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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.
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The Hagen-Poiseuille law as shown in equation 3 is used to relate the change in pressure
across a vessel to its bulk volumetric flow rate. These two quantities are related through the
vascular hydrolysis resistance, which in turn is in terms of blood plasma viscosity (µ), the vessel
length (L), and the vessel radius (R). The remaining constitutive equation used in the model is
the plasma skimming law. Countless observation have shown that daughter vessels with smaller
radii receives more plasma than RBCs; therefore, the RBC phase volumetric flow fraction of the
daughter vessel may be expressed as the discharge hematocrit of the parent vessel, H1, minus a
depletion term. However the inclusion of the depletion term introduces more degrees of freedom
since its value would vary from each daughter vessel to another. Therefore, to reduce the degrees
of freedom the daughter RBC phase fraction may be written in terms of an adjusted hematocrit
value, H*, and a plasma skimming coefficient, θ, as in equation 4.
The plasma skimming coefficient may be further expressed in terms of the ratio of the
cross-sectional area of the parent (A1) and daughter (A2, A3) vessels and the drift parameter, M –
equation 5. Now the volumetric flow rate conservation equation of the RBC phase may be
rewritten with the substitution of the plasma skimming coefficient and the adjusted hematocrit
value, as given in equation 6 for a vessel bifurcation. Since the flow rates, Q, the parent
hematocrit value, H1, and the plasma skimming coefficients are already known or defined, the
equation may be rearranged to explicitly solve for the adjusted hematocrit. The adjusted
hematocrit value may then be used to calculate the hematocrit value of the daughter vessels.
The steps described above were applied to a single bifurcation (Fig. 1A) and a large
network with multiple bifurcations (Fig. 1B) to understand the effects of changing the plasma
skimming coefficients and the red blood cell concentration. It is important to note that when the
model was applied to the single bifurcation, values that are easy to compute were assigned as the
volumetric flow rates of the of the parent and daughter vessels. Therefore, the Hagen-Poiseuille
law was not used. However, the assigned flow rates satisfied the conservation equation of the
volumetric flow rates at the site of bifurcation. For the larger network, the equations listed were
directly applied. Values for the plasma viscosity, radii, vessel length, and the pressure drops were
from the literature or were chosen to mirror values established by studies, such that the network
may be a good representation of real blood vessel networks. When using the Hagen-Poiseuille
law, blood plasma is assumed to have an ideal viscosity and isn’t corrected for the nonideal
blood rheology for simplicity.
In order to determine to the optimal parametric value for the drift parameter M, data
fitting procedure was done on previously collected bifurcation data by Pries et al. [2]. The data
provided values of the fractional red cell flow for each daughter vessels as a function of the
vessel’s fractional flow.
Figure 1A: Schematic of a vessel bifurcation. The second daughter vessel is bigger than the third daughter
vessel. The subscripted variables are positions adjacent to their own representative vessels.
Q1, H1, A1
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Equations:
∇⃑⃑ ∙ 𝑄 = 0 (1)
∇ ∙ 𝑄𝑅𝐵𝐶 = ∇ ∙ (𝑄𝐻𝑑) = 0 (2)
∆𝑃 = 𝑄
8µ𝐿
𝜋𝑅4
(3)
𝐻2 = 𝐻1 − ∆𝐻 = 𝜃2 ∙ 𝐻∗ 𝐻3 = 𝜃3 ∙ 𝐻∗ (4)
𝜃2 = (𝐴2
𝐴1)
1𝑀
𝜃3 = (𝐴3
𝐴1)
1𝑀
(5)
𝑄1𝐻1 = 𝑄2𝐻2 + 𝑄3𝐻3 = 𝑄2𝜃2𝐻∗ + 𝑄3𝜃3𝐻
∗ (6)
3. Results
Figure 1B: Schematic of a network with multiple bifurcations. The vessels become smaller the further away it is
from the main parent vessel marked with Q1 and H1. The numbers are assigned for the purpose of making
identifying a specific vessel in the network easier.
Figure 2: The hematocrit values of the daughter
vessels plotted against the parent hematocrit values
with different drift parameters ranging from 1 to 10
at increments of 0.5. Blue lines correspond to the
bigger daughter vessel and the green lines
correspond to the smaller daughter vessel that
resulted from the bifurcation. The arrows points in
the direction in which the M is increasing. As the
parent hematocrit value increase the difference in
the hematocrit values of the daughter vessels
increase. An increase in the drift parameter
decreases the difference found in the hematocrit
values of the daughter vessels.
