key: cord-1036946-0czphube
authors: Varfolomeev, S. D.; Panin, A. A.; Bykov, V. I.; Tsybenova, S. B.; Shogenova, L. V.; Chuchalin, A. G.
title: Kinetic model of development of acute viral infection in the human body. Critical conditions, control mechanisms, “thermoheliox”
date: 2020-07-20
journal: Russ Chem Bull
DOI: 10.1007/s11172-020-2886-4
sha: ab1c69ec557548c778c67321a8e00a174c95fe59
doc_id: 1036946
cord_uid: 0czphube

A kinetic model of the development of acute viral infection is proposed and the dynamic behavior of key variables, including the concentrations of viral particles, infected cells, and pathogenic microorganisms, is described. The change in the hydrogen ion concentration in the lungs and pH dependence of the activity of carbonic anhydrase, a key respiration enzyme, are critical factors. An acute bifurcation transition determining either the life or collapse of the system is demonstrated. The transition is associated with exponential increase in the concentrations of participants in the process and with functioning of the key enzyme, carbonic anhydrase. A physicochemical interpretation is given for the therapeutic effect of temperature rise and potential therapeutic effect of “thermoheliox”, that is, breathing by heated helium-oxygen mixture.

The infection of a living organism with viral particles, resulting in development of a clinical presentation potentially culminating in a collapse (death) of the organism, is a complex dynamic process. The most important parameters determining the process dynamics are the concentration of the infecting viral agent, the concentration of pathogenic microfl ora, which develops symbiotrophically on the aff ected cells, and physical conditions of the process such as temperature and pH of the medium. With the goal to create conditions for suppression of the viral infection and for controlling the treatment process, it appears highly expedient to construct and analyze kinetic models describing the development of the pathological process.

Quite a few publications of the last decade describe modeling of the dynamics of viral growth in the body, taking account of the production of pathogenic microfl ora and human immune system response. A review 1 surveys the mathematical models of the dynamics of infl uenza within an organism, including and not including the immune response. 1 Baccam et al. 2 modifi ed the simplest model of infection with the infl uenza virus by adding equations that take account of the delay of viral production (which is 6-8 h), interferon induction to inhibit the viral replication in an infected cell, and the process of treatment. Most of mathematical models describe the incubation, the viral growth, activation of the immune system, and treatment of infection, but they do not consider the causes for collapse of the system, that is, molecular causes for the death of an organism related to the disease. The mathematical models neglect the disturbance of the acid-alkaline balance (pH), 3 which plays an important role in the functioning of respiratory tract and metabolism. Increasing amount of carbon dioxide and accumulation of acid metabolites decreases pH of the blood and reduces the catalytic activity of carbonic anhydrase, which corresponds to respiratory acidosis.

Kinetic model. Here we present the mathematical model describing the growth dynamics of the viral infection, formation of the pathogenic microfl ora, and change in the pH in the lesion. The kinetic model and its parametric analysis are presented below. The model is based on the kinetic equations reported in our previous studies 4, 5 on the growth and evolution of microbial and viral populations.

The basic set of equations can be represented in the form

where [N] is the concentration of the infecting virus; k 1 is the specifi c viral replication rate in the body; α(Т) is the parameter characterizing the rate of virus eradication by thermal inactivation, immune response, etc., [N] 0 is the initial infecting concentration of the virus. The virus penetrating into the host cells, fi rst of all, into lung cells, replicates and destroys the infected cells. The metabolically destroyed cells represent a favorable medium for the growth of pathogenic microorganisms. The products are actually dead cells, they acts as a substrate for the growth of pathogenic microfl ora: 

where μ m is the maximum specifi c growth rate; K P is the pathogen affi nity to the substrate (product), β(T) is the parameter characterizing the thermal death of microorganisms.

The pathogenic mcrofl ora, e.g., pneumococci, are facultative anaerobes and, under limited aeration conditions, in poor air exchange areas, they utilize the anaerobic mechanism for ATP synthesis and thus produce organic acids. This is a way to local respiratory acidosis and collapse (death) of the organism.

