Bio-Simulation of the Induction of Forced Resonance Mechanical Oscillations to Virus Particles by Non-Ionizing Electromagnetic Radiation: Prospects as an Anti-Virus Modality

Abstract

The induction of acoustic-mechanical oscillations to virus particles by illuminating them with microwave signals is analyzed theoretically. Assuming the virus particle is of spherical shape, its capsid consisting primarily of glycoproteins, a viscous fluid model is adopted while the outside medium of the sphere is taken to be the ideal fluid. The electrical charge distribution of virus particles is assumed to be spherically symmetric with a variation along the radius. The generated acoustic-mechanical oscillations are computed by solving a boundary value problem analytically, making use of Green’s function approach. Resonance conditions to achieve maximum energy transfer from microwave radiation to acoustic oscillation to the particle are investigated. Estimation of the feasibility of the technique to compete with virus epidemics either for sterilization of spaces or for future therapeutic applications is examined briefly.

1. Introduction

The study of the physical properties of various types of viruses has attracted significant interest from several interdisciplinary research groups during the last 10 years [1]. Mechanical properties of virus particles with diameters of 10–300 nm, in particular the capsids enclosing the virus gene structures (DNA, RNA both single- and double-stranded), have been studied experimentally using atomic force and electron microscopy (AFM) [2, 3]. Also, elasticity theory methods are applied to conclude the mechanical properties of virus particles [4]. Electric charge distributions of the virus have also been studied by several researchers [5, 6, 7, 8, 9]. It is observed that under physiological conditions of salinity and acidity, virus capsid assembly requires the presence of genomic material that is oppositely charged to the core proteins [10]. Furthermore, few researchers have focused their research on the possibility of inducing photon-phonon interactions [11] in virions, which are the virus causing infections [12, 13, 14, 15, 16, 17]. Already resonance phenomena of the H3N2 and H1N1 viruses have been demonstrated experimentally [18] , leading to a high rate extinction of them at a resonance microwave frequency near 8 GHz. The physical phenomenon attributed to this interaction is the separation of positive-negative electric charges on the body of the virus particles and the coupling of microwave energy through the interaction with the three-dimensional bipolar electric charge distributions, generating mechanical oscillations at the same frequency. At specific microwave frequencies depending on the diameter and other properties of the particle [19, 20], primarily the dipole acoustic mode, have been claimed and strong coupling leading to high level virus killing rates have been demonstrated recently [19, 20]. The effects of hydration levels on the bandwidth of microwave resonant absorption induced by confined acoustic vibrations have also been studied [21]. It should be stated that the involved phenomenon is of non-thermal nature related to non-ionizing radiation, in this case being the I band microwaves (6–10 GHz). Raman scattering phenomena [22] have also verified the existence of acoustic-mechanical resonance phenomena in virus particles [23].

The recent ongoing Covid-19 worldwide pandemic [24] and its severe consequences make it attractive to investigate the possibility of utilizing the above-mentioned resonance phenomenon either in sterilization of spaces [25] such as clinics, public venues, hospitals or in the future as a therapeutic modality in some cases. In this direction, the possibility of utilizing similar methods used in microwave-induced hyperthermia, to raise the temperature of malignant tumors inside the human body, could be envisaged as a therapeutic modality. In fact, contrary to hyperthermia where usually lower frequencies of 27–2450 MHz are used [26], in this case much higher frequencies (6–8 GHz) were needed to be used. In some cases, ultrasound and laser radiation modalities have also been used, in clinical hyperthermia, along the low-frequency microwave radiation using endo-cavitary radiators. However, in the present case, the interaction of non-ionizing radiation with tissues will be entirely different from hyperthermia to compete with the virus populations. The rather high frequencies used in the present resonance phenomenon pose a challenging problem to penetrate with high-intensity electric fields inside the human body such as in the case of the lungs. However, in the case of the larynx and throat and even some parts of the lungs, endo-cavitary radiators [27], as done in hyperthermia, could be used. Finally, since in the present case, the action of microwave radiation has the character of a resonance interaction, it is foreseen not to need longtime irradiation, contrary to hyperthermia, which in order to raise the tissues from 43 to 45^oC needs usually 45–60 minutes. This principle allows to propose in present case short duration pulsed-periodic high-intensity microwave signals [28, 29]. This is expected to alleviate to some degree the penetration problem of electromagnetic energy to the human body.

