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Over the last several decades, it has become increasingly accepted that the term xenobiotic relates to environmental impact, since environmental xenobiotics are understood to be substances foreign to a biological system, which did not exist in nature before their synthesis by humans. In this context, xenobiotics are persistent pollutants such as dioxins and polychlorinated biphenyls, as well as plastics and pesticides. Dangerous and unstable situations can result from the presence of environmental xenobiotics since their harmful effects on humans and ecosystems are often unpredictable. For instance, the immune system is extremely vulnerable and sensitive to modulation by environmental xenobitics. Various experimental assays could be performed to ascertain the immunotoxic potential of environmental xenobiotics, taking into account genetic factors, the route of xenobiotic penetration, and the amount and duration of exposure, as well as the wave shape of the xenobiotic. In this paper, we propose an approach for the analysis of xenobiotic metabolism using mathematical models and corresponding methods. This study focuses on a pattern depicting mathematically modeled processes of resonant absorption of a xenobiotic harmonic oscillation by an organism modulated as an absorbing oscillator structure. We represent the xenobiotic concentration degree through a spatial concentration vector, and we model and simulate the oscillating regime of environmental xenobiotic absorption. It is anticipated that the results could be used to facilitate the assessment of the processes of environmental xenobiotic absorption, distribution, biotransformation and removal within the framework of compartmental analysis, by establishing appropriate mathematical models and simulations.

Environmental xenobiotics are substances, which did not exist in nature before their synthesis by humans. They are becoming increasingly problematic in medicine and environmental systems, since they are relatively new substances and difficult to categorize, and since it is challenging to assess their effects on human health and the environment.

A xenobiotic is defined as a chemical, which is found in an organism but which is not normally produced or expected to be present in it [

Over the last decades, xenobiotics have been extended to the environment. Many studies [

Over the last decades, several xenobiotics have been found to be toxic to the immune system [

Many reports in the literature [

Furthermore, persistent environmental contaminants such as dioxins and polychlorinated biphenyls (PCBs) have been shown to modulate the activities of several different hormones [

Similarly, problems caused by environmental emissions during the operation of coal-fired power stations, beyond the environmental depletion, are also related to and impact human health [_{2}, NO_{x} as well as gases that undergo chemical reactions to form fine particles in the atmosphere [

As a consequence, the pollutant load caused by environmental xenobiotics concerns researchers in medical and environmental fields.

A body removes xenobiotics by xenobiotic metabolism [

Studies [

A better understanding may be related to the wave shape of a xenobiotic oscillation, if we accept a holistic view of xenobiotic metabolism. Following this idea, we here study a superposition of two oscillations, one corresponding to the biological organism system and the other to the environmental xenobiotic. A myriad of assays could be experimentally run to improve understanding, but such an approach is time and effort consuming and usually yields case-specific information. The more abstract view adopted here seeks an analogy between a living system (an organism) and an electrical system, based on the understanding that any biological organism may be seen as a system oscillating at its own frequency (or having an own wavelength).

Consequently, the processes of absorption, distribution, biotransformation and removal of environmental xenobiotics could be studied through mathematical models and methods. In related research, compartmental analysis has been applied to the body [

These ideas [

According to the theory of complex systems analysis, one could apply the mathematical pattern method for the assessment of the behavior of an ecosystem or biological organism. A mathematical model represents a transformation of the relations between the system variables in appropriate mathematical structures. The mathematical patterns are usually expressed by algebraic equations, differential or logical, with their form, structure and parameters depending on the real system [

This study deals with a pattern depicting the mathematical representation of the process of environmental xenobiotic absorption. To fully understand xenobiotic processes, mathematical patterns also need to be established for describing the xenobiotic entrance into the system, circulation and distribution to organs and tissues where metabolism occurs, and subsequent excretion, but that work is beyond the scope of the present undertaking.

