detection of diabetes with matlab

1

Simulation in Biomedicine

Detection of Diabetes

Kévin MEFFRAY

Biomedical Engineering Master

Hochschule Furtwangen University

Villingen-Schwenningen, Germany

Rahman MIZANUR

Biomedical Engineering Master

Hochschule Furtwangen University

Villingen-Schwenningen, Germany

2

Table of Contents

Table of Contents..................................................................................................................... 2

Executive summary................................................................................................................. 3

Task.......................................................................................................................................................... 3

Method .................................................................................................................................................... 3

Results ..................................................................................................................................................... 5

Matlab code .............................................................................................................................. 7

Function “main” .................................................................................................................................... 7

Function “err” ........................................................................................................................................ 8

Function “gluc”...................................................................................................................................... 8

Function “errorfm” ............................................................................................................................... 8

Presentation.............................................................................................................................. 9

3

Executive summary/Abstract

Task

Statement of the task

Thus, )sincos()( 21 tctcetgt

ωωα

+=− and )sincos()( 210 tctceGtG

tωω

α

++=− .

Research has shown that lab results of 24/ω > 4 indicate a mild case of diabetes. For example, suppose

that we ha given the GTT to four patients we suspect of having a mild case of diabetes. The results for

each patient are shown in the following table. Which patients, if any, have a mild case of diabetes?

Patient 1 Patient 2 Patient 3 Patient 4

G0 80.00 90.00 100.00 110.00

t = 1 85.32 91.77 103.35 114.64

t = 2 82.54 85.69 98.26 105.89

t = 3 78.25 92.39 96.59 108.14

t = 4 76.61 91.13 99.47 113.76

Analysis of the task

Regarding the references added to the statement, g(t) and G(t) are respectively the excess glucose

concentration and the total glucose concentration of the patient. It’s also explained that 24/ω is a time

constant in hours named resonant period. Akkerman et al. claim that a resonant period over 4 hours is

a sign of diabetes.

The data are the results of a glucose tolerance test applied on 4 patients. Each patient received an

injection of glucose and then their glucose concentration has been measured every hour during 4

hours.

The task is to identify the parameters which set our model to fit the experimental data. When the

parameters are identified, the resonant period is calculated and compare to 4 hours. If the resonant

period is higher than 4 hours, the program should point out a case of diabetes.

Method

g(t) and G(t) being non-linear, the problem is to fit a non-linear equation. This has been already

done in class through the Levenberg-Marquardt algorithm. The same algorithm has been used in our

task.

Implementation

The Levenberg-Marquardt algorithm is implemented in a function given in class named

“SMarquardt”.

4

This implementation is described so:

[X,info,perf] = SMarquardt(fun,par,x0,opts)

where:

- fun is the function which needs to be minimized through the least-square fitting. In this

implementation, the function needed to be minimized is the function giving the difference

between the experimental data and the data calculated by the model at the same point. This

function is implemented in the Matlab function “err”.

- par is a vector of eventual parameters involved in the function to minimized. In this

implementation, this vector contains the experimental data for one patient.

- x0 is the initial vector of the implementation. It contains normal values of the unknown

parameter.

- otps is a vector of options. The values of this vector used in this implementation are the same

than the one used in class.

- X is the matrix of the approximation of the unknown parameter for each iteration of the

algorithm. The first column of the vector will be x0 and the last column is the final estimation

of the unknown parameter.

- info is a matrix containing different informations about the process of the algorithm. The only

value of this vector used in this implementation is info(5) which is the number of iteration

done by the algorithm.

- perf is a matrix containing different informations about the performances of the algorithm.

The only informations used in this implementation are the values of the first line of this vector

which are the results of half of the square residual sum. In the results, the first value of this

line is compared to the last value of this line to get an idea about the efficiency of the

algorithm.

The next step was to implement the “err” function. As it has been already explain, the SMarquardt

function already minimize the square residual sum of the function. So the task of the “err” function is

to evaluate the values of the estimated glucose concentration through the function “gluc” and to

calculate the difference with the experimental data.

The last step was the implementation of the function “gluc” which evaluate the value of the model

at a time t depending of the different parameters c1, c2, α, and ω. The equation of the excess glucose

concentration has been chosen in the implementation of this function to reduce the number of

parameters and avoid mistakes mixing the G0 of the different patients.

The general implementation of the detection is divided in 4 step.

The first step is to insert the data.

The second step is to transform the matrix of total glucose concentration in a matrix of excess glucose

concentration allowing avoiding the parameter G0.

The third step is the processing of the Levenberg-Marquardt algorithm.

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The final step is a shaping of the results in order present the results. In this step is included the if-loop

identifying if the patient has diabetes or not. In order to get a good graphic shape of the model, a new

time matrix has been define and the values of the model have been calculated for this time matrix.

Problems

The major problem in this implementation is that the Levenberg-Marquardt algorithm is very

sensitive to the initial vector and no normal values of the parameters were available in the references.