It is known that pH ∼7.15 is the critical value in the blood circulation system. At lower pH in the blood, metabolism of the organism is disturbed. This is due to the fact that the active sites of many enzymes contain the imidazole group of histidine, the pK a of which is 7.0-7.2, 6 depending on the protein structure. Protonation of the imidazole group leads to complete loss of enzyme activity.

The key enzyme of functioning of the respiration system is carbonic anhydrase, which catalyzes carbonic acid transfer from the liquid to the gas phase as gaseous carbon dioxide:

The catalytic activity of this enzyme is determined by the imidazole group of histidine with pK a ∼7.0. Blocking the activity of this enzyme means complete termination of respiration.

Presumably, under conditions of considerable lung damage by pathogenic microfl ora, pH change in lungs by 0.2-0.3 units is the major factor of collapse of the respiratory system.

In terms of the considered model, the dynamics of pH variation in the aff ected area can be described by Eq. (5) with the assumption that the proton production rate is proportional to the concentration of pathogenic microfl ora:

The coeffi cient δ characterizes the productivity of microorganisms for proton generation and the buff er properties of the blood system. The term γ([H + ] 0 -[H + ]) describes the "openness" of the system for protons, where [H + ] 0 corresponds to the concentration of protons entering the damaged area, [H + ] is the averaged proton concentration in the lesion, γ is the mass transfer coeffi cient. The parameter v c is the rate of the carbonic anhydrase enzymatic reaction, i.e., the rate of production of hydroxyl ions by the carbonic anhydrase-catalyzed reaction. Considering the pH dependence of the rate of enzymatic reaction, the rate equation can be presented in the form , (6) where V m is the maximum rate of the catalytic reaction, K M is the Michaelis constant. Provided that the bicarbonate concentration is constant (relatively small portion of process development), the constant factor in Eq. (6) will be denoted by A. Solution of the system of Eqs (1)-(6) gives a kinetic description for the observed development of the pathology upon infection with a coronavirus.

One of the scenarios of process development is shown in Fig. 1 . Equations (1)-(6) were integrated using the following parameters: The obtained solution qualitatively describes the infection phenomenon and development of the disease.

1. The incubation period, in which there are practically no signs of the disease. From Fig. 1 (dashed line), it is observed that with the parameters given above, this period is ∼150 h (approximately 6.5 days).

2. At the end of the incubation period (induction period), fast symbiotrophic increase in the virus concentration and pathogenic microfl ora concentration with signifi cant accumulation of the virus and microorganisms is observed.

3. In the primary periods of pathology development (incubation period and initial period of active increase in the virus and pathogenic microfl ora concentrations), pH value in the damaged area is in the range of 7.2-7.4. In the absence of therapy or deliberate infl uence on the system behavior, bifurcation growth and pH-based suppression of carbonic anhydrase activity take place. This should result in complete respiratory arrest. Using the above-indicated parameters, the model predicts that respiratory arrest will occur on the 15th or 16th day after infection. The collapse is of the bifurcation nature, and the transition point is easily identifi ed in the plot (see Fig. 1 ). The bifurcation point is a crucially important characteristic of the process. This is the point of no return, actually, the time of death of the organism. When this point has been passed, the release of CO 2 from the liquid phase to the gas phase is stopped. The uncontrolled increase in the bicarbonate concentration in blood takes place, i.e., breathing is blocked.

It is of interest to study the eff ect of variation of the parameters on the system behavior in order to identify the most sensitive elements and process control methods. Below we consider several important cases.

