3/25/2015 1 Effects of Plasma Skimming Coefficients and RBC Concentration on RBC Spatial Distribution Jagan Jimmy [email protected]This report is produced under the supervision of BIOE310 instructor Prof. Linninger. Abstract A model has been proposed to analyze the spatial distribution of red blood by using a plasma skimming coefficient. Further analysis needs to be carried out to understand how the changes in the plasma skimming coefficients or a decrease in the red blood cell concentration affects the spatial distribution of RBC. In order to understand the changes those variables may cause, the change in the hematocrit value of a vessel caused by those variables in question needs to be modeled and understood by looking at the impact of each variable. Nonetheless, the model proposed to predict the RBC spatial distribution is able to predict the distribution of red blood cells if it assumes certain values for some of its variables. The model is significant for it is applicable to various systems and networks, especially in understanding the dynamics of oxygen delivery to tissues supplied by small arteriolar structures. This may be applied to various studies to optimize systems that depends on oxygen delivery by red blood cells, etc. 1. Introduction Modern imaging techniques can provide great insight into how the blood flows within small vessels in the body and the impact it has on tissue oxygenation. It is known that blood behaves as a bi-phasic fluid, where the two phases are the blood plasma and the erythrocytes. However, in large vessels the effects of the bi-phasic behavior of the blood flow may be ignored since the erythrocyte phase is significantly larger than the plasma phase. But, in smaller vessels such as the capillaries the bi-phasic behavior of blood flow must be accounted for since it greatly affects how the erythrocytes are distributed further along the vessel. It is noted that when such vessels are split into multiple daughter vessels of various sizes, the largest daughter vessel gets a higher portion of the erythrocyte from the original parent vessel, whereas the smaller vessels are primarily provided with the plasma. This uneven splitting of the red blood cells is known a plasma skimming, and it could eventually lead to tissue damage due to limited oxygen distribution [1]. Therefore, it is important that the bi-phasic flow of the blood be modeled to gain a better understanding of the oxygenation efficiency. A model has been proposed to predict the distribution of RBC as the vessel branches off. The model makes use of a plasma skimming coefficient which represents the attraction of RBCs to the center of the vessel when plasma skimming takes place. Nonetheless, a better understanding of RBC distribution as a result of varying plasma skimming coefficient and systematic decrease in RBC concentration has yet to be understood. This report hopes to explore further into the relationship between RBC distribution, RBC concentration, and the plasma skimming coefficient. 2. Methods The model which predicts the distribution of RBC as the parent vessel branches off uses two conservation laws and two constitutive equations. The first conservation equation pertains to the conservation of the volumetric blood flow, Q, at the branching site of any of the vessel as shown in equation 1. The second conservation equation pertains to the conservations of the volumetric flow rate of the erythrocyte phase, Q RBC , at the branching sites, as shown in equation 2 . The volumetric flow rate of the erythrocytes in a vessel is the product of the total volumetric flow in a vessel and the flow rate fraction of the erythrocyte phase – the hematocrit value, H d .
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3/25/2015 1
Effects of Plasma Skimming Coefficients and RBC Concentration on RBC Spatial
Distribution Jagan Jimmy
[email protected] This report is produced under the supervision of BIOE310 instructor Prof. Linninger.
Abstract
A model has been proposed to analyze the spatial distribution of red blood by using a
plasma skimming coefficient. Further analysis needs to be carried out to understand how the
changes in the plasma skimming coefficients or a decrease in the red blood cell concentration
affects the spatial distribution of RBC. In order to understand the changes those variables may
cause, the change in the hematocrit value of a vessel caused by those variables in question needs
to be modeled and understood by looking at the impact of each variable. Nonetheless, the model
proposed to predict the RBC spatial distribution is able to predict the distribution of red blood
cells if it assumes certain values for some of its variables. The model is significant for it is
applicable to various systems and networks, especially in understanding the dynamics of oxygen
delivery to tissues supplied by small arteriolar structures. This may be applied to various studies
to optimize systems that depends on oxygen delivery by red blood cells, etc.