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Figure 3: The plasma skimming coefficient of the
two daughter vessels are plotted with respect to
the drift parameter. The two daughter plasma
skimming coefficient begin to come close after the
initial rapid increase at the small drift parameter.
Figure 4: The hematocrit values of
the vessels in the large network
plotted against different parent
hematocrit along varying drift
parameters. (Drift parameters greater
than one.) Each set of grouped points
that expands in in the x and y axis are
the hematocrit values of the 23 values.
As the parent hematocrit increases the
difference between the hematocrit
values of the daughter vessels
increase. For a given parent
hematocrit value, as the drift
parameter increase the difference in
the hematocrit among the daughter
vessels decrease.
Figure 5: Fractional red cell flow in the daughter
vessels at a single bifurcation expressed as a
function of the fractional bulk blood flow. The
original data is scattered on the graph and the
model for each of the daughter vessel’s
hematocrit is shown. The blue values and line
pertains to the smaller daughter vessel with a
diameter of 6µm and the green values and line
pertains to the larger daughter vessel with a
diameter of 8µm. Note that the parent hematocrit
was 0.43 with a diameter of 7.5µm.
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Table 1: A sample of the values used and yielded for the simple bifurcation.
Daughter
Vessel 1 (DV 1)
Parent
Hematocrit
DV Hd at M = 2 DV Hd at M = 4 DV Hd at M = 6 DV Hd at M = 8
0.4 0.4824 0.4402 0.4266 0.4199
Q2 = 3
A2/A1 = 0.7
0.6 0.7236 0.6603 0.6399 0.6298
0.8 0.9648 0.8805 0.8532 0.8397
Daughter
Vessel 2 (DV 2)
Parent
Hematocrit
DV Hd at M = 2 DV Hd at M = 4 DV Hd at M = 6 DV Hd at M = 8
0.4 0.3647 0.3828 0.3886 0.3915
Q3 = 7
A3/A1 = 0.4
0.6 0.5470 0.5741 0.5829 0.5872
0.8 0.7294 0.7655 0.772 0.7830
Table 2: A sample of the hematocrit values of the large network.
Hematocrit Values of the vessels
Ves
sel
#
Ves
sel
Dia
met
er
(µm
)
M =4 H1 = 0.45
H1 = 0.45 H1 = 0.55 M = 3 M = 7
1 14 0.4500 0.5500 0.4500 0.4500
2 13.75 0.4480 0.5476 0.4474 0.4489
3 13.5 0.4506 0.5508 0.4509 0.4504
4 13.25 0.4464 0.5457 0.4453 0.4480
5 13 0.4534 0.5541 0.4545 0.4519
6 12.75 0.4490 0.5487 0.4486 0.4494
7 12.5 0.4561 0.5575 0.4581 0.4535
8 12.25 0.4515 0.5518 0.4520 0.4509
9 12 0.4584 0.5603 0.4612 0.4548
10 11.75 0.4536 0.5544 0.4548 0.4521
11 14 0.4521 0.5525 0.4528 0.4512
12 13.75 0.4543 0.5553 0.4557 0.4525
13 13.5 0.4501 0.5502 0.4502 0.4501
14 13.25 0.4522 0.5527 0.4529 0.4513
15 13 0.4479 0.5475 0.4472 0.4488
16 12.75 0.4558 0.5570 0.4577 0.4533
17 12.5 0.4513 0.5515 0.4517 0.4507
18 12.25 0.4536 0.5544 0.4548 0.4521
19 12 0.4490 0.5487 0.4486 0.4494
20 11.75 0.4513 0.5515 0.4517 0.4507
21 11.5 0.4464 0.5456 0.4452 0.4480
22 11.25 0.4560 0.5574 0.4581 0.4534
23 11 0.4509 0.5512 0.4512 0.4505
4. Discussion
Since the hematocrit values are an intensive property they are not conserved across any
bifurcation or division, this can be observed upon inspection of the values listed in either Table 1
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or Table 2. However, the volumetric bulk flow and the volumetric flow of the RBC are both
conserved throughout all the simulated bifurcations. The results show that an increase in the
parent hematocrit value or the RBC concentration yielded an increase in the difference between
the hematocrit values of the daughter vessels. In Fig. 2, this trend can be seen as the gap between
the lines representing the hematocrit values of the two daughter vessels increase as the parent
hematocrit value increases. Similarly, in Fig. 4, the same conclusion may be obtained since the
variation among the hematocrit values of the vessels increase as the parent hematocrit increases.