The solution to this problem was to use the fitting algorithm implemented in the function fminsearch

in Matlab which is less sensible to the initial vector. Regarding the fact that the fminsearch algorithm

doesn’t include the evaluation of the square residual sum, this calculation has to be included in the

“errorfm” function. This algorithm led to results for each patient which were averaged to create an

initial vector for the Levenberg-Marquardt algorithm.

Results

*************** Patient 1 *******************

Fitting results after 25 iterations

Square sum of residuals for initial guess / last iterate : 54.726224 /

0.257647

Parameters: alpha=0.173469, omega=1.221912, C1=0.102495,

C2=6.484800

The time constant is 5.142092 so the patient has a mild case of

diabetes.

*************** Patient 2 *******************



0.637985

Parameters: alpha=-0.184088, omega=2.279678, C1=0.208101,

C2=2.612618

The time constant is 2.756173 so the patient doesnt have diabetes

0 0.5 1 1.5 2 2.5 3 3.5 485

86

87

88

89

90

91

92

93

94

95

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

0 0.5 1 1.5 2 2.5 3 3.5 476

77

78

79

80

81

82

83

84

85

86

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

6

*************** Patient 3 *******************



2.640664

Parameters: alpha=-0.043125, omega=1.518054, C1=0.711780,

C2=3.037021

The time constant is 4.138973 so the patient has a mild case of

diabetes

*************** Patient 4 *******************



0.556820

Parameters: alpha=0.099449, omega=1.968686, C1=0.164680,

C2=5.966058

The time constant is 3.191563 so the patient doesnt have diabetes

0 0.5 1 1.5 2 2.5 3 3.5 4104

106

108

110

112

114

116

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

0 0.5 1 1.5 2 2.5 3 3.5 496

97

98

99

100

101

102

103

104

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

7

Matlab code/Abstract

Function “main”

clear all clc

%% Data

global t

data = [ 80 90 100 110 85.32 91.77 103.35 114.64 82.54 85.69 98.26 105.89 78.25 92.39 96.59 108.14 76.61 91.13 99.47 113.76]; %Column : Patients %Lines : G0-G4 : total glucose concentration at t=0-4 t=0:1:4;

%% Test

for p=1:1:length(data) g(p,:)=data(p,:)-data(1,:); end

global t gd

Tr = 0:0.1:4;

for k=1:1:length(g(1,:)) Gc = 0; gd=g(:,k); % par(k,:)=fminsearch('errorfm',[0.5 0.5 1 1]); [par,info,perf]=SMarquardt('err',gd,[0.01 1.7 0.3 4.5],[1e-3 1e-4 1e-8

200 1e-8]);

fprintf('*************** Patient %d *******************\n',k); fprintf('Fitting results after %d iterations\n', info(5)); fprintf('Square sum of residuals for initial guess / last iterate : %f

/ %f\n',2*perf(1,1),2*perf(1,end)); fprintf('Parameters: alpha=%f, omega=%f, C1=%f,

C2=%f\n',par(1,end),par(2,end),par(3,end),par(4,end)); if (2*pi()/par(2,end))>4 fprintf('The time constant is %f so the patient has a mild case of

diabetes\n\n',2*pi()/par(2,end)); else fprintf('The time constant is %f so the patient doesnt have

diabetes\n\n',2*pi()/par(2,end)); end

Gc = gluc(Tr,par(:,end))+data(1,k);

figure

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plot(t,data(:,k),'x','LineWidth',1.5,'MarkerSize',10); hold on plot(Tr,Gc,'r','LineWidth',1.5); xlabel('time [hour]'); ylabel('glucose level [mg/dL]'); legend('measured data','model result');

end

Function “err”

function err=err(param,gd)

global t

gc=gluc(t,param);

err=(gd-gc);

end

Function “gluc”

function g=gluc(t,param)

alpha=param(1); omega=param(2); c1=param(3); c2=param(4);

for k=1:1:length(t) g(k,1)=exp(-alpha*t(k))*((c1*cos(omega*t(k)))+(c2*sin(omega*t(k)))); end

end

Function “errorfm”

function err=errorfm(param)

global t gd

gc=gluc(t,param);

err=sum((gd-gc).^2);

end

9

Presentation/Abstract

Biomedical Engineering Master, Furtwangen University, Villingen-Schwenningen, Germany


K. Meffray, R. Mizanur



Introduction

Methods

Results

Materiel

Testing for diabetes

=> Fitting a non-linear equation

)sincos()(21

tctcetgt

ωωα

+=−

)sincos()( 210 tctceGtGt

ωωα

++=−

• Glucose Tolerance Test (GTT)

• Model is a differential equation

• Characteristic solutions of the equation

- Total glucose concentration :

- Excess glucose concentration :

