It looks like MATLAB, Octave and Python seem to be the preferred tools for scientific and engineering analysis (especially those involving physical models with differential equations). However as part of my learning R experience, I wanted to check out some of R tools for parameter fitting of models involving ordinary differential equations. R has packages deSolve for solving differential equations and FME for parameter fitting. The specific example here is taken from the computational appendix (A.6) of the book Chemical Reactor Analysis and Design Fundamentals by Rawlings and Ekerdt. In fact, all examples in this book are available in Octave and MATLAB.

This example involves two reactions in series

where we need to estimate the rate constants of the the two reactions and from data

Here the required libraries are loaded

# set working directory setwd("~/R/wkspace") # load libraries library(ggplot2) #library for plotting library(reshape2) # library for reshaping data (tall-narrow <-> short-wide) library(deSolve) # library for solving differential equations library(minpack.lm) # library for least squares fit using levenberg-marquart algorithm

The data available is concentration of A, B, and C over time and is loaded below and plotted

#load concentration data df=read.table("ABC_data.dat") names(df)=c("time","ca","cb","cc") # plot data tmp=melt(df,id.vars=c("time"),variable.name="species",value.name="conc") ggplot(data=tmp,aes(x=time,y=conc,color=species))+geom_point(size=3)

The system being solved is

with inital concentrations being .

The rate equations are captured in a function that is an input parameter to the ODE solver

# rate function rxnrate=function(t,c,parms){ # rate constant passed through a list called parms k1=parms$k1 k2=parms$k2 # c is the concentration of species # derivatives dc/dt are computed below r=rep(0,length(c)) r[1]=-k1*c["A"] #dcA/dt r[2]=k1*c["A"]-k2*c["B"] #dcB/dt r[3]=k2*c["B"] #dcC/dt # the computed derivatives are returned as a list # order of derivatives needs to be the same as the order of species in c return(list(r)) }

Computing the predicted concentration for a given set of rate constants involves just solving the ODEs with intial conditions. This is illustrated for the parameter values (which is the actual parameters based on which data was generated).

# predicted concentration for a given parameter set cinit=c(A=1,B=0,C=0) t=df$time parms=list(k1=2,k2=1) out=ode(y=cinit,times=t,func=rxnrate,parms=parms) head(out) time A B C [1,] 0.000 1.00000000 0.0000000 0.00000000 [2,] 0.263 0.59096447 0.3555550 0.05348051 [3,] 0.526 0.34923934 0.4834497 0.16731099 [4,] 0.789 0.20638798 0.4958219 0.29779013 [5,] 1.053 0.12172441 0.4543303 0.42394528 [6,] 1.316 0.07193491 0.3925423 0.53552284

This is wrapped into a function whose input is the parameters to be estimated and the output is the residuals. Here there are three concentrations that are fitted. In general, we may want to put different weights to these but in this example, they have the same weight. Also, the package FME has several nice features for fitting and uncertainty estimation. Here I have not used FME but directly tried to do the fit more to learn how this works.

ssq=function(parms){ # inital concentration cinit=c(A=1,B=0,C=0) # time points for which conc is reported # include the points where data is available t=c(seq(0,5,0.1),df$time) t=sort(unique(t)) # parameters from the parameter estimation routine k1=parms[1] k2=parms[2] # solve ODE for a given set of parameters out=ode(y=cinit,times=t,func=rxnrate,parms=list(k1=k1,k2=k2)) # Filter data that contains time points where data is available outdf=data.frame(out) outdf=outdf[outdf$time %in% df$time,] # Evaluate predicted vs experimental residual preddf=melt(outdf,id.var="time",variable.name="species",value.name="conc") expdf=melt(df,id.var="time",variable.name="species",value.name="conc") ssqres=preddf$conc-expdf$conc # return predicted vs experimental residual return(ssqres) }

The parameter fitting is done using levenberg-marquardt routine in package minpack.lm.

