MATHEMATICAL MODELING AND COMPUTATIONAL ANALYSIS OF ANTIMICROBIAL RESISTANCE WITHIN HOST CELLS

​ABSTRACT

Antimicrobial resistance, AMR, is one of the global healthcare issues worldwide. AMR is a state which happens when infected by pathogen cells are no longer responding to drugs. One approach is to mathematically design suitable model to predict the behavior of infected by AMR cells, including physical differences between healthy and infected cells. This study examines how AMR affects the change in velocity of drugs within cells. It was shown that AMR infected cells have lower velocity in comparison to healthy cells.

Keywords: antimicrobials, mathematical design, infected cells.

Introduction

Among the all problems related to public health sector governance worldwide currently, an increasing number of people misusing and overusing antibiotics is considered one of the most dangerous threats to human health for present and future generations. The reasons behind the statement are the increasing number of death cases due to improper use of antibiotics.

It is estimated that improper use of drugs causes directly 1.27 million deaths and indirectly 4.95 million deaths, implying exacerbating the conditions of patients diagnosed with primary issues other than those treated by the antibiotics [6, p. 629-655]. Additionally, the number of direct death cases was estimated by different regions: 11.2 death cases per 100 000 people in North and South Africa, 11.7 death cases per 100 000 people in East Asia and Oceania, 14.4 death cases per 100 000 people in Latin America, 17.6 death cases per 100 000 in Central and Eastern Europe and Central Asia, including Kazakhstan, 21.5 death cases per 100 000 people in South Asia, and 23.7 death cases per 100 000 people in Sub-Saharan Africa [11, p. 658]. Moreover, it was calculated that AMR resulted in annually 5 million death cases, emphasizing that every fifth death was a death of a child [6, p. 629-655].

The misuse or overuse of antibiotics causes the state of antimicrobial resistance, AMR. Antimicrobial resistance is a condition when bacteria and viruses within an affected human body are no longer successfully defeated by antibiotics [4]. Such situations happen when the bacteria and viruses evolve mechanisms that protect them from antibiotics treatment which once was effective against these pathogens [2, p. 606].

Even though such biological processes of the bacteria and viruses learning how to combat antibiotics occur through the generations of people through inevitable and independent from poor use of antibiotics genetical changes within the bacteria and viruses themselves, poor use of the treatment – in other words, mentioned before misuse and overuse of antibiotics – accelerates the genetic change rate and, thus, has been aggravating the problems related to AMR, primarily increasing the number of deaths – both of direct and indirect types [1, p. 2055944].

Alleviating the consequences of AMR is of great importance for the fields of the global medicine, public health, and national economies since those consequences threaten our general abilities to defeat common (seasonal, chronicle, or well-studied) infections, to combat at successful rates which humanity has made a lot of progress in for the recent centuries, meaning for present times reversing the global notable efforts and work done to treat the patients; it also complicates the performance of surgery operations due to exacerbating health conditions of the patients making them unable to go through surgical occasions [8, p. 11387]; failures in attempting to resolve the global AMR trends additionally costs national economies financial losses due to the emerging need and search for new and more expensive ways to treat common diseases [6, p. 629-655].

The prosperous research activities are related to better diagnosis of AMR conditions’ presence in the patients, creating and designing the models of the bacteria and viruses that have evolved AMR abilities, creating and designing biomaterials for improving antibiotics detective properties, providing modeling approaches and strategies to describe mathematically and computationally the behavior of the pathogens under different conditions causing by different antibiotics treatment respectively, and designing the possible scenarios for antibiotics-free treatment.

The scientific novelty of the mentioned earlier research areas has been undoubtedly contributed to generating and implementing prosperous solutions. However, the focus of this work is explicitly on mathematical and computational modeling of the pathogenic abilities to resist antibiotics.

The goals of this paper are to design mathematical and computational models of antibiotics; movement across the pathogens and to examine related to antibiotics flow within the affected region physical properties of AMR enhanced pathogens. The objects of the work are the pathogens living within the affected human body, possessing AMR properties and experiencing the antibiotics invasion.

Physical Problem

When the drug is consumed, the drug molecules, by blood vessels, are going through cells – it is the way drug molecules are reaching cells and having an impact on infected cells. However, there is a liquid space between cells, so-called extracellular matrix. The function of the space are to deliver different on purpose molecules to cells, to ensure cells are apart from each other, but at the same time the cells are in the tissue [9, p. 4-27]. We can define the cells and extracellular matrix surroung it a space of our research interests.

Since we are interested in the cell’s ability to consume drugs, the cells can be represented as a porous media and drugs molecules as a flow going through the porous medium. Porous media is a space where some parts of it are represented as void, pore, that does not consume or let anything go through it [10]. Indeed, the AMR infected cells do not have the same permeability, an ability to intake molecules through it, and, therefore, there would be much more void than in healthy cells [5, p. 1555-1623].

Mathematical Model

To explain infected by AMR cells mathematically, a two-dimensional situation is examined. Below, the given system of equations represents the laminar flow of drug molecules of a certain concentration through the space of the cells. The model is in accordance with Darcy law, which states that the value of the fluid going through porous medium is directly proportional to the permeability of the medium and is opposity to the viscosity of the medium [3, p. 429-443].

                                                                                             (1)

                                                    (2)

                                                   (3)

                                                                (4)

                                                             (5)

 – horizontal component of the drug velocity, m/s;

 – vertical component of the drug velocity, m/s;

 – time, s;

 – mass density, kg/m^3;

 – pressure, Pa;

 – Darcy law’s variable in porous media, m/s;

 – concentration, mol/kg;

 – kinematic viscosity, m^2 /s;

 - dynamic viscosity, Pa*s;

 – angle, degree;

 – length, m;

– concentration constant;

Computational Analysis

To represent velocity’s dependence on the porosity of the medium, the following computational analysis was conducted: setting time and pressure to 0 s, the velocity values were varied.

For analysis to be sophisticated, the space is divided into five parts. The left part is inlet, the right part is outlet, the middle part is divided further inti three parts – three porous media with different porosity values. The health cells’ space is represented by space with porosity value of 0.3, it is located in the upper part. The porosity value of 0.3 implies that 30% of the space is void. The infected cells’ space is represented with porosity value of 0.7, it is located in the lower part. The porosity value of 0.3 implies that 70% of the space is void. The middle part is represented by space with porosity value of 0.1 and is put into analyusis in order to show the flow through cell-free space. The porosity value of 0.1 implies that 10% of the space is void.

Picture 1. Velocity distribution with the initial value of 1 m/s.

Picture 2. Velocity distribution with the initial value of 2 m/s.

Picture 3. Velocity distribution with the initial value of 0.5 m/s.

As it is evident from the pictures above, the velocity is higher when the drug flow is going through the healthy cells, in comparison to AMR infected cells, independently from different scenarios.

Conclusion

To summarize, the mathematical model for explaining the drug flow through AMR infected cells was given, and the computational analysis was done to prove that velocity is lower in the space of AMR infected cells. However, the studies done only depicts velocity as the main physical parameter when describing AMR infected cells. Further research is recommended.

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