Basic pharmacodynamic models for agents that alter the lifespan distribution of natural cells

Abstract

A new class of basic indirect pharmacodynamic models for agents that alter the loss of natural cells based on a lifespan concept are presented. The lifespan indirect response (LIDR) models assume that cells (R) are produced at a constant rate (k(in)), survive during a certain duration T(R), and finally are lost. The rate of cell loss is equal to the production rate but is delayed by T(R). A therapeutic agent can increase or decrease the baseline cell lifespan to a new cell lifespan, T(D), by temporally changing the proportion of cells belonging to the two modes of the lifespan distribution. Therefore, the change of lifespan at time t is described according to the Hill function, H(C(t)), with capacity (E(max)) and sensitivity (EC(50)), and the pharmacokinetic function C(t). A one-compartment cell model was examined through simulations to describe the role of pharmacokinetics, pharmacodynamics and cell properties for the cases where the drug increases (T(D) > T(R)) or decreases (T(D) < T(R)) the cell lifespan. The area under the effect curve (AUCE) and explicit solutions of LIDR models for large doses were derived. The applicability of the model was further illustrated using the effects of recombinant human erythropoietin (rHuEPO) on reticulocytes. The cases of both stimulation of the proliferation of bone marrow progenitor cells and the increase of reticulocyte lifespans were used to describe mean data from healthy subjects who received single subcutaneous doses of rHuEPO ranging from 20 to 160 kIU. rHuEPO is about 4.5-fold less potent in increasing reticulocyte survival than in stimulating the precursor production. A maximum increase of 4.1 days in the mean reticulocyte lifespan was estimated and the effect duration on the lifespan distribution was dose dependent. LIDR models share similar properties with basic indirect response models describing drug stimulation or inhibition of the response loss rate with the exception of the presence of a lag time and a dose independent peak time. The current concept can be applied to describe the pharmacodynamic effects of agents affecting survival of hematopoietic cell populations yielding realistic physiological parameters.

Fig. 1

Schematic diagram of basic LIDR models that alter the lifespan distribution. The response R is produced at a zero-order rate constant k_in. The removal rate is determined by the baseline lifespan T_R and the maximum or minimum lifespan T_D. The drug changes the distribution of cell lifespans between T_R and T_D according to the Hill function characterized by E_max and EC₅₀ (open box). C(t) denotes time-dependent drug concentrations at the effect site

Fig. 2

Schematic diagram of the PK/PD model of rHuEPO effect on reticulocyte counts in healthy subjects. Processes, variables, and parameters presented in this diagram are explained under Methods

Fig. 3

Simulated profiles of drug plasma concentrations (upper panel) and corresponding responses for the LIDR model T_D > T_R (middle panel) and model T_D < T_R (lower panel) for various doses

Fig. 4

Sensitivity of LIDR models that alter the lifespan distribution to E_max (upper panels) and to EC₅₀ (lower panels). The response curves were simulated for Dose = 10⁴

Fig. 5

Sensitivity of LIDR models that alter the lifespan distribution to T_D (upper panels) and to R₀ (lower panels). The response curves were simulated for Dose = 10⁴

Fig. 6

Sensitivity of LIDR models that alter the lifespan distribution to T_R. For the responses shown in the lower panels k_in was set to 2.083 unit/h and R₀ was calculated according to Eq. 15

Fig. 7

Time courses of the mean lifespans, MLR(t) for LIDR models that alter the lifespan distribution: model T_D > T_R (upper panel) and model T_D < T_R (lower panel)

Fig. 8

Frequency distributions of estimates of the PD parameters for model T_D > T_R. One hundred individual responses were simulated with the residual error model described by Eq. 19. The estimates of EC₅₀, E_max , T_D, and T_R obtained from successful minimizations are shown as histograms. The histogram for R₀ is omitted. The value of parameters used for simulations were Dose = 10⁴ μg, EC₅₀ = 100 ng/ml, E_max = 0.8, T_D = 72 h, T_R = 48 h, and R₀ = 100 units

Fig. 9

Observed (open symbols) and model predicted (solid lines) mean rHuEPO serum concentrations after subcutaneous administration in healthy subjects

Fig. 10

Observed (open symbols) and model predicted (solid lines) mean reticulocyte counts for healthy subjects who received subcutaneous doses of rHuEPO

Fig. 11

Time courses of the model-predicted mean reticulocyte lifespan ML_RET(t) in healthy subjects after subcutaneous administration of rHuEPO at indicated doses (kIU)