Comparison of hematocrit values of the daughter vessels from tables 1 and 2 for various parent
hematocrit value shows this relationship. Nonetheless, the trend suggest that an increase in the
concentration of RBCs cause the larger vessels to be hematocrit concentrated while smaller
vessels to be hematocrit diluted and magnifies the difference in the hematocrit value among the
daughter vessels; however, an increase in the drift parameter decreases the difference among the
hematocrit values of the daughter vessels considerably. In Fig. 2, the relationship between the
drift parameter and the hematocrit values can be seen as the hematocrit profile of the daughter
vessels comes closer with an increase in the drift parameter. The same effect can be observed in
Fig. 4, as the spread of the hematocrit values of the daughter vessels decreases as the drift
parameter increase.
Furthermore, Fig. 3 shows that as the drift parameter increases, the difference between
the plasma skimming coefficients of the daughter vessels begins to become insignificant. This
trend explains why the daughter vessels’ hematocrit values began to approach one another with
an increase in the drift parameter. After data fitting procedures were carried out on the
bifurcation data obtained by Pries et al. it was determined that the drift parameter that best fits
the data and models the data of fractional red cell flow as a function of fraction blood flow is
1.18, as shown in Fig. 5. This value differs considerably from the value of M = 5.25 reported by
Gould and Linninger [1], even though the graphical representation of fitted model with M = 5.25
is extremely alike to the one in Fig. 5. However, upon looking at the effect of the M value of
approximately 1.18 in Fig. 2, the obtainment of M = 1.18 isn’t quite reasonable because for
parent hematocrit values that are close to unity the largest daughter vessel’s hematocrit value
seem to go above unity. Therefore, it is possible that the drift parameter that best fits here does so
only for the specific data set or is due to other miscellaneous error in the data.
Nonetheless, an increase in the RBC concentration can lead to an increased uneven
distribution of the RBC among the daughter vessels, whereas a decrease in the RBC
concentration expressed through a considerably low parent hematocrit leads to RBC being
distributed without much significant difference among the daughter vessels. Interestingly
enough, the effect of the change in the plasma skimming coefficient on RBC distribution is a
rather strange one. An extremely high drift parameter results in a model with an inaccurate
distribution that underestimates the difference in the hematocrit values of the daughter vessels.
5. Perspective
Modeling the distribution of RBCs at bifurcations and other branching sites are important
for many applications, especially when dealing with oxygenation of various organs. The model’s
use of the plasma skimming coefficient is effective to a great extent in predicting the RBC
distributions at bifurcations. The predictions reflect prior observations and may be applied to
innovative applications that makes use of the plasma skimming coefficient to improve oxygen
treatment, etc.
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Intellectual Property
Biological and physiological data and some modeling procedures provided to you from Dr. Linninger’s lab are
subject to IRB review procedures and Intellectual property procedures.
Therefore, the use of these data and procedures are limited to the coursework only. Publications need to be approved
and require joint authorship with staff of Dr. Linninger’s lab.
References
[1] Gould, I.G., Linninger A. L., “Hematocrit distribution and tissue oxygenation in large
microcirculatory networks.” Microcirculation, (2014): epub.
[2] Pries Ar, Ley K, Claassen M, Gaehtgens P. Red-cell distribution at microvascular
bifurcations Microvasc Res 38: 81 – 101, 1989.
