Solution and implementation of distributed lifespan models

Abstract

We consider a population where every individual has a unique lifespan. After expiring of its lifespan the individual has to leave the population. A realistic approach to describe these lifespans is by a continuous distribution. Such distributed lifespan models (DLSMs) were introduced earlier in the indirect response context and consist of the mathematical convolution operator to describe the rate of change. Therefore, DLSMs could not directly be implemented in standard PKPD software. In this work we present the solution representation of DLSMs with and without a precursor population and an implementation strategy for DLSMs in ADAPT , NONMEM and MATLAB . We fit hemoglobin measurements from literature and investigate computational properties.

Appendices

Appendices

Appendix 1

To estimate the amount of individuals at the initial population state N(0) for the DLSM (5), (6) we define the production term

$$ \widetilde{k_{in}}(t) =\left\{ \begin{array}{ll} k_{in}(t) & t \leq 0, \\ 0 & t > 0. \end{array}\right. $$

(24)

Then the appropriate initial value N(0) is determined by the claim $\lim \nolimits_{t \rightarrow \infty} N(t) = 0.$ Substituting (24) into (5) results in

$$ \frac{d}{dt} N(t) = \widetilde{k_{in}}(t) - \int \limits_0^\infty \widetilde{k_{in}}(t-x) l_N(x) dx . $$

(25)

The general solution of (25) reads

$$ \begin{aligned} N(t) &= N(0) + \int \limits_0^t \widetilde{k_{in}}(s)ds - \int \limits_0^t \int \limits_0^\infty \widetilde{k_{in}}(s-x) l_N(x) dx ds \\ &= N(0) - \int \limits_0^\infty l_N(x) \int \limits_0^t \widetilde{k_{in}}(s-x) ds dx. \end{aligned} $$

Then

$$ \begin{aligned} \lim \limits_{t \rightarrow \infty} N(t) &= 0 \\ \Leftrightarrow N(0) &= \int \limits_0^\infty l_N(x) \int \limits_0^\infty \widetilde{k_{in}}(s-x) ds dx. \end{aligned} $$

Using (24)

$$ \begin{aligned} N(0) &= \int \limits_0^\infty l_N(x) \int \limits_0^x k_{in}(s-x) ds dx \\ &= \int \limits_0^\infty l_N(x) \int \limits_{-x}^0 k_{in}(s) ds dx . \end{aligned} $$

Appendix 2

We consider a constant past k _in (t) = $k_{in}^{0}$ for t ≤ 0 and obtain for the initial values

$$ \begin{aligned} N(0) &= \int \limits_0^\infty l_N(x) \int \limits_{-x}^0 k_{in}^0 ds dx \\ &= k_{in}^0 \int \limits_0^\infty l_N(x) x dx = k_{in}^0 T_N, \end{aligned} $$

and

$$ \begin{aligned} M(0) &= \int \limits_0^\infty l_M(y) \int \limits_{-y}^0 \left(k_{in}^0*l_N\right)(s) ds dy \\ &= \int \limits_0^\infty l_M(y) \int \limits_{-y}^0 k_{in}^0 \int \limits_0^\infty l_N(x) dx ds dy \\ &= \int \limits_0^\infty l_M(y) \int \limits_{-y}^0 k_{in}^0 ds dy = k^0_{in} T_M. \end{aligned} $$

Appendix 3

To estimate the amount M(0) of the follow-up population for the DPLSM (7), (8) a similar approach as in Appendix 1 is applied. We define the production

$$ \widetilde{k_{in}}(t) =\left\{ \begin{array}{ll} \left(k_{in}*l \right)(t) & t \leq 0, \\ 0 & t > 0, \end{array}\right. $$

(26)

and use the condition $\lim \nolimits_{t \rightarrow \infty} M(t) = 0$ to determine M(0). Substituting (26) into (7) results in

$$ \begin{aligned} M(t) &= M(0) + \int \limits_0^t \widetilde{k_{in}}(s) ds - \int \limits_0^t \int \limits_0^\infty \widetilde{k_{in}}(s-y)l_M(y) dy ds \\ &= M(0) - \int \limits_0^\infty l_M(y) \int \limits_0^t \widetilde{k_{in}}(s-y) ds dy, \end{aligned} $$

and therefore,

$$ \begin{aligned} M(0) & = \int \limits_0^\infty l_M(y) \int \limits_0^\infty \widetilde{k_{in}}(s-y) ds dy \\ &=\int \limits_0^\infty l_M(y) \int \limits_{-y}^0 \left(k_{in}*l_N\right)(s) ds dy. \end{aligned} $$

(27)

