Intermittent Adaptation: A Mathematical Model of Drug Tolerance, Dependence and Addiction

Abstract

A model of drug tolerance, dependence and addiction is presented. The model is essentially much more complex than the commonly used model of homeostasis, which is demonstrated to fail in describing tolerance development to repeated drug administrations. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a process of intermittently developing adaptation. The oral detection and analysis of endogenous substances is proposed to be the primary stimulus triggering the adaptation process. Anticipation and environmental cues are considered secondary stimuli, becoming primary only in dependence and addiction or when the drug administration bypasses the natural—oral—route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking to be autonomously functioning adaptation processes. Simulations with the mathematical model demonstrate the model’s behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence.

Appendix

The model is a non-linear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus—the drug intake—and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process; adaptation to the effect of a drug and adaptation to the interval between drug taking are considered autonomous tolerance processes.

The mathematical model is derived in detail in Peper (2004b). In the present appendix the equations are summarised. A block diagram of the model is shown in Fig. 2.25. For the sake of brevity, the index ‘(t)’ in time signals is omitted.

Fig. 2.25

Block diagram of the mathematical model

1.1 A.1 The Digestive Tract

The digestive system plays no role in the regulation loop. Drug transport through the digestive tract is modeled as a first order function:

$$ S_{\mathrm{digest}} = \int_{0}^{t} \mathit{drug}\,dt- \frac{1}{T_{\mathrm{digest}}}\int_{0}^{t}S_{\mathrm{digest}}\,dt$$

(1)

The input to the block is the drug administration, drug. The input signal is integrated to obtain the drug level when it enters the bloodstream, the output of the block S _digest. A fraction 1/T _digest of the output signal is subtracted from the input to account for the distribution of the drug in the digestive tract. T _digest is the time constant of this process.

1.2 A.2 The Bloodstream

After digestion, the drug enters the bloodstream where it is dispersed. In the present configuration of the model, the drug and the substance produced by the process are assumed to be identical in composition and consequently add in the bloodstream. The amount of the total substance in the bloodstream will be reduced by the body’s metabolism. The processes are modeled by a first order function:

$$ S_{\mathrm{blood}} = \int_{0}^{t} (S_{\mathrm{process}} + S_{\mathrm{digest}})\,dt -\frac{1}{T_{\mathrm{blood}}}\int_{0}^{t} S_{\mathrm{blood}}\, dt$$

(2)

The input signals—the drug as it moves from the digestive tract into the bloodstream, S _digest, and the substance produced by the process, S _process—are added and integrated, yielding the output of the block, the blood drug level S _blood. To account for the body’s metabolism, a fraction 1/T _blood of the output signal is subtracted from the input.

1.3 A.3 The Adaptive Regulator

The input signals of the adaptive regulator are the drug administration and the sensor signal, processed by the loop control block. The sensor signal provides the information about the drug effect. The output of the adaptive regulator counteracts the disturbance by lowering the process output during the drug’s presence. The adaptive regulator comprises a fast and a slow regulator. The fast regulator consists of the blocks “drug regulator”, “interval regulator” and “model estimation”. The slow regulator suppresses the slow changes in the input signal, its output being the average of the input signal. As the fast regulator reacts to fast changes only, the output of the slow regulator is subtracted from its input. It is assumed that the body more or less separately develops tolerance to the drug’s presence and to the intervals between drug administrations. The fast regulator therefore consists of a regulator which provides the adaptation to the drug’s direct effect and a regulator which provides adaptation to the interval between drug taking. The output of the complete adaptive regulator is a combination of signals from its individual components.

The model assumes the body to anticipate the effect of a drug to which it has developed tolerance. This implies that the body has made an estimate of what is going to happen when the drug is administered: it has a model of it. The organism has also made an estimate of the magnitude of the drug effect at the given state of tolerance development. These two entities are the main factors determining the functioning of the fast regulator: the level of tolerance development and the course of the drug effect.

1.3.1 A.3.1 The Fast Regulator

The fast regulator consists of the blocks “drug regulator”, “interval regulator” and “model estimation” (Fig. 2.26). The input signal of the drug regulator S _d is multiplied by M _drug, which represents the course of the drug level in the input signal over time. This signal is integrated (1/s) with a time constant T _drug, yielding its average. The resulting value is a slowly rising signal, L _drug. Multiplying L _drug by M _drug yields the output signal S _drug.

Fig. 2.26

Block diagram of the adaptive regulator

Because of the slow response of the circuit, changes in the input magnitude will be followed only slowly by the output. The speed of change of the output magnitude—representing the slow development of tolerance by the organism—depends on the frequency of occurrence of the drug signal and the amplification of the feedback loop: 1/T _drug. The relation between the signals is

$$ S_{\mathrm{drug}} = M_{\mathrm{drug}}\cdot\frac{1}{T_{\mathrm{drug}}}\int_{0}^{t} (S_{d} -S_{\mathrm{drug}}) \cdot M_{\mathrm{drug}}\,dt$$

(3)

and

$$ S_{\mathrm{drug}} = L_{\mathrm{drug}} \cdot M_{\mathrm{drug}}$$

(4)

The input to the interval regulator is obtained when the output signal of the drug regulator—S _drug—is subtracted from its top value L _drug. The model of the interval is M _int.

