Fitting Monod Equation with ODE45 to data using lsqcurvefit function

Hello!

I am fitting Monod equation to a data containing substrate (s), biomass (x), and ethanol (p) concentration against time. The objective is to get the parameters: 1) umax, 2) ks, 3) Yxs, and 4)Yps that will best represent the data. The differential equations are:

Here is my initial code using assumed values of the four parameters:

umax = 0.5;
ks = 6.5;
Yxs = 0.2;
Yps = 1.2;
%a(1) = x
%a(2) = s
%a(3) = p
f = @(t,a) [umax*a(1)*a(2)/(ks + a(2)); -(1/Yxs)*umax*a(1)*a(2)/(ks + a(2)); (1/Yps)*umax*a(1)*a(2)/(ks + a(2))];
xt0 = [0.0904,9.0115,0.0151];
[tspan,a] = ode45(f,[0 25],xt0);    
figure
plot(tspan,a(:,1),tspan,a(:,2),tspan,a(:,3))

Here is the code for trying to fit it into the actual data (script file):

function pos = paramfun1(x,tspan)

umax = x(1);
ks = x(2);
Yxs = x(3);
Yps = x(4);
xt0 = x(5:7);
f = @(t,a) [umax*a(1)*a(2)/(ks + a(2)); -(1/Yxs)*umax*a(1)*a(2)/(ks + a(2)); (1/Yps)*umax*a(1)*a(2)/(ks + a(2))];
[~,pos] = ode45(f,tspan,xt0);

Here is my call function (in the command window):

xt0 = zeros(1,7);
xt0(1) = umax;
xt0(2) = ks;
xt0(3) = Yxs;
xt0(4) = Yps; 
data =[0 3 5 8 9.5 11.5 14 16 18 20 25 27, 0.0904 0.1503 0.2407 0.3864 0.5201 0.6667 0.8159 0.9979 1.0673 1.1224 1.1512 1.2093; 0 3 5 8 9.5 11.5 14 16 18 20 25 27, 9.0115 8.8088 7.9229 7.2668 5.3347 4.911 3.5354 1.4041 0 0 0 0; 0 3 5 8 9.5 11.5 14 16 18 20 25 27, 0.0151 0.0328 0.0621 0.1259 0.2949 0.3715 0.4199 0.522 0.5345 0.6081 0.07662 0.7869];
%time =[0 3 5 8 9.5 11.5 14 16 18 20 25 27];
[pbest,exitflag,output] = lsqcurvefit(@paramfun,xt0,tspan,data);
fprintf('New parameters: %f, %f, %f, %f',pbest(1:4));

The error is function value not equal to YDATA. Btw, this code was based from an example in MATLAB. (https://www.mathworks.com/help/optim/ug/fit-differential-equation-ode.html)

My data is:

time = [0 3 5 8 9.5 11.5 14 16 18 20 25 27]

x = 0.0904 0.1503 0.2407 0.3864 0.5201 0.6667 0.8159 0.9979 1.0673 1.1224 1.1512 1.2093

s = 9.0115 8.8088 7.9229 7.2668 5.3347 4.911 3.5354 1.4041 0 0 0 0

p = 0.0151 0.0328 0.0621 0.1259 0.2949 0.3715 0.4199 0.522 0.5345 0.6081 0.07662 0.7869

Please help! I do not know how to input my data into the lsqcurvefit function.

Thanks in advance!

[Star Strider]

Monod kinetics and curve fitting

Answer 1

[David Wilson]

Here's my solution.

First set up the measured data, plot it, and look to see that it is all OK.

t = [0 3 5 8 9.5 11.5 14 16 18 20 25 27]' 
x = [0.0904 0.1503 0.2407 0.3864 0.5201 0.6667 0.8159 0.9979 1.0673 1.1224 1.1512 1.2093]'; 
s = [9.0115 8.8088 7.9229 7.2668 5.3347 4.911 3.5354 1.4041 0 0 0 0]';
p = [0.0151 0.0328 0.0621 0.1259 0.2949 0.3715 0.4199 0.522 0.5345 0.6081 0.07662 0.7869]';

plot(t,[x,s,p],'o-')

No problems there. Note that the measured data is in columns.

