A distributed delay approach for modeling delayed outcomes in pharmacokinetics and pharmacodynamics studies

Abstract

A distributed delay approach was proposed in this paper to model delayed outcomes in pharmacokinetics and pharmacodynamics studies. This approach was shown to be general enough to incorporate a wide array of pharmacokinetic and pharmacodynamic models as special cases including transit compartment models, effect compartment models, typical absorption models (either zero-order or first-order absorption), and a number of atypical (or irregular) absorption models (e.g., parallel first-order, mixed first-order and zero-order, inverse Gaussian, and Weibull absorption models). Real-life examples were given to demonstrate how to implement distributed delays in Phoenix^® NLME™ 8.0, and to numerically show the advantages of the distributed delay approach over the traditional methods.

Appendices

Appendix 1: Relationship between the discrete delay and the gamma distributed delay

Let ${\mathcal {T}}$ be a random variable that is gamma distributed with rate parameter $\kappa$ and shape parameter $\nu$. Then its mean $\mu$ and variance $\sigma ^{2}$ are respectively given by

$$\mu = \frac{\nu }{\kappa }, \quad \sigma ^{2} = \frac{\mu ^{2}}{\nu }.$$

(56)

If ${\mathcal {T}}$ is viewed as a delay time, then $\mu$ represents the mean delay time, and $\sigma ^{2}$ gives a measure of the degree of concentration of the delay about its mean.

Equation (56) implies that, for the given mean delay time $\mu$, $\sigma ^{2} \rightarrow 0$ as the shape parameter $\nu \rightarrow +\infty$. Thus, the discrete delay can be recovered as a limit of gamma distributed delays (see Fig. 5 for a visualization of this).

Fig. 5

Probability density and cumulative distribution functions of gamma distributions with different values for shape parameter $\nu$ and same value for the mean ($\mu = 10$)

Appendix 2: Using the linear chain trick to reduce (3) to a system of ODEs in the case where g is the PDF of an Erlang distribution

Since the linear chain trick is an elegant method and is rarely known in the PKPD literature, we demonstrate how one can use this approach to reduce (3) to a system of ODEs in the case where g is the PDF of an Erlang distribution given by

$$g(t;\kappa ,n) = \frac{\kappa ^{n}t^{n-1}}{(n- 1)!}\exp (-\kappa t).$$

(57)

Here $\kappa$ is the rate parameter (a positive number), $n$ denotes the shape parameter (a positive integer).

By using the transformation $s = t-\tau$, we see that (3) can be equivalently written as

$${\mathcal{S}}(t) = \int _{-\infty }^{t}g(t-s;\kappa ,n)S(s)ds.$$

(58)

Hence, to show that (3) can be reduced to a system of ODEs, it is equivalent to show that (58) can be reduced to a system of ODEs. To do that, we let

$$x_{j}(t)= \int _{-\infty }^{t}g(t-s;\kappa ,j)S(s)ds, \quad j=1,2,\ldots ,n;$$

that is,

$$x_{j}(t) = \int _{-\infty }^{t}\frac{\kappa ^{j}(t-s)^{j-1}}{(j - 1)!}\exp (-\kappa (t-s))S(s)ds$$

(59)

for $j=1,2,\ldots ,n$, and

$${\mathcal{S}}(t) =x_{n}(t) = \int _{-\infty }^{t}\frac{\kappa ^{n}(t-s)^{n-1}}{(n- 1)!}\exp (-\kappa (t-s))S(s)ds.$$

(60)

Equation (59) implies that

$$x_{1}(t) = \int _{-\infty }^{t}\kappa \exp (-\kappa (t-s))S(s)ds.$$

Differentiating both sides of the above equation with respect to t yields that

$$\begin{aligned} \dot{x}_{1}(t) &= \kappa S(t) - \kappa \int _{-\infty }^{t}\kappa \exp (-\kappa (t-s))S(s)ds \\ &= \kappa S(t) - \kappa x_{1}(t). \end{aligned}$$

(61)

For any $j\ge 2$, differentiating both sides of (59) with respect to t gives that

$$\begin{aligned} \dot{x}_{j}(t) &= \int _{-\infty }^{t}\frac{\kappa ^{j}(t-s)^{j-2}}{(j-2)!}\exp (-\kappa (t-s))S(s)ds \\ & - \kappa \int _{-\infty }^{t}\frac{\kappa ^{j}(t-s)^{j-1}}{(j - 1)!}\exp (-\kappa (t-s))S(s)ds \\ &= \kappa x_{j-1}(t) - \kappa x_{j}(t). \end{aligned}$$

(62)

By (59), (61) and (62), we see that with an Erlang distributed delay having rate parameter $\kappa$ and shape parameter $n$, (58) reduces to the following system of ODEs

$$\begin{aligned} \dot{x}_{1}(t) &= \kappa S(t) - \kappa x_{1}(t), \\ \dot{x}_{i}(t) &= \kappa x_{i-1}(t) - \kappa x_{i}(t), \quad i=2,3,\ldots , n, \\ x_{i}(0) &= \int _{-\infty }^{0}\frac{\kappa ^{i}(-s)^{i-1}}{(i-1)!}\exp (\kappa s)S_{0}(s)ds, \quad i=1,2,\ldots ,n, \end{aligned}$$

with ${\mathcal{S}} = x_{n}$, where $S_{0}$ denotes the history function (that is, $S(t) = S_{0}(t)$ for $t\le 0$).

