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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.

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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)