Six parameters are common to one, two or three compartment models:
There are two parameterisations implemented in PFIM for one-compartment models, (V and k) or (V and CL). The equations are given for the first parameterisation (V,k). For extra-vascular administration, V and CL are apparent volume and clearance. The equations for the second parameterisation (V,CL) are derived using k=CLV.
C(t)=DVe−k(t−tD)
C(t)=n∑i=1DiVe−k(t−tDi)
C(t)=DVe−k(t−tD)1−e−kτ
C(t)={DTinf1kV(1−e−k(t−tD))if t−tD≤Tinf,DTinf1kV(1−e−kTinf)e−k(t−tD−Tinf)if not.
C(t)={n−1∑i=1DiTinfi1kV(1−e−kTinfi)e−k(t−tDi−Tinfi)+DnTinfn1kV(1−e−k(t−tDn))if t−tDn≤Tinfn,n∑i=1DiTinfi1kV(1−e−kTinfi)e−k(t−tDi−Tinfi)if not.
C(t)={DTinf1kV[(1−e−k(t−tD))+e−kτ(1−e−kTinf)e−k(t−tD−Tinf)1−e−kτ]if (t−tD)≤Tinf,DTinf1kV(1−e−kTinf)e−k(t−tD−Tinf)1−e−kτif not.
C(t)=DVkaka−k(e−k(t−tD)−e−ka(t−tD))
C(t)=n∑i=1DiVkaka−k(e−k(t−tDi)−e−ka(t−tDi))
C(t)=DVkaka−k(e−k(t−tD)1−e−kτ−e−ka(t−tD)1−e−kaτ)
For two-compartment model equations, C(t)=C1(t) represent the drug concentration in the first compartment and C2(t) represents the drug concentration in the second compartment.
As well as the previously described PK parameters, the following PK parameters are used for the two-compartment models:
There are two parameterisations implemented in PFIM for two-compartment models: (V, k, k12 and k21), or (CL, V1, Q and V2). For extra-vascular administration, V1 (V), V2, CL, and Q are apparent volumes and clearances.
The second parameterisation terms are derived using:
For readability, the equations for two-compartment models with linear elimination are given using the variables α, β, A and B defined by the following expressions:
α=k21kβ=QV2CLV1β
β={12[k12+k21+k−√(k12+k21+k)2−4k21k]12[QV1+QV2+CLV1−√(QV1+QV2+CLV1)2−4QV2CLV1]
The link between A and B, and the PK parameters of the first and second parameterisations depends on the input and are given in each subsection.
For intravenous bolus, the link between A and B, and the parameters (V, k, k12 and k21), or (CL, V1, Q and V2) is defined as follows:
A=1Vα−k21α−β=1V1α−QV2α−β
B=1Vβ−k21β−α=1V1β−QV2β−α
C(t)=D(Ae−α(t−tD)+Be−β(t−tD))
C(t)=n∑i=1Di(Ae−α(t−tDi)+Be−β(t−tDi))
C(t)=D(Ae−αt1−e−ατ+Be−βt1−e−βτ)
For infusion, the link between A and B, and the parameters (V, k, k12 and k21), or (CL, V1, Q and V2) is defined as follows:
A=1Vα−k21α−β=1V1α−QV2α−β
B=1Vβ−k21β−α=1V1β−QV2β−α
C(t)={DTinf[Aα(1−e−α(t−tD))+Bβ(1−e−β(t−tD))]if t−tD≤Tinf,DTinf[Aα(1−e−αTinf)e−α(t−tD−Tinf)+Bβ(1−e−βTinf)e−β(t−tD−Tinf)]if not.
C(t)={n−1∑i=1DiTinfi[Aα(1−e−αTinfi)e−α(t−tDi−Tinfi)+Bβ(1−e−βTinfi)e−β(t−tDi−Tinfi)]+DTinfn[Aα(1−e−α(t−tDn))+Bβ(1−e−β(t−tDn))]if t−tDn≤Tinf,n∑i=1DiTinfi[Aα(1−e−αTinfi)e−α(t−tDi−Tinfi)+Bβ(1−e−βTinfi)e−β(t−tDi−Tinfi)]if not.
For first order absorption, the link between A and B, and the parameters (ka, V, k, k12 and k21), or (ka, CL, V1, Q and V2) is defined as follows:
A=kaVk21−α(ka−α)(β−α)=kaV1QV2−α(ka−α)(β−α)
B=kaVk21−β(ka−β)(α−β)=kaV1QV2−β(ka−β)(α−β)
C(t)=D(Ae−α(t−tD)+Be−β(t−tD)−(A+B)e−ka(t−tD))
C(t)=n∑i=1Di(Ae−α(t−tDi)+Be−β(t−tDi)−(A+B)e−ka(t−tDi))
C(t)=D(Ae−α(t−tD)1−e−ατ+Be−β(t−tD)1−e−βτ−(A+B)e−ka(t−tD)1−e−kaτ)
The list of PK models with Michaelis-Menten elimination implemented in PFIM are summarised in Appendix I.2. Presently, there is no implementation for multiple dosing with IV bolus administration in the PFIM software. For infusion and oral administration, the implementation in PFIM does not allow designs with different groups of doses as the dose is included in the model.