By analogy with thermal physics, where the temperature difference gives the sense and magnitude of the transferred energy, the main vector of behavioral analysis in the situation of an environmental xenobiotic “attack” is represented by the xenobiotic concentration

To create a homogenous framework and problem definition, we address the percentage representation of the xenobiotic concentration through the spatial concentration vector

Consider a mathematical model with the input quantities _{i}_{i}_{1}(_{2}(_{1}(_{2}(_{i}_{i}

We hope to later show that a mathematical model described by a differential equation of order two with concentrated parameters could be accepted for a complex process of an environmental xenobiotic absorption by a linear structure. In line with this idea, we define a hypothetical situation in which, from an environmental xenobiotic source with harmonic behavior, the xenobiotic is absorbed by the biologic organism modeled as a system with a linear structure. The xenobiotic concentration is subsequently denoted by

Within the structure of a modulated absorption system corresponding to the biological organism (the target of a xenobiotic), one can identify specific elements of xenobiotic compounds that are of a dissipating type or an accumulating type. As stated earlier, a mathematical model depicting a xenobiotic absorption process could be a differential equation of order 2. Consequently, we consider an analogous pattern of physics, namely an electrical structure type RLC series circuit, where the accumulating elements are described by the capacity

The differential equation corresponding to this transient regime is:

The solution of the differential Equation (1) is expressed by the relationship:

Here:

δ is the damping factor of the circuit:

ω_{0} is the resonant pulsation:

ω_{e} is the own pulsation of the biologic organism system:

and

If the damping factor has a small value,

This expression emphasizes the superposition of two oscillating components, with the harmonic xenobiotic pulsation ω and the biologic system pulsation ω_{0} = ω_{e}, respectively.

The mathematical modeling stage is followed by the simulation of specific phenomena by using appropriate computer software. For instance, one could use MATLAB software with SIMULINK and SimPowerSystems extensions. For this purpose we developed a SIMULINK model entailing specific blocks generated by the SIMULINK library.

Based on the MATLAB-SIMULINK utility, the explicit function (15) leads to obtaining the simulation model for the spatial vector of concentration

Simulation model for spatial vector of concentration

Subsequently, based on this simulation pattern, the representations of

_{0}.

In

Modulating signal obtained on basis of absorption circuit elements. Case 1.

Harmonic oscillation of xenobiotic. Case 1.

Simulation diagram for spatial vector of concentration

The majority of research carried out on environmental contaminants has shown that these pollutants, following low level exposure to humans and animals, cause unexplained and irregular concentrations of xenobiotics to be accumulated in biological systems [_{02} of the biological system. The simulation model for the spatial vector of concentration _{02}. In

Simulation model for spatial vector of concentration

Through this second case we seek to ascertain that the natural resonant frequency, as a characteristic parameter of any biologic entity, plays an important role in the evolution law of the absorption process of environmental xenobiotics. Since this could be a plausible explanation for the unpredictable results of experimental tests, it is likely that mathematical modeling and simulation represent a good additional approach in the study of absorption and removal processes for environmental xenobiotics.

Modulating signal with lower resonant frequency obtained on basis of absorption circuit elements. Case 2.

Harmonic oscillation of xenobiotic. Case 2.

Simulation diagram for spatial vector of concentration

The study of any system, including biological systems, usually entails an analysis of inputs and outputs, and system behavior can be assessed on the basis of mathematical modeling and simulation. Here, for an environmental xenobiotic source with an assumed harmonic behavior, the xenobiotic concentration evolution within a biological system is determined, assuming an analogy with a linear structure characterized by xenobiotic compounds of both dissipating and accumulating types.

Based on MATLAB software with SIMULINK and SimPowerSystems extensions, a SIMULINK model is developed entailing specific blocks generated with the SIMULINK library. The results determine the variation in time of the spatial vector of concentration

This study also aims to ascertain if the natural resonant frequency, as a characteristic parameter of any biologic entity, can provide a plausible explanation of the unpredictable results of experimental tests on the absorption process of environmental xenobiotics. Mathematical modeling and simulation could provide an appropriate additional approach for studying absorption and removal processes of environmental xenobiotics.

Through consideration of hypothetic simulation diagrams for the spatial vector of xenobiotic concentration

The authors declare no conflict of interest.