The relation between the signals in the fast regulator describing the drug’s presence is then

$$ \begin{aligned}[b]S_{\mathrm{drug}} &= M_{\mathrm{drug}}\cdot\frac{1}{T_{\mathrm{drug}}}\int_{0}^{t} (S_{d} -S_{\mathrm{drug}}) \cdot M_{\mathrm{drug}}\,dt\\&\quad {}- M_{\mathrm{drug}}\cdot\frac{1}{T_{\mathrm{decline}}}\int_{0}^{t}\frac{S_{\mathrm{drug}}}{M_{\mathrm{drug}}} \,dt\end{aligned}$$

(5)

and

$$ S_{\mathrm{drug}} = L_{\mathrm{drug}} \cdot M_{\mathrm{drug}}$$

(6)

Similarly, the equation describing the interval regulator is

$$ \begin{aligned}[b]S_{\mathrm{int}} &= M_{\mathrm{int}} \cdot\frac{1}{T_{\mathrm{int}}}\int_{0}^{t} (L_{\mathrm{drug}} -S_{\mathrm{drug}} - S_{\mathrm{int}}) \cdot M_{\mathrm{int}}\,dt\\&\quad {} - M_{\mathrm{int}}\cdot\frac{1}{T_{\mathrm{decline}}}\int_{0}^{t} \frac{S_{\mathrm{int}}}{M_{\mathrm{int}}}\,dt\end{aligned}$$

(7)

and

$$ S_{\mathrm{int}} = L_{\mathrm{int}} \cdot M_{\mathrm{int}}$$

(8)

The output of the interval regulator is S _int. The output signal of the total fast regulator is obtained by subtracting the interval signal from the top level of the drug signal:

$$ S_{\mathrm{out}} = L_{\mathrm{drug}} - S_{\mathrm{int}}$$

(9)

Figure 2.27 shows the implementation of the fast regulator in the mathematical simulation program Simulink (see Peper 2004b).

Fig. 2.27

Fast regulator implemented in Simulink

1.3.2 A.3.2 Estimation of the Drug Effect in the Adaptive Regulator

As the duration of the drug administration is relatively short in most cases, it may be represented by a short pulse. The model of the course of the drug concentration when it enters the bloodstream—M _drug—is then computed by calculating the effect of a pulse with a magnitude of 1 on the digestive tract’s transfer function. The input of the interval is acquired when the signal “drug” is subtracted from its top value of 1. Multiplying this signal by the transfer of the digestive tract yields the model of the interval M _int:

$$ M_{\mathrm{drug}} = \int_{0}^{t} \mathit{drug} \,dt- \frac{1}{T_{\mathrm{digest}}}\int_{0}^{t}M_{\mathrm{drug}} \,dt$$

(10)

and

$$ M_{\mathrm{int}} = \int_{0}^{t} (1 - \mathit{drug})\,dt- \frac{1}{T_{\mathrm{digest}}}\int_{0}^{t}M_{\mathrm{int}} \,dt$$

(11)

T _digest is the time constant of the digestive system.

1.3.3 A.3.3 The Slow Regulator

The slow regulator models the long term adaptation to the drug effect. In the tolerant state, the slow adaptation causes the magnitude of the negative reaction after the drug effect to depend on the interval between drug administrations: an infrequently taken drug has a small effect during the interval, while a frequently taken drug causes a large rebound. The slow regulator counteracts the disturbance by lowering the level of the process by the average of the drug effect. Its input signal—the sensor signal, processed by the loop control block—provides the information about the drug effect. The average of the input signal is obtained by a low pass filter with a time constant T _slow:

$$ S_{\mathrm{slow}} = \int_{0}^{t} S_{\mathrm{contr}} \,dt- \frac{1}{T_{\mathrm{slow}}}\int_{0}^{t}S_{\mathrm{slow}} \,dt$$

(12)

1.4 A.4 The Process

The model does not incorporate the characteristics of the process and the process regulator. In a specific model of drug tolerance where the process is included, the effect of the process transfer on loop stability has to be controlled by the loop control block.

1.5 A.5 Loop Control

A loop control is an essential element in any regulated system. It incorporates the open loop amplification, which determines the accuracy of the regulation, and it provides the necessary conditions for stable operation of the negative feedback system. For stable operation, the regulation loop has to contain compensation for the effect of superfluous time constants: their effect on the signals in the loop has to be counteracted by circuits with an inverse effect. In the present form of the model, only the effect of the bloodstream on the regulation loop is counteracted as the transfer of the process and its regulator and the transfer function of the sensor are set at unity. The relation between the input and the output of the loop control is

$$ S_{\mathrm{sens}} = \int_{0}^{t} S_{\mathrm{contr}} \,dt - \frac{1}{T_{\mathrm{blood}}}\int_{0}^{t}S_{\mathrm{sens}} \,dt$$

(13)

1.6 A.6 The Sensor

The sensor transforms the chemical signal S _blood—the blood drug level—into the signal S _sense. In the present model, this transformation is assumed to be linear and is set at 1. In specific models of physiological processes, this complex mechanism can be described more accurately. Stable operation then requires that the effect of its transfer on loop stability is controlled by the loop control block.