# PharmPK Discussion - Solution to a differential equation for a PD model

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• On 30 Jun 2010 at 11:28:47, "Samtani, Mahesh [PRDUS]" (bourne5321.aaa.gmail.com) sent the message
`The following message was posted to: PharmPKHello,I am trying to find the analytical solution to a PD model that has thedifferential equation:dR/dt = k1*R*(1-(R/Rmax)^k2); R(0) = Rok1, k2, and Rmax are parameters that need to be fitted, while Ro is thebaseline value (which can be fitted or fixed). The response (R)increases initially at an exponential rate governed by the rate constantk1. The PD profile has a S-shaped curve as a function of time and itapproaches the value of Rmax at time approaches infinity.If there is an analytical solution to this differential equation then itmakes my life easier when trying to perform a PK-PD analysis (of courseI can fit the equation using differential equations but the model willrun much faster with an explicit equation).Kindly provide the integration process so I can learn how to do itmyself for future reference. I believe that this is a Bernoullidifferential equation, which could help with integration. I guess theharder way would be to use integration by parts (I tried to find thesolution but keep getting stuck). I guess Laplace transforms wouldn'twork either because this isn't a linear differential equation.Please help,MNS`
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• On 30 Jun 2010 at 19:59:06, Sri Duvvuri (bourne5321.-at-.gmail.com) sent the message
`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) ThanksSridhar`
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• On 1 Jul 2010 at 10:21:18, bourne5321.aaa.gmail.com sent the message
`The following message was posted to: PharmPKHi 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 allk2. What you will get most probably is a general solution of the formt=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 estimationproblems with functions like these.All in all, if your software can fit parameters in differential equationmodels, 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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• On 1 Jul 2010 at 10:15:15, "Samtani, Mahesh [PRDUS]" (bourne5321.aaa.gmail.com) sent the message
`The following message was posted to: PharmPKHello,I am interested in only positive values of k2. Rmax and Ro are of coursepositive, while k1 can be positive or negative. In most subjects theresponse increases with time but there are a few individuals where theresponse decreases with time. I think with decreasing R values anegative k1 would be useful.Thanks,Mahesh`
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