Supplementary material for:
Oscillation Dynamics of Embolic Microspheres in Flows with Red Blood
Cell Suspensions
Tamal Dasa,d, Dario Carugob, Xunli Zhangb, and Suman Chakrabortyc,1
a. Department of Biotechnology, Indian Institute of Technology Kharagpur, India – 721302
b. Bioengineering Group, Faculty of Engineering and the Environment, University of
Southampton, Southampton SO17 1BJ, UK
c. Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, India
– 721302
d. Present Address: Max Planck Institute for Intelligent Systems, Stuttgart – 70569,
Germany
1
Corresponding Author, Email: suman@mech.iitkgp.ernet.in
A] Power Law Model for Blood Rheology
In the Power Law model, the shear stress (  ) is related to the rate of deformation ( γ )
as τ  aγ n , where the parameters a and n are functions of the blood hematocrit fraction. In
particular, one may write [A1]:
a  C1 exp(C2 H C )
and
n  1  C3 H C
where H C is the hematocrit fraction in the blood sample and C1 , C 2 , C3 are species specific
coefficients that can be experimentally determined. The dependence of C1 on plasma
composition and on hematocrit can be summarized by an empirical equation [A1]:
C1  C1* exp[C4 (TPMA) / H C 2 ]
where TPMA is total protein minus the albumin fraction of blood, C1*  0.00797dPa( s) n and
C4  145.85d lg 1 . Considering established practical values of the relevant parameters, we
have made estimates of the various coefficients describing the blood rheology as follows
[A2]: [C1, C2, C3] = [1.48mPa·sn, 0.0512, 0.00499]
[A1] D. J. Schneck and C. L. Lucas, Biofluid mechanics (New York University Press, New
York, 1990)
[A2] G. P. Galdi, R. Rannacher, A. M. Robertson, and S. Turek, Hemodynamical flows:
modeling, analysis and simulation (Birkhauser Basel, 2008)
B] Determination of the Reynolds number
Reynolds number (Re) in the inertial range 2-30 was varied within the capillary. The
volumetric flow rate to be imposed at the capillary inlet (Qin) in order to achieve the desired
value for Re was calculated. As described in [A], we considered blood as a viscoelastic fluid
which manifests shear-thinning behavior (i.e. the apparent viscosity decreases with increasing
the shear rate), and a Power Law model was employed to describe the dependence of fluid
shear stress on shear rate. For this reason, a generalized Reynolds number for Power Law
fluids defined by Metzner and Reed [B1] was employed in the present study, which
accounted for fluid shear-thinning behavior:
Re 
U 2n Dhn
n
a n   n1
Where the consistency index (a) and the power law index (n) were determined as described in
[A]. ρ is the density of the RBC suspension, and was calculated by assuming a density of
1025 kg/m3 for plasma and 1125 kg/m3 for RBC [B2]. U is the mean fluid velocity and Dh is
capillary hydraulic diameter (Dh = 1.1 mm). ɛ is the product of f/2 and the Reynolds number
for a Newtonian fluid under laminar flow regime, where f is Fanning’s friction factor. ɛ is
thus equal to 8 for flow within circular ducts [B3]. ϕ(n) is an hyperbolic function of the
Power Law index and, for circular ducts, it is equal to [B3]:
 n  
3n  1
4n
[B1] A. B. Metzner and J. C. Reed, AIChe Journal 1, 434 (1955)
[B2] Y. Fung, Biomechanics: circulation. Second Edition (Springer, New York, 1997)
[B3] F. Delplace and J. C. Leuliet, The Chemical Engineering Journal and The Biochemical
Engineering Journal 56, 33 (1995)
C] Detection of Bead Position and Quantification of Bead Velocity
Supplemental Figure C1. Schematic Representation of Image Analysis. In Step 1, images
were captured at a regular time interval. In Step 2, sharpness of particle edges (L-Left, RRight) was evaluated to determine which particles were in-focus. In Step 3, coordinates of infocus particles relative to channel center were evaluated. Particle velocity was then
determined from these coordinates obtained at successive images frame.
Description of Experimental Setting, Image Acquition, and Image Analysis:
For each experiment, at first, we set the microscope focus in the middle plane (in z-direction)
of the channel. We attained this focus by marking the z-positions of top (ztop-wall) and bottom
(zbottom-wall) walls of channel first and subsequently, by setting the focal plane at the mean zposition (zfocus = (ztop-wall + zbottom-wall ) / 2 ). We then started the flow, and after a waiting time
of 30 seconds, we started capturing the images. For recording the particle velocity, images
were taken at different time points, with fixed time-gap between successive frames. At each
time frame then image was analyzed to ensure that only properly focused beads were counted
for further data processing. This was done by analyzing a line scan of intensity, passing
through the middle of each bead in the frame. For example, in Suppl. Fig. 1, lines 1, 2 and 3
were selected to analyze the focus of particles 1, 2 and 3, respectively. The postulate is that if
the particle is properly focused a sharp jump of intensity should be observed at the left (L)
and right (R) edges. If the absolute difference of intensity between neighboring pixels is
calculated, one should then see a sharp peak over these edges (Step 2 of Suppl. Fig. 1). To
judge the acceptable level of focus, we have set a cut-off value of 25. Particles were taken
into consideration if only both L and R peaks were found to be above this cut-off value. Once
one particle is identified to be in focus (e.g. Particle 3 in the Suppl. Fig. 1), its (x,y)
coordinates relative to the channel center were determined. Velocity of this particle was then
calculated from the position of the particle in successive image frames and applying the
formula vp = (xn+1-xn)/Δt, where Δt is the time gap between two successive image frames. In
the number ensemble representation, the positions and mean velocities of all particles from
all images were taken together while plotting a data (e.g. Fig. 2a and 4). In the time ensemble
method, position and velocity of a single particle obtained at different time frames were
plotted (e.g. Fig. 3a). We noted that we could apply this way of determining the particle
velocity in axial direction as well to the cases where particles moved in helical trajectories. In
such condition, the particle did have a radial velocity component, but since this radial
component was perpendicular to the axial component of the partical velocity, it did not affect
the estimation of vp in x-direction.