In all the mentioned publications of this resonance phenomenon, the virus particle is assumed of being an elastic particle, as was modeled by H. Lamb in 1887 [30] for the oscillations of an ideal spherical isolated in space. In the present chapter, the mathematical analysis is carried out also considering the surrounding medium of the virus particle and taking into account the interaction of external microwave radiation with the electric charge distribution of the virus particle. Based on the recently published data on the structure [31] of the Covid-19 virion being 100–150 nm in diameter, because of its reach liposome capsid with few proteins on it, the present work leads us to adopt the model of the spherical virus particle as a viscous fluid while the outer space is taken to be an ideal fluid, with different acoustic characteristics of the spherical particle. Furthermore, vortex phenomena in modeling the viscous virus structure are neglected, since it is assumed that these are very weak and they have no effect on the resonance phenomenon to be studied.

The bio-simulation of the forced oscillations of a virus particle is carried out in the following mathematical steps presented with a flow diagram (Figure 1).

Figure 1.

A flow chart diagram of the bio-simulation analysis of the interaction of virus particles with electromagnetic waves inducing acoustic oscillations inside the virus particle.

2. Mathematical formulation of the phenomenon

A spherical particle of radius α, shown in Figure 2, is assumed to pose a continuous electric charge distribution with spherical symmetry defined by the equation

Figure 2.

Spherical model for the virus particle.

$${\rho }_q\left(\mathbf{r}\right)=\frac{Q}{\sigma }\left(1-\frac{5}{3}{\left(\frac{r}{\alpha }\right)}^2\right)$$E1

where Q is the total positive electric charge in the center of the sphere, σ = 8πα³(3/5)^3/2/15 is a normalization constant. The term 5/3 in the

Above Eq. (1) was selected to have the total charge of the particle to be zero, that is to have a balance between the positive (inner r<α(3/5)^1/2 region) and negative (towards the external surface) charge distributions. It is evident that the particle could have the opposite charge distribution and the same analysis is valid. The proposed method is extendable to the case of non-symmetric charge distribution, and then higher order modes will be excited.

The spherical model of the virus is assumed to be a compressible fluid, characterized by its homogenous mass density ρ₁ acoustic wave propagation speed c₁, and total viscosity constant (dynamic and bulk) χ. Then, assuming $e^{\mathit{j\omega t}}$ as time dependence, the propagation of acoustic wave phenomena is described by the following field equations [32]:

3. Newton law

$${\rho }_1\mathit{j\omega }{\mathit{v}}_{\mathbf{1}}\left(\mathit{r}\right)=-\nabla P_1\left(\mathit{r}\right)+x\nabla \left(\nabla .{\mathit{v}}_1\left(\mathit{r}\right)\right)+\mathit{f}$$E2

where ${\mathit{v}}_{\mathbf{1}}$ is the velocity, $P_1$ the pressure field and $\mathit{f}$ is the force density (N/m³) because of the electric charge distribution inside the sphere.

4. Mass continuity equation

$$\mathit{j\omega }{P}_1\left(\mathit{r}\right)=-c_1^2{\rho }_1\nabla .v_1\left(\mathit{r}\right)$$E3

The force density term $\mathit{f}$ in Eq. (2), taking into account the charge distribution given in Eq. (1) and the incident electric field $\mathit{E}\left(r\right)=E_o\widehat{z}$ of the microwave radiation propagating along the x-axis and polarized parallel to the z-axis (see Figure 2), considering the size of the particle to be extremely small compared to microwave radiation wavelength, is obtained to be:

$$\mathit{f}\mathit{=}{\rho }_1\left(\mathit{r}\right)E_o\widehat{z}$$E4

Operating on the Eq. (2) the $\nabla .$ operator from the left-hand side, substituting Eq. (3), Eq. (4) and rearranging the terms the following wave equation is obtained:

$${\nabla }_{.}^2{P}_1+k_1^2{P}_1=E_o{u}_{o}z$$E5

where

$$k_1=\frac{\omega /c_1}{\sqrt{1+\mathit{j\epsilon }}},=\frac{\mathit{\omega \chi }}{c_1^2{\rho }_1},u_o=-\frac{10Q}{3{\mathit{\sigma \alpha }}^2}$$E6

