# Bounded parameter distribution

A bounded parameter distribution, for example between a and b, can not be set directly through the interface, but have to be defined in two steps: (1) an auxiliary parameter and its distribution choice in the GUI, and (2) a transformation of the auxiliary parameter into the parameter of interest in the structural model file.

Let’s take a simple PK example where a volume V is constrained. The structural model for this example is:

[LONGITUDINAL]
input = {V, k}
EQUATION:
; PK model definition
Cc = pkmodel(V, k)


• Thus, to have a parameter V between two bounds a=1 and b=10, you have to define the structural model as below
[LONGITUDINAL]
input = {V_logit, k}
EQUATION:
; PK model definition
a = 1
b = 10
V_bound = a+V_logit*(b-a)
Cc = pkmodel(V=V_bound, k)


In the “Statistical model & Tasks” tab of the GUI, the distribution for V_logit should be set to LOGIT.

• To have a parameter V larger than a=1 (with ‘a’ different from 0), you have to define the structural model as below
[LONGITUDINAL]
input = {V_log, k}
EQUATION:
; PK model definition
a = 1
V_bound = a+V_log
Cc = pkmodel(V=V_bound, k)


In the “Statistical model & Tasks” tab of the GUI, the distribution for V_log should be set to LOGNORMAL.

• To have a parameter V smaller than b=10, you have to define the structural model as below
[LONGITUDINAL]
input = {V_log, k}
EQUATION:
; PK model definition
b = 10
V_bound = b-V_log
Cc = pkmodel(V=V_bound, k)


In the “Statistical model & Tasks” tab of the GUI, the distribution for V_log should be set to LOGNORMAL.

Notice that, using that transformation, you have to multiply the standard error of V_logit by (b-a) in the first case to have the standard error of the initial V_bound parameter. It is not necessary for the two other cases as it is an offset. In addition, you can output V_bound for each individual using the table statement.