**Basic PK/PD principles of proliferative and circular systems**

Philippe Jacqmin (1), Lynn McFadyen (2) and Janet R. Wade (1)

(1) Exprimo NV, Lummen, Belgium; (2) Clinical Pharmacology, Sandwich Laboratories, Pfizer Inc., UK.

Mathematical modelling is increasingly being applied to interpret and predict the dynamics of diseases (1,2,3). Within the infectious disease field it is commonly accepted that viral dynamics (VD) are characterized by a fundamental biological principle: the basic reproductive ratio (4) (RR0). RR0 is a derived model parameter that gives the average number of offspring generated by a single virus during its entire life span, in the absence of constraints. When RR0 is higher than 1, the system grows, when RR0 is lower than 1 the system goes to extinction. At RR0 of 1, production and elimination are in equilibrium and the system just survives. By extension and under certain assumptions, it can be proposed that proliferative systems such as bacteria, fungi and cancer cells share the same fundamental principle.

The goal of therapeutic agents used to treat proliferative systems is to bring RR0 below the break point of 1 and eradicate the disease. The inhibitor concentration (IC) that decreases RR0 to 1 can be called the reproduction minimum inhibitory concentration (RMIC). Assuming non competitive inhibition, it can be demonstrated that RMIC is equal to (RR0-1)*IC_{50} where RR0 is system specific and IC_{50} (concentration of inhibitor that gives 50% of the maximum inhibition) is compound specific. Across a population, the RMIC has a joint distribution arising from both RR0 and IC_{50}. For a particular individual, when their IC is higher than their RMIC, the proliferative system goes to extinction. If IC is lower than the RMIC, the proliferative system will eventually grow (after a transient decrease in some cases: e.g. viral dynamics).

Based on these two basic model parameters: RR0 and RMIC, several PK-PD principles of proliferative systems can be derived, such as:

- System survival (i.e. RR0
_{INH}=1) can occur at different levels of inhibition depending on RR0 (5): for example, when RR0=2, RMIC=IC_{50}. When RR0=10, RMIC=9*IC_{50}= IC_{90}. - When
*in vitro*and*in vivo*RR0 are different (which is often the case),*in vitro*and*in vivo*RMIC will also be different. Only when both*in vitro*and*in vivo*RR0 are known can*in vitro*RMIC be scaled to*in vivo*RMIC and use to predict efficacious inhibitor exposure. - Mechanistically, logistic regression of binary outcomes such as failure/success rates as a function of drug exposure is an expression of the RMIC distribution across the population.
- Time of failure or success is a function of the IC/RMIC ratio: e.g. for failure (ratio is below 1), then the further below 1 the ratio is, the earlier the failure.
- In order to be equally efficacious at steady state (i.e. same proliferation rate), two treatments (e.g. qd vs bid) should give rise to the same
__average__RR0_{INH}. This can easily be calculated using PK-PD simulations. Indirectly, it indicates that successful C_{avg}/RMIC, C_{min}/RMIC and C_{max}/RMIC ratios are PK (e.g. half-life) and schedule dependent. It also supports the concept of time above MIC (TAM) for time-dependent antibiotics (6). - Time varying inhibition of proliferative systems can be handled by calculating the equivalent effect constant concentration (7). (ECC) which is equal to IC
_{50}*INH_{avg}/(1-INH_{avg}) where INH_{avg}is the area under the inhibition-time curve divided by the dosing interval.

These fundamental and derived basic PK-PD principles will be illustrated using a simple PK-PD model for proliferative systems and a more complex PK-PD model for viral dynamics (3,8,9). based on the Lotka-Volterra principle. Finally, some considerations on the application of these concepts to circular (disease) systems such as inflammation, allergy and bone turnover will be briefly mentioned.

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