The new nonlocal delayed feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion-reaction equations. The control loop defines the changes of the substrate concentration delivered into the bioreactor at the external boundary of the bioreactor depending on the difference of measurements of the produced drug delivered into the body and the flux of the drug prescribed by a doctor in accordance with the therapeutic protocol. The system of PDEs is solved by using the finite difference method, the control loop parameters are defined from the analysis of stationary linearized equations. The stability of the algorithm for the inverse boundary condition is investigated. Results of computational experiments are presented and analysed.

Mathematical problems of biological systems are attracting a lot of attention from specialists in many fields. In this paper, from mathematical point of view we restrict to models described by non-stationary and non-linear diffusion–reaction equations. The dynamics of their solutions can be very complicated, the interaction of different physical processes can lead to development of spatial and temporal patterns and instabilities (Murray,

The delayed feedback control mechanism is used in many technological applications (Pyragas,

The rest of this paper is organized as follows. In Section

In Section

The finite volume method is used to approximate the diffusion process in Section

In Section

In this paper, we consider a simple model used to simulate dynamics of various bioreactors (Hillen and Painter,

It is interesting also to consider more complicated bio-reaction processes (Hillen and Painter,

In order to define a full mathematical model we formulate initial conditions

Such a combination of boundary conditions is not defining a classical well-posed boundary value problem. In order to use such bioreactors in real life applications, we propose to find the equivalent boundary condition for the substrate function

In general the inverse problems belong to the class of ill-posed problems (Aster

The additional boundary condition (

In this section we use the delayed feedback control loop technology (controllers) to achieve the desired regime of drug production (Kok Kiong

A classical proportional–integral–derivative controller (PID controller) is used in paper (Ivanauskas

Our approach to construct the proportional controller is based on the following ideas. The reformulated initial boundary value problem (

We are interested to control a so-called steady-state error. Thus the asymptotic analysis of stationary (limit) system of equations is done and the nonlinear interaction term is linearized around some constant value of

Using this information a simple definition of the proportional controller algorithm is obtained. In order to follow the dynamics of drug flux prescribed by a doctor, the required supply of the substrate into the bioreactor is defined as:

The smart bioreactors have a possibility to perform the electrochemical monitoring of the enzymatic reaction. Let us assume that we can measure the concentration of the produced drug flux

In many cases it is important also to control the total amount of the drug produced during the bioreaction. This additional objective function can be included into the control algorithm by adding the correction into the definition of function

Let us consider the second case of boundary conditions used in the control system

We consider Taylor series expansion of function

As an example we present errors

In this section we consider the discrete approximation of the problem (

For functions defined on the grid

We approximate the differential problem (

The nonlinear boundary value problem (

Next we consider in more details the boundary condition (

Both boundary conditions with approximations (

The accuracy of the proposed control algorithm depends on the accuracy of approximation (

In this section we present results of some computational experiments. The model constants are selected as in Ivanauskas

The results of computational experiments have shown that the control quality of boundary condition (

First we illustrate the important fact that the drug production process reacts with a fixed delay to the changes of the substrate boundary condition (

Product (

It follows from the presented results that the response of the product flow rate to the stepwise change of the substrate concentration is delayed approximately 0.25 seconds.

Next we give a brief theoretical justification of the obtained result. and restrict to differential case of models. One general technique to analyse the stability for non-stationary differential problems is to apply the eigenvalue criterion for the space depending operators. Thus we solve the eigenvalue problem

The influence of time delay to the stability and efficiency of the control algorithm will be much more important when a convection transport mechanism is included into the mathematical model. Such models will be investigated in a separate paper.

One important recommendation follows, that the treatment procedures defined by doctors should follow smooth changes of the drug concentration.

In this section we consider the accuracy of the proposed delayed feedback control algorithm. We use this algorithm to reconstruct two typical in real-world applications boundary conditions for the substrate

Then the direct problem (

The fluxes of the produced drug for the specified boundary conditions (

Then the feedback control algorithm (

Reconstructed boundary conditions

The main conclusion from these results is that reconstructed boundary conditions are approximating the exact boundary conditions sufficiently accurately. It is also clearly seen that the proposed dynamical control of total amount of produced drugs influences the control procedure.

In general the inverse problems belong to the class of ill-posed problems (Aster

In order to test the sensitivity of the inverse reconstruction procedure with respect to perturbations of the function

Reconstructed drug fluxes

In Fig.

In this section we have investigated the stability of the control algorithm. The standard test is to analyse the reaction of the controlled function

Reconstructed drug flux

In this section the proposed feedback control algorithm is applied for two different treatment protocols, when a short-time treatment process is considered.

The first protocol uses the piecewise linear changes of the drug flow over time, i.e.

Application of the proportional control algorithm: a) piecewise linear treatment protocol, red colour function defines the theoretical treatment function

The second treatment protocol defines the drug flow which changes exponentially over time. In Fig.

Robustness of the proposed feedback control method is investigated experimentally. We tested the accuracy of the proposed control algorithm for a fixed value of the control parameter

In this paper we have proposed a new delayed feedback control algorithm for a mathematical model which describes the drug delivery system. The system simulates the enzyme-containing bioreactor and the prodrug is converted into an active drug during the reaction. The finite volume method is used to approximate the given nonstationary reaction-diffusion equations. It approximates the system of partial differential equations with the second order in space and time.

The proposed delayed feedback control algorithm is based on solution of two inverse boundary condition problems. The stability of this algorithm is investigated for the case of the stationary solution. This analysis enables us to formulate all parameters of the control algorithm. Results of computational experiments show that the proposed control algorithm is accurate and robust.

Two drug treatment protocols, linear stepwise and exponential, are used to investigate the efficiency of the inverse control algorithm. It is proved that the produced drug flows approximate both investigated treatment protocols with high accuracy. Thus the proposed feedback control algorithm can be recommended to be used in medical practices.

Authors would like to thank Prof. J. Janno for fruitful discussions on ill-posedness of the obtained inverse problems.