Now, we need a function to export the predicted measurements as a function of the unknown parameters and initial conditions. My function is:

function ypred = paramfun1(p,t)
umax = p(1); ks = p(2); Yxs = p(3); Yps = p(4); % use disperse here ...
xt0 = p(5:7); % initial conditions 
f = @(t,a) [umax*a(1)*a(2)/(ks + a(2)); ... 
            -(1/Yxs)*umax*a(1)*a(2)/(ks + a(2)); ...
            (1/Yps)*umax*a(1)*a(2)/(ks + a(2))];
[~,ypred] = ode45(f,t,xt0);

end 

May I suggest that you think about naming your functions sensible names. Also it is more elegant to use disperse (from the user's group) here.

Now we are ready to test it, and make sure our output is the same shape as the actual data.

xt0 = [0.1;9;0.1]; % read off from data plot [x0 ,S0, p0]';   
p0 = [umax; ks; Yxs; Yps; xt0] 

Ypred = paramfun1(p0,t); % measurement predictions

Now we are ready for the fit. I'm using your initial conditions, and we can get reasonable start values from the data.

pfit = lsqcurvefit(@paramfun1,p0,t,[x,s,p])

Now plot the fitted curve and the actual data. They should line up.

umax = pfit(1);  ks = pfit(2);Yxs = pfit(3); Yps = pfit(4);

f = @(t,a) [umax*a(1)*a(2)/(ks + a(2)); -(1/Yxs)*umax*a(1)*a(2)/(ks + a(2)); (1/Yps)*umax*a(1)*a(2)/(ks + a(2))];
xt0 = pfit(5:7); 
[tspan,a] = ode45(f,[0 25],xt0);    

plot(tspan,a(:,1),tspan,a(:,2),tspan,a(:,3), '-', ...
     t,x,'o',t,s,'+',t,p,'s')

Plot shows the regressed fit is not too bad:

[StarSign1997]

Thank you so much! Actually, I'll be doing a model discrimination among three models. One of them is Monod. Okay I understand now how the flow of passing the argument is supposed to be.

Thank you again!

[StarSign1997]

Is it possible for the initial values of the parameters to change as well? Because at different initial condition, the final parameter values also change. I am looking for the parameters that fit the data the best. Maybe an R^2 value =0.98.

[StarSign1997]

I got it. I used the finite difference step size found in the example. The final parameter values are not anymore so sensitive to the initial guess and almost has the same value regardless of my initial guess.

Thanks!

[Vinoj Liyanaarachchi]

First of all, I would like to thank StarSign1997 and David Wilson

StarSign1997 Can you kindly provide the final code.... It would be a great help.

I also have to model the same set of equations. Experimental data are as follows:

t = [0 2 4 6 8 10 12 14 16]';

x = [0.06 0.11 0.46 0.78 1.42 2.36 2.49 2.49 2.54]';

s = [9.54 9.33 9.52 9.06 8.05 7.27 6.03 5.80 5.51]';

p = [0.00 0.00 0.00 0.01 0.18 0.35 0.49 0.53 0.55]';

I used the code provided by David Wilson (I don't know how to disperse the intial values, So I used same code provided)

Curves were not fitted to the experimental data well.

Please help!

Thanks in advance!

[StarSign1997]

Ahh yes. I tried your data and we have the same results.

