MATLAB: What model would fit a set of values that increase then decrease over time

best fitmodelling

Trying to model the evolution of a product that is first produced (increase in concentration) then it is consumed/lost (decrease in concentration). Does anyone know which model/equation would fit this set of data?
if true
  % code
Days  Concentration
0  6.38
1  39.81
2  63.13
3  25.09
6  14.93
9  8.78
12  6.01
15  2.47
18  0.31
end

Best Answer

Solving for the second compartment in a two-compartment model comes close.
Model:
[C1] =k1=> [C2] =k3=>
[  ] <=k2= [  ]
First, derive symbolically:
syms C1(t) C2(t) k1 k2 k3 t C10 C20 
Eq1 = diff(C1) == C1*k2 - C1*k1;
Eq2 = diff(C2) == C1*k1 - C2*k3;
Csol = dsolve(Eq1, Eq2, C1(0) == C10, C2(0) == 0);
C1 = Csol.C1
C2 = Csol.C2
f = matlabFunction(C2)
Then use fminsearch or another solver to estimate the parameters:
A = [0  6.38
    1  39.81
    2  63.13
    3  25.09
    6  14.93
    9  8.78
    12  6.01
    15  2.47
    18  0.31];
f = @(in1,t)-(in1(:,1).*in1(:,2).*exp(-in1(:,4).*t))./(-in1(:,2)+in1(:,3)+in1(:,4))+(in1(:,1).*in1(:,2).*exp(-t.*(in1(:,2)-in1(:,3))))./(-in1(:,2)+in1(:,3)+in1(:,4))
b0 = [90 1 1 1];
B = fminsearch(@(b) norm(A(:,2) - f(b,A(:,1))), b0);
xv = linspace(min(A(:,1)), max(A(:,1)));
figure
plot(A(:,1), A(:,2), 'p')
hold on
plot(xv, f(B,xv), '-r')
hold off
grid
The best model describes the system that produced your data.
Experiment to get the result you want.
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