Closed-form

Closed-form models

The PML contains built-in support for closed-form models with up to three compartments, and with optional first order input, optional lag time, and optional bioavailability. The models are implemented recursively so they can handle any combination of dosing scenarios. It should be cautioned that closed-form models can give incorrect results if time-varying model parameters are present.

Closed-form example (micro-constant parameterization):

   cfMicro(A1, Ke)

Specifies a 1-compartment model. A1 is the amount in the central compartment, and Ke is the elimination rate parameter.

   cfMicro(A1, Ke, K12, K21)

Specifies a 2-compartment model, same as the 1-compartment model, but with two additional parameters K12 and K21.

   cfMicro(A1, Ke, K12, K21, K13, K31)

Specifies a 3-compartment model, same as the 2-compartment model, but with two additional parameters K13 and K31.

   cfMicro(A1, Ke, first=(Aa=Ka))

Specifies first-order input to any of the models above. Aa is the amount in the absorption compartment, and Ka is the absorption rate.

Closed-form example (macro-constant parameterization):

   cfMacro(A1, C1, A1Dose, A, Alpha, strip=A1Strip)
   cfMacro(A1, C1, A1Dose, A, Alpha, B, Beta, strip=A1Strip)
   cfMacro(A1, C1, A1Dose, A, Alpha, B, Beta, C, Gamma, strip=A1Strip)

Specifies 1, 2, and 3-compartment models, in which observed concentration C1 is modeled as the exponential sum A*exp(-t*Alpha)+B*exp(-t*Beta)+C*exp(-t*Gamma). A1 is the dosing target but is not a variable that can be referred to in the model. A1Strip is the name of a covariate specifying the “stripping dose.” A1Dose is a variable that records the initial bolus amount. A1Strip is the name of a covariate specifying the “stripping dose.” A1Strip should be set equal to A1Dose when fitting a model. However, when simulating a model, A1Strip should correspond to the original dose given when the model was estimated, while A1Dose can vary to whatever one wishes to simulate.

If the optional argument strip=A1Strip is not given, the initial bolus amount is used as the stripping dose. The model can be used with any dosing sequence, but it is an error if there is no specified stripping dose and no initial bolus.

   cfMacro(Aa, C1, AaDose, A, Alpha, Ka, strip=A1Strip)

This model is the same as above except for the additional final parameter Ka, signifying first-order absorption. In this case, the model without first-order absorption is convolved with the one-term first-order absorption term, resulting in the final model. Everything else is the same as above.

Closed-form example (macro-constant parameterization, simple form):

   cfMacro1(A, Alpha)

Specifies a 1-compartment model. A is the amount in the central compartment, and Alpha is the elimination rate parameter. It can be used with any dosing sequence but its response to a bolus dose is A=D*exp(-t*Alpha).

   cfMacro1(A, Alpha, B, Beta)

Specifies a 2-compartment model. A is the amount in the central compartment. It can be used with any dosing sequence but its response to a bolus dose is D*[(1-B)*exp(-t*Alpha)+ B*exp(-t*Beta)].

   cfMacro1(A, Alpha, B, Beta, C, Gamma)

Specifies a 3-compartment model. A is the amount in the central compartment. It can be used with any dosing sequence but its response to a bolus dose is D*[(1-B-C)*exp(-t*Alpha)+ B*exp(-t*Beta)+C*exp(-t*Gamma)].

   cfMacro1(A, Alpha, first=(Aa=Ka))

Any of the above models can be converted to first-order absorption by putting the following after the other arguments.

   , first=(Aa=Ka)

Aa is the amount in the absorption compartment, and Ka is the absorption rate. As above, A is the amount in the central compartment. It can be used with any dosing sequence, and it allows dosing to both Aa and A. (The model is two models superimposed, one is the base model, and the other is the base model convolved with a first-order model.)

See the “Closed-Form” section for more information on this method.