# parameter fitting using levenberg marquart algorithm # initial guess for parameters parms=c(k1=0.5,k2=0.5) # fitting fitval=nls.lm(par=parms,fn=ssq)

The fitval object has information on estimated parameter and variance covariance matrix

# Summary of fit summary(fitval) Parameters: Estimate Std. Error t value Pr(>|t|) k1 2.01906 0.04867 41.49 <2e-16 *** k2 0.99297 0.01779 55.82 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.0212 on 58 degrees of freedom Number of iterations to termination: 7 Reason for termination: Relative error in the sum of squares is at most `ftol'. # Estimated parameter parest=as.list(coef(fitval)) parest $k1 [1] 2.019065 $k2 [1] 0.992973 # degrees of freedom: # data points - # parameters dof=3*nrow(df)-2 dof [1] 58 # mean error ms=sqrt(deviance(fitval)/dof) ms [1] 0.02119577 # variance Covariance Matrix S=vcov(fitval) S k1 k2 k1 0.0023685244 -0.0003605831 k2 -0.0003605831 0.0003164724

The estimated parameters are which is close to the real parameters used to generate the data. The predicted profiles is overlaid with experimental data

# plot of predicted vs experimental data # simulated predicted profile at estimated parameter values cinit=c(A=1,B=0,C=0) t=seq(0,5,0.2) parms=as.list(parest) out=ode(y=cinit,times=t,func=rxnrate,parms=parms) outdf=data.frame(out) names(outdf)=c("time","ca_pred","cb_pred","cc_pred") # Overlay predicted profile with experimental data tmppred=melt(outdf,id.var=c("time"),variable.name="species",value.name="conc") tmpexp=melt(df,id.var=c("time"),variable.name="species",value.name="conc") p=ggplot(data=tmppred,aes(x=time,y=conc,color=species,linetype=species))+geom_line() p=p+geom_line(data=tmpexp,aes(x=time,y=conc,color=species,linetype=species)) p=p+geom_point(data=tmpexp,aes(x=time,y=conc,color=species)) p=p+scale_linetype_manual(values=c(0,1,0,1,0,1)) p=p+scale_color_manual(values=rep(c("red","blue","green"),each=2))+theme_bw() print(p)

## Estimation of Parameter Uncertainty

Usually though the model is nonlinear, the confidence region is approximated using the following ellipsoid

where is the estimated value of parameter, is the variance covariance matrix of estimated parameters and is the number of parameters and is the total number of data points.

# Get the 95% confidence region # Inverse of covariance matrix Sinv=solve(S) # get points for a circle with radius r r=sqrt(qf(0.95,2,58)*2) theta=seq(0,2*pi,length.out=100) z=cbind(r*cos(theta),r*sin(theta)) # transform points of circle into points of ellipse using # svd of inverse covariance matrix Sinv_svd=svd(Sinv) # inverse of covariance matrix xt=t(Sinv_svd$v)%*%diag(1/sqrt(Sinv_svd$d))%*%t(z) # transform from circle to ellispse x=t(xt) # translate the ellipse so that center is the estimated parameter value x=x+matrix(rep(as.numeric(parest),100),nrow=100,byrow=T) plot(x[,1],x[,2],type="l",xlab="k1",ylab="k2",lwd=2) points(parest$k1,parest$k2,pch=20,col="blue",cex=2)

Another way of estimating the uncertainty is using bootstrapping procedure. Here several simulated datasets are generated using the current estimated model and adding random normal noise to each data point (with mean=0 and variance = mean square error from model). Then the parameter is estimated for each simulated dataset. The set of parameters thus generated indicate the uncertainty in parameter estimates.

