4.2. The model

ChemReac can be seen as a physical process involving a set of regrouping atoms with the rearrangement of electronic shells of reacting participants, giving place to the generation of new molecular structures called reaction products. The new ways of controlling ChemReac have their basis on the selectivity of spin, a process involving the spins of molecules, electrons, and nuclei of all participants. For this reason, the rate of spin-selective processes is dependent on MF, which alters the spins of the participants, changing partially or wholly the spin selectivity [9, 89, 90]. Thus, to reveal the interaction to explain the cytoprotective effect of ELF-EMF in CYP450, we must define the conditions where the quantum measurement is performed. The first condition is that all quantum states participating in the hepatocytes-RP system in the enzymatic reaction are singlets, because of their high reactivity. The second condition is during the enzymatic procarcinogen activation of CYP450 when the xenobiotics are metabolized, in which appear the RP when is generated the OS. The RP intermediaries are produced in this step, and they are responsible for the insult to hepatocytes, which become the future preneoplastic lesions after to finish the ChemIndHep. The third condition is daily MF stimulation during all ChemIndHep. Nonetheless, the spin evolution of RP is driven by the MF through the HypInt and their reactivity is controlled by spin dynamics, converting nonreactive triplets into reactive singlets through quantum measurement. We showed the way in which the MF modulates the charges in migration evaluating the recombination probability to exemplify. In this respect, when the RP interacts with another electrons' spin, this interaction acts as a catalyst, increasing the recombination probability and accelerating the S !T interconversion [9, 90, 91].

#### 4.2.1. Haberkorn approach

To study the hepatocytes-RP system, we use the information of all parameters employed in the evaluation of MFE, the recombination yield, and the singlet population, which are all included in the Haberkorn approach; this is the most common theory used for spin dynamics studies. This approach is obtained using the spin density matrix in the framework of the Liouville–von Neumann equation involving the rate at which singlets disappear, called kS; it is involved in an unnormalized wave function [9, 80] <sup>∥</sup>ψ><sup>¼</sup> cSe�kSt=<sup>2</sup>∥<sup>S</sup> <sup>&</sup>gt; <sup>þ</sup>cT∥<sup>T</sup> > : Here the amplitudes for the singlet disappear at the rate �kS=2, provoked by the interaction between RP and a third electron for the electron configuration of the hepatocytes (see reference [9] for details). We have written the evolution of the standard density matrix as [9]:

$$
\hat{\rho}\_{\text{in}} = \begin{pmatrix} \mathsf{C}\_{\text{S}} \mathsf{C}\_{\text{S}}^{\*} e^{-k \circ t} & \mathsf{C}\_{\text{S}} \mathsf{C}\_{\text{T}}^{\*} e^{-k \circ t/2} \\ \mathsf{C}\_{\text{T}} \mathsf{C}\_{\text{S}}^{\*} e^{-k \circ t/2} & \mathsf{C}\_{\text{T}} \mathsf{C}\_{\text{T}}^{\*} \end{pmatrix} \equiv \hat{\rho}\_{0} \rightarrow \hat{\rho} = \begin{pmatrix} \mathsf{C}\_{\text{S}} \mathsf{C}\_{\text{S}}^{\*} \left[1 - e^{-k \circ t}\right] & \widetilde{\mathcal{Q}} \\ \stackrel{\sim}{\mathcal{Q}}^{\dagger} & \hat{\rho}\_{0} \end{pmatrix}. \tag{1}
$$

Moreover, to satisfy the unicity of the trace in the density matrix, we include the third electron. Obtaining the so-called reaction products, ^r, according to the rule of conservation of the number of entities participating, i.e., for the generation of some product population, ∅ � ¼ ð Þ 0 0 is a null vector [9, 79].

