How to fit kinetic model to estimate kinetic parameter from experimental data
I am getting errors in the output. Please help be solve the problem. I have attached the Excel file of the experimental data.
Thank you.
%Fermentation data
Xdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’e2:e13′);
Gdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’b2:b13′);
Bdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’c2:c13′);
Edata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’d2:d13′);
% Group all data into yariable y
yinv =[Xdata’; Gdata’; Bdata’; Edata’];
%Data for time
timeex = readmatrix(‘batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’a2:a13′);
%Set ODE solver
% Initial cell biomass concentration (Initial condition)
y0=[1.04 0.78 0.00 0.00];
%Fermentation time
tspan = linspace(0,72);
%Set optimization problem
%Let b = matrix of paramters to be determined
% b= [um ks k1 K1G K1B k2 K2G YXG m k3]
b = optimvar(‘b’,10,"LowerBound",0,"UpperBound",72);
%Set functions for ODE solver for solving ODE
function solferment = toODE(b,tspan,y0)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
solferment = deval(sol,tspan);
end
%Convert function for ODE solving to optimization expression
%To use RtoODE in an objective function, convert the function to an
%optimization expression by using fcn2optimexpr.
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
%Express the objective function as the sum of squared differences between
%the ODE solution and the solution with true parameters.
SSE = sum(sum((myfcn-yinv).^2));
%Create an optimization problem with the objective function SSE.
prob = optimproblem ("Description","Fit ODE parameters",’ObjectiveSense’,’min’);
%Objective function (to be minimized)
prob.Objective = SSE;
%Show structure of problem
show(prob)
%Solve Problem
%To find the best-fitting parameters x, give an initial guess
%x0 for the solver and call solve.
% Set initial guesses of parameters
initialGuess.b = [0.18 1.0 0.61 0.18 5.85 3.20 16.25 0.11 3.40 3.02];
%Solve optimization problem
[sol,optval] = solve(prob,initialGuess);
%Extract the results
%Fitted Parameters
bfinal =sol.b;
%Sum of square Error
SSEfinal = optval;
%Plot the simulated data and experimental data
%Call ODE to solve an equation using Final Fitted Parameters (bfinal)
solysim = ode45(@(t,y)batchferment(t,y,bfinal),tspan,y0);
%Evaluate the simulated results at specified tspan
ysim = deval(solysim,tspan);
%Plot graphs
figure;
plot(timeex,Xdata,’*r’,tspan,ysim(1,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘X (g/L)’)
figure;
plot(timeex,Gdata’,’*r’,tspan,ysim(2,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘G (g/L)’)
figure;
plot(timeex,Bdata’,’*r’,tspan,ysim(3,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘B (g/L)’)
figure;
plot(timeex,Edata’,’*r’,tspan,ysim(4,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘E (g/L)’)
%Equations for Batch
%y(1) = X = Biomass Concentration (g/l)
%y(2) = G = Glucose Concentration (g/l)
%y(3) = B = Cellobiose Concentration (g/L)
%y(4) = E = Ethanol Concentration (g/l)
function dydt = batchferment(t,y,b)
y(1) = X;
y(2) = G;
y(3) = B;
y(4) = E;
%%%Growth equations
%dx/dt = (um*X*G)/(KG+G)
u = (b(1)*y(1)*y(2))/(b(2)+y(2));
%u = (b(1)*X*G)/(b(2)+G);
%parameters
b(1) = um (1/h);
b(2) = ks (g/L);
%%%Cellobiose equations
C = b(11)-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04));
r1 = (b(3)*C)/(1+(y(2)/b(4))+(y(3)/b(5)));
r2 = (b(6)*y(3))/(1+(y(2)/b(7)));
b(3) = k1;
b(4) = K1G;
b(5) = K1B;
b(6) = k2;
b(7) = K2G;
%C = Cellulose concentration (g/l)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%r1 = (k1*C)/(1+(G/K1G)+(B/K1B))
%r2 = (k2*B)/(1+(G/K2G)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%C = 83.65-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04))
%%% Glucose equations
rG = ((1/b(8))*(dx/dt))-(b(9)*y(1));
b(8) = YXG;
b(9) = m;
%rG = ((1/YXG)*(dX/dt))-(m*X)
%%% Ethanol Concentration
Et = k3*(u*X)*(1/YXG);
b(10) = k3;
%Material balance equations
dXdt = u*X;
dGdt = (r2/0.95)-rG;
dBdt = (r1/0.947)-r2;
dEdt = Et;
dydt = [dXdt;dGdt;dBdt;dEdt];
end
This is the output after running:
>> Batch1
Unrecognized function or variable ‘X’.
