A cell-level model of pharmacodynamics-mediated drug disposition

Abstract

We aimed to develop a cell-level pharmacodynamics-mediated drug disposition (PDMDD) model to analyze in vivo systems where the PD response to a drug has an appreciable effect on the pharmacokinetics (PK). An existing cellular level model of PD stimulation was combined with the standard target-mediated drug disposition (TMDD) model and the resulting model structure was parametrically identifiable from typical in vivo PK and PD data. The PD model of the cell population was controlled by the production rate k _in and elimination rate k _out which could be stimulated or inhibited by the number of bound receptors on a single cell. Simulations were performed to assess the impact of single and repeated dosing on the total drug clearance. The clinical utility of the cell-level PDMDD model was demonstrated by fitting published data on the stimulatory effects of filgrastim on absolute neutrophil counts in healthy subjects. We postulated repeated dosing as a means of detecting and quantifying PDMDD as a single dose might not be sufficient to elicit the cellular response capable of altering the receptor pool to visibly affect drug disposition. In the absence of any PD effect, the model reduces down to the standard TMDD model. The applications of this model can be readily extended to include chemotherapy-induced cytopenias affecting clearance of endogenous hematopoietic growth factors, different monoclonal antibodies and immunogenicity effects on PK.

Appendices

Appendix 1: Clearance of total drug

Since A _tot  = A _P  + BV _c one can combine Eqs. (2) and (35) to obtain a differential equation

$$\frac{{dA_{tot} }}{dt} = In(t) - \left( {k_{el} + k_{pt} } \right)A_{P} + k_{tp} \cdot A_{T} - k_{int} \cdot B \cdot V_{c} + k_{in} \cdot \xi \cdot b\left( {1 + H_{in} (b)} \right) - k_{out} \cdot \left( {1 + H_{out} (b)} \right)B \cdot V_{c} .$$

(41)

Therefore the elimination rate of the total drug from the body is

$${\text{elimination}}\;{\text{rate}} = k_{el} \cdot A_{P} + k_{int} \cdot B \cdot V_{c} + k_{out} \left( {1 + H_{out} (b)} \right)B \cdot V_{c} .$$

(42)

Dividing (42) by C yields (40).

Appendix 2: Rapid binding approximation

In many instances receptor binding and dissociation are much faster than the remaining processes described by the model. This results in the inability to uniquely identify the binding parameters. To resolve this problem the rapid binding approximation of the full model (2)–(8) can be applied based on the quasi-equilibrium assumption [16]:

$$\frac{C\cdot r}{b} = \frac{{k_{on} }}{{k_{off} }} = K_{D} .$$

(43)

To be able to eliminate the binding parameters k _on and k _off from the model equations new variables are introduced, the total receptor number/cell (r _tot ) and total amount of the drug in the central compartment (A _tot ):

$$r_{tot} = r + b\quad {\text{and}}\quad A_{tot} = A_{P} + \xi \cdot b \cdot N.$$

(44)

Then the model equations become:

$$\frac{{dA_{tot} }}{dt} = In(t) - \left( {k_{el} + k_{pt} } \right)C \cdot V_{c} + k_{tp} \cdot A_{T} - k_{int} \cdot \xi \cdot b \cdot N + \xi \cdot b \cdot \frac{dN}{dt},$$

(45)

$$\frac{{dA_{T} }}{dt} = k_{pt} \cdot C \cdot V_{c} - k_{tp} \cdot A_{T} ,$$

(46)

$$\frac{{dr_{tot} }}{dt} = k_{syn} - k_{deg } \cdot r_{tot} + \left( {k_{deg } - k_{int} } \right) \cdot b,$$

(47)

where the derivative dN/dt is defined by (7),

$$b = \frac{{r_{tot} \cdot C}}{{K_{D} + C}},$$

(48)

and

$$C \,=\, \frac{1}{2}\left( {C_{tot} - R_{tot} - K_{D} + \sqrt {\left( {C_{tot} - R_{tot} - K_{D} } \right) + 4 \cdot K_{D} \cdot C_{tot} } } \right).$$

(49)

The total drug and total receptor concentrations in the central compartment are defined as follows

$$C_{tot} = \frac{{A_{tot} }}{{V_{c} }} = C + B \quad {\text{and}}\quad R_{tot} = \frac{{r_{tot} \cdot \xi \cdot N}}{{V_{c} }} = R + B.$$

(50)

Given that In(t) describes the input rate to the A _tot compartments (including bolus doses), the initial conditions for (45)–(47) are

$$A_{tot} (0) = 0,\quad A_{T} (0) = 0,\quad r_{tot} (0) = \frac{{k_{syn} }}{{k_{deg } }},\quad N(0) = \frac{{k_{in} }}{{k_{out} }}.$$

(51)