Control mechanisms. Virus destruction (immune response). The most sensitive parameter determining the system behavior is characteristics of the viral destruction rate (parameter α). Under natural conditions, the destruction is based on activation of the immune system, which produces antibodies against viral proteins. In terms of our model, τ c , that is, the time of system existence before the collapse (life span after infection) is a highly convenient parameter describing the main process characteristics. This time can be visualized by integrating the set of equations and computations. The life span is the period before the bifurcation transition of the hydrogen ion concentration, the break point, and experimentally observed fast increase in [H + ] (see Fig. 1 ). The dependence of τ c on the parameter α was studied by mathematical modeling methods. Figure 2 shows the time dependence of [H + ] at various α and the derived dependence of τ c on α.

The point α = k 1 is the critical point in which the rate of growth of the virus concentration in the organism turns into zero. This is the condition of full recovery.

Antibiotics (suppression of the growth of the pathogenic microfl ora). The development of the infectious process is markedly aff ected by the growth rate of the pathogenic fl ora (see Eq. (4)). In the therapy, this process is controlled by administration of antibiotics, which inhibit the growth of microorganisms to one or another extent.

The results of a mathematical experiment studying the eff ect of the parameters μ m and K P on the system life span are presented in Fig. 3 . It can be seen that a decrease in the growth rate of microorganisms (decrease in μ m and increase in K P ) has a benefi cial eff ect on the life span, that is, the time before the bifurcation transition (pH jump).

Key role of carbonic anhydrase. The carbonic anhydrase enzyme plays a fundamentally important role in the respiration mechanism, as it "discharges" the biochemically formed bicarbonate ion to gaseous CO 2 and hydroxyl ion. Under normal physiological conditions (pH ∼7.4), this reaction is virtually irreversible (exhalation; the system is open for CO 2 ). The catalytic activity of carbonic anhydrase is controlled by the ionogenic group with pK a ∼7 (see Eq. (6)). The pH shift (even a minor one) towards the acid region (decrease in the pH) decreases the catalytic activity of the enzyme; the rate of production of OHions decreases in parallel. The process is self-accelerating and leads to the bifurcation pH jump.

The dependence of τ c on the catalytic activity of carbonic anhydrase (A) is depicted in Fig. 4 . It can be seen that the life span τ c can be considerably increased by increasing the enzyme activity. Carbonic anhydrase has a Zn 2+ ion in the active site. This means that effi cient therapy of a viral infection requires saturation of patient´s metabolism with Zn 2+ ions.

Therapeutic eff ect of high temperature. The natural development of the disease is associated with the rise of body temperature. The infl ammatory response is initiated by a large set of biochemical reactions, including the synthesis of prostaglandin type infl ammatory media-tors 7,8 and thermal shock proteins and activation of the immune system.

Temperature rise aff ects most appreciably the rate of thermal death of microorganisms and viruses. It is known that with increasing temperature, the concentration of viruses ([N]) and microorganisms decreases exponentially due to thermal death where ΔH N * and ΔH M * are the activation energies of the thermal death of viruses and microorganisms, respectively. Microorganisms, including pathogens, are very sensitive to temperature rise. Indeed, the rate of death of Escherichia coli increases 14.3-fold as the temperature increases from 54 to 60 °C, and the rate of death of Staphylococcus aureus grows almost 5-fold as the temperature increases from 53 to 57 °C. In view of the published data, 9 one can estimate ΔH M * = 100 kcal mol -1 (thermal death of S. aureus) and ΔH M * = 118 kcal mol -1 (thermal death of E. coli). If we assume that the thermal death of the viruses is determined by the thermal destruction of capsule proteins, then for rough calculations it can be taken that ΔH N * = 40-50 kcal mol -1 .

Within the framework of the model, it appears possible to consider direct inactivation of viruses and pathogenic microorganisms on going from the normal body temperature of 36 °C to the fever temperature of 41 °C. If the activation energy of the thermal destruction of viruses is conventionally taken to be 40 kcal mol -1 , transition from the normal temperature to the fever temperature causes the parameter α to increase 2.8-fold. According to calculations, the bifurcation collapse point shifts in this case from 360 to 420 h. For the activation energy of 50 kcal mol -1 , the α value increases 3.5-fold, and the collapse shifts to 440 h. Thus, the patient gets an additional portion of time to fi ght the disease.