Error in Batch1>batchferment (line 110)
y(1) = X;
Error in Batch1>@(t,y)batchferment(t,y,b) (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in odearguments (line 92)
f0 = ode(t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 104)
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);
Error in Batch1>toODE (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in optim.problemdef.fcn2optimexpr
Error in optim.problemdef.fcn2optimexpr
Error in fcn2optimexpr
Error in Batch1 (line 38)
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
Caused by:
Function evaluation failed while attempting to determine output size. The function might contain an error, or might not be well-defined at
the automatically-chosen point. To specify output size without function evaluation, use ‘OutputSize’.
>>I am getting errors in the output. Please help be solve the problem. I have attached the Excel file of the experimental data.
Thank you.
%Fermentation data
Xdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’e2:e13′);
Gdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’b2:b13′);
Bdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’c2:c13′);
Edata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’d2:d13′);
% Group all data into yariable y
yinv =[Xdata’; Gdata’; Bdata’; Edata’];
%Data for time
timeex = readmatrix(‘batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’a2:a13′);
%Set ODE solver
% Initial cell biomass concentration (Initial condition)
y0=[1.04 0.78 0.00 0.00];
%Fermentation time
tspan = linspace(0,72);
%Set optimization problem
%Let b = matrix of paramters to be determined
% b= [um ks k1 K1G K1B k2 K2G YXG m k3]
b = optimvar(‘b’,10,"LowerBound",0,"UpperBound",72);
%Set functions for ODE solver for solving ODE
function solferment = toODE(b,tspan,y0)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
solferment = deval(sol,tspan);
end
%Convert function for ODE solving to optimization expression
%To use RtoODE in an objective function, convert the function to an
%optimization expression by using fcn2optimexpr.
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
%Express the objective function as the sum of squared differences between
%the ODE solution and the solution with true parameters.
SSE = sum(sum((myfcn-yinv).^2));
%Create an optimization problem with the objective function SSE.
prob = optimproblem ("Description","Fit ODE parameters",’ObjectiveSense’,’min’);
%Objective function (to be minimized)
prob.Objective = SSE;
%Show structure of problem
show(prob)
%Solve Problem
%To find the best-fitting parameters x, give an initial guess
%x0 for the solver and call solve.
% Set initial guesses of parameters
initialGuess.b = [0.18 1.0 0.61 0.18 5.85 3.20 16.25 0.11 3.40 3.02];
%Solve optimization problem
[sol,optval] = solve(prob,initialGuess);
%Extract the results
%Fitted Parameters
bfinal =sol.b;
%Sum of square Error
SSEfinal = optval;
%Plot the simulated data and experimental data
%Call ODE to solve an equation using Final Fitted Parameters (bfinal)
solysim = ode45(@(t,y)batchferment(t,y,bfinal),tspan,y0);
%Evaluate the simulated results at specified tspan
ysim = deval(solysim,tspan);
%Plot graphs
figure;
plot(timeex,Xdata,’*r’,tspan,ysim(1,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘X (g/L)’)
figure;
plot(timeex,Gdata’,’*r’,tspan,ysim(2,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘G (g/L)’)
figure;
plot(timeex,Bdata’,’*r’,tspan,ysim(3,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘B (g/L)’)
figure;
plot(timeex,Edata’,’*r’,tspan,ysim(4,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘E (g/L)’)
%Equations for Batch
%y(1) = X = Biomass Concentration (g/l)
%y(2) = G = Glucose Concentration (g/l)
%y(3) = B = Cellobiose Concentration (g/L)
