Delayed logistic indirect response models: realization of oscillating behavior

Abstract

Indirect response (IDR) models are probably the most frequently applied tools relating the effect of a signal to a baseline response. A response modeled by such a classical IDR model will always return monotonously to its baseline after drug administration. We extend IDR models with a delay process, i.e. a retarded response state, that leads to oscillating response behavior. First, IDR models with a first-order production and second-order loss term based on the famous logistic equation are constructed. Second, a delay process similar to the delayed logistic equation is included. Relations of the classical IDR model with our extended IDR model concerning response and model parameters are revealed. Simulations of typical response profiles are presented and data fitting of a model for leptin and cholesterol dynamics after administration of methylprednisolone is performed. The influence of the delay parameter on the other model parameters is discussed.

Appendices

Appendix 1: Computation of the PD effect $PD_{eff}$

We have introduced the classical IDR model and the logistic equation based LoIDR and DLoIDR models. Depending on the action of the drug we have four different realizations $f=f_i$, $i=I,II,III,IV$ for each model. We now show the proposed PD effect $PD_{eff}$ for these models only when the drug inhibits the inflow term, i.e., $f=f_{I}$. A slightly modified proof also works for the three other realizations. In case of $f=f_{I}$ we have to calculate

$$\begin{aligned} PD_{eff}&= \int _0^{\infty } R_{Base} -R(t) \, dt \end{aligned}$$

with $R_{Base}=R^0$ for the classical IDR model and $R_{Base}=R^0_L$ in case of the LoIDR and DLoIDR model.

• (i) IDR model PD effect:

Using Eq. (11) we obtain

$$\begin{aligned} PD_{eff}= \, & {} \frac{1}{k_{out}} \int _0^{\infty } k_{in} -k_{out} R(t) \, dt \\= \, & {} \frac{1}{k_{out}} \int _0^{\infty } \frac{d}{dt} R(t) + k_{in} e(c(t)) \,dt \end{aligned}$$

(18)

$$\begin{aligned}= \, & {} \frac{1}{k_{out}} \int _0^{\infty } \frac{d}{dt} R(t) \, dt \; + \int _0^{\infty } R^0 e(c(t)) \, dt . \end{aligned}$$

(19)

Due to $R^0 = R(0) = \lim _{t \rightarrow \infty } R(t)$ the left integral on the right hand side of Eq. (19) vanishes and we end up with

$$\begin{aligned} PD_{eff}&= \int _0^{\infty } R^0 e(c(t)) \, dt . \end{aligned}$$

(20)

• (ii) LoIDR model PD effect:

With the model Eq. (14) we can calculate

$$\begin{aligned} PD_{eff}= \, & {} \int _0^{\infty } R^0_L - R(t) \, dt \\= \, & {} \int _0^{\infty } \frac{1}{k_{outL} R(t)} \frac{d}{dt} R(t)+ R^0_L e(c(t))\,dt \end{aligned}$$

(21)

$$\begin{aligned}= \, & {} \int _0^{\infty } \frac{1}{k_{outL}} \frac{d}{dt} ({\text{ ln }}(R(t))) \, dt \\&+ \int _0^{\infty } R^0_L e(c(t)) \, dt \\= \, & {} \frac{1}{k_{outL}} \lim _{b \rightarrow \infty } \left[ {\text{ ln }}(R(t)) \right] _{t=0}^{t=b} \\&+ \int _0^{\infty } R^0_L e(c(t)) \, dt \, . \end{aligned}$$

(22)

Again, using $R^0_L = R(0) = \lim _{t \rightarrow \infty } R(t)$ the first term on the right hand side of Eq. (22) vanishes and we end up with the proposed result Eq. (20).

• (iii) DLoIDR model PD effect:

In presence of a delay we use the initial condition $R(s) =R^0_L$, $-T \le s \le 0$ and rewrite the PD effect term to

$$\begin{aligned} PD_{eff}= \, & {} \int _0^{\infty } R^0_L -R(t) \, dt \\= \, & {} \int _T^{\infty } R^0_L -R(t-T) \, dt \\= \, & {} \int _0^{\infty } R^0_L -R(t-T) \, dt \, . \end{aligned}$$

(23)

We use the representation Eq. (23) of the PD effect as well as Eq. (16) and compute

$$\begin{aligned} PD_{eff}= \, & {} \int _0^{\infty } R^0_L - R(t-T) \, dt \\= \, & {} \int _0^{\infty } \frac{1}{k_{outL} R(t)} \frac{d}{dt} R(t)+ R^0_L e(c(t)) \, dt \, . \end{aligned}$$

(24)

The Eq. (24) and Eq. (21) are identical and we can mimic the computation of part (ii) and obtain the final result Eq. (20).

Appendix 2: Stability of the equilibria $R^0$ and $R^0_L$

The base equations ($c=0$) to the proposed models read

$$\begin{aligned} \frac{d}{dt} R(t)&= k_{in} -k_{out} R(t) \end{aligned}$$

(25)

for Eqs. (11)–(12),

$$\begin{aligned} \frac{d}{dt} R(t)&=k_{inL} R(t) -k_{outL}R(t) R(t-T) \end{aligned}$$

(26)

for Eqs. (14)–(15) ($T=0$) and Eqs. (16)–(17) ($T>0$).

Linearizing Eqs. (25)–(26) at their baseline $R^0$, $R^0_L$ respectively, leads to

$$\begin{aligned} \frac{d}{dt} h(t)&= -k_ {out} h(t) \end{aligned}$$

(27)

for the IDR model, and

$$\begin{aligned} \frac{d}{dt} h(t)&= - k_{inL} h(t-T) \end{aligned}$$

(28)

for the LoIDR ($T =0$) and the DLoIDR Eqs. ($T >0$). Thus, the equilibria $R^0$, $R^0_L$ are stable for the IDR and the LoIDR model. Moreover, due to Hairer et al. [22] the baseline $R^0_L$ is stable for the DLoIDR model if $0< T < \pi /(2 k_{inL})$. Finally, comparing Eq. (27) with Eq. (28) ($T=0$) shows that the solutions R(t) of the classical IDR and the LoIDR model approximate the equilibria with the same exponential rate, if $R^0 =R^0_L$ and $k_{out} = k_{inL}$ holds.

Appendix 3: Linearized LoIDR model

Linearizing the LoIDR model Eq. (14) of type I, III at its baseline $\bar{R} = R_L^0 = k_{inL}/k_{outL}$, $\bar{c} = 0$ yields

$$\begin{aligned} \frac{d}{dt} h(t)= \, & {} \left( f_{I,III}(\Theta _L, 0, R^0_L) + R^0_L \frac{\partial f_{I,III}}{\partial R}( \Theta _L,0, R^0_L)\right) \\&\cdot (h(t)-R^0_L )\\= \, & {} - R^0_L k_{outL} (h(t)-R^0_L )\\= \, & {} - k_{inL}h(t)+ k_{inL} R^0_L , \end{aligned}$$

which is the IDR model at $\bar{c} =0$ with production rate $k_{in} = k_{inL} R^0_L$ and loss rate $k_{out}= k_{inL}$. Finally, a similar argument holds for type II, IV.