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Appendix A: Coding
Q1 = 10; Q2 = 3; Q3 = Q1 - Q2;
k = (0.40:.05:.85); p = (1:.5:10);
for j = 1:length(p);
for i = 1:length(k);
H(i,1,j) = k(i);
A = [1 0.7 0.4];
PSC2(j) =
(A(2)/A(1))^(1/p(j)); PSC3(j) =
(A(3)/A(1))^(1/p(j));
HAdj =
(Q1*H(i,1,j))/(Q2*PSC2(j) +
Q3*PSC3(j));
H(i,2,j) = PSC2(j)*HAdj; H(i,3,j) = PSC3(j)*HAdj;
end
end
figure;
for j = 1:length(p);
plot(H(:,1,j),H(:,2,j),H(:,1,j),H(:
,3,j)) hold on;
end xlabel('H_1'); ylabel('H_d (daughter
vessels)');
figure; plot(p,PSC2,p,PSC3) ylabel('Plasma Skimming
Coefficients (\theta)'); xlabel('M (Drift Parameter)');
legend('Daughter Vessel
1','Daughter Vessel
2','Location','Southeast');
Plotting the Larger Network:
close all; clear all;
ptCoordMx = [2 2; 2 4; 2 6; 4 6; 2 8; 4 8; 2 10; 4 10; 2 12; 4 12; 2 14; 5 4; 5 6; 7 6; 5 8; 8 4; 8 6; 10 6; 8 9; 12 6; 10 8; 10 9; 9 11; 11 4; ];
faceMx = [1 2; 2 3; 3 5; 3 4; 5 7; 5 6; 7 9; 7 8; 9 11; 9 10; 2 12; 12 16; 12 13; 13 14; 13 15; 16 24; 16 17;
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17 18; 17 19; 19 22; 19 23; 18 21; 18 20; ];
pointMx = [-1 0 0; 1 -2 -11; 2 -3 -4; 4 0 0; 3 -5 -6; 6 0 0; 5 -8 -7; 8 0 0; 7 -9 -10; 10 0 0; 9 0 0; 11 -12 -13; 13 -14 -15; 14 0 0; 15 0 0; 12 -16 -17; 17 -18 -19; 18 -23 -22; 19 -20 -21; 23 0 0; 22 0 0; 20 0 0; 21 0 0; 16 0 0; ];
%Diameter =
[12,15,16,9,13,12,8,11,9,15,10,12,9
,13,10,13,14,14,12,8,10,16,9]*(10^-
6); %Diameter = [linspace(14,19,10)
linspace(14,9,13)]*(10^-6); %Diameter = [linspace(14,14,10)
linspace(13,13,13)]*(10^-6);
i = 2; D(1) = 14;
while i <= 10 D(i) = D(i-1) - .25; i = i +1; end
D(11) = 14; i = 12; while i <=23
D(i) = D(i-1) - 0.25;
i = i +1; end
Diameter = D*(10^-6);
alpha = 128*(1.5/1000)*150*10^-
6./(pi*Diameter.^4);
[row1 col1] = size(faceMx); [row2 col2] = size(pointMx); [row3 col3] = size(ptCoordMx);
E = [100 5 5 5 5 5 5 5 5 5 5 5
5]*133.322368; %enter the given
initial conditions in the matrix
starting with P1... In our case P1
= 100
c = 1;
for i = 1:row2 if col2-
length(find(pointMx(i,:))) == col2
- 1 A(i, row1+i) = 1; b(i,1) = E(c); c = c + 1; end if col2-
length(find(pointMx(i,:))) < col2 -
1 for j = 1: col2; if pointMx(i,j) > 0 A(i,pointMx(i,j)) =
1; else if pointMx(i,j) <
0
A(i,abs(pointMx(i,j))) = -1; end end end b(i,1) = 0; end end
for i = 1:row1; A(i+row2,faceMx(i,1) + row1) =
1; A(i+row2,faceMx(i,2) + row1) =
-1; A(i+row2,i) = -alpha(i); b(i+row2,1) = 0; end x = A\b;
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% Note that the values in the x
vector correspond to the variables
in the % following order: x =
[F1,F2,...,F15,F16,P1,P2,...,P12,P1
3]'
%% P = sym('P', [row2 1]); for i = 1:row1; for j = 1:col1; Matrix1(i,j) =
P(faceMx(i,j)); end end
F = sym('F', [row1 1]); for i = 1:row2; if col2-
length(find(pointMx(i,:))) < col2 -
1 for j = 1:col2; if pointMx(i,j) > 0 Matrix2(i,j) =
F(pointMx(i,j)); end if pointMx(i,j) < 0 Matrix2(i,j) = -
