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:
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:
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:
Key features characterizing models labeled as “QSP” models:
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
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 (Rp) and Rp) 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
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:
QSP model has more than one steady state constrains:
ssRp*Vp+ ssRbm*Vbm = k0*TR,
Q*ssRbm = k0,
ssRBCp*Vp = k0*TRBC
6. Initial conditions for “mechanism based PK/PD” model
Differ from that for QSP model
7. Parameter values taken from literature are ssRp, ssRBCp , TRBC for “mechanism based PK/PD” model and ssRp, ssRBCp, TRBC, Vbm, Vp for “QSP” model
8. Parameter for fitting are Smax, SC50, TP1, TP2, TR, Vd for “mechanism based PK/PD” model and Smax, SC50, TP1, TP2, 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
As a result 3 sets of parameters are available. Each set allow to reproduce PK of corresponding specie with reasonable quality (see Fig 2A).
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.