Effects of Plasma Skimming Coefficients and RBC Concentration on RBC Spatial
Distribution
Jagan Jimmy
jjimmy2@uic.edu
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.
2/9/2016
1
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.
Q1, H1, A1
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.
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2
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.
Equations:
⃑ ∙𝑄 =0
∇
(1)
∇ ∙ 𝑄𝑅𝐵𝐶 = ∇ ∙ (𝑄𝐻𝑑 ) = 0
(2)
∆𝑃 = 𝑄
𝐻2 = 𝐻1 − ∆𝐻 = 𝜃2 ∙ 𝐻 ∗
1
𝐴2 𝑀
𝜃2 = ( )
𝐴1
8µ𝐿
𝜋𝑅 4
(3)
𝐻3 = 𝜃3 ∙ 𝐻 ∗
(4)
1
𝐴3 𝑀
𝜃3 = ( )
𝐴1
𝑄1 𝐻1 = 𝑄2 𝐻2 + 𝑄3 𝐻3 = 𝑄2 𝜃2 𝐻 ∗ + 𝑄3 𝜃3 𝐻 ∗
(5)
(6)
3. Results
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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3
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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4
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
0.8
0.7236
0.9648
0.6603
0.8805
0.6399
0.8532
0.6298
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
Q3 = 7
A3/A1 = 0.4
0.4
0.6
0.8
0.3647
0.5470
0.7294
0.3828
0.5741
0.7655
0.3886
0.5829
0.772
0.3915
0.5872
0.7830
Vessel #
Vessel
Diameter
(µm)
Table 2: A sample of the hematocrit values of the large network.
Hematocrit Values of the vessels
M =4
H1 = 0.45
H1 = 0.45
H1 = 0.55
M=3
M=7
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
14
13.75
13.5
13.25
13
12.75
12.5
12.25
12
11.75
14
13.75
13.5
13.25
13
12.75
12.5
12.25
12
11.75
11.5
11.25
11
0.4500
0.4480
0.4506
0.4464
0.4534
0.4490
0.4561
0.4515
0.4584
0.4536
0.4521
0.4543
0.4501
0.4522
0.4479
0.4558
0.4513
0.4536
0.4490
0.4513
0.4464
0.4560
0.4509
0.5500
0.5476
0.5508
0.5457
0.5541
0.5487
0.5575
0.5518
0.5603
0.5544
0.5525
0.5553
0.5502
0.5527
0.5475
0.5570
0.5515
0.5544
0.5487
0.5515
0.5456
0.5574
0.5512
0.4500
0.4474
0.4509
0.4453
0.4545
0.4486
0.4581
0.4520
0.4612
0.4548
0.4528
0.4557
0.4502
0.4529
0.4472
0.4577
0.4517
0.4548
0.4486
0.4517
0.4452
0.4581
0.4512
0.4500
0.4489
0.4504
0.4480
0.4519
0.4494
0.4535
0.4509
0.4548
0.4521
0.4512
0.4525
0.4501
0.4513
0.4488
0.4533
0.4507
0.4521
0.4494
0.4507
0.4480
0.4534
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
or Table 2. However, the volumetric bulk flow and the volumetric flow of the RBC are both
2/9/2016
5
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.
2/9/2016
6
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.
2/9/2016
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Appendix A: Coding
Q1 = 10;
Q2 = 3;
Q3 = Q1 - Q2;
legend('Daughter Vessel
1','Daughter Vessel
2','Location','Southeast');
k = (0.40:.05:.85);
p = (1:.5:10);
Plotting the Larger Network:
for j = 1:length(p);
close all;
clear all;
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;
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;
];
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)');
2/9/2016
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;
8
17
17
19
19
18
18
];
18;
19;
22;
23;
21;
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;
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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 col2length(find(pointMx(i,:))) == col2
- 1
A(i, row1+i) = 1;
b(i,1) = E(c);
c = c + 1;
end
if col2length(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;
9
% 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 col2length(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 col2length(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: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];
% 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
HSys = [0.45:0.05:0.85];
figure;
PSC(1*pointMx(i,2)) = (A(-
2/9/2016
for k = 1:size(M,2);
for j = 1:size(HSys,2);
H(1) = HSys(j);
for i = 1:row2
if col2length(find(pointMx(i,:))) == 0
10
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));
0.433510638
0.406914894
0.47606383
0.507978723
0.507978723
];
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)));
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
];
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
2/9/2016
h1
pd
d1
d2
=
=
=
=
0.319444444
0.35
0.383333333
0.405555556
0.452777778
0.43;
7.5; %microMeters
6;
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));
plot(data3(:,1),Ratio(:,i)); hold
on;
end
RSQR = rsqr + Rsqr;
figure;
plot(m, RSQR);
[V I] = min(RSQR)
m(I)
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);
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