#### 4.2.2. Quantum measurements

are responsible for a few phenomena such as chemical polarization of electrons and nuclei, and the influence of static and pulsating MF [9, 79]. An RP can decay by recombination, or pull apart the radical by diffusion, or react with other radicals. One of the properties of RP is that recombination probability depends on spin multiplicity, and it varies during RP lifetime. Such variations, as an interesting detail, are manifested as dynamic quantum oscillations, the socalled quantum beats between (S, T) spin states of the RP. The quantum beats modulate the probability of appearance of some reaction channels of the RP that at the time affect the MFE. By studying these quantum beats, one can reveal valuable information concerning the structure, reaction, molecular, and spin dynamics of RP [9, 63, 79, 87]. The RP spin correlation is formed in the coherent state, which oscillates between the S and T spin state, an oscillation that depends on the spin Hamiltonian operator parameters (see references [8, 9] for details), in particular that of the HypInt. The period of the oscillation on organic radicals is in the range of nanoseconds, making RP recombination a plausible test that suggests that small EMF affects

ChemReac can be seen as a physical process involving a set of regrouping atoms with the rearrangement of electronic shells of reacting participants, giving place to the generation of new molecular structures called reaction products. The new ways of controlling ChemReac have their basis on the selectivity of spin, a process involving the spins of molecules, electrons, and nuclei of all participants. For this reason, the rate of spin-selective processes is dependent on MF, which alters the spins of the participants, changing partially or wholly the spin selectivity [9, 89, 90]. Thus, to reveal the interaction to explain the cytoprotective effect of ELF-EMF in CYP450, we must define the conditions where the quantum measurement is performed. The first condition is that all quantum states participating in the hepatocytes-RP system in the enzymatic reaction are singlets, because of their high reactivity. The second condition is during the enzymatic procarcinogen activation of CYP450 when the xenobiotics are metabolized, in which appear the RP when is generated the OS. The RP intermediaries are produced in this step, and they are responsible for the insult to hepatocytes, which become the future preneoplastic lesions after to finish the ChemIndHep. The third condition is daily MF stimulation during all ChemIndHep. Nonetheless, the spin evolution of RP is driven by the MF through the HypInt and their reactivity is controlled by spin dynamics, converting nonreactive triplets into reactive singlets through quantum measurement. We showed the way in which the MF modulates the charges in migration evaluating the recombination probability to exemplify. In this respect, when the RP interacts with another electrons' spin, this interaction acts as a catalyst, increasing the recombination probability and accelerating the S !T inter-

To study the hepatocytes-RP system, we use the information of all parameters employed in the evaluation of MFE, the recombination yield, and the singlet population, which are all included in the Haberkorn approach; this is the most common theory used for spin dynamics studies.

BS [9, 88].

4.2. The model

52 Vitamin E in Health and Disease

conversion [9, 90, 91].

4.2.1. Haberkorn approach

When we have a ChemReac with only a singlet spin state as in our case, we can consider it as a quantum measurement [80]; the amplitudes for the singlet disappear at the rate of � kS=2 provoked by the interaction of the third spin electron that can be studied. During their evolution, the RP can change their spin multiplicity. Through the use of electron spin resonance spectroscopy (ESR) studies, such spin changes are simply called beats, meaning dynamical quantum oscillation between the ∥S > and ∥T > spin states of the RP. Using them, we study the behavior of the spin dynamics of RP. A crucial issue here is that RP appears in the coherent state, which permits the oscillations between ∥S > and ∥T > spin states of the RP, commanded by HypInt. Measured at a quantum level, these beats represent the manifestation of the RP in ESR studies. Tacitly, the beats correspond to S \$ T spin-flip transitions generated by HypInt. The behavior of an unpaired electron under MF, or without MF, determines the influence of HypInt so we can measure the MFE. Eq. (1) expresses how singlet disappears at the desired rate kS. Nevertheless, the off-diagonal terms represent the coherent superposition decaying at a rate of kS=2. This expresses the motion equation for the density matrix with H^ int as the Hamiltonian interaction operator as (see reference [9], for details):

$$\frac{d\hat{\rho}}{dt} = -i\left[\hat{H}\_{int}, \hat{\rho}\right] - \frac{1}{2}k\_{\mathcal{S}}\left(\hat{\rho}\hat{Q}\_{\mathcal{S}} + \hat{Q}\_{\mathcal{S}}\hat{\rho}\right),\tag{2}$$