A fundamentally important issue is the structure of Eq. (1). The exponential growth of the virus concentration is possible only for k 1 > α. In terms of the basic parameters Therapeutic eff ect of the thermoheliox. The use of thermoheliox, that is, a thermolyzed mixture of helium and oxygen appears to be the most promising therapeutic means to suppress the viral growth. The essence of this approach is the action on the patient´s respiratory system with a thermolyzed mixture of oxygen and helium at a relatively high temperature. The therapeutic procedure is to make the patient breathe with a mixture of helium and oxygen (80-60% helium, 20-40% oxygen) with a temperature of 50-90 °C. The strategy of using a thermolyzed helium and oxygen mixture was scientifi cally substantiated in detail; it proved to be effi cient in the treatment of respiratory pathologies, ischemic stroke, and pathologies of pregnancy. This method was successfully used for the treatment of more than 2500 patients of the pulmonary and neurology departments of the D. D. Pletnev State Clinical Hospital. 10-12 A. A. Panin developed a unique medical device for controlling the composition and temperature of the helium-oxygen mixture and administration of medical drugs into lungs. 13 The available experimental and practical experience (the use in sauna baths for inhaling air at temperatures up to 100 °C) demonstrates that the thermolyzed oxygen and nitrogen mixture is virtually harmless to the human body within 20-30 minutes of exposure. The thermolyzed helium-oxygen mixture behaves in a similar way. Helium, which has a high diff usion ability, effi ciently drains and shunts all tissues of the body and markedly increases the microcirculation in all organs and tissues. Thermoheliox substantially increases the oxygen delivery, decreases the resistance of respiratory tract, improves the ventilation/perfusion ratio via the alveolar capillary membrane in lungs, and maintains the acid-alkaline balance. The eff ect is thermoheliox is much more effi cient than the eff ect of a room-temperature mixture of oxygen and helium. Mammalian and human cells use specialized protection mechanisms from temporary overheating (heat-shock proteins). Meanwhile, the virus is subjected to effi cient destruction via protein and nucleic acid denaturation. Indeed, the infl uenza virus remains intact for only several minutes at 50-60 °C, the HIV is inactivated 100-fold within 30 min at 56 °C, the hepatitis virus loses activity within 2 min at 100 °C, and the foot-and-mouth disease virus is destroyed within 5-10 min at 50-60 °C.

The essential instability of the viruses against elevated temperature accounts for the seasonal nature of airborne transmitted viral infections. A. A. Panin (January 2020) proposed using thermoheliox as a potential agent fi r killing coronaviruses.

It is of interest to analyze the infl uence of thermoheliox in terms of the model of development of viral and bacterial damage discussed here. If the activation energy of the thermal destruction of the virus is taken to be 50 kcal mol -1 , the exposure of the damaged area for 30 min at 60 °C may decrease the concentration of the virus several-fold (Fig. 6) . The subsequent exposures lead to a dramatic decrease in the concentrations of the pathogenic virus and microorganisms in the body.

It is possible to estimate the temperature dependence of the degree of destruction of the pathogenic virus ([N]/[N] 0 ) on a 30-min respiratory exposure: , (7) where α(T 36 ) is the kinetic parameter at a normal body temperature (in this case, 5•10 -3 h -1 ), Δt exp is the respiratory exposure time (usually 0.5 h), T w is the "operating" temperature of the damage medium. With the use of ther- moheliox, the gas mixture enters the respiratory system at the rated temperature T N , while the outlet temperature of the mixture is usually 10-20 °C lower. It follows from Eq. (7) that at the "operating" temperature of 50 °C, (1) and with periodic exposure to thermoheliox at the "operating" temperature T w = 55 (2) and 60 °C (3).

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Kaskad arakhidonovoi kisloty

XXVIII Natsional'nyi Kongress Organov Dykhaniya

Proc. Int. Conf. of ERS

Registration Certifi cate for the Medical Device RNZ 2016\3988