%y(4) = E = Ethanol Concentration (g/l)
function dydt = batchferment(t,y,b)
y(1) = X;
y(2) = G;
y(3) = B;
y(4) = E;
%%%Growth equations
%dx/dt = (um*X*G)/(KG+G)
u = (b(1)*y(1)*y(2))/(b(2)+y(2));
%u = (b(1)*X*G)/(b(2)+G);
%parameters
b(1) = um (1/h);
b(2) = ks (g/L);
%%%Cellobiose equations
C = b(11)-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04));
r1 = (b(3)*C)/(1+(y(2)/b(4))+(y(3)/b(5)));
r2 = (b(6)*y(3))/(1+(y(2)/b(7)));
b(3) = k1;
b(4) = K1G;
b(5) = K1B;
b(6) = k2;
b(7) = K2G;
%C = Cellulose concentration (g/l)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%r1 = (k1*C)/(1+(G/K1G)+(B/K1B))
%r2 = (k2*B)/(1+(G/K2G)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%C = 83.65-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04))
%%% Glucose equations
rG = ((1/b(8))*(dx/dt))-(b(9)*y(1));
b(8) = YXG;
b(9) = m;
%rG = ((1/YXG)*(dX/dt))-(m*X)
%%% Ethanol Concentration
Et = k3*(u*X)*(1/YXG);
b(10) = k3;
%Material balance equations
dXdt = u*X;
dGdt = (r2/0.95)-rG;
dBdt = (r1/0.947)-r2;
dEdt = Et;
dydt = [dXdt;dGdt;dBdt;dEdt];
end
This is the output after running:
>> Batch1
Unrecognized function or variable ‘X’.
Error in Batch1>batchferment (line 110)
y(1) = X;
Error in Batch1>@(t,y)batchferment(t,y,b) (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in odearguments (line 92)
f0 = ode(t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 104)
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);
Error in Batch1>toODE (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in optim.problemdef.fcn2optimexpr
Error in optim.problemdef.fcn2optimexpr
Error in fcn2optimexpr
Error in Batch1 (line 38)
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
Caused by:
Function evaluation failed while attempting to determine output size. The function might contain an error, or might not be well-defined at
the automatically-chosen point. To specify output size without function evaluation, use ‘OutputSize’.
>> I am getting errors in the output. Please help be solve the problem. I have attached the Excel file of the experimental data.
Thank you.
%Fermentation data
Xdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’e2:e13′);
Gdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’b2:b13′);
Bdata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’c2:c13′);
Edata = readmatrix(‘Batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’d2:d13′);
% Group all data into yariable y
yinv =[Xdata’; Gdata’; Bdata’; Edata’];
%Data for time
timeex = readmatrix(‘batch1.xlsx’,’Sheet’,’sheet1′,’Range’,’a2:a13′);
%Set ODE solver
% Initial cell biomass concentration (Initial condition)
y0=[1.04 0.78 0.00 0.00];
%Fermentation time
tspan = linspace(0,72);
%Set optimization problem
%Let b = matrix of paramters to be determined
% b= [um ks k1 K1G K1B k2 K2G YXG m k3]
b = optimvar(‘b’,10,"LowerBound",0,"UpperBound",72);
%Set functions for ODE solver for solving ODE
function solferment = toODE(b,tspan,y0)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
solferment = deval(sol,tspan);
end
%Convert function for ODE solving to optimization expression
%To use RtoODE in an objective function, convert the function to an
%optimization expression by using fcn2optimexpr.
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
%Express the objective function as the sum of squared differences between
%the ODE solution and the solution with true parameters.
SSE = sum(sum((myfcn-yinv).^2));
%Create an optimization problem with the objective function SSE.
prob = optimproblem ("Description","Fit ODE parameters",’ObjectiveSense’,’min’);
%Objective function (to be minimized)
prob.Objective = SSE;
%Show structure of problem
show(prob)
%Solve Problem
%To find the best-fitting parameters x, give an initial guess
%x0 for the solver and call solve.
