Exploration of structural differences between mechanism based PK/PD and QSP models: implication in translational modeling

Introduction

There are lots of mathematical models describing disease pathogenesis and effect of different therapies published in literature. Most of the published models were referred by authors either  “mechanism based PK/PD” or ‘Quantitative Systems Pharmacology” (“QSP”) models. Both mechanism-based PK/PD and QSP models are applied to describe PK and PD of a drug and explore translational problems.

Possible criteria to distinguish between these types of models may include:

1. purpose of model development (for example, optimal dose prediction vs studying mechanism of action)
2. type of data used to identify model parameters (in vitro vs ex vivo vs in vivo)
3. way of model analysis/processing (identification of parameters, typical model analysis, etc)
4. model structure (what physiology is taken into account, etc)

In this study we will analyze “model structure” only as possible criteria to distinguish between the types of models. In particular, we will address following questions:

• Are there any principles underlying model structure and ODE system construction which are different for “mechanism-based PK/PD” and “QSP” models?
• Is there any way to transform “mechanism-based PK/PD” model to “QSP” one?

Analysis of right-hand side structure of PD part of literature available models

Several mechanism-based PK/PD models and QSP models available in literature were analyzed [PMID: 22442825, PMID: 23903463, PMID: 23568527, PMID: 24918743, PMID: 20232116, PMID: 12676891, PMID: 16973883, PMID: 15317827]. Results of the analysis are presented below:

Key findings specific for “mechanism based PK/PD” models:

• there are no “physical compartments” (tissues, organs etc) in the PD model and, consequently, volumes of the “physical compartments” are not considered
• there is no assignment of PD variables (amounts or concentrations of molecules, cell numbers) to real “physical compartments”
• there is no any mass transfer between “physical compartments”
• interaction of variables biologically located in different “physical compartments” is described in terms of influences (without any mass transfer!) and does not take into account volumes of “physical compartments”

Key features characterizing models labeled as “QSP” models:

• several models take into account “physical compartments” and mass transfer between them in their PD part
• several models do not take “physical compartments” into account 

Results of the analysis we propose to distinguish “mechanism based PK/PD” and “QSP” models in following manner:

“Mechanism based PK/PD” models are ODE based models describing PD in terms of variables potentially located in different “physical compartments” BUT do not taking into account volumes of the “physical compartments” and mass transfer between them.

“System pharmacology” models are ODE based models describing PD in terms of variables potentially located in different “physical compartments” AND taking into account volumes of the “physical compartments” and mass transfer between them.

Transformation of PK/PD to QSP model and translation of newly baked QSP model from rat to monkey and human

In this section we will apply principles allowing us to distinguish between “mechanism based PK/PD” and “QSP” formulated above and transform particular “mechanism based PK/PD” model found in literature to “QSP” model. After such transformation we will apply allometric scaling approach to translate the newly baked QSP model from rat to monkey and human.

Workflow of transformation of PK/PD model to systems model

• Let us consider “mechanism-based PK/PD” model of recombinant human erythropoietin (rHuEPO) in rats published in [PMID: 16973883]
• Transform it to “QSP” model of rHuEPO in rats
• Calibrate it against “rat” data + PK reproduction for all species
• Allometrically scale it to monkey and human
• Compare predictions of scaled QSP model with monkey and human data published in [PMID: 12676891] and [PMID: 15317827], correspondently.

Transformation of “mechanism based PK/PD” model oif rHuEpo in rat to “QSP” model of rHuEPO in rats

Fig. 1 (left part) demonstrates structure of the mechanism based PK/PD model published in [PMID: 16973883]. It includes 2 compartmental PK sub-model of rHuEpo and PD model describing its effect on Red Blood Cells (RBC) maturation and death. Indeed, the PD sub-model includes 2 variables (reticulocyte count (R_p) and R_p) located in one compartment/volume entitled as Vd (volume of distribution). PD sub-model includes 3 processes: (1) appearance of reticulocytes in Vd, (2) maturation of R to RBC and (3) RBC death. rHuEpo stimulates appearance of R in Vd.

Fig. 1.  Schemes of variables, compartments/”physical volumes” and processes included in “mechanism based PK/PD” and “QSP” models

Fig 1 (right) represent how “mechanism based PK/PD” model was transformed to “QSP” model in accordance with the principles declared in section “Analysis of right-hand side structure of PD part of literature available models”. Indeed, following changes were introduced

1. Instead of 1 physical compartment V_d in “mechanism based PK/PD” model there were introduced 2 physical compartments PL (plasma) and BM (bone marrow) and they were assigned with corresponding physiological values (V_p, V_bm)
2. New variable representing reticulocyte count in BM (R_bm) was introduced
3. New process representing transport of reticulocytes from BM to PL was introduced. Rate law of the process was represented by following equation:

v₄=Q*R_p

       4. As a result of changes (1) – (3) PD part of ODE system of “mechanism based PK/PD” model:

      5. Steady state constraints for “mechanism based PK/PD” model looks like following:

^ssR_p=k₀*T_R

QSP model has more than one steady state constrains:

^ssR_p*V_p+ ^ssR_bm*V_bm = k₀*T_R,

Q*^ssR_bm = k₀,

^ssRBC_p*V_p = k₀*T_RBC

     6. Initial conditions for “mechanism based PK/PD” model

R_p(0)=^ssR_p, RBC_p(0)=^ssRBC_p

Differ from that for QSP model

R_p(0)=^ssR_p, RBC_p(0)=^ssRBC_p,

R_bm(0)=^ssR_bm

     7. Parameter values taken from literature are ^ssR_p, ^ssRBC_{p ,} T_RBC for “mechanism based PK/PD” model and ^ssR_p, ^ssRBC_p, T_RBC, V_bm, V_p for “QSP” model

     8. Parameter for fitting are S_max, SC₅₀, T_P1, T_P2, T_R, V_d for “mechanism based PK/PD” model and S_max, SC₅₀, T_P1, T_P2, Q for “QSP” model. We can see that in framework of QSP model we need fit less parameters.  

Calibration PK sub-model against data for all species and calibration of PD sub-model against “rat” data

PK sub-model was calibrated separately against

• rat PK data [PMID: 16973883]
• monkey PK data [PMID: 12676891]
• human PK data [PMID: 15317827]

As a result 3 sets of parameters are available. Each set allow to reproduce PK of corresponding specie with reasonable quality (see Fig 2A).

A

B

Fig. 2. Quality of calibration of (A) PK sub-model against PK data measured for rat, monkey and human, (B) PD sub-model of newly baked QSP model (see parameters specified clause (8) of sub-section “Transformation of “mechanism based PK/PD” model oif rHuEpo in rat to “QSP” model of rHuEPO in rats”).

Allometric scaling of PD sub-model of QSP model of rat to monkey and human

We have used following expression to allometrically scale model rates and times:

Scaled “rat” model was then applied to predict measured response in monkey and human

Comparison of predictions of scaled QSP model with monkey and human data

Fig. 3 demonstrates that monkey and human PD data measured for single and multiple dose administration ([PMID: 12676891] and [PMID: 15317827]) is satisfactory predicted by QSP model of the corresponding specie allometrically scaled from rat QSP model.

A

B 

Fig. 3. Quality of prediction of PD data measured for (A) monkey and (B) human with QSP model allometrically scaled from rat QSP model.

Conclusions.

• Criteria distinguishing “mechanism-based PK/PD” and “QSP” models is proposed  
• Example of transformation of “mechanism-based PK/PD” to “QSP” model is presented

Possible advantages QSP models vs “mechanism based PK/PD” models in translational research are demonstrated