I want to do a parameter sensitivity analysis to my model

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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
  3 Comments
Esraa Abdelkhaleq
Esraa Abdelkhaleq on 4 Jan 2017
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
Thanks in advance.

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