The pressure field outside of the particle assuming an ideal fluid is described by two respective equations of Eqs. (2) and (3):

$$\mathit{j\omega }{\rho }_o{\mathit{v}}_o\left(\mathit{r}\right)=-\nabla P_o\left(\mathit{r}\right)$$E7

$$\mathit{j\omega }{P}_o\left(\mathit{r}\right)=-c_o^2{\rho }_o\nabla \mathbf{.}{\mathit{v}}_o\left(\mathit{r}\right)$$E8

where $P_o\left(\mathit{r}\right)$ is the pressure, ${\mathit{v}}_o\left(\mathit{r}\right)$ the velocity, ${\rho }_o$the mass density and $c_o$ the acoustic speed. Also combining Eqs. (7) and (8):

$${\nabla }^2{P}_o\left(\mathit{r}\right)+k_o^2{P}_o\left(\mathit{r}\right)=0$$E9

where $k_o=\omega /c_o$ is the wave constant of the infinite space outside of the sphere.

5. Solution of the boundary value problem

5.1 The field expression inside the sphere r<α

The acoustic pressure inside the spherical particle being excited by the interaction of microwave electric field component acting to electric charges, being inside the spherical volume, could be described in terms of the primary ($P_{10}\left(\mathit{r}\right))$and secondary ($P_{11}\left(\mathit{r}\right)$) pressure fields [33].

In the analysis to follow spherical coordinates are used r, θ and φ being the radial distance from the origin, θ being the angle measured from z-axis and φ the azimuth angle.

The primary field $P_{10}\left(\mathit{r}\right)$ should satisfy Eq. (4) with the right-hand side inhomogeneous term. Based on Green’s theory, assuming the outside medium being infinite, the primary pressure is determined by using the equation:

$$P_{10}\left(\mathit{r}\right)=u_o{\iiint }_{\mathit{\text{Sphere}}}{G}_1\left(\mathit{r},{\mathit{r}}^{\prime }\right)z^{’}d{\mathit{r}}^{’}$$E10

inserting z’=r’cos(θ’) and the expansion [34].

$$G_1\left(\mathit{r},{\mathit{r}}^{\prime }\right)=-j{k}_1\sum _{n=0}^{\infty }{j}_n\left(k_1{r}_{<}\right)\underset{n}{\overset{\left(2\right)}{h}}\left(k_1{r}_{>}\right)\sum _{m=-n}^n{Y}_n^m\left(\theta ,\phi \right)Y_n^{\ast m}\left({\theta }^{\prime },{\Phi }^{\prime }\right)$$E11

where $j_n\left(.\right)$ and $h_n^{\left(2\right)}\left(.\right)$are the spherical Bessel and Hankel (second type) functions,$r_{<}=min\left(r,r^{\prime }\right)$ and $r_{>}=max\left(r,r^{\prime }\right)$, the angular spherical wave function

$$Y_n^m\left(\theta ,\phi \right)=j^n\sqrt{\frac{2n+1}{4\pi }\frac{\left(n-m\right)!}{\left(n+m\right)!}}{e}^{\mathit{jm\phi }}{P}_n^m\left(\theta ,\phi \right)$$E12

$$\text{and}{P}_n^m\left(\theta ,\phi \right)\text{being the Legendre function}.$$

Substituting Eq. (11) into Eq. (10) and z’=r’cos(θ’), the fact the double summation in Eq. (10) being limited on the terms m=0 and n=1 after the orthogonality of the angular wave functions and the Bessel functions integral [35]

$${\int }_{r=0}^{\alpha }{j}_1\left(k_1{r}_{<}\right)h_1^{\left(2\right)}\left(k_1{r}_{>}\right)r^{\prime 3}d{r}^{\prime }=a^4{j}_1\left(k_{1}r\right)w_o+\mathit{jr}/k_1^3$$

$$\text{whith}{w}_o=\left(3h_1^{\left(2\right)}\left(k_{1}a\right)-k_1\alpha h_0^{\left(2\right)}\left(k_{1}a\right)\right)/\left(a^2{k}_1^2\right)$$

leads to the result of the primary pressure field

$$P_{10}\left(\mathit{r}\right)=j{k}_1{u}_o{E}_{o}cos\left(\theta \right)\left(a^4{j}_1\left(k_{1}r\right)w_o+\frac{\mathit{jr}}{k_1^3}\right)$$E13