1*F(abs(pointMx(i,j))); end end elseif col2-
length(find(pointMx(i,:))) == col2
-1 Matrix2(i,1) = P(i); end end
M = [sum(Matrix2,2); Matrix1(:,1) -
Matrix1(:,2) - F.*alpha'];
% for i = 1:length(M) % disp([char(M(i)),' =
',num2str(b(i))]); % end % % Symbols = [F;P]; % for i = 1:(row1+row2) % disp([char(Symbols(i)), ' = '
num2str(x(i))]); % end
figure;
for i = 1:row1 Xcoord =
[ptCoordMx(faceMx(i,:),1)]; Ycoord =
[ptCoordMx(faceMx(i,:),2)]; plot(Xcoord,Ycoord,'*-
','Color',[0.75 .5 .25],'LineWidth'
,Diameter(i)/(10^-6)/4); hold on;
FLabelX = mean(Xcoord); FLabelY = mean(Ycoord);
text(FLabelX,FLabelY,['\bf','\color
{red}','Q',int2str(i),',
','\bf','\color{blue}','H',int2str(
i)]);
end
hold off; xlim([min(ptCoordMx(:,1))-7,
max(ptCoordMx(:,1))+7]) ylim([min(ptCoordMx(:,2))-7,
max(ptCoordMx(:,2))+7]) set(gca,'XTickLabel',[]);
set(gca,'YTickLabel',[]); set(gca,'XTick',[]);
set(gca,'YTick',[]);
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(-
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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
Data fitting
data2 = [0.074468085 0.002777778 0.103723404 0.002777778 0.220744681 0.091666667 0.242021277 0.113888889 0.401595745 0.272222222
0.433510638 0.319444444 0.406914894 0.35 0.47606383 0.383333333 0.507978723 0.405555556 0.507978723 0.452777778 ];
data3 = [0.470744681 0.544444444 0.484042553 0.6 0.510638298 0.605555556 0.579787234 0.655555556 0.553191489 0.686111111 0.585106383 0.725 0.747340426 0.897222222 0.771276596 0.911111111 0.882978723 0.988888889 0.904255319 0.997222222 0.92287234 0.997222222 ];
h1 = 0.43; pd = 7.5; %microMeters d1 = 6; d2 = 8;
m = [5.25];
X = [0:0.005:1];
% for i = 1:length(m); % % psc1 = ((d1/pd)^2)^(1/m(i)); % psc2 = ((d2/pd)^2)^(1/m(i)); % % hadj = h1./(X*psc1 + (1 -
X)*psc2); % % % h2 = psc1*hadj; % % Hadj = h1./((1-X)*psc1 +
X*psc2); % % h3 = psc2*Hadj; % %
scatter(data2(:,1),data2(:,2),'fill
ed') % hold on;
scatter(data3(:,1),data3(:,2),'fill
ed'); hold on; % plot(X , (h2.*X)/(0.43),X,
h3.*X/0.43); hold on; % % end %
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% plot([0 1],[0
1],'r','LineStyle','--') % % axis([0 1 0 1]) % ylabel('Fractional red cell
flow'); % xlabel('Fractional flow'); % grid on;
figure; scatter(data2(:,1),data2(:,2));
hold on; scatter(data3(:,1),data3(:,2));
clear hadj
for i = 1:length(m);
psc1 = ((d1/pd)^2)^(1/m(i)); psc2 = ((d2/pd)^2)^(1/m(i));
hadj = h1./(data2(:,1)*psc1 + (1 -
data2(:,1))*psc2);
h2(:,i) = psc1.*hadj;
ratio(:,i) =
data2(:,1).*h2(:,i)/h1;
rsqr(i) = sum((ratio(:,i) -
data2(:,2)).^2);
plot(data2(:,1),ratio(:,i)); hold
on;
end
for i = 1:length(m);
psc1 = ((d1/pd)^2)^(1/m(i)); psc2 = ((d2/pd)^2)^(1/m(i));
hadj = h1./((1 - data3(:,1))*psc1 +
data3(:,1)*psc2);
h3(:,i) = psc2*hadj;
Ratio(:,i) =
data3(:,1).*h3(:,i)/h1;
Rsqr(i) = sum((Ratio(:,i) -
data3(:,2)).^2);
plot(data3(:,1),Ratio(:,i)); hold
on;
end
RSQR = rsqr + Rsqr;
figure; plot(m, RSQR);
[V I] = min(RSQR)
m(I)