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Estimation of Physiological Parameters Using a Physiologically Based Model

Overview

To evalute glucose trasnport and phosphorylation in skeletal muscle,a physiologically based model has been proposed by our group. Herein, we show an example of implementation of the proposed model using COMKAT software. In particular, the below example shows the implementation of the proposed model for a phosphorylatable glucsoe analog (e.g. ¹⁸F-labeled 2-fluoro-2-deoxy-D-glucose). The below figure shows the kinetics of [¹⁸F]2FDG in skeletal muscle. The model has three tissue compartments with five rate constants.

Rate constants k₁ and k₂ describe the passive exhange of the glucose analog between plasma and interstitial space.

Rate constants k_3a and k_4a describe the inward and backward transport of the glucose analog via glucose transporters.

Rate constant k_5a describe the phosphorylation of the intracellular glucose analog catalyzed by hexokinase.

COMKAT Implementation of the Physiologically Based Model

Here, we show an example of implementation of the proposed model using COMKAT software.

cm=compartmentModel;

% Define parameters
cm=addParameter(cm,'k1',0.1);                                             
cm=addParameter(cm,'k2','k1/Fis');                                        
cm=addParameter(cm,'k3','VG/(Fis*KGa+Fis*ISg*KGa/KGg)'); 
cm=addParameter(cm,'k4','VG/(Fic*KGa+Fic*ICg*KGa/KGg)'); 
cm=addParameter(cm,'k5','VH/(Fic*KHa+Fic*ICg*KHa/KHg)');  

cm=addParameter(cm,'Fb',0.02);         % fraction of total space occupied by blood
cm=addParameter(cm,'Fis',0.3);          % fraction of total space occupied by interstitial space
cm=addParameter(cm,'Fic','1-Fis-Fb');  % fraction of total space occupied by intracellular space
cm=addParameter(cm,'F',1);

cm=addParameter(cm,'Pg',6);     % Plasma glucose concentration (mM)
cm=addParameter(cm,'ISg',5.4); % Interstitial glucose concentration (mM)
cm=addParameter(cm,'ICg',0.2); % Intracellular glucose concentration (mM)

cm=addParameter(cm,'KGg',3.5); % Michaelis constant (mM) of glucose transporter (GLUT) for glucose 
cm=addParameter(cm,'KHg',0.13); % Michaelis constant (mM) of hexokinase (HK) for glucose 

cm=addParameter(cm,'KGa',14); % Michaelis constant (mM) of GLUT for glucose analog 
cm=addParameter(cm,'KHa',0.17); % Michaelis constant (mM) of HK for glucose analog 

cm=addParameter(cm,'VG','(k1*Pg-k1*ISg)/(ISg/(KGg+ISg)-ICg/(KGg+ICg))'); % Maximal velocity of glucose transport for glucose and its analogs
cm=addParameter(cm,'VH','(k1*Pg-k1*ISg)/(ICg/(KHg+ICg))'); % Maximal velocity of glucose phosphorylation for glucose and its analogs

% Specific activity (sa): if the unit of image data is the same with that of input function, the specific activity is 1.
cm=addParameter(cm,'sa',1); 
% Usually, the decay time correction of image data is performed. So, dk is zero. 
cm=addParameter(cm,'dk',0);

% Define scan time (minutes)
delay = 0.0;
scanduration = 120; 

t=[ones(5,1)/30;ones(10,1)/12;ones(12,1)*0.5;ones(8,1);ones(21,1)*5];
scant = [[0;cumsum(t(1:(length(t)-1)))] cumsum(t)];
scanTime  = [scant(:,1),scant(:,2)];

cm = set(cm, 'ScanTime', scanTime);

lambda = [-12.02 -2.57 -0.02];

a = [1771.5 94.55 14.27];

cm = addParameter(cm, 'pfeng', [delay a lambda]');

% Define input function
cm = addInput(cm, 'Cp', 'sa', 'dk', 'fengInputByPar','pfeng'); % Cp is the plasma input function
cm = addInput(cm, 'Ca',1,0, 'fengInputByPar', 'pfeng'); % Ca is the decay-corrected whole-blood input function. Herein, we assume that Cp=Ca.

% Define compartment
cm=addCompartment(cm,'IS'); % interstitial 
cm=addCompartment(cm,'IC'); % intracellular
cm=addCompartment(cm,'IP'); % intracellular phosphorylated 
cm=addCompartment(cm,'Junk');

% Define link
cm=addLink(cm,'L','Cp','IS','k1');
cm=addLink(cm,'K','IS','Junk','k2');
cm=addLink(cm,'K','IS','IC','k3');
cm=addLink(cm,'K','IC','IS','k4');
cm=addLink(cm,'K','IC','IP','k5');

% Define output obtained from each normalized compartment
wlistTotal={'IS','F';'IC','F';'IP','F'};
xlistTotal={'Ca','Fb'};
cm=addOutput(cm,'TissueTotal',wlistTotal,xlistTotal);

[PET,PETIndex,Output,OutputIndex]=solve(cm);

% Define model parameters to be estimated
cm=addSensitivity(cm,'k1','ISg','ICg','Fis','Fb'); 

noise_level=0.05;

sd=noise_level*sqrt(PET(:,3)./(PET(:,2)-PET(:,1)));

data=sd.*randn(size(PET(:,3)))+PET(:,3);

cm=set(cm,'ExperimentalData',data);

cm = set(cm,'ExperimentalDataSD',sd);

optsIRLS = setopt('SDModelFunction', @IRLSnoiseModel);
cm = set(cm, 'IRLSOptions', optsIRLS);

oo = optimset('TolFun', 1e-8, 'TolX', 1e-4,'Algorithm','interior-point');
cm = set(cm, 'OptimizerOptions', oo);

% Set initial conditions and bounds
%        k1,    ISg,   ICg,     Fs,     Fb
pinit = [0.01  ; 4.4 ; 0.1    ;0.15 ; 0.01];
plb   = [0.001; 1    ; 0.001;0.10 ; 0     ];
pub  = [0.5   ; 6    ; 1       ;0.60 ; 0.04];

[pfit, qfit, modelfit, exitflag, output, lambda, grad, hessian, objfunval] = fitGen(cm, pinit, plb, pub, 'IRLS'); 

figure;
t = 0.5*(PET(:,1)+PET(:,2));
plot(t,PET(:,3),'g-',t,data,'ro','LineWidth',2);
xlabel('Time (minutes)');
ylabel('Concentrations (uCi/mL)');
legend('Noisy data','Noise-free data');

figure;
t = 0.5*(PET(:,1)+PET(:,2));
plot(t,data,'ro',t,modelfit,'b-','LineWidth',2);
xlabel('Time (minutes)');
ylabel('Concentrations (uCi/mL)');
legend('Noisy data', 'Model fit');

Pg=6;KGa=14;KHa=0.17;KGg=3.5;KHg=0.13; % Units are mM

k1=pfit(1);k2=pfit(1)/pfit(4);ISg=pfit(2);ICg=pfit(3);Fis=pfit(4);Fb=pfit(5);

% Physiological parameters 

VG=(k1*ISg-k1*Pg)/(ICg/(KGg+ICg)-ISg/(KGg+ISg))      
 
VH=(k1*Pg-k1*ISg)/(ICg/(KHg+ICg))            
           
CI=VG*ISg/(KGg+ISg)           
                           
CE=VG*ICg/(KGg+ICg)    
                               
PR=VH*ICg/(KHg+ICg) 
                            

Results