Noticing that the primary field depends only to $P_1^o\left(\mathit{\text{cosθ}}\right)=cos\left(\theta \right)$ angular function (n=1 and m=0 terms), the secondary pressure field is written easily:

$$P_{11}\left(\mathit{r}\right)\mathbf{=}A{j}_1\left(k_{1}r\right)cos\left(\theta \right)$$E14

5.2 The field expression outside the sphere ( r>α)

Considering the excitation of only the wave with cos(θ) dependence and the necessity of radiation condition to be valid for $r\to +\infty$ we can write easily:

$$P_o\left(\mathrm{r}\right)=B{h}_1^{\left(2\right)}\left(k_{1}r\right)cos\left(\theta \right)$$E15

In Eqs. (14), (15), the unknown coefficients A and B are determined by imposing the validity of the boundary conditions at the spherical surface r=α:

Continuity of pressure fields $P_{10}+P_{11}=P_0$

Continuity of radial velocities $\widehat{\mathit{r}}\mathit{.}\left({\mathit{v}}_{\mathbf{1}}\left(\mathit{r}\right)\mathit{-}{\mathit{v}}_{\mathbf{0}}\left(\mathit{r}\right)\right)\mathit{=}0$

Then the A and B coefficients are calculated easily after some algebraic operations.

The final solution for the secondary field inside the sphere and in particular for the total pressure is obtained to be:

$$P_1=a^{-2}Q{E}_{o}cos\left(\theta \right)W$$E16

$$W=-j2.569k_{1}a\left[\left(j_1\left(k_{1}r\right)w_o+\frac{\mathit{jr}}{a^4{k}_1^3}\right)+j_1\left(k_{1}r\right)S\right]$$

$$S=T/R$$

$$T=\left(a{k}_1{j}_1^{\prime }\left(k_{1}a\right)w_o+\frac{j}{a^3{k}_1^3}\right)h_1^{\left(2\right)}\left(k_{o}a\right)a^{-1}{k}_o^{-1}-\left(j_1\left(k_{1}a\right)w_o+j{a}^{-3}{k}_1^{-3}\right)h_1^{\prime \left(2\right)}\left(k_{o}a\right)$$

$$R=j_1\left(k_{1}a\right)h_1^{\prime \left(2\right)}\left(k_{o}a\right)-\left(1+\mathit{j\epsilon }\right)j_1^{\prime }\left(k_{1}a\right)h_1^{\left(2\right)}\left(k_{o}a\right)\frac{{\rho }_o{k}_1}{{\rho }_1{k}_o}$$

6. Numerical calculations

After computing the pressure field as given in Eq. (16), it is shown that the “form factor” W is a function of the dimensionless quantities $k_{1}a,k_{o}a,\left({\rho }_o{k}_1\right)/({\rho }_1{k}_0$), $r/\alpha$ and the parameter related to the viscosity of the virus particle $\epsilon =\frac{\mathit{\omega \chi }}{c_1^2{\rho }_1}=\left(\frac{k_{o}a{c}_o^2}{c_1^2}\right)\delta$, where $\delta =\chi /\left(c_o{\rho }_1\alpha \right)$ is a quantity related to the total viscosity of the spherical virus particle. Remembering that $\chi =\frac{4\eta }{3}+\kappa$ [36], where $\eta ,\kappa$are the shear and bulk viscosity coefficients of a Newtonian fluid, we take the quantity $\delta$ as a “measure of the degree of viscosity” in our calculations. Furthermore, the interest being on the maximum pressure on the particle, we take $\theta =0\text{or}\pi$ on the two poles of the sphere where the rupture of the virus capsid is sought.

In Figure 3, numerical results of the W “form factor” are given in the range $0.1<k_{o}a<3.0$ for various parameters of the ratios $\frac{{\rho }_o}{{\rho }_1}=1.05,1.1,1.2\text{and}1.3$, c₀=1560m/s (speed of sound in the outer space), c₁ = 1950 m/s (speed of sound inside the sphere) . Meanwhile, the viscosity parameter is taken χ = 0.01 and χ = 0.001, which corresponds to the total viscosity constant corresponding to δ = 0.1 and δ = 0.01 (N$\bullet s/m^2)$. The numerical results show interesting resonance behavior when viscosity is δ = 0.01. The phenomenon is stronger as δ decreases.

