I want to do a parameter sensitivity analysis to my model

Question

Esraa Abdelkhaleq 2017-1-4

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https://ww2.mathworks.cn/matlabcentral/answers/319019-i-want-to-do-a-parameter-sensitivity-analysis-to-my-model

编辑： dpb 2017-1-31

Hello,

I have a model equation contains some parameters, I want to do a parameter sensitivity analysis to some parameters to justify the values of the parameters. How can I do that using Matlab?

I am doing two things:

The first is solving the model equation for qi(t) and set the solution equal to (CxVpl= 3.8 * 3150) to get a value for "t" which must be positive.

But what I obtained is:

sol =

Empty sym: 0-by-1

So, I want to do a sensitivity analysis to the parameters to get a valid solution.

The model equation,

dqidt=(fi_Ri*Nc0*ai\(Kgr+di))*(exp(Kgr*t)-exp(-di*t))+(fi_Ri*Ni0*exp(-di*t))+fih_Rih_Nih-Kei*qi

The solution for qi(t), by taking Laplace Transform,

qi(t) =
(Nih*Rih*fih)/Kei + (exp(Kgr*t)*(Kgr + di))/(Nc0*Ri*ai*fi*(Kei + Kgr)) - (exp(-di*t)*(- Nc0*Ni0*ai*Ri^2*fi^2 + Kgr + di))/(Nc0*Ri*ai*fi*(Kei - di)) + (exp(-Kei*t)*(Nc0*ai*qi0*Kei^3*Ri*fi + Nc0*ai*qi0*Kei^2*Kgr*Ri*fi - Nc0*Ni0*ai*Kei^2*Ri^2*fi^2 - Nc0*ai*qi0*Kei^2*Ri*di*fi - Nc0*Nih*Rih*ai*fih*Kei^2*Ri*fi + Kei*Kgr^2 - Nc0*Ni0*ai*Kei*Kgr*Ri^2*fi^2 - Nc0*ai*qi0*Kei*Kgr*Ri*di*fi - Nc0*Nih*Rih*ai*fih*Kei*Kgr*Ri*fi + 2*Kei*Kgr*di + Nc0*Nih*Rih*ai*fih*Kei*Ri*di*fi + Kei*di^2 + Nc0*Nih*Rih*ai*fih*Kgr*Ri*di*fi))/(Kei*Nc0*Ri*ai*fi*(Kei - di)*(Kei + Kgr))

The parameters:

fi_Ri (Immune biomarker shedding rate) =10.925*10^(-6) ;
ai (Immune cell activation rate) =4.74 ; 
Nc0 (Initial number of tumor cells) =1 ; 
Kgr (Tumor growth rate) = 5.78*10^(-3);
di (Immune cell death rate) =11.31 ;
Ni0 (Initial number of immune cells) =1 ;
fih_Rih_Nih (Immune biomarker healthy influx) =7.16*10^(4) ; 
Kei (Immune biomarker elimination rate) = 2.14
C (cutoff limit) = 3.8;

The code for obtaining the value of "t":

syms q(t)
q(t) = ((10.925*10^(-6)*4.74*1)\((5.78*10^(-3)+11.31)*(5.78*10^(-3)+2.14)))*exp(5.78*10^(-3)*t)+(((4.74*1)\((5.78*10^(-3)+11.31)*(2.14-11.31)))-((4.74*1)\((5.78*10^(-3)+11.31)*(5.78*10^(-3)+2.14)))-(1\(2.14-11.31)))*(10.925*10^(-6)*exp(-2.14*t))+(7.16*10^(4)*exp(-2.14*t))+(1-((4.74*1)\(5.78*10^(-3)+11.31)))*((10.925*10^(-6)\(2.14-11.31))*exp(-11.31*t))-(3150*3.8)==0 ;
sol = vpasolve(q)

The second thing is plotting a relation between "Blood biomarker concentration" and "Time".

The code:

function [t,qi] = call_dstate()
 tspan = [0 900]; % set time interval
 qi0 = 0; % set initial condition
 %cutoff=c*vpl;
 cutoff=3.8*3150;
 threshold=1*3150;
 % dstate evaluates r.h.s. of the ode
 [t,qi] = ode45( @dstate ,tspan ,qi0);
 %plot(t,qi)
 plot(t,qi,'-b')
 hline1=refline(0,cutoff);
 hline1.Color='g';
 hline2=refline(0,threshold);
 hline2.Color='r';
 xlabel ('Time')
 ylabel ('Blood Biomarker Concentration')
 title ('Immune Biomarker Shedding by Immune & Healthy cells')
 disp([t,qi]) % displays t and qi(t)
       function dqidt = dstate (t,qi)
             fi_Ri=10.925*10^(-6) ; ai=4.74 ; Nc0=1 ; Kgr= 5.78*10^(-3);di=11.31 ;Ni0=1 ;fih_Rih_Nih =7.16*10^(4) ; Kei= 2.14 ; 
             dqidt=(fi_Ri*Nc0*ai\(Kgr+di))*(exp(Kgr*t)-exp(-di*t))+(fi_Ri*Ni0*exp(-di*t))+fih_Rih_Nih-Kei*qi;
       end
end

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Esraa Abdelkhaleq 2017-1-4

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I am doing two things:

The first is solving the model equation for qi(t) and set the solution equal to (CxVpl= 3.8 * 3150) to get a value for "t" which must be positive.

