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The following message was posted to: PharmPK
Hello,
I am trying to find the analytical solution to a PD model that has the
differential equation:
dR/dt = k1*R*(1-(R/Rmax)^k2); R(0) = Ro
k1, k2, and Rmax are parameters that need to be fitted, while Ro is the
baseline value (which can be fitted or fixed). The response (R)
increases initially at an exponential rate governed by the rate constant
k1. The PD profile has a S-shaped curve as a function of time and it
approaches the value of Rmax at time approaches infinity.
If there is an analytical solution to this differential equation then it
makes my life easier when trying to perform a PK-PD analysis (of course
I can fit the equation using differential equations but the model will
run much faster with an explicit equation).
Kindly provide the integration process so I can learn how to do it
myself for future reference. I believe that this is a Bernoulli
differential equation, which could help with integration. I guess the
harder way would be to use integration by parts (I tried to find the
solution but keep getting stuck). I guess Laplace transforms wouldn't
work either because this isn't a linear differential equation.
Please help,
MNS
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Mahesh,
I got the following equation. Please check it before you use it. Brief graphical exploration of the equation was very similar to what you mentioned. Let me know if you want the exact derivation. I used partial fractions after re-organizing the equation
R = Rmax *(M/(M-1))^(1/K2)
where M = ((R0^K2)/((R0^K2)-(Rmax^K2)))*exp(K1*K2*time)
Thanks
Sridhar
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The following message was posted to: PharmPK
Hi Mahesh,
It looks like you're trying to find a solution of the form R=R(t, k1,
k2,Rmax). I have some doubts that a solution of this form exists for all
k2. What you will get most probably is a general solution of the form
t=t(R, k1, k2,Rmax), or even more general, a solution of the form F(R,t,
k1, k2,Rmax)=0. Again you need a software that can handle estimation
problems with functions like these.
All in all, if your software can fit parameters in differential equation
models, and if the parameters can be identified at all, based on pairs (R,
t), I would follow this way.
Regards,
Peter
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The following message was posted to: PharmPK
Hello,
I am interested in only positive values of k2. Rmax and Ro are of course
positive, while k1 can be positive or negative. In most subjects the
response increases with time but there are a few individuals where the
response decreases with time. I think with decreasing R values a
negative k1 would be useful.
Thanks,
Mahesh
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