# Simulation based estimation of uncertainty # store original experimental data in a separate dataframe dforig=df # conc profile based on estimated k1 and k2 cinit=c(A=1,B=0,C=0) t=dforig$time parms=parest out=ode(y=cinit,times=t,func=rxnrate,parms=parms) outsim=matrix(0,nrow=nrow(dforig),ncol=4) outsim[,1]=out[,1] # number of simulations nsim=1000 parsim=matrix(0,nrow=nsim,ncol=2) colnames(parsim)=c("k1","k2") for (i in 1:nsim){ # Simulate data set by adding normal random variable with mean 0 and stdev from fit outsim[,2:4]=out[,2:4]+matrix(rnorm(3*nrow(dforig)),nrow=nrow(dforig),ncol=3)*ms df=data.frame(outsim) names(df)=c("time","ca","cb","cc") # get parameter estimate for the simulated dataset parms=as.numeric(parest) fitsim=nls.lm(par=parms,fn=ssq) # store estimated parameters in the ith row parsim[i,]=coef(fitsim) } # plot the parameter estimates from the 1000 simulations plot(parsim[,1],parsim[,2],xlab="k1",ylab="k2") # overlay the 95% ellipse computed previously lines(x[,1],x[,2],col="blue",lwd=2)

Next, the percentage of parameter estimates that fall within the ellipse is computed and found to be 93% (expected is 95%). In this case, the ellipsoidal approximation of parameter uncertainty seems adequate. But this might not be the case in general.

# percentage of parameters from simulation within the 95% ellipse tmp=rep(0,length.out=nsim) for(i in 1:nsim){ tmp[i]=(parsim[i,]-as.numeric(parest))%*%Sinv%*%(parsim[i,]-as.numeric(parest)) } sum(tmp <= qf(0.95,2,58)*2)/nsim [1] 0.933

As I mentioned previously, the package FME has several functions for parameter fitting, MCMC simulation, sensitivity analysis but was not used in this post. Also, when running multiple simulations using ODE models, it might be better to use the C compiled version of reaction rate function. deSolve package provides that ability and describes it in this vignette. The data and R code is available here.

# session Info sessionInfo() R version 3.0.1 (2013-05-16) Platform: i386-w64-mingw32/i386 (32-bit) locale: [1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252 [3] LC_MONETARY=English_United States.1252 LC_NUMERIC=C [5] LC_TIME=English_United States.1252 attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] minpack.lm_1.1-7 deSolve_1.10-6 reshape2_1.2.2 ggplot2_0.9.3.1 loaded via a namespace (and not attached): [1] colorspace_1.2-2 dichromat_2.0-0 digest_0.6.3 grid_3.0.1 gtable_0.1.2 [6] labeling_0.2 MASS_7.3-26 munsell_0.4 plyr_1.8 proto_0.3-10 [11] RColorBrewer_1.0-5 scales_0.2.3 stringr_0.6.2 tools_3.0.1

A nice example, thanks! One note: It looks like you left out the line to calculate the inverse of S. I assume you did this:

Sinv = solve(S)

Thanks for the catch. Yes I used solve(S) for computing the inverse. I will fix it. I guess I illustrated the perils of doing copy paste:)

Shankar

Nice work! I am advocating R for this kind of modelling in the hard sciences, based on Karline Soeraert’s book on ecological modelling. Your example shows how it should be done.

First time I have seen R but I would like to give this a go. Is there a quick start guide to how to do the first few steps. I installed R but cannot even do the first part.

Great example!!!

Just I was looking for the File: ABC_data.dat and I did not found it.

Could you indicate me where is it?

Carlos,

Thanks for your interest. The data and code is available at the following location.

THANKS A LOT BEST REGARDS!!! Carlos

> El 12/11/2014, a las 12:51, Notes of a Dabbler escribió: > >

Dear Dabbler I wonder if you have other examples parameter fitting ode with optim function? Or can you recommend me any lecture or handouts?

I am learning R and your example was pretty good!!!

Best regards Carlos

> El 12/11/2014, a las 12:51, Notes of a Dabbler escribió: > >

Dear Carlos,

I don’t have other examples that I have done in R. But one of the reference I find useful is:

Chemical Reactor Analysis and Design

It uses Octave as language but hopefully the method in this post can be used to apply to other problems.

Regards,

Shankar

THANKS A LOT I am sure it should be useful to me.