• Criteria for diabetes : 2π/ω > 4

Questions

•

=> Levenberg-Marquardt algorithm

•

10


Patient data

113.7699.4791.1376.61t = 4

108.1496.5992.3978.25t = 3

105.8998.2685.6982.54t = 2

114.64103.3591.7785.32t = 1

110.00100.0090.0080.00G0

Patient 4Patient 3Patient 2Patient 1

• GTT of 4 patient taken at 1 hour intervals during 4 hours

Introduction

Materiel

Methods

•

Results

•

Questions


Implementation of the Levenberg-Maquart Algorithm

• Implementation of the function “err”

function g=gluc(t,param)

alpha=param(1);

omega=param(2);

c1=param(3);

c2=param(4);

for k=1:1:length(t)

g(k,1)=exp(-alpha*t(k))*((c1*cos(omega*t(k)))+(c2*sin(omega*t(k))));

end

end

• Implementation of the function “gluc”

function err=err(param,gd)

global t

gc=gluc(t,param);

err=(gd-gc);

end

• The function “SMarquardt”

[X,info,perf] = SMarquardt(fun,par,x0,opts)

)sincos()( 21 tctcetgt

ωωα

+=−

Introduction

Materiel

Methods

•

Results

•

Questions

11


Implementation of the Levenberg-Maquart Algorithm

• Implementation of the parameters identificationclear allclc

%% Data

global t

data = [ 80 90 100 11085.32 91.77 103.35 114.64 82.54 85.69 98.26 105.8978.25 92.39 96.59 108.1476.61 91.13 99.47 113.76];

t=0:1:4;

%% Test

for p=1:1:length(data)g(p,:)=data(p,:)-data(1,:);

end

global t gd

Tr = 0:0.1:4;

for k=1:1:length(g(1,:))Gc = 0;gd=g(:,k);

% X0=fminsearch('errorfm',[0.5 0.5 1 1]);[par,info,perf]=SMarquardt('err',gd,[0.01 1.7 0.3 4.5],[1e-3 1e-4 1e-8 200 1e-8]);

fprintf('*************** Patient %d *******************\n',k);fprintf('Fitting results after %d iterations\n', info(5)); fprintf('Square sum of residuals for initial guess / last iterate : %f / %f\n',2*perf(1,1),2*perf(1,end)); fprintf('Parameters: alpha=%f, omega=%f, C1=%f, C2=%f\n',par(1,end),par(2,end),par(3,end),par(4,end)); if (2*pi()/par(2,end))>4

fprintf('The time constant is %f so the patient has a mild case of diabetes\n\n',2*pi()/par(2,end));else

fprintf('The time constant is %f so the patient doesnt have diabetes\n\n',2*pi()/par(2,end));end

Gc = gluc(Tr,par(:,end))+data(1,k);

figureplot(t,data(:,k),'x','LineWidth',1.5,'MarkerSize',10); hold onplot(Tr,Gc,'r','LineWidth',1.5); xlabel('time [hour]'); ylabel('glucose level [mg/dL]'); legend('measured data','model result');

end

Introduction

Materiel

Methods

•

Results

•

Questions

0 0.5 1 1.5 2 2.5 3 3.5 476

77

78

79

80

81

82

83

84

85

86

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

0 0.5 1 1.5 2 2.5 3 3.5 485

86

87

88

89

90

91

92

93

94

95

time [hour]

glu

cos

e le

vel

[mg

/dL

]

measured data

model result


Results

*************** Patient 1 *******************


Square sum of residuals for initial guess / last iterate :

54.726224 / 0.257647

Parameters: alpha=0.173469, omega=1.221912,

C1=0.102495, C2=6.484800

The time constant is 5.142092 so the patient has a mild

case of diabetes.

*************** Patient 2 *******************



56.862800 / 0.637985

Parameters: alpha=-0.184088, omega=2.279678,

C1=0.208101, C2=2.612618

The time constant is 2.756173 so the patient doesnt have

diabetes

Introduction

Materiel

Methods

•

Results

•

Questions

12

0 0.5 1 1.5 2 2.5 3 3.5 496

97

98

99

100

101

102

103

104

time [hour]

glu

cose level [m

g/d

L]

measured data

model result

0 0.5 1 1.5 2 2.5 3 3.5 4104

106

108

110

112

114

116

time [hour]

glu

cose level [m

g/d

L]

measured data

model result


Results

*************** Patient 3 *******************



10.040399 / 2.640664

Parameters: alpha=-0.043125, omega=1.518054,

C1=0.711780, C2=3.037021

The time constant is 4.138973 so the patient has a mild

case of diabetes

*************** Patient 4 *******************



13.622471 / 0.556820

Parameters: alpha=0.099449, omega=1.968686,

C1=0.164680, C2=5.966058

The time constant is 3.191563 so the patient doesnt have

diabetes

Introduction

Materiel

Methods

•

Results

•

Questions

Biomedical Engineering Master, Furtwangen University, Villingen-Schwenningen, Germany


K. Meffray, R. Mizanur

Detection of DiabetesQuestions ?

detection of diabetes with matlab

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