Intermittent Adaptation

Abstract

A mathematical model of drug tolerance and its underlying theory is presented. The model is essentially more complex than the generally 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 regulated adaptive process. The oral detection and analysis of endogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model 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 autonomous tolerance 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. Simulations of different ways withdrawal can be accomplished, demonstrates the practical applicability of the model.

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Appendix

In a previous paper, the mathematical implementation of the model and the derivation of the formulae describing its components is extensively discussed [37]. In this appendix, a summary is given of the formulae. For the sake of brevity, the index ‘(t)’ in time signals is omitted. [Fig. A1] shows a block diagram of the mathematical model.

Fig. A1 Block diagram of the mathematical model.

1. The digestive tract

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

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 spread in drug distr ibution in the diges tive tract. T_digest is the time constant of this process.

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 modelled with a first order function:

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.

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 adaptive regulator comprises a fast and a slow regulator.

3a. The fast regulator

[Fig. A2] shows a block diagram of the fast regulator. The fast regulator consists of the blocks ‘drug regulator’, ‘interval regulator’ and ‘model estimation’. [Fig. A3] shows the implementation of the fast regulator in the mathematical simulation program Simulink. 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.

The relation between the signals is:

Fig. A2 Block diagram of the adaptive regulator.

Fig. A3 The fast regulator implemented in Simulink.

And

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:

and

Similarly, the equation describing the interval regulator is:

and

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:

3b. Estimation of the drug effect in the adaptive regulator

The model of the course of the drug concentration when it enters the bloodstream – M_drug – is 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: 1. Multiplying this signal with the transfer of the digestive tract yields the model of the interval M_int:

And

T_digest is the time constant of the digestive system.

3c. The slow regulator

The slow regulator counteracts the disturbance by lowering the level of the process with the average of the drug effect. This is obtained by a low pass filter with a time constant T_slow:

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 “The loop control” block.

5. Loop control

A loop control provides the necessary conditions for stable operation of the negative feedback system. In the present form of the model, the effect of the bloodstream on the regulation loop is counteracted. The relation between the input and the output of the loop control is:

6. The sensor

The sensor transforms the chemical signal S_blood – the blood drug level – into the signal S_sense. This transformation is in the present model 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 “The loop control” block.

Disclosure

The author declares that there are no financial interests to disclose.

Correspondence