where <sup>Q</sup>^ <sup>S</sup> <sup>¼</sup> <sup>∥</sup><sup>S</sup> >< <sup>S</sup><sup>∥</sup> is the projection operator for the singlet state. The yield of recombination calculated from the singlet state of the RP can be evaluated by [9]:

$$\boldsymbol{\Phi}\_{\rm S} = \mathbf{k}\_{\rm S} \int\_{0}^{\infty} \mathrm{Tr} \Big[ \hat{\mathbf{Q}}\_{\rm S} \hat{\boldsymbol{\rho}}(\mathbf{t}) \Big] d\mathbf{t},\tag{3}$$

In fact, with Eq. (3), we evaluate the effect of MF on the yields of the diamagnetic products involved, and in those RP that do not participate in the recombination process. Furthermore, in the exponential approach, the Φ<sup>S</sup> represents the effect of all reencounter times for the reencounter probability of a diffusive geminate RP when it describes the time evolution after their formation, and <sup>τ</sup>�<sup>1</sup> <sup>¼</sup> kS is the average reencounter time when kS <sup>¼</sup> kT [9]. We use as an initial condition the fact that the population is born in a singlet state ∥Ψð Þ t ¼ 0 >¼ ∥S >. The evolution time wave function reads <sup>∥</sup>Ψð Þ<sup>t</sup> <sup>&</sup>gt;<sup>¼</sup> <sup>P</sup> <sup>n</sup> APn ð Þt ∥Pn > þ ASð Þt ∥S > þA<sup>T</sup><sup>0</sup> ð Þt ∥T<sup>0</sup> >. In this sense, the quantum measurements [92] give us the formation of products and then the effect of singlets in the hepatocytes with an intensity <sup>∥</sup>ASð Þ<sup>t</sup> <sup>∥</sup><sup>2</sup> [9, 79]. With use of the spinbased quantum mechanical model, we perform the calculation of singlet spin population and determine the MFE, obtaining a result of 61% compared with the experimental findings of 56% and 58%. Evaluating the quantum yield for the RP intermediaries in the substrate-product system of the CYP450, it is interesting to illustrate the cytoprotective mechanism, which consists of the diminution of the singlet population, responsible for diminishing the number of initiated cells, and, therefore, the preneoplastic lesion formation. To study the spin population behavior of the system, we diagonalize the interaction spin Hamiltonian in the superstate representation, applying the Lanczos method [9]. Beyond the mathematical model, the biology of the problem concerns the action of the EMF on RP affecting the hepatocytes during the enzymatic procarcinogen activation of the CYP450, precisely modulating the charges that are in migration during the electron transfer reactions generated by the interaction of the CYP450 in their substrate-producing electrophilic species and ROS. The intermediaries generated during this process are the source of the first insult to hepatocytes on their way to becoming preneoplastic lesions in the ChemIndHep protocol. We used three assumptions: (1) the ChemReac are spin selective, (2) ChemReac are nuclear spin selective, and (3) ChemReac are selective with the spin of the electron. Under such circumstances, only the reactions with singlets favored the formation of standard molecules. The reactions with triplet RP are forbidden. Under this outline, the spin of the electron controls the generation of the magnetic spin effects [9]. One of the keys of the model is to consider the role of the enzymatic protein as a molecular motor, catalyzing electrons to the reaction, where the RP is generated into the substrate of CYP450 when it is metabolizing the xenobiotics used in the MHRM. The hepatocytes are in contact with the RP in the liver as in a thermal bath, and we assume a Gibbsian distribution, interacting harmonically with them. This strategy includes the very tough dissipation problem, whose quantization process involves some difficulties. To do this, we employed the Caldera–Legget model [93, 94], which explicitly includes it. We use the path integral method in the Feynman–Vernon functional approach to describe the time evolution of the spin population of the system. Also, we employ the influence-functional technique to incorporate the Brownian motion at any temperature. Thus, our system can be considered close and we can apply the traditional quantization method [93]. We consider conservation of energy, and to give an exact treatment of the quantum dissipation dynamics, we use the hierarchical equation of motion, which is more tractable from a numerical point of view. We use for simplicity, Hp <sup>¼</sup> <sup>ω</sup>osc^<sup>b</sup> † ^b modeling the ChemRec and consider the thermal bath as a set of harmonic oscillators with equally spaced energy levels at the frequency ωosc. This expresses the complete system to study as <sup>H</sup>^ Sb <sup>¼</sup> <sup>P</sup> αμ ^a<sup>þ</sup> <sup>μ</sup> <sup>F</sup>^� αμ <sup>þ</sup> <sup>F</sup>^<sup>þ</sup> αμ ^a� μ � �, representing ^a<sup>þ</sup> <sup>μ</sup> � ^a† <sup>μ</sup> ^a� <sup>μ</sup> � ^a<sup>μ</sup> � �, the creation (annihilation) operator of the electron in some specified spin-orbit state. Moreover,