Best ragards Carlos

> El 12/12/2014, a las 7:27, Notes of a Dabbler escribió: > >

Dear rdabbler,

first of all, thank you very much for your little walkthrough, very helpful!! thanks!

i have a short question, may be you have an idea ….

I have 4 parameters to fit (5 rate constants) but i only have a dataset for one of the entities (like the chemical entities A, B in your example).

so that the ODE system looks like that:

r=rep(0,length(c))

r[1]=-kinf*c[“inf”]

r[2]=+kinf*c[“inf”]-k12*c[“A”]+k21*c[“B”]-ke*c[“C”]

r[3]=+k12*c[“A”]-k21*c[“B”]

r[4]=+ke*c[“A”]

when I come to the

Summary(fitval)

i just gives me this error:

“Error in chol.default(object$hessian) :

the leading minor of order 1 is not positive definite”

Do you have a clue on what is going wrong and on how to fix it?

Thanks in advance,

best regards

Darax

Darax,

I don’t know why this might be happening. One thought as I was looking through the equations was following. It looks like the set of reactions you are modeling is:

Inf -> A (forward rate kinf)

A = B (forward rate k12, reverse rate k21)

A -> C (forward rate ke)

So I am wondering if the expression for r(2) is the following (c[“A”] in the last term instead of c[“C”]) so that mole balance will be satisfied:

r[2]=+kinf*c[“inf”]-k12*c[“A”]+k21*c[“B”]-ke*c[“A”]

Maybe that change might fix things. If you find a solution, please post it in the comment here.

Thanks,

Shankar

Hi Shankar,

unfortunately i couldn’t fix the problem. Nevertheless, maybe you can help me with a different problem.

Do you have an idea on how to use concentrations instead of amounts in the ODEs?

In my Code below this would be (c[“Aarterial”]/Var) for example. c[“Aarterial”] is the amount of a drug in this case and Var is the volume of a compartment (organ).,

Maybe you have an idea on how to do that…

My example is as follows:

pharmacokinetics <- function(t,c,parms){

r=rep(0,length(c))

Qad=parms$Qad

Qbr=parms$Qbr

Qhe=parms$Qhe

Qkid=parms$Qkid

Qhep=parms$Qhep

Qmu=parms$Qmu

Var=parms$Var

Vad=parms$Vad

Vbr=parms$Vbr

Vhe=parms$Vhe

Vkid=parms$Vkid

Vhep=parms$Vhep

Vmu=parms$Vmu

Kpad=parms$Kpad

Kpbr=parms$Kpbr

Kphe=parms$Kphe

Kpkid=parms$Kpkid

Kphep=parms$Kphep

Kpmu=parms$Kpmu

BP=parms$BP

CLrenal=parms$CLrenal

CLmet=parms$CLmet

fup=parms$fup

r[1] = Qad*((c["Aarterial"]/Var)-((c["Aadipose"]/Vad)/Kpbr*BP)) # adipose = FM

r[2] = Qbr*((c["Aarterial"]/Var)-((c["Abrain"]/Vbr)/Kpbr*BP)) # brain

r[3] = Qhe*((c["Aarterial"]/Var)-((c["Aheart"]/Vhe)/Kphe*BP)) # heart

r[4] = Qkid*((c["Aarterial"]/Var)-((c["Akid"]/Vkid)/Kpkid*BP))-CLrenal*Ckidneyfree # kidney

r[5] = Qhep*((c["Aarterial"]/Var)-((c["Aliv"]/Vhep)/Kphep*BP))-CLmet*Cliverfree # liver

r[6] = Qmu*((c["Aarterial"]/Var)-((c["Amu"]/Vmu)/Kpmu*BP)) # muscle

r[7] = (c["Aadipose"]/Vad)/Kpad*BP + (c["Abrain"]/Vbr)/Kpbr*BP +(c["Aheart"]/Vhe)/Kphe*BP+(c["Akid"]/Vkis)/Kpkid*BP