Appendix 4

Differentiation of (10) with respect to time t gives

$$ \begin{aligned} \frac{d}{dt}N(t) &= \int \limits_0^\infty \left(1-L_N(s)\right) k'_{in}(t-s) ds \\ &= \left[ -\left(1-L_N(s)\right) k_{in}(t-s) \right]_{s=0}^{s=\infty} \\ &\quad - \int \limits_0^\infty \left(-l_N(s)\right) \left(-k_{in}(t-s)\right) ds \\ &= k_{in}(t) - \int \limits_0^\infty l_N(s) k_{in}(t-s) ds \\ &= k_{in}(t)-\left(k_{in}*l_N\right)(t). \end{aligned} $$

With (10) we obtain for the initial value

$$ \begin{aligned} N(0) &= \int \limits_0^\infty \left(1-L_N(s)\right) k_{in}(-s) ds \\ &= \left[ \left( 1-L_N(s) \right) \int \limits_0^s k_{in}(-x) dx \right]_{s=0}^{s=\infty} \\ & \quad - \int \limits_0^\infty \left(-l_N(s)\right) \int \limits_0^s k_{in}(-x) dx ds \\ &= \int \limits_0^\infty l_N(s) \int \limits_{-s}^0 k_{in}(x) dx ds. \end{aligned} $$

Appendix 5

Generally, for an arbitrary continuous function f, the trapezoidal rule reads

$$ \int \limits_{\tau_0}^{\tau_{end}} f(s) ds \approx \frac{h}{2} \sum \limits_{i=1}^m \left(f\left(s_{i+1}\right)+ f\left(s_i\right)\right), $$

(28)

with the equidistant partition s _j  = τ ₀ + (j − 1)h with $h = \frac{\tau_{end}-\tau_0}{m}$ for j = 1,…,m + 1 of the lifespan interval [τ ₀, τ _end ].

Appendix 6

Differentiation of (13) with respect to time t gives

$$ \begin{aligned} \frac{d}{dt} M(t) &= \int \limits_0^\infty \left(1-L_M(x)\right) \int \limits_0^\infty k_{in}^{'}(t-x-s) l_N(s) ds dx \\ &= \int \limits_0^\infty l_N(s) \int \limits_0^\infty \left(1-L_M(x) \right) k_{in}^{'}(t-x-s) dx ds. \end{aligned} $$

(29)

The inner integral in (29) could be written as

$$ \begin{aligned} &\int \limits_0^{\infty} \left(1-L_M(x)\right){k}_{in}^{'}(t-x-s) dx \\ &= \left[-\left(1-L_M(x)\right)k_{in}(t-x-s)\right]_{x=0}^{x=\infty} \\ & \quad - \int\limits_0^{\infty} k_{in}(t-x-s) l_M(x) dx \end{aligned} $$

(30)

$$ = k_{in}(t-s) - \int \limits_0^\infty k_{in}(t-x-s) l_M(x) dx. $$

(31)

Substitution of (31) into (29) gives

$$ \begin{aligned} \frac{d}{dt} M(t) &= \int \limits_0^\infty l_N(s) k_{in}(t-s) ds \\ & \quad - \int \limits_0^\infty l_N(s) \int \limits_0^\infty k_{in}(t-x-s) l_M(x) dx ds \\ &= \left(k_{in} * l_N \right)(t) - \int \limits_0^\infty l_N(s) \left(\left(k_{in}*l_M\right)(t-s)\right) ds \\ &= \left(k_{in} * l_N \right)(t) - \left( k_{in}*l_N*l_M \right) (t). \end{aligned} $$

For the initial value we obtain

$$ \begin{aligned} M(0) &= \int \limits_0^\infty \left(1-L_M(x)\right) \int \limits_0^\infty k_{in}(-x-s) l_N(s) ds dx \\ &= \left[ \left(1-L_M(x)\right) \int \limits_0^x \int \limits_0^\infty k_{in}(-y-s) l_N(s) ds dy \right]_{x=0}^{x=\infty}\\ &\quad + \int \limits_0^\infty l_M(x) \int \limits_0^x \int \limits_0^\infty k_{in}(-y-s) l_N(s) ds dy dx \\ &= \int \limits_0^\infty l_M(x) \int \limits_0^x\left( k_{in}*l_N \right) (-y) dy dx \\ &= \int \limits_0^\infty l_M(x) \int \limits_{-x}^0 \left( k_{in}*l_N\right)(y) dy dx. \end{aligned} $$

Appendix 7

In the following we present the implementation of the trapezoidal rule for (10). Only additional functions and modified ADAPT subroutines are presented. For the complete code see the supplemental material.

Logarithm of the gamma function $\Gamma(x)$ based on [14]: CDF L(x) of the WBD: Multiple dosing function for c(t): Production k _in (t): Integrand in (10): Trapezoidal rule: The ADAPT subroutine OUTPUT calls the trapezoidal rule for the different dosing groups.