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Library of models

Pharmacokinetic models

Compartmental models and parameters

Six parameters are common to one, two or three compartment models:

One-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.

Models with linear elimination

One-compartment models

Intravenous bolus

  • single dose

C(t)=DVek(ttD)

  • multiple doses

C(t)=ni=1DiVek(ttDi)

  • steady state

C(t)=DVek(ttD)1ekτ

Infusion

  • single dose

C(t)={DTinf1kV(1ek(ttD))if ttDTinf,DTinf1kV(1ekTinf)ek(ttDTinf)if not.

  • multiple doses

C(t)={n1i=1DiTinfi1kV(1ekTinfi)ek(ttDiTinfi)+DnTinfn1kV(1ek(ttDn))if ttDnTinfn,ni=1DiTinfi1kV(1ekTinfi)ek(ttDiTinfi)if not.

  • steady state

C(t)={DTinf1kV[(1ek(ttD))+ekτ(1ekTinf)ek(ttDTinf)1ekτ]if (ttD)Tinf,DTinf1kV(1ekTinf)ek(ttDTinf)1ekτif not.

First order absorption

  • single dose

C(t)=DVkakak(ek(ttD)eka(ttD))

  • multiple doses

C(t)=ni=1DiVkakak(ek(ttDi)eka(ttDi))

  • steady state

C(t)=DVkakak(ek(ttD)1ekτeka(ttD)1ekaτ)

Two-compartment models

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:

  • V2, the volume of distribution of second compartment
  • k12, the distribution rate constant from compartment 1 to compartment 2
  • k21, the distribution rate constant from compartment 2 to compartment 1
  • Q, the inter-compartmental clearance
  • α, the first rate constant
  • β, the second rate constant
  • A, the first macro-constant
  • B, the second macro-constant

There are two parameterisations implemented in PFIM for two-compartment models: (Vkk12 and k21), or (CLV1Q and V2). For extra-vascular administration, V1 (V), V2, CL, and Q are apparent volumes and clearances.

The second parameterisation terms are derived using:

  • V1=V
  • CL=k×V1
  • Q=k12×V1
  • V2=k12k21×V1

For readability, the equations for two-compartment models with linear elimination are given using the variables αβA and B defined by the following expressions:

α=k21kβ=QV2CLV1β

β={12[k12+k21+k(k12+k21+k)24k21k]12[QV1+QV2+CLV1(QV1+QV2+CLV1)24QV2CLV1]

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.

Intravenous bolus

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=1Vαk21αβ=1V1αQV2αβ

B=1Vβk21βα=1V1βQV2βα

  • single dose

C(t)=D(Aeα(ttD)+Beβ(ttD))

  • multiples doses

C(t)=ni=1Di(Aeα(ttDi)+Beβ(ttDi))

  • steady state

C(t)=D(Aeαt1eατ+Beβt1eβτ)

Infusion

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=1Vαk21αβ=1V1αQV2αβ

B=1Vβk21βα=1V1βQV2βα

  • single dose

C(t)={DTinf[Aα(1eα(ttD))+Bβ(1eβ(ttD))]if ttDTinf,DTinf[Aα(1eαTinf)eα(ttDTinf)+Bβ(1eβTinf)eβ(ttDTinf)]if not.

  • multiple doses

C(t)={n1i=1DiTinfi[Aα(1eαTinfi)eα(ttDiTinfi)+Bβ(1eβTinfi)eβ(ttDiTinfi)]+DTinfn[Aα(1eα(ttDn))+Bβ(1eβ(ttDn))]if ttDnTinf,ni=1DiTinfi[Aα(1eαTinfi)eα(ttDiTinfi)+Bβ(1eβTinfi)eβ(ttDiTinfi)]if not.

  • steady state

First-order absorption

For first order absorption, the link between A and B, and the parameters (ka, V, k, k12 and k21), or (kaCLV1Q and V2) is defined as follows:

A=kaVk21α(kaα)(βα)=kaV1QV2α(kaα)(βα)

B=kaVk21β(kaβ)(αβ)=kaV1QV2β(kaβ)(αβ)

  • single dose

C(t)=D(Aeα(ttD)+Beβ(ttD)(A+B)eka(ttD))

  • multiple doses

C(t)=ni=1Di(Aeα(ttDi)+Beβ(ttDi)(A+B)eka(ttDi))

  • steady state

C(t)=D(Aeα(ttD)1eατ+Beβ(ttD)1eβτ(A+B)eka(ttD)1ekaτ)

Models with Michaelis-Menten elimination

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.