+(c["Aliv"]/Vhep)/Kphep*BP +(c["Amu"]/Vmu)/Kpmu*BP # venous

Ckidneyfree = r[4]*fup

Cliverfree = r[5]*fup

return(list(r))

}

cinit=c(

Aarterial = 100,

Aadipose = 0.1,

Abrain = 0,

Aheart = 0,

Akid = 0,

Aliv = 0,

Amu = 0

)

BW <- 70

#Fractional tissue volumes

FVad = 0.213

FVar = 0.0257

FVbr = 0.02

FVhe = 0.0047

FVkid = 0.0044

FVhep = 0.021

FVmu = 0.4

Var = BW * FVar

Vad = BW * FVad

Vbr = BW * FVbr

Vhe = BW * FVhe

Vkid = BW * FVkid

Vhep = BW * FVhep

Vmu = BW * FVmu

parms=list(

Qad=19.4994,

Qbr=46.79856,

Qhe=15.59952,

Qkid=74.09772,

Qhep=83.99756538,

Qmu=66.29796,

Var,

Vad,

Vbr,

Vhe,

Vkid,

Vhep,

Vmu,

Kpad=1,

Kpbr=1,

Kphe=1,

Kpkid=1,

Kphep=1,

Kpmu=1,

BP=1,

CLrenal=0.5,

CLmet=0.5,

fup=1

)

t=seq(0,24,1)

out=ode(y=cinit,times=t,func=pharmacokinetics,parms=parms)

out

Thanks in advance!!

Reblogged this on the math researcher and commented:

A good tutorial on how to do parameter fitting for models using differential equations. It has helped me with my research, so I am here to pass along the information.

Hope you enjoy!

-Krystal

Dear rdabbler,

Thanks for this great example and I have been using it quite a lot with my work. I have run into a unique problem in that your example works great for long time series but one of the new data sets I have uses a lot short time series data (n=6) that all needs to be fitted with the same function.

My low tech solution to this has been to create columns for the subsequent time series and add them to the code:

# rate function

rxnrate=function(t,c,parms){

# rate constant passed through a list called parms

a=parms$a

h=parms$h

# c is the concentration of species

# derivatives are computed below

r=rep(0,length(c))

r[1]=-c[“B”]*a*c[“A”]/(c[“B”]+a*h*c[“A”])# prey equation 1

r[2]=0# predator equation 1

r[3]=-c[“D”]*a*c[“C”]/(c[“D”]+a*h*c[“C”])# prey equation 2

r[4]=0# predator equation 2

r[5]=-c[“F”]*a*c[“E”]/(c[“F”]+a*h*c[“E”])# prey equation 3

r[6]=0# predator equation 3

# and so on

return(list(r))

}

The problem with this is that I quickly run out of letters (e.g. c[“A”]) in addition to the fact that it is extremely inefficient to hard code all of these time series (over 100 in total).

My question is because the paired equations are all the same, is there a solution where I can write them once and have the function apply it to all the subsequent paired time series. I have tested this method against other analytical methods for solving differential equations and the outputs are pretty comparable.

If you have a solution to this I would greatly appreciate it.

Thanks

Kevin

Great example. I am also trying to use R in PK/PD modeling and simulations. This is a great starting point for me.

One question:

When dealing with data from multiple subjects there’s two types of uncertainties: the inter-individual uncertainty and residual error. When we solve a differential equation, is there any good way to add uncertainty to a specific parameter, and add residual error also?

For example: we assume drug concentration dy/dt = -k*y + ε. When we simulate for y changes over time for several subjects, how shall we let k change randomly? Is it also possible to generate a random number (ε) for each observation (this would be each time t).

Thank you.

Thanks for your interest. I don’t have much experience with PK/PD modeling. But it seems like some kind of bayesian modeling would help with what you are looking to do. I came across this youtube video (https://www.youtube.com/watch?v=Q_8NSvPtehc) on a package called PMXStan which does bayesian modeling for PK. But I couldn’t find the actual package in CRAN.