Initial conditions: {C(t)=0 for t<tDC(tD)=DVdCdt=−Vm×CKm+C
Initial conditions: C(t)=0 for t<tDdCdt=−Vm×CKm+C+inputinput(t)={DTinf1Vif 0≤t−tD≤Tinf0if not.
Initial conditions: C(t)=0 for t<tD1dCdt=−Vm×CKm+C+inputinput(t)={DiTinfi1Vif 0≤t−tDi≤Tinfi,0if not.
Initial conditions: C(t)=0 for t<tDdCdt=−Vm×CKm+C+inputinput(t)=DVkae−ka(t−tD)
Initial conditions: C(t)=0 for t<tD1dCdt=−Vm×CKm+C+inputinput(t)=n∑i=1DiVkae−ka(t−tDi)
Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for t≤tDC1(tD)=DVdC1dt=−Vm×C1Km+C1−k12C1+k21V2VC2dC2dt=k12VV2C1−k21C2
Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for t≤tDdC1dt=−Vm×C1Km+C1−k12C1+k21V2VC2+inputdC2dt=k12VV2C1−k21C2input(t)={DTinf1Vif 0≤t−tD≤Tinf0if not.
Initial conditions: {C1(t)=0 for t<tD1C2(t)=0 for t≤tD1dC1dt=−Vm×C1Km+C1−k12C1+k21V2VC2+inputdC2dt=k12VV2C1−k21C2input(t)={DiTinfi1Vif 0≤t−tDi≤Tinfi,0if not.
Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for t≤tDdC1dt=−Vm×C1Km+C1−k12C1+k21V2VC2+inputdC2dt=k12VV2C1−k21C2input(t)=DVkae−ka(t−tD)
Initial conditions: {C1(t)=0 for t<tD1C2(t)=0 for t≤tD1dC1dt=−Vm×C1Km+C1−k12C1+k21V2VC2+inputdC2dt=k12VV2C1−k21C2input(t)=n∑i=1DiVkae−ka(t−tDi)
For these response models, the effect E(t) is expressed as:
E(t)=A(t)+S(t)
where A(t) represents the model of drug action and S(t) corresponds to the baseline/disease model. A(t) is a function of the concentration C(t) in the central compartment.
The drug action models are presented in section Drug action models for C(t). The baseline/disease models are presented in section Baseline/disease models. Any combination of those two models is available in the PFIM library.
Parameters
NB: Vm is in concentration per time unit and Km is in concentration unit.
linear model A(t)=AlinC(t)
quadratic model A(t)=AlinC(t)+AquadC(t)2
logarithmic model A(t)=Aloglog(C(t))
Emax model A(t)=EmaxC(t)C(t)+C50
sigmoïd Emax model A(t)=EmaxC(t)γC(t)γ+Cγ50
Imax model A(t)=1−ImaxC(t)C(t)+C50
sigmoïd Imax model A(t)=1−ImaxC(t)γC(t)γ+Cγ50
full Imax model A(t)=−C(t)C(t)+C50
sigmoïd full Imax model A(t)=−C(t)γC(t)γ+Cγ50
S(t)=0
S(t)=S0
S(t)=S0+kprogt
S(t)=S0e−kprogt
S(t)=S0(1−e−kprogt)
In these models, the drug is not acting on the effect E directly but rather on Rin or kout.
Thus the system is described with differential equations, given dEdt as a function of Rin, kout and C(t) the drug concentration at time t.
The initial condition is: while C(t)=0, E(t)=Rinkout.
Parameters
Emax model dEdt=Rin(1+EmaxCC+C50)−koutE
sigmoïd Emax model dEdt=Rin(1+EmaxCγCγ+Cγ50)−koutE
Imax model dEdt=Rin(1−ImaxCC+C50)−koutE
sigmoïd Imax model dEdt=Rin(1−ImaxCγCγ+Cγ50)−koutE
full Imax model dEdt=Rin(1−CC+C50)−koutE
sigmoïd full Imax model dEdt=Rin(1−CγCγ+Cγ50)−koutE
Emax model dEdt=Rin−kout(1+EmaxCC+C50)E
sigmoïd Emax model dEdt=Rin−kout(1+EmaxCγCγ+Cγ50)E
Imax model dEdt=Rin−kout(1−ImaxCC+C50)E
sigmoïd Imax model dEdt=Rin−kout(1−ImaxCγCγ+Cγ50)E
full Imax model dEdt=Rin−kout(1−CC+C50)E
sigmoïd full Imax model dEdt=Rin−kout(1−CγCγ+Cγ50)E