Figure 3.

Dependence of W (Eq. (16)) function to k_oα. The W function is the “form factor” of the resonating spherical virus particle and the resonance takes place when the R term takes maximum value at a specific microwave frequency, which is inducing mechanical-acoustic oscillation through electric force interaction between electromagnetic wave and electric charges.

It is well known that the scattering of incident waves to a sphere (acoustic or electromagnetic waves) shows resonance phenomena when the refractive index of the spherical scatter has a large value. This phenomenon is well known in classical and quantum physics (Regge poles). As mentioned in the introduction section several researchers have foreseen this phenomenon. However, in the present analysis, the adopted model and analysis takes into account, although in a simplified form, all the involved mechanisms. The resonance is occurring near the angular frequency ω=πc_o/(2a), which corresponds to the “dipole mode” of the spherical particle.

In order to assess the feasibility of utilizing the phenomenon to compete with the virus populations, we need to calculate the pressure being developed at the spherical surface. Placing in Eq. (16)r = α, 2α = 100 nm and Q = Ne_o, e_o=1.62 x 10⁻¹⁹ Cb (electron charge), N being the number of + or – charges we obtain after Eq. (16) the pressure P₁ =4$\bullet 10^{-5}\bullet$N∙E_o∙W, since $W\sim 2.000(\mathrm{see}$Figure 3) following the data given in ref. [9], the surface electric charge being $\sigma \le 0.5\mathit{eo}/n{m}^2$ the virus area being A_s = 4πα², we obtain $N\sim 3\bullet 10^4$ and $P_1\sim$2.400 E_o (Pa). Then, if the imposed electric field at microwave frequency is E_o = 1.000 (V/m) (this corresponds to a power density of 130 mW/cm² much less used in hyperthermia treatments many times being 15.000 mW/cm² as a continuous wave signal), we arrive to the estimation that the pressure oscillation amplitude exerted on the two poles of the virus will be $P_1\sim 2.4\mathrm{MPa}$. The mentioned microwave field-generated pressure wave on the capsid surface seems to be comparable with the bulk Young’s module being equal to 5 MPa (see the end of the section ‘Methods’ of ref. [8]).

In Figure 4, computations in the case of zero viscosity are presented for various ratios ${\rho }_1/{\rho }_o$. The strong resonance phenomenon of the ratio ${\rho }_1/{\rho }_o$ is 1.3 while the lowering of the resonance frequency as this ratio decreases while the peak value of W has some variation.

Figure 4.

Dependence of W [Eq. (16)] function to k_oα in case of absence of viscosity.

The above initial results show that the argument expressed by the National Taiwan University in their seminal paper of ref. [18] is verified theoretically with the present model.

The microwave resonance frequency is computed easily known the value of k*= k_oa the peak value is attained, that is f_resonance (Hz)=k*c/(2πα) where c=3.10⁸ (m/s) is the speed of electromagnetic waves in vacuum.

7. Conclusions

A simplified model of a virus particle allowed to analyze the coupling phenomena between microwave (electromagnetic) radiation and acoustic waves generated inside the particle. Based on recent publications on virus physical and electronic properties of viruses, similar to Covid-19, computations show the possibility of strong interactions to generate rupture or capsid of the viruses. This action is based on the Coulomb force exerted by the oscillating field on the inhomogeneous electric charges within the spherical particle. The microwave resonance frequency—which is identical to the acoustic wave—is in the region of 6–10 GHz.

The prospect of using the presented principle to sterilize public spaces, hospitals, clinics etc. is an attractive proposition. The present microwave technology is available for the development of this type of portable device. Moreover, the existing more than 40 years of experience in clinical hyperthermia, which is based on the use of low microwave frequencies as an adjuvant therapy to treat many cancer diseases, makes it attractive to investigate the possibility of developing technologies to implement the mentioned idea in the future to depopulate virus populations inside the human body. Before this, extensive in vitro trials in virus and cell cultures need to be carried out to follow with animal trials as well.