Bioequivalence and Bioavailability Forum • dose selection to investigate dose-proportionality

martin
★★

Austria,
2009-12-23 11:12
(5231 d 09:52 ago)

Posting: # 4520
Views: 5,840

dose selection to investigate dose-proportionality [Surveys]

Dear colleagues!

Motiviated by a recent discussion with HS I would like to set up a simulation study to evaluate power properties of different approaches to investigate dose-proportionality. To mimic the real world situation I would like to start a little survey and discussion regarding dose selection for these kind of studies.

I would be happy to get your points of view regarding the following questions:

how many doses are usually used in these kind of studies?
how doses are selected? I encountered that doses are often doubled in these kind of studies (eg. 0.5 mg, 1 mg, 2 mg, 4 mg).

best regards

martin

Helmut
★★★

Vienna, Austria,
2009-12-23 17:16
(5231 d 03:49 ago)

@ martin
Posting: # 4521
Views: 5,606

Dose-proportionality

Post reply

Dear Martin,

following our inspiring yesterday's discussion some points:

At least three ... In my experience four is most commonly employed.
Not necessarily doubled, but equally spaced in the logarithmic domain (modifying your example: 4 levels, lowest 0.5 mg, highest 8 mg: intermediates at 1.26 mg, 3.17 mg).

Have a look at two classical papers:

Gough K, Byrom B, Ellis S, Lacey L, McKellar J, Hutchison M, and O Keene
Assessment of Dose Proportionality: Report From the Statisticians in the Pharmaceutical Industry/Pharmacokinetics UK Joint Working Party
Drug Information Journal 29(3), 1039–48 (1995)
Smith BP, Vandenhende FR, DeSante KA, Farid NA, Welch PA, Callaghan JT, and ST Forgue
Confidence Interval Criteria for Assessment of Dose Proportionality
Pharm Res 17(10), 1278–83 (2000)

I came across some Fibonacci-stuff; too lazy to search.

It should be noted that the power model is a purely empirical one - i.e., is not based on physiological grounds. Unlike dose-corrected BE-assessments between two dose levels, any deviation of the exponent (beta>1: enzyme limited excretion, saturation; beta<1: enzyme induction) is not described in PK-models. Pharmacokineticists are not happy with the power model anyway. To quote Nick Holford (PKPD-List, 2006):

I am aware of the whacky power function used by non-pharmacological statistians to 'declare dose proportionality'.
Any rational attempt to understand kinetics in relation to dose would use PK theory not empirical ad hoc 'beat it to death with a P value' statistics.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin
★★

Austria,
2009-12-29 23:47
(5224 d 21:17 ago)

(edited by martin on 2009-12-30 16:37)
@ Helmut
Posting: # 4534
Views: 5,658

Dose-proportionality

Post reply

Dear HS !

thank you for your input and the references!

below some code to play around; it is an optimal contrast problem – the power to detect a deviation from dose-proportionality depend on the unknown monotone dose-response shape. Theoretically, the hypothesis regarding dose-proportionality should be specified in detail to “maximise power”. For example, the hypothesis

H0: all 4 doses are within the proportional range
H1: 3 highest doses are outside the proportional range

would suggest application of the reverse helmert contrast. Additionally, one-sided step-down testing allows a formal estimation of the minimum dose where we can not reject the null hypothesis of dose-proportionality – similar as estimation of the minimum effective dose.

best regards

martin


rm(list=ls(all=TRUE))

# function to simulate power for an overall assessment of dose-proportionality based on the power-law model and 4 dose levels

sim.dose4prop <- function(nsim, cv, n, a, b, mindose, maxdose, range, alpha=0.05){ 



# input parameters:

# nsim ... number of simulation runs

# cv ... coefficient of variation of random variables (e.g. AUCs)

# n ... n per dose level

# a, b ... parameters for power law model: AUC = a*dose^b 

# mindose ... lowest dose

# maxdose ... highest dose 

# range ... doses (index) for which parameter b of power law model is applied (e.g. b=1.1 and range=c(2:4) specifies that doses 2 to 4 are outside proportional range) 

# alpha ... type I error, two-sided (default = 0.05)



   # function to define AUC using power law model

   powermod <- function(dose, a, b, range){return(c(a*dose[-range]^1, a*dose[range]^b))}



   # function to generate log-normal distributed RVs (e.g. AUCs) with identical CVs per dose. See e.g. Jaki et al. (2009) for more detail.

   # Jaki T, Wolfsegger MJ, Ploner M (2009). Confidence intervals for ratios of AUCs in the case of serial sampling: A comparison of seven methods. 

   # Pharmaceutical Statistics, 8(1):12-24. DOI:10.1002/pst.321. 

   data.gen <- function(dose, mue, n, cv){

      k <- length(dose)   

      sd.log <- sqrt(log(1+cv^2))

      res <- as.data.frame(matrix(nrow=k*n, ncol=6))

      res[,1] <- sort(rep(dose,n))

      res[,2] <- sort(rep(mue,n))

      res[,3] <- log(res[,2]) - 1/2 * log(1+cv^2)

      res[,4] <- sd.log

      res[,5] <- rlnorm(n=k*n, meanlog=res[,3], sdlog=res[,4])

      res[,6] <- res[,5]/res[,1]
      

      colnames(res) <- c('dose', 'mue', 'mue.log', 'sd.log', 'auc', 'auc.adj')

      return(res)

   }



   # definition of contrasts see e.g. Bretz and Hothorn (2003) for more detail

   # Bretz F and Hothorn LA (2003). Statistical Analysis of Monotone or Non-monotone Dose–Response Data from In Vitro Toxicological Assays.