Here is the script I used in the command window. The function parafun1 is the same as the one posted.

t = [0 2 4 6 8 10 12 14 16]';
x = [0.06 0.11 0.46 0.78 1.42 2.36 2.49 2.49 2.54]'; 
s = [9.54 9.33 9.52 9.06 8.05 7.27 6.03 5.80 5.51]';
p = [0.00 0.00 0.00 0.01 0.18 0.35 0.49 0.53 0.55]';
plot(t,[x,s,p],'o-')
umax = 0.5;
ks = 2.5;
Yxs = 5;
Yps = 1.2;
xt0 = [0.06;9.54;0.00]; % read off from data plot [x0 ,S0, p0]';   
p0 = [umax; ks; Yxs; Yps; xt0] 
ypred = paramfun1(p0,t); % measurement predictions

options = optimoptions ('lsqcurvefit','FiniteDifferenceStepSize',1e-4,'FiniteDifferenceType','central');
[pfit, presnorm, presidual] = lsqcurvefit(@paramfun1,p0,t,[x,s,p],[],[],options);
pfit = lsqcurvefit(@paramfun1,p0,t,[x,s,p])
umax = pfit(1);  ks = pfit(2);Yxs = pfit(3); Yps = pfit(4);
f = @(t,a) [umax*a(1)*a(2)/(ks + a(2)); -(1/Yxs)*umax*a(1)*a(2)/(ks + a(2)); (1/Yps)*umax*a(1)*a(2)/(ks + a(2))];
xt0 = pfit(5:7); 
[tspan,a] = ode45(f,[0 16],xt0);    
plot(tspan,a(:,1),'-',tspan,a(:,2),'--',tspan,a(:,3),'k-.', ...
     t,x,'o',t,s,'+',t,p,'s')
title('Monod Equation Simulation')
xlabel('Time')
ylabel('Concentration')
legend('Simulated Biomass','Simulated Substrate','Simulated Ethanol','Biomass (Actual)','Substrate (Actual)','Ethanol (Actual)')

[Vinoj Liyanaarachchi]

Thank you StarSign1997 !!

Are there any techniques to improve the parameter estimation? Above shapes of the curves are not acceptable in my case (microalgae cultivation).

Thanks in advance!

[Antara Mazumder]

Hi this was really helpful, but I dont know I worte exactlr same code, but it showing me an error for using options saying " Empty keys are not allowed in this container.", dint understand this.

Answer 2

[Alex Sha]

The follow results obtained from 1stOpt may be used for comparison:

Code:

Variable t,x,s,p;
ODEFunction x'=umax*s*x/(ks + s);
            s'=-(1/Yxs)*umax*s*x/(ks + s);
            p'=(1/Yps)*umax*s*x/(ks + s);
Data;
t = [0 2 4 6 8 10 12 14 16];
x = [0.06 0.11 0.46 0.78 1.42 2.36 2.49 2.49 2.54];
s = [9.54 9.33 9.52 9.06 8.05 7.27 6.03 5.80 5.51];
p = [0.00 0.00 0.00 0.01 0.18 0.35 0.49 0.53 0.55];

Results:

Root of Mean Square Error (RMSE): 0.250520910244571
Sum of Squared Residual: 1.50625743527444
Correlation Coef. (R): 0.981509121545543
R-Square: 0.963360155677103
Adjusted R-Square: 0.914727231437843
Determination Coef. (DC): 0.942399473562309
F-Statistic: 14.7361249635671

Parameter          	Best Estimate
--------------------	-------------
umax	0.117013263727751
ks	-5.83263810513223
yxs	0.699511367324087
yps	5.01252941213249

[Vinoj Liyanaarachchi]

Thank you Alex Sha. Actually I'm new to matlab.

Can you share the code..

Thanks in advanse!

[Rifat Hasan]

Hi ! I have tried to run the code by David welson. I am getting an error.

Solver stopped prematurely.

lsqcurvefit stopped because it exceeded the function evaluation limit,

options.MaxFunctionEvaluations = 7.000000e+02.

How can I solve this issue?

Answer 3

[Alex Sha]

Hi, Vinoj Liyanaarachchi, the code have been provided above alreadly, but note, that code is not for Matlab, but for another package called 1stOpt. In Matlab, no matter lsqcurvefit or fminsearch functions, are all to adopt local optimization algorithms, it is why those functions depended haveily on initial start-values. Global optimization toolbox in Matlab (like GA) is not good enough to ensure the global solution.