Evaluation of Assumptions Underpinning Pharmacometric Models

Abstract

Assumptions inherent to pharmacometric model development and use are not routinely acknowledged, described, or evaluated. The aim of this work is to present a framework for systematic evaluation of assumptions. To aid identification of assumptions, we categorise assumptions into two types: implicit and explicit assumptions. Implicit assumptions are inherent in a method or model and underpin its derivation and use. Explicit assumptions arise from heuristic principles and are typically defined by the user to enable the application of a method or model. A flowchart was developed for systematic evaluation of assumptions. For each assumption, the impact of assumption violation (‘significant’, ‘insignificant’, ‘unknown’) and the probability of assumption violation (‘likely’, ‘unlikely’, ‘unknown’) will be evaluated based on prior knowledge or the result of an additional bespoke study to arrive at a decision (‘go’, ‘no-go’) for both model building and model use. A table of assumptions with standardised headings has been proposed to facilitate the documentation of assumptions and evaluation of results. The utility of the proposed framework was illustrated using four assumptions underpinning a top-down model describing the warfarin-coagulation proteins’ relationship. The next step of this work is to apply the framework to a series of other settings to fully assess its practicality and its value in identifying and making inferences from assumptions.

Appendix

Assumptions underpinning a joint model for six vitamin K-dependent coagulation proteins (18) were evaluated. The joint model is described by the following system of ordinary differential equations:

Pharmacokinetic model:

$$ \frac{dD}{dt}=-{k}_a\cdotp D;\kern1.25em \left\{\begin{array}{c}\ t<\mathrm{LAG},{D}_{t=0}=0\kern1.75em \\ {}\ t\ge \mathrm{LAG},{D}_{t=0}=\mathrm{Dose}\end{array}\right. $$

$$ \frac{dA}{dt}={k}_a\cdotp D-\frac{\mathrm{CL}}{V}\cdotp A;\kern1.25em {A}_{t=0}=0 $$

Pharmacodynamic model:

$$ \frac{d\mathrm{II}}{dt}={R}_{\mathrm{in},\mathrm{II}}\cdotp {I}_{II}-{k}_{\mathrm{out},\mathrm{II}}\cdotp \mathrm{II};\kern1.25em {\mathrm{II}}_{t=0}=\frac{R_{\mathrm{in},\mathrm{II}}}{k_{\mathrm{out},\mathrm{II}}} $$

$$ \frac{d\mathrm{VII}}{dt}={R}_{\mathrm{in},\mathrm{VII}}\cdotp {I}_{\mathrm{VII}}-{k}_{\mathrm{out},\mathrm{VII}}\cdotp \mathrm{VII};\kern1.25em {\mathrm{VII}}_{t=0}=\frac{R_{\mathrm{in},\mathrm{VII}}}{k_{\mathrm{out},\mathrm{VII}}} $$

$$ \frac{d\mathrm{IX}}{dt}={R}_{\mathrm{in},\mathrm{IX}}\cdotp {I}_{\mathrm{IX}}-{k}_{\mathrm{out},\mathrm{IX}}\cdotp \mathrm{IX};\kern1.25em {\mathrm{IX}}_{t=0}=\frac{R_{\mathrm{in},\mathrm{IX}}}{k_{\mathrm{out},\mathrm{IX}}} $$

$$ \frac{dX}{dt}={R}_{\mathrm{in},X}\cdotp {I}_X-{k}_{\mathrm{out},X}\cdotp X;\kern1.25em {X}_{t=0}=\frac{R_{\mathrm{in},X}}{k_{\mathrm{out},X}} $$

$$ \frac{d\mathrm{PC}}{dt}={R}_{\mathrm{in},\mathrm{PC}}\cdotp {I}_{\mathrm{PC}}-{k}_{\mathrm{out},\mathrm{PC}}\cdotp \mathrm{PC};\kern1.25em {\mathrm{PC}}_{t=0}=\frac{R_{\mathrm{in},\mathrm{PC}}}{k_{\mathrm{out},\mathrm{PC}}} $$

$$ \frac{d\mathrm{PS}}{dt}={R}_{\mathrm{in},\mathrm{PS}}\cdotp {I}_{\mathrm{PS}}-{k}_{\mathrm{out},\mathrm{PS}}\cdotp \mathrm{PS};\kern1.25em {\mathrm{PS}}_{t=0}=\frac{R_{\mathrm{in},\mathrm{PS}}}{k_{\mathrm{out},\mathrm{PS}}} $$

$$ P=\left\{\mathrm{II},\mathrm{VII},\mathrm{IX},\mathrm{X},\mathrm{PC},\mathrm{PS}\right\} $$

$$ {I}_P(t)=1-\frac{I_{\max, P}\cdotp A(t)}{I{A}_{50,P}+A(t)} $$

$$ {R}_{\mathrm{in},P}={P}_{t=0}\cdotp {k}_{\mathrm{out},P}. $$

Here, LAG, general lag parameter; CL , clearance of warfarin; D, warfarin dose; II, factor II; IA_50,P , warfarin amount in the body that gives half the maximum inhibitory effect; I_max,P, maximum inhibitory effect; IX, factor IX; k_a, first-order absorption rate constant of warfarin; k_out,P, first-order coagulation protein degradation rate constant; PC, protein C; PS, protein S; R_in,P, zero-order functional coagulation protein production rate; t, time; V, volume of distribution of warfarin; VII, factor VII; X, factor X; P_t = 0, coagulation protein concentration at baseline.