1. Introduction
Modern imaging techniques can provide great insight into how the blood flows within
small vessels in the body and the impact it has on tissue oxygenation. It is known that blood
behaves as a bi-phasic fluid, where the two phases are the blood plasma and the erythrocytes.
However, in large vessels the effects of the bi-phasic behavior of the blood flow may be ignored
since the erythrocyte phase is significantly larger than the plasma phase. But, in smaller vessels
such as the capillaries the bi-phasic behavior of blood flow must be accounted for since it greatly
affects how the erythrocytes are distributed further along the vessel. It is noted that when such
vessels are split into multiple daughter vessels of various sizes, the largest daughter vessel gets a
higher portion of the erythrocyte from the original parent vessel, whereas the smaller vessels are
primarily provided with the plasma. This uneven splitting of the red blood cells is known a
plasma skimming, and it could eventually lead to tissue damage due to limited oxygen
distribution [1]. Therefore, it is important that the bi-phasic flow of the blood be modeled to gain
a better understanding of the oxygenation efficiency. A model has been proposed to predict the
distribution of RBC as the vessel branches off. The model makes use of a plasma skimming
coefficient which represents the attraction of RBCs to the center of the vessel when plasma
skimming takes place. Nonetheless, a better understanding of RBC distribution as a result of
varying plasma skimming coefficient and systematic decrease in RBC concentration has yet to
be understood. This report hopes to explore further into the relationship between RBC
distribution, RBC concentration, and the plasma skimming coefficient.
2. Methods
The model which predicts the distribution of RBC as the parent vessel branches off uses
two conservation laws and two constitutive equations. The first conservation equation pertains to
the conservation of the volumetric blood flow, Q, at the branching site of any of the vessel as
shown in equation 1. The second conservation equation pertains to the conservations of the
volumetric flow rate of the erythrocyte phase, QRBC, at the branching sites, as shown in equation
2. The volumetric flow rate of the erythrocytes in a vessel is the product of the total volumetric
flow in a vessel and the flow rate fraction of the erythrocyte phase – the hematocrit value, Hd.
Creating the 3d graph: Q = x(1:23)*(1000^3); %% mm3/s
A = pi*(Diameter/2).^2; PSC = zeros(1,row1);
M = [0:0.1:1];
HSys = [0.45:0.05:0.85];
for k = 1:size(M,2);
for j = 1:size(HSys,2);
H(1) = HSys(j);
for i = 1:row2 if col2-
length(find(pointMx(i,:))) == 0
PSC(-
1*pointMx(i,2)) = (A(-
3/25/2015 12
1*pointMx(i,2))/A(pointMx(i,1)))^(1
/M(k)); PSC(-
1*pointMx(i,3)) = (A(-
1*pointMx(i,3))/A(pointMx(i,1)))^(1
/M(k));
HAdj =
(Q(pointMx(i,1))*H(pointMx(i,1)))/(
Q(-1*pointMx(i,2))*PSC(-
1*pointMx(i,2)) + Q(-
1*pointMx(i,3))*PSC(-
1*pointMx(i,3)));
H(-1*pointMx(i,2))
= HAdj*PSC(-1*pointMx(i,2)); H(-1*pointMx(i,3))
= HAdj*PSC(-1*pointMx(i,3));
end
end
CumulatH(:,j) = H; end
CHH(:,:,k) = CumulatH;
end
for j = 1:size(CHH,3); for i = 1:size(CHH,1);
scatter3(HSys,CHH(i,:,j),linspace(M
(j),M(j),length(HSys)),'*'); hold on; end end hold off; xlabel('Parent Vessel Hematocrit') zlabel('Drift Parameter (M)'); ylabel('Discharge Hematocrit'); grid on