Dear researchers,
I like to follow the microscopic mass balance approach for predicting intestinal absorption. Three dimensionless numbers viz., dose number, absorption number, and dissolution number can be calculated with the appropriate parameters. Then these dimensionless numbers can be substituted in the differential equations and it can be solved together.
I found an article published in Springer, AAPS PharmSciTech (attached).
I followed the same values and try to solve with R program, I can't able to solve (Authors used Matlab to solve the differential equations). Please give an idea to solve these differential equations.
Here is some r code examples to solve ODE's.
https://cran.r-project.org/web/packages/deSolve/vignettes/deSolve.pdf
https://journal.r-project.org/archive/2010-2/RJournal_2010-2_Soetaert~et~al.pdf
Or use Runge Kutta in Excel
see http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1130&context=ejsie
Or more simpler is Eulers method in excel here (but not as accurate)
https://www.youtube.com/watch?v=Nawwx0QoWn8
https://www.youtube.com/watch?v=vXWb1Kspx8s
Dear Loopinder,
Thanks for your reply.
Rscript I developed to solve the above the differential equation (please see the attachment for differential equation) is,
yini
Prof. Anwar,
Thanks for the immediate reply.
Mathematical models cannot be developed without assumptions. Each model built on some assumptions. In the microscopic model, intestine is only considered. So, the model was developed with the assumption for orally administered drugs.
Still, I asked the authors regarding that model.
After seeing your reply, I searched for the manuscripts which published after 1993, (Oh et al. Estimating the fraction dose absorbed from suspensions of poorly soluble compounds in humans: a mathematical model). I couldn't find much research work carried out with the developed microscopic model.
Using Euler and Excel (as always a good idea to use a different method and different software and do things from the basics) I get the attached result in Excel 2003. There are problems at z=0.01 as the derivative becomes infinite for y1. However y2 remains well behaved and rises to 0.01.
When y1 becomes zero I assume it stays at zero and cannot become negative. So artificially put that condition in. Then dy2=1.54 * y[1] * (1-y[2]) - 2.14 * y[2]
becomes dy2 =0 - 2.14 * y[2] and decays.
If you look at the excel sheet I put y1=0 at cell B989, because if you do not it will blow up as the y1 derivative becomes infinite. That is why this is a stiff equation.
So modify your code in R to make y1=0 when y1 becomes less than 0.001 or maybe less than 0.01 say to stop y1 derivative blowing up to infinity. This will take care of the stiffness, though perhaps not how a proper mathematician would take care of it. The excel calculation will not be as accurate as Runge Kutta being based on Euler, but it gives an understanding of the problem.
Dear Loopinder,
Thank you so much for the excel calculations. It was very clear to calculate the differential equations.
I'm trying to write the R code for solving differential equations. R code given below (in bold)
yini
Here I used if, else statement. Its working better. But still I'm not able to get dy1 as zero.
yini
Thanks for the reply.
I tried with the code you given. I attached the results.
After fifth iteration (@ z = 0.01), I'm getting the dy1 as -0.2625 instead of zero. But the results of dy2 as similar as it solve by Euler's method (you solved in excel).
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