Appendix 3: Dynamic behavior of the logistic growth model with a gamma distributed delay

Notice that the logistic growth model with a gamma distributed delay (42) is extended from the following well-known logistic growth model

$$\begin{aligned} \dot{S}(t) & = rS(t)\left( 1 - \frac{S(t)}{K} \right) , \\ S(0) & = S_{0}, \end{aligned}$$

(63)

where r denotes the intrinsic growth rate, K is the carrying capacity, and $S_{0}$ is a positive constant. As demonstrated in Fig. 6, the solution to (63) has a sigmoidal shape and is asymptotic to the carrying capacity K. To have some idea of the effect of the delay on the dynamics, the solutions to (42) obtained with the same shape parameter value (ShapeParam = 2.5) but different mean delay times are plotted against time t, and are shown in Fig. 6. This figure indicates that introducing a delay in (63) produces an oscillation around the carrying capacity. Specifically, when the mean delay time is two, the oscillation is damped in time with the solution eventually approaching to the carrying capacity. As the mean delay time increases to four, the oscillation is sustained. This example clearly shows that DDEs can exhibit richer dynamics than their corresponding ODEs. In addition, one can adjust the mean delay time to achieve desired type of oscillations.

Fig. 6

Plot for the solution to (63) against time t (depicted by the dotted line with legend “No delay”), and the ones for the solutions to (42) obtained with the same shape parameter value (ShapeParam = 2.5) but different mean delay times, where $r = 0.8$, $K = 1$ and $S_{0} = 0.5$

Appendix 4: PML source code for example 1

Appendix 5: PML source code for example 2

Appendix 6: Using the linear chain trick to reduce time-varying distributed delay (51) to a system of ODEs in the case where g is given by (53)

Here we demonstrate how to use the linear chain trick to reduce (51) to a system of ODEs in the case where g is given by (53). By using the transformation $s = t-\tau$, we see that (51) can be equivalently written as

$${\mathcal{S}}(t) = \int _{-\infty }^{t}g(s, t-s; n)S(s)ds.$$

(64)

Hence, to show that (51) can be reduced to a system of ODEs, it is equivalent to show that (64) can be reduced to a system of ODEs. To do that, we let

$$z_{j}(t)= \frac{1}{\psi (t)}\int _{-\infty }^{t}g(s, t-s; j)S(s)ds, \quad j=1,2,\ldots ,n;$$

that is,

$$z_{j}(t) = \frac{1}{(j - 1)!}\int _{-\infty }^{t}\left( \int _{s}^{t}\psi (\xi )d\xi \right) ^{j - 1} \exp \left( -\int _{s}^{t}\psi (\xi )d\xi \right) S(s)ds$$

(65)

for $j=1,2,\ldots ,n$, and

$${\mathcal{S}}(t) = \psi (t) z_{n}(t).$$

(66)

Equation (65) implies that

$$z_{1}(t) = \int _{-\infty }^{t}\exp \left( -\int _{s}^{t}\psi (\xi )d\xi \right) S(s)ds.$$

Differentiating both sides of the above equation with respect to t yields that

$$\begin{aligned} \dot{z}_{1}(t) &= S(t) - \psi (t)\int _{-\infty }^{t}\exp \left( -\int _{s}^{t}\psi (\xi )d\xi \right) S(s)ds \\ &= S(t) - \psi (t) z_{1}(t). \end{aligned}$$

(67)

For any $j\ge 2$, differentiating both sides of (65) with respect to t gives that

$$\begin{aligned} \dot{z}_{j}(t) &= \frac{\psi (t)}{(j-2)!}\int _{-\infty }^{t}\left( \int _{s}^{t}\psi (\xi )d\xi \right) ^{j - 2} \exp \left( -\int _{s}^{t}\psi (\xi )d\xi \right) S(s)ds \\ & -\frac{\psi (t)}{(j - 1)!}\int _{-\infty }^{t}\left( \int _{s}^{t}\psi (\xi )d\xi \right) ^{j - 1} \exp \left( -\int _{s}^{t}\psi (\xi )d\xi \right) S(s)ds \\ &= \psi (t) z_{j-1}(t) - \psi (t) z_{j}(t). \end{aligned}$$

By the above Eqs. (65) and (67), Eq. (64) reduces to the following system of ODEs

$$\begin{aligned} \dot{z}_{1}(t) &= S(t) - \psi (t) z_{1}(t), \\ \dot{z}_{i}(t) &= \psi (t) (z_{i-1}(t) - z_{i}(t)), \quad i=2,3,\ldots , n, \\ z_{i}(0) &=z_{i}^{0}, \quad i=1,2,\ldots ,n, \end{aligned}$$

with ${\mathcal{S}}(t) = \psi (t) z_{n}(t)$, and $z_{i}^{0}$ given by

$$z_{i}^{0} = \tfrac{1}{(i-1)!}\int _{-\infty }^{0} \left( \int _{s}^{0}\psi (\xi )d\xi \right) ^{i - 1} \exp \left( -\int _{s}^{0}\psi (\xi )d\xi \right) S_{0}(s)ds$$

for $i=1,2,\ldots , n$, where $S_{0}$ denotes the history function (that is, $S(t) = S_{0}(t)$ for $t\le 0$).