## Soft Matter

Red blood cells, a major component of blood (a shear thinning fluid), are usually large enough that thermal fluctuation effects are negligible meaning their equilibrium diffusivity (<math>D_{thermal}</math>) is small:

<math>D_{thermal} = \frac{kT}{f} \sim 0.1 \frac{\mu m^2}{s}</math> (for a 4 μm flat disk at room temperature)

where <math>f</math> is the viscous drag coefficient. However, these soft cell suspensions when driven by pressure gradients or subject to shearing will result, via concentration variations and velocity gradients, in complex multi-particle interactions causing fluctuations in particle movements. Thus it is necessary to study these microscopic interactions to understand larger-scale observables such as clotting.

The observed fluctuations in cell movement seen in the experimental plane can be described in terms of the mean-squared displacement:

<math>\langle\Delta r^{2}(\tau)\rangle = \langle\big(r_{bulk}(\tau)-r_{cell}(\tau)^{2}\big)\rangle = D\tau</math>

where the diffusion constant <math>D</math> is larger than the equilibrium diffusivity. This shows that the movement of a cell, in relation to the bulk flow, becomes rapidly decorrelated at any instant from its subsequent movement.

This diffusive process has a characteristic length scale (i.e. mean free path length between interactions) and time scale (i.e. time between interactions). At the Reynolds number regimes associated with microvasculature (<math>Re</math> <math>O(0.01)</math>), the timescale is the inverse of the local shear rate <math>\dot{\gamma}</math>. Thus the diffusivity scales as:

<math>D = C\dot{\gamma}\lambda^2</math>

where <math>C</math> is a dimensionless constant. It was also experimentally shown that for stiff deoxygenated sickle cells that <math>C_{deoxygenated} < C_{oxygenated}</math> thus cell stiffness influences the suspension's dynamics independent of bulk flow behavior. This can also be thought of in terms of "effective temperature" based on the mean square of the cellular velocity fluctuations so that blood that is "hot" is less likely to coagulate or "freeze" as compared to "cold" blood with fewer cellular fluctuations resulting in stagnant blood flow (thus leading to vessel occlusion).

For more information about rheology, go to Rheological behavior.