One-compartment models

Intravenous bolus

  • single dose

Initial conditions: {C(t)=0 for t<tDC(tD)=DVdCdt=Vm×CKm+C

Infusion

  • single dose

Initial conditions: C(t)=0 for t<tDdCdt=Vm×CKm+C+inputinput(t)={DTinf1Vif 0ttDTinf0if not.

  • multiple doses

Initial conditions: C(t)=0 for t<tD1dCdt=Vm×CKm+C+inputinput(t)={DiTinfi1Vif 0ttDiTinfi,0if not.

First order absorption

  • single dose

Initial conditions: C(t)=0 for t<tDdCdt=Vm×CKm+C+inputinput(t)=DVkaeka(ttD)

  • multiple doses

Initial conditions: C(t)=0 for t<tD1dCdt=Vm×CKm+C+inputinput(t)=ni=1DiVkaeka(ttDi)

Two-compartment models

Intravenous bolus

  • single dose

Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for ttDC1(tD)=DVdC1dt=Vm×C1Km+C1k12C1+k21V2VC2dC2dt=k12VV2C1k21C2

Infusion

  • single dose

Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for ttDdC1dt=Vm×C1Km+C1k12C1+k21V2VC2+inputdC2dt=k12VV2C1k21C2input(t)={DTinf1Vif 0ttDTinf0if not.

  • multiple doses

Initial conditions: {C1(t)=0 for t<tD1C2(t)=0 for ttD1dC1dt=Vm×C1Km+C1k12C1+k21V2VC2+inputdC2dt=k12VV2C1k21C2input(t)={DiTinfi1Vif 0ttDiTinfi,0if not.

First order absorption

  • single dose

Initial conditions: {C1(t)=0 for t<tDC2(t)=0 for ttDdC1dt=Vm×C1Km+C1k12C1+k21V2VC2+inputdC2dt=k12VV2C1k21C2input(t)=DVkaeka(ttD)

  • multiple doses

Initial conditions: {C1(t)=0 for t<tD1C2(t)=0 for ttD1dC1dt=Vm×C1Km+C1k12C1+k21V2VC2+inputdC2dt=k12VV2C1k21C2input(t)=ni=1DiVkaeka(ttDi)

Pharmacodynamic models

Immediate response models

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.

Drug action models

  • 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)=1ImaxC(t)C(t)+C50

  • sigmoïd Imax model A(t)=1ImaxC(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

Baseline/disease models

  • null baseline

S(t)=0

  • constant baseline with no disease progression

S(t)=S0

  • linear disease progression

S(t)=S0+kprogt

  • exponential disease increase

S(t)=S0ekprogt

  • exponential disease decrease

S(t)=S0(1ekprogt)

Turnover response models

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

Models with impact on the input (Rin)

  • Emax model dEdt=Rin(1+EmaxCC+C50)koutE

  • sigmoïd Emax model dEdt=Rin(1+EmaxCγCγ+Cγ50)koutE

  • Imax model dEdt=Rin(1ImaxCC+C50)koutE

  • sigmoïd Imax model dEdt=Rin(1ImaxCγCγ+Cγ50)koutE

  • full Imax model dEdt=Rin(1CC+C50)koutE

  • sigmoïd full Imax model dEdt=Rin(1CγCγ+Cγ50)koutE

Models with impact on the output (kout)

  • Emax model dEdt=Rinkout(1+EmaxCC+C50)E

  • sigmoïd Emax model dEdt=Rinkout(1+EmaxCγCγ+Cγ50)E

  • Imax model dEdt=Rinkout(1ImaxCC+C50)E

  • sigmoïd Imax model dEdt=Rinkout(1ImaxCγCγ+Cγ50)E

  • full Imax model dEdt=Rinkout(1CC+C50)E

  • sigmoïd full Imax model dEdt=Rinkout(1CγCγ+Cγ50)E