Porosity Modeling of an Inexpensive Device for the Rapid Detection
of Sickle Cell Disease
Christopher Brown, Alexey Aprelev, and Frank A. Ferrone
Department of Physics, Drexel University
Motivation
Sickle cell disease is a genetic condition that causes
hemoglobin in red blood cells (RBCs) to associate,
creating rigid polymers in the cell when
deoxygenated. Because of this reaction, sickled cells
lose their flexibility obstruct flow through the
microcirculation.
Image 1 – The absence of oxygen removes the cell’s flexibility, becoming rigid. The
process is reversible.
The genetic disease affects nearly 100,000 people in
America. In Africa, where the disease is most
prevalent, roughly 300,000 children with sickle cell
disease are born yearly, many of which do not have
access to resources required for modern detection
techniques.
The Bead-Chamber Device
We have designed and built a device
that is sensitive to the presence of
sickled cells using commercially
available materials. The device consists
of two small capillaries that wedge a
column of packed glass spheres in a
larger glass capillary. The device allows
blood to flow through the bead matrix
and speed of the output flow indicates
whether the blood is sickled or not.
Figure 1 – Rise time differs
between normal (AA) blood
and sickle (SS) blood. Because
oxygenated SS blood reacts
like AA blood, we can use it
as our local control test. Thus
our diagnostic test is a
comparison between the rise
times of a person’s blood
oxygenated and
deoxygenated
The Important Role of Porosity
Our device works well with our initial specifications, but we would
like to understand how the process of sickling makes blood flow
slower through the device, and upon what variables in the system is
this dependent. This understanding could help us design and build
better, and possibly more sensitive, devices. The flow velocity of a
fluid through a porous bed of spheres can be expressed, according to
Carman, as,
The Ratio of Rise Times
Unsickled RBC
ro
rd
Sickled RBC
Figure 3 – To achieve the
rise-time ratio observed
in the data presented in
Figure 1, the offset radius
of the sickled red blood
cells needs to be
significantly large. The
yellow band shows the
range of experimental
ratios in rise time we have
observed. Our bead sizes
range from 125-150 μm in
diameter.
where ε is the porosity or void volume of the porous bed. As
viscosity, η, is a macroscopic characteristic of a fluid, we consider a
different approach, using the red blood cells’ microscopic changes to
affect the fluid kinetics. The calculation of void volume is somewhat
subtle. The full void volume on the left is an overestimate, because
finite spheres would really only occupy the volume in the picture on
the right.
When a red blood cell becomes deoxygenated and sickles, the long
polymer bundles of hemoglobin stretch the cell membrane,
effectively increasing the mean radius of the cells. This shrinks the
net pore size between the beads, reducing the porosity of the
system. These polymer strands also make the cell inflexible, unlike an
oxygenated unsickled cell. As such, we can consider an offset radius
for the cells within which no cell can get closer to the sphere surface.
This effectively reduces the porosity by decreasing the volume
available to the cells, while holding the unit volume constant.
Full Void Volume
Effective Void Volume
Image 2 (above) – Our model is based on the idea that a
sickled red blood cell with a finite radius cannot get as close
to the surface of the sphere as an unsickled red blood cell,
having a smaller effective radius. As such, we consider the
offset radii rd and ro , for a deoxygenated cell and the
oxygenated cell, respectively. Note that the sickled cell is not
spherical, but rather rigid and abnormally shaped. We are
approximating them to be spheres for simplicity’s sake. The
net volume of the pores in turn shrink due to this offset
radii, effectively lowering the porosity of the bead-chamber
system.
Because there is a non-linear dependence on porosity in Carman’s
equation, this reduction may be able to explain why we see such a
dramatic decrease in flow when the blood sickles.
By confining the bead matrix into a finite volume, we
introduce edge effects. Carman introduces a correction
to his velocity expression that deals with the edge
effects.
References:
- Carman P. C., "Fluid flow through granular beds." Transactions, Institution of Chemical Engineers, London, 15, 150-166 (1937)
- White, J. G., “Ultrastructural features of erythrocyte and hemoglobin sickling.” Arch. Intern. Med. 133, 545-562 (1974)
The typical RBC membrane has a surface area of
130 μm2 which can yield a maximum spherical
radius of 3.2 μm. This is not enough to give the 4-6
increase in rise time that we observed in Figure 1.
Rather, from the figure above, we need an offset
radius (i.e. effective cell radius) of 4.9 – 5.1 μm.
Figure 4 – Assuming a
constant effective radius for
the sickled cell, we can
estimate the number of
spikes afforded by extra
surface area, given a pseudospherical radius, smaller than
that of the effective radius.
Sacrificing only a few tenths
of a micron can give enough
surface area to spawn a
significant number of cellular
protrusions due to the
polymer strands of
hemoglobin.
A cell size of near 5 µm radius seems large, but if we
consider the physiological parameters, we realize
very few sickle cells look like a sphere, as shown in
Image 3 to the left. Many have spiky protrusions. If
the cell is able to sacrifice some surface area to a
number of small protrusions, the cell is theoretically
able to achieve this effective radius.
Cylinder Effects on Porosity
The correction now allows us to properly calculate flow velocities
through the bead-chamber and the ratio of rise times, which is our
measured quantity.
The measured rise time of the blood sample is
inversely proportional to the fluid’s velocity. To
compare the rise times of oxygenated and
deoxygenated blood, we can set up a ratio of these
rise times, which depends primarily on porosity.
Image 3 (above) – Sickled cells come in many different
shapes, not just the stereotypical sickle shape that may
come to mind. The first picture in the top right corner is the
only cell not sickled.
This large increase in size required leaves us a little
wary. Other models may be able to match the
observed data with fewer assumptions. The porosity
model is within the realm of possibility, but we are
investigating other models as well to understand our
system better.