Indirect pharmacodynamic models for responses with circadian removal

Abstract

Rhythmicity in baseline responses over a 24-h period for an indirect pharmacological effect R(t) can arise from either a periodic time-dependent input rate $k_{in} \left( t \right)$ or a periodic time-dependent loss constant $k_{out} \left( t \right)$. If either $k_{in} \left( t \right)$ or $k_{out} \left( t \right)$ follows some nonstationary biological rhythm (e.g., circadian), then the response R(t) also displays a periodic behavior. Indirect response models assuming time-dependent input rates $\left[ {k_{in} \left( t \right)} \right]$ have been utilized to capture drug effects on various physiological responses such as hormone suppression, immune cell trafficking, and gene expression in tissues. This paradigm was extended to consider responses with circadian-controlled loss $\left[ {k_{out} \left( t \right)} \right]$ mechanisms. Theoretical equations describing this model are presented and simulations were performed to examine expected response behaviors. The model was able to capture the chronobiology and pharmacodynamics of applicable drug responses, including the uricosuric effects of lesinurad in humans, suppression of the beta amyloid (Aβ) peptide by a gamma-secretase inhibitor in mouse brain, and the modulation of extracellular dopamine by a dopamine transporter inhibitor in rat brain. This type of model has a mechanistic basis and shows utility for capturing drug responses displaying nonstationary baselines controlled by removal mechanism(s).

Appendix

Circadian input

In many pharmacodynamic systems the input rate of measured variables is regulated by endogenous biorhythmic processes such as circadian rhythms [1]. Then $k_{in}$ varies with time $k_{in} = k_{in} \left( t \right)$ and consequently the response also exhibits a similar biorhythmic pattern. The simplest biorhythmic response $R_{b} \left( t \right)$ can be described by the cosine function:

$$ R_{b} \left( t \right) = R_{m} + R_{a} \cos \left( {\frac{2\pi }{T}(t - t_{p} )} \right) $$

(13)

where $R_{m}$ is the mean baseline (mesor), $R_{a}$ is the amplitude $\left( {R_{m} > R_{a} } \right)$, $t_{p}$ is the peak time (acrophase), and $T = 24\,{\text{h}}$.In the case of a circadian input mechanism, the indirect response model in Fig. 1 is given by

$$ \frac{{dR_{b} \left( t \right)}}{dt} = k_{in} \left( t \right) - k_{out} \cdot R_{b} \left( t \right) $$

(14)

where $k_{in} \left( t \right)$ is circadian function of time and $k_{out}$ is first-order constant. Equation (14) can be re-arranged to express $k_{in} \left( t \right)$ as

$$ k_{in} \left( t \right) = k_{out} \cdot R_{b} \left( t \right) + \frac{{dR_{b} }}{dt}\left( t \right) $$

(15)

Substituting Eqs. (13) into (15),

$$ k_{in} \left( t \right) = k_{out} R_{m} + k_{out} R_{a} cos\left( {\frac{2\pi }{T}\left( {t - t_{p} } \right)} \right) + \frac{d}{dt} \left( R_{m} + R_{a} cos\left( {\frac{2\pi }{T}(t - t_{p} )} \right) \right) $$

(16)

Taking the derivative of $\frac{{dR_{b} }}{dt}\left( t \right)$ in Eq. (16) yields,

$$ k_{in} \left( t \right) = k_{out} R_{m} + k_{out} R_{a} \cos \left( {\frac{2\pi }{T}\left( {t - t_{p} } \right)} \right) - \frac{2\pi }{T}R_{a} \sin \left( {\frac{2\pi }{T}\left( {t - t_{p} } \right)} \right) $$

(17)

Circadian output

Some mechanisms controlling indirect responses and endogenous markers used in pharmacodynamic modeling can exhibit circadian variations in their rate of removal from the system [1]. Then $k_{out}$ varies with time $k_{out}$ = $k_{out} \left( t \right).$ Unlike the case of systems with $k_{in} \left( t \right)$, however, the pattern of the circadian response is not expected to follow a biorhythmic pattern similar to $k_{out} \left( t \right).$ In the case of a circadian removal mechanism, the indirect response model in Fig. 1 is given by

$$ \frac{{dR_{b} \left( t \right)}}{dt} = k_{in} - k_{out} \left( t \right) \cdot R_{b} \left( t \right) $$

(18)

where $k_{out} \left( t \right)$ is a circadian function of time and $k_{in}$ a zero-order constant. Equation (18) can be re-arranged to express $k_{out} \left( t \right)$ as

$$ k_{out} \left( t \right) = \frac{{k_{in} - \frac{{dR_{b} }}{dt}\left( t \right)}}{{R_{b} \left( t \right)}} $$

(19)

Substituting Eq. (13) into (19),

$$ k_{out} \left( t \right) = \frac{{k_{in} - \frac{d}{dt} \left( R_{m} + R_{a} \cos \left( {\frac{2\pi }{T}(t - t_{p} )} \right) \right)}}{{R_{m} + R_{a} \cos \left( {\frac{2\pi }{T}(t - t_{p} )} \right)}} $$

(20)

Taking the derivative of $\frac{{dR_{b} }}{dt}\left( t \right)$ in Eq. (20),

$$ k_{out} \left( t \right) = \frac{{k_{in} + \frac{2\pi }{T}R_{a} \sin \left( {\frac{2\pi }{T}\left( {t - t_{p} } \right)} \right)}}{{R_{m} + R_{a} \cos \left( {\left. {\frac{2\pi }{T}(t - t_{p} } \right)} \right)}} $$

(21)