THETA
Please note that THETA in NONMEM can be any parameter that is being fixed or estimated.
However, many times THETA does represent the parameters themselves: i.e., Clearance, Volume of distribution or peripheral tissue volume, absorption, etc.
Population Parameter = THETA*exp(ETA)
Uppsala Method of Using THETAs for EPSs
Note that the reason I mentioned that THETA can be any parameter that is being fixed or estimated because many times people (i.e., Uppsala University teaches) do not use the EPSs within the $ERROR block within nonmem in the way in which it was designed to be used because by building the $SIGMA block by means of THETA, the model can be more stable. For example, do the following:
$SIGMA 1 FIX
The above says SIGMA is being fixed to 1 but elsewhere in the script, you can make it so that SIGMA is not actually fixed and this method can help with the stability of your model (i.e., whether the model converges).
To implement this: simply have one or more of your THETAs represent the additive or proportional portions of SIGMA. For example:
$THETA
(0, 0.5); THETA(7) prop_error
(0, 1.5); THETA(8) add_error
$ERROR
IPRED = F
W1=IPRED*IPRED*THETA(7)*THETA(7)+THETA(8)*THETA(8) ; Proportional and Additive Error
W=SQRT(W1)
IF (W.EQ.0) W=1
IRES=DV – IPRED
IWRES=IRES/W
Y=IPRED +EPS(1)*W; note that here EPS(1) is fixed to 1 but you could weight it differently if desired.
$SIGMA 1 FIX
ETA
ETA = Has a mean of 0 and a variance of OMEGA (ETA ~ norm(mean=0,variance=OMEGA^2). THETA is used to describe the Parameter distribution on an individual level.
EPSILON
EPSILON = Random variability associated with individuals with a mean of 0 and variance SIGMA (could be due to assay errors, and other unexplained errors). This is similar to the equation before where: Deviations EPSILON ~ norm(mean=0, variance=SIGMA=standard deviation^2).
The $ERROR block is where the EPSs can be defined. For example:
$ERROR
IPRED = F
IRES = DV – F
W = F
IF(W.EQ.0) W = 1
IWRES = IRES/W
Y = IPRED + W*EPS(1)
However, if you are using the Uppsala method for using THETAs for SIGMAs you will want to use THETAs when defining the W as shown earlier within the Uppsala Method-based section above.
Effects on parameters
Covariates can help describe variation in parameters. For example, age or gender may help dictate what a parameter is on the individual level.
Weight (WT) effect on clearance can be written as so:
For example: CLpopulation = CLi*exp(WTi/WTmean)^THETA
Relative Standard Error
RSE=100*Standard Error/Estimate
RSE=100*standard deviation/Estimate
RSE=100*sqrt(variance)/Estimate
Coefficient of Variation (CV)
%CV = 100*Standard Deviation/mean
%CV = 100*sqrt(OMEGA)
%CV = 100*sqrt(Variance)/mean
PopPK Software
NONlinear Mixed Effects Modeling (NONMEM)
Monolix
NLME (S-Plus and R)
S-ADAPT
Phoenix NLME
WinBUGS, OpenBUGS (with BUGSModelLibrary)
SAS PROC NLMIXED (SAS)
https://ascpt.onlinelibrary.wiley.com/doi/epdf/10.1002/psp4.12404
https://ascpt.onlinelibrary.wiley.com/doi/epdf/10.1002/psp4.12422