But what I obtained is:

sol =

Empty sym: 0-by-1

So, I want to do a sensitivity analysis to the parameters to get a valid solution.

The model equation,

dqidt=(fi_Ri*Nc0*ai\(Kgr+di))*(exp(Kgr*t)-exp(-di*t))+(fi_Ri*Ni0*exp(-di*t))+fih_Rih_Nih-Kei*qi

The solution for qi(t), by taking Laplace Transform,

qi(t) =

(Nih*Rih*fih)/Kei + (exp(Kgr*t)*(Kgr + di))/(Nc0*Ri*ai*fi*(Kei + Kgr)) - (exp(-di*t)*(- Nc0*Ni0*ai*Ri^2*fi^2 + Kgr + di))/(Nc0*Ri*ai*fi*(Kei - di)) + (exp(-Kei*t)*(Nc0*ai*qi0*Kei^3*Ri*fi + Nc0*ai*qi0*Kei^2*Kgr*Ri*fi - Nc0*Ni0*ai*Kei^2*Ri^2*fi^2 - Nc0*ai*qi0*Kei^2*Ri*di*fi - Nc0*Nih*Rih*ai*fih*Kei^2*Ri*fi + Kei*Kgr^2 - Nc0*Ni0*ai*Kei*Kgr*Ri^2*fi^2 - Nc0*ai*qi0*Kei*Kgr*Ri*di*fi - Nc0*Nih*Rih*ai*fih*Kei*Kgr*Ri*fi + 2*Kei*Kgr*di + Nc0*Nih*Rih*ai*fih*Kei*Ri*di*fi + Kei*di^2 + Nc0*Nih*Rih*ai*fih*Kgr*Ri*di*fi))/(Kei*Nc0*Ri*ai*fi*(Kei - di)*(Kei + Kgr))

The parameters:

fi_Ri (Immune biomarker shedding rate) =10.925*10^(-6) ;

ai (Immune cell activation rate) =4.74 ;

Nc0 (Initial number of tumor cells) =1 ;

Kgr (Tumor growth rate) = 5.78*10^(-3);

di (Immune cell death rate) =11.31 ;

Ni0 (Initial number of immune cells) =1 ;

fih_Rih_Nih (Immune biomarker healthy influx) =7.16*10^(4) ;

Kei (Immune biomarker elimination rate) = 2.14

C (cutoff limit) = 3.8;

The code for obtaining the value of "t":

syms q(t)
q(t) = ((10.925*10^(-6)*4.74*1)\((5.78*10^(-3)+11.31)*(5.78*10^(-3)+2.14)))*exp(5.78*10^(-3)*t)+(((4.74*1)\((5.78*10^(-3)+11.31)*(2.14-11.31)))-((4.74*1)\((5.78*10^(-3)+11.31)*(5.78*10^(-3)+2.14)))-(1\(2.14-11.31)))*(10.925*10^(-6)*exp(-2.14*t))+(7.16*10^(4)*exp(-2.14*t))+(1-((4.74*1)\(5.78*10^(-3)+11.31)))*((10.925*10^(-6)\(2.14-11.31))*exp(-11.31*t))-(3150*3.8)==0 ;
sol = vpasolve(q)

The second thing is plotting a relation between "Blood biomarker concentration" and "Time".

The code:

function [t,qi] = call_dstate()
 tspan = [0 900]; % set time interval
 qi0 = 0; % set initial condition
%cutoff=c*vpl;
 cutoff=3.8*3150;
 threshold=1*3150;
 % dstate evaluates r.h.s. of the ode
 [t,qi] = ode45( @dstate ,tspan ,qi0);
 %plot(t,qi)
 plot(t,qi,'-b')
 hline1=refline(0,cutoff);
 hline1.Color='g';
 hline2=refline(0,threshold);
 hline2.Color='r';
 xlabel ('Time')
 ylabel ('Blood Biomarker Concentration')
 title ('Immune Biomarker Shedding by Immune & Healthy cells')
 disp([t,qi]) % displays t and qi(t)
       function dqidt = dstate (t,qi)
             fi_Ri=10.925*10^(-6) ; ai=4.74 ; Nc0=1 ; Kgr= 5.78*10^(-3);di=11.31 ;Ni0=1 ;fih_Rih_Nih =7.16*10^(4) ; Kei= 2.14 ; 
             dqidt=(fi_Ri*Nc0*ai\(Kgr+di))*(exp(Kgr*t)-exp(-di*t))+(fi_Ri*Ni0*exp(-di*t))+fih_Rih_Nih-Kei*qi;
       end
end