1. QL (Quantification limits) handling in Population Analysis: Special case of right censoring (AQL)

2. Context Background
Pharmacokinetic data consist of drug concentration measurements, as well as reports of some measured concentrations being below the quantification limit of the assay (BQL) or after quantification limit of the assay (AQL). A pharmacokinetic model may be fit to these data, and for this purpose, the BQL/AQL observations must be either discarded or handled in a special way.
The referenced paper called “Journal of Pharmacokinetics and Pharmacodynamics, Vol. 28, No. 5, October 2001 ( 2001)”Ways to Fit a PK Model with Some Data Below the Quantification Limit” Stuart L. Beal” analyzed 7 methods but all linked to left censoring.
In this example, we explain the generic strategy to deal with BQL/AQL data

3. Theoretical Background
Normalizing a normal distribution
Suppose we have a normal distribution with mean mu and standard deviation sigma. Let us assume a random smaple from that distribution and let us call it x
If we perform the transformation z=(x-mu)/sigma, z will be normally distributed with mean 0 and standard deviation 1 (N(0,1)). This is called the standard normal distribution.

4. Theoretical Background
Usually in modeling the mean is associated with the predicted response while x is the actual observation
Suppose we have an observation called EObs and the corresponding prediction is E. The error model is defined by a standard deviation stdev. Therefore we will have z=(Eobs-E)/stdev being N(0,1)
Suppose we have an Emax model; E=Emax*t/(EC50+t)
Suppose now that E (the response) has been transformed at each time into a N(0,1). Therefore any value of E must also be transformed. Suppose we define the maximum value (Emax) for E to be 100. If E has been transformed into N(0,1), we must change the units of Emax using the same transformation which is
Emaxnormalized=(Emax-E)/stdev

5. Cumulative normal distribution
Suppose a N(0,1) distribution, the integral from –infinity to x of the N(0,1), is called the cumulative distribution function (CDF) of the standard normal distribution. We will denote it phi(x).
Now since the overall integral from –infinity to +infinity of the N(0,1) distribution =1, the integral from x to infinity of the N(0,1) distribution =1-phi(x)
Suppose now we want to know the integral from 40 to 80 on the N(0,1) distribution. It is easy to show that it is equal to the integral from –infinity to 80 – the integral from –infinity to 40=phi(80)-phi(40)

6. Special functions in Phoenix
Cumulative distribution of the N(0,1) from –infinity to x is called phi(x)
phi(x)=integral from –infinity to x of the N(0,1) distribution
Lognormal density at x, associated with the normal density with mean 0 and standard deviation stdev is called lnorm(x,stdev)
Note that lnorm uses requires to center the distribution to a mean of 0

7. Likelihood computations for special QL (quantification limits) cases
If an observation is known, then the likelihood is given by the normal density at the observation value, given the predicted response and its corresponding variance.
The interface provides such calculation
If the observation is unknown but below the limit of quantification (LOQ), then the likelihood is obtained by calculating the probability given the predicted responses and its corresponding variance to have the observation between –infinity and LOQ. After transforming the response into a N(0,1) and getting LOQnormalized, this is just phi(LOQnormalized).

8. Likelihood computations for special QL (quantification limits) cases
If the observation is unknown but larger than a specific value (ALQ), then the likelihood is obtained by calculating the probability given the predicted responses and its corresponding variance to have the observation between ALQ and +infinity. After transforming the response into a N(0,1), let say you get ALQnormalized, then that probability is just 1-phi(ALQnormalized)

9. Test Case: The data

10. C is assumed to the independent variable in a simple Emax model
Data Information
EObs is the observed response that I assumed to be right censored above 80 with the maximum response assumed to be 100
It means that any recorded value of 80 means in fact that the response is between 80 and 100
C is assumed to the independent variable in a simple Emax model
E = Emax * C / (EC50 + C)
Eobscovariate is a copy of Eobs and is needed when calculating the different likelihoods
An observation cannot be read when defined in an equation

11. CODE AND EXPLANATION
test(){
# upper limit of quantification
UOQ is the upper limit of quantification
UOQ=80
C is read as a covariate
covariate(C)
This defines the Emax model
E = Emax * C / (EC50 + C)

12. CODE AND EXPLANATION
We define multiplicative error
stdev=cv*E
We read the observations as covariate
covariate(EObscovariate)
We normalize the response into a N(0,1) and call it nE
nE=(EObscovariate-E)/stdev
We define cv as a fixed effect
fixef(cv=c(,0.1,))

13. CODE AND EXPLANATION
We normalize now the 2 parameters used in the likelihood computation which are the UOQ(upper limit of quantification) and the UOQmax (maximum value of the response which is 100)
nUOQ=(UOQ-E)/stdev
nUOQmax=(100-E)/stdev

14. ? log(phi(nUOQmax)-phi( nUOQ )) : lnorm( EObs - E, stdev) )
CODE AND EXPLANATION
We calculate now the log-likelihood depending if the response is <80 which is uncensored or larger than 80 (censored)
Note that lnorm is the lognormal distributed with mean 0. Therefore we need to transform the observed response here to have a mean of 0. This is done by shifting the observation (Eobs) by the corresponding prediction(E). Note also that EObs can be read in a LL statement
LL(EObs
, (EObs >= UOQ
? log(phi(nUOQmax)-phi( nUOQ ))
: lnorm( EObs - E, stdev) )

15. CODE AND EXPLANATION
We define EC50 as a structural parameter with both fixed and random effect while Emax is frozen to 100
stparm(EC50 = tvEC50 * exp(nEC50))
stparm(Emax = tvEmax)
fixef(tvEC50 = c(, 1, ))
fixef(tvEmax(freeze) = c(, 100, ))
ranef(diag(nEC50) = c(1))

16. Results
The model can be run with either Laplacian or QRPEM as the likelihood is defined through a LL statement