   # ATLA-ALTERN LAB ANIM 31:81-96 Suppl. 1 JUN 2003. URL: http://ecvam.jrc.it/publication/Bretz-Hothorn.pdf last visited at 2009-12-27

   numdose <- 4

   contr.lowhigh <- c(-1,0,0,1) # t-test of highest versus lowest dose (pairwise contrast)

   contr.helmert <- c(-1,-1,-1,3) # helmert contrast   

   contr.revhelm <- c(-3,1,1,1) # reverse helmert contrast   

   contr.linearc <- c(-3,-1,1,3) # linear contrast

   contr.jonckht <- c(-1, -0.67, -0.17, 1.83) # Jonkheere-Terpstra analogon contrast

   contr.maximum <- c(-0.87, -0.13, 0.13, 0.87) # Maximum contrast

   contr.linr24t <- c(-12, -2, 2, 12) # Linear 2–4 contrast



   # define doses: to be equidistant on log-scale

   step <- (log(maxdose)-log(mindose))/(numdose-1)

   log.dose <- c(log(mindose), rep(NA, numdose-1))

   for(i in 2:numdose){log.dose[i] <- log.dose[i-1]+step}

   dose <- exp(log.dose)

   mue <- powermod(dose=dose, a=a, b=b, range=range)

   k <- length(dose)

   power <- rep(0,9)



   for(i in 1:nsim){



      # generate log-normal distributed random variables 

      data <- data.gen(dose=dose, mue=mue, n=n, cv=cv)



      # log-log regression approach: two-sided CI

      res1 <- confint(lm(log(data$auc) ~ log(data$dose)), level=1-alpha)[2,]


      

      # ANOVA approach 

      temp <- summary(lm(log(data$auc.adj) ~ as.factor(data$dose)))

      res2 <- 1-pf(q=temp$fstatistic[1], df1=temp$fstatistic[2], df2=temp$fstatistic[3])

   

      # contrast approaches: two-sided hypothesis

      xq <- tapply(log(data$auc.adj), data$dose, mean)

      sd.pooled <- sqrt(sum((n-1)*tapply(log(data$auc.adj), data$dose, var))/(n*k-k))

      df <- n*k - k 

   

      nom3 <- sum(contr.lowhigh*xq)

      den3 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.lowhigh)^2))



      nom4 <- sum(contr.helmert*xq)

      den4 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.helmert)^2))



      nom5 <- sum(contr.revhelm*xq)

      den5 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.revhelm)^2))



      nom6 <- sum(contr.linearc*xq)

      den6 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.linearc)^2))



      nom7 <- sum(contr.jonckht*xq)

      den7 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.jonckht)^2))



      nom8 <- sum(contr.maximum*xq)

      den8 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.maximum)^2))



      nom9 <- sum(contr.linr24t*xq)

      den9 <- (sd.pooled/sqrt(n))*sqrt(sum((contr.linr24t)^2))



      res3 <- 2*(1-pt(q=abs(nom3/den3), df=df))

      res4 <- 2*(1-pt(q=abs(nom4/den4), df=df))

      res5 <- 2*(1-pt(q=abs(nom5/den5), df=df))

      res6 <- 2*(1-pt(q=abs(nom6/den6), df=df))

      res7 <- 2*(1-pt(q=abs(nom7/den7), df=df))

      res8 <- 2*(1-pt(q=abs(nom8/den8), df=df))

      res9 <- 2*(1-pt(q=abs(nom9/den9), df=df))



      if(res1[1] > 1 | res1[2] < 1){power[1] <- power[1] + 1}

      if(res2 < alpha){power[2] <- power[2] + 1}

      if(res3 < alpha){power[3] <- power[3] + 1}

      if(res4 < alpha){power[4] <- power[4] + 1}

      if(res5 < alpha){power[5] <- power[5] + 1}

      if(res6 < alpha){power[6] <- power[6] + 1}

      if(res7 < alpha){power[7] <- power[7] + 1}

      if(res8 < alpha){power[8] <- power[8] + 1}

      if(res9 < alpha){power[9] <- power[9] + 1}

   }



   res <- rep(NA, 11)

   res[1] <- mindose

   res[2] <- maxdose

   res[c(3:11)] <- power/nsim

   names(res) <- c('mindose', 'maxdose', 'log-log reg', 'anova', 't-test', 'helmert', 'revhelm', 'linearc', 'jonckht', 'maximum', 'linr24t')

   return(res)

}





# set parameters for simulations

nsim <- 1E4

cv <- 0.5

n <- 10

a <- 1 



# dose-proportionality for complete dose range (simulate type I error) 

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1, mindose=10, maxdose=20, range=c(1:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1, mindose=10, maxdose=100, range=c(1:4))



# all 4 doses outside proportional range

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=20, range=c(1:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=100, range=c(1:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=100, range=c(1:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=1000, range=c(1:4))



# last 3 doses outside proportional range 

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=20, range=c(2:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=100, range=c(2:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=100, range=c(2:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=1000, range=c(2:4))



# last 2 doses outside proportional range 

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=20, range=c(3:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=100, range=c(3:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=100, range=c(3:4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=1000, range=c(3:4))



# only highest dose outside proportional range

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=20, range=c(4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.1, mindose=10, maxdose=100, range=c(4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=100, range=c(4))

sim.dose4prop(nsim=nsim, cv=cv, n=n, a=a, b=1.15, mindose=10, maxdose=1000, range=c(4))