% Set initial guesses of parameters
initialGuess.b = [0.18 1.0 0.61 0.18 5.85 3.20 16.25 0.11 3.40 3.02];
%Solve optimization problem
[sol,optval] = solve(prob,initialGuess);
%Extract the results
%Fitted Parameters
bfinal =sol.b;
%Sum of square Error
SSEfinal = optval;
%Plot the simulated data and experimental data
%Call ODE to solve an equation using Final Fitted Parameters (bfinal)
solysim = ode45(@(t,y)batchferment(t,y,bfinal),tspan,y0);
%Evaluate the simulated results at specified tspan
ysim = deval(solysim,tspan);
%Plot graphs
figure;
plot(timeex,Xdata,’*r’,tspan,ysim(1,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘X (g/L)’)
figure;
plot(timeex,Gdata’,’*r’,tspan,ysim(2,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘G (g/L)’)
figure;
plot(timeex,Bdata’,’*r’,tspan,ysim(3,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘B (g/L)’)
figure;
plot(timeex,Edata’,’*r’,tspan,ysim(4,:),’-b’)
legend(‘exp’,’sim’)
xlabel(‘Time (h)’)
ylabel(‘E (g/L)’)
%Equations for Batch
%y(1) = X = Biomass Concentration (g/l)
%y(2) = G = Glucose Concentration (g/l)
%y(3) = B = Cellobiose Concentration (g/L)
%y(4) = E = Ethanol Concentration (g/l)
function dydt = batchferment(t,y,b)
y(1) = X;
y(2) = G;
y(3) = B;
y(4) = E;
%%%Growth equations
%dx/dt = (um*X*G)/(KG+G)
u = (b(1)*y(1)*y(2))/(b(2)+y(2));
%u = (b(1)*X*G)/(b(2)+G);
%parameters
b(1) = um (1/h);
b(2) = ks (g/L);
%%%Cellobiose equations
C = b(11)-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04));
r1 = (b(3)*C)/(1+(y(2)/b(4))+(y(3)/b(5)));
r2 = (b(6)*y(3))/(1+(y(2)/b(7)));
b(3) = k1;
b(4) = K1G;
b(5) = K1B;
b(6) = k2;
b(7) = K2G;
%C = Cellulose concentration (g/l)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%r1 = (k1*C)/(1+(G/K1G)+(B/K1B))
%r2 = (k2*B)/(1+(G/K2G)
%C = C0-(0.9*G)-(0.947*B)-(0.9*E/0.511)-(1.137*(X-X0))
%C = 83.65-(0.9*y(2))-(0.947*y(3))-(0.9*y(4)/0.511)-(1.137*(y(1)-1.04))
%%% Glucose equations
rG = ((1/b(8))*(dx/dt))-(b(9)*y(1));
b(8) = YXG;
b(9) = m;
%rG = ((1/YXG)*(dX/dt))-(m*X)
%%% Ethanol Concentration
Et = k3*(u*X)*(1/YXG);
b(10) = k3;
%Material balance equations
dXdt = u*X;
dGdt = (r2/0.95)-rG;
dBdt = (r1/0.947)-r2;
dEdt = Et;
dydt = [dXdt;dGdt;dBdt;dEdt];
end
This is the output after running:
>> Batch1
Unrecognized function or variable ‘X’.
Error in Batch1>batchferment (line 110)
y(1) = X;
Error in Batch1>@(t,y)batchferment(t,y,b) (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in odearguments (line 92)
f0 = ode(t0,y0,args{:}); % ODE15I sets args{1} to yp0.
Error in ode45 (line 104)
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);
Error in Batch1>toODE (line 30)
sol = ode45(@(t,y)batchferment(t,y,b),tspan,y0);
Error in optim.problemdef.fcn2optimexpr
Error in optim.problemdef.fcn2optimexpr
Error in fcn2optimexpr
Error in Batch1 (line 38)
myfcn = fcn2optimexpr(@toODE,b,timeex,y0);
Caused by:
Function evaluation failed while attempting to determine output size. The function might contain an error, or might not be well-defined at
the automatically-chosen point. To specify output size without function evaluation, use ‘OutputSize’.
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