the bath operators F^�

<sup>r</sup>^cðÞ¼ <sup>t</sup> <sup>U</sup>^ <sup>0</sup>ð Þ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>r</sup>^cð Þ<sup>t</sup> <sup>U</sup>^ †

4.2.3. Results on hepatocytes

represented according to [9, 79]:

<sup>r</sup>^ð Þ¼ <sup>0</sup> <sup>∥</sup>AS∥<sup>2</sup> ASA<sup>∗</sup>

AT0A<sup>∗</sup>

T0

<sup>S</sup> ∥AT0∥<sup>2</sup>

!

density functions Jαμνð Þ¼ ω π

by the reaction operator <sup>R</sup>^ <sup>S</sup> <sup>¼</sup> <sup>α</sup>^<sup>b</sup>

<sup>r</sup><sup>T</sup>0T<sup>0</sup> ðÞ¼ <sup>t</sup> <sup>r</sup><sup>T</sup>0T<sup>0</sup> ð Þ<sup>0</sup> , <sup>r</sup>ST<sup>0</sup> ðÞ¼ <sup>t</sup> <sup>r</sup>ST<sup>0</sup> ð Þ<sup>0</sup> <sup>e</sup>�kS

matic reaction rPPðÞ¼� t rSS ½ � ð Þ� t rSSð Þ0 [9].

αμ <sup>¼</sup> <sup>P</sup>

<sup>k</sup>tαμ<sup>k</sup> ^aα<sup>k</sup> <sup>¼</sup> <sup>F</sup>^αμ � �†

the singlet state. To evaluate the recombination process, which is responsible for the time evolution (see details in Ref. [9]), we express the Liouville equation (Eq. 2) in the quantum interaction

Cytoprotective Effect of 120 Hz Electromagnetic Fields on Early Hepatocarcinogenesis: Experimental…

the system. Formally, we evaluate all operator quantities involved in the interaction representation. Once we calculate the thermal bath's degree of freedom contributions and apply the detailed balance principle, we arrive at the solution to the Haberkorn approach, <sup>r</sup>SSðÞ¼ <sup>t</sup> <sup>r</sup>SSð Þ<sup>0</sup> <sup>e</sup>�kSt

From the last equation, we can present the form in which products are formed. We suppose that in the initial process at t ¼ 0, rPPð Þ¼ 0 0, there are no damaged hepatocytes. The behavior of the generation of damaged hepatocytes will depend on the initial singlet spin population rSSð Þ0 , which means that the medium absorbs all that is produced by the enzy-

We described the effect through the following dynamical mapping [9, 79] ∥S> ∥0 >Hep ! η1∥P > ∥ σ >Hep þ η2∥S > ∥0 >Hep, ∥T<sup>0</sup> > ∥0 >Hep ! ∥T<sup>0</sup> > ∥0 >Hep: When we normalize it, measurement of the generation of some reaction product on the hepatocytes ∥P > is

> η∗ <sup>2</sup>AT0A<sup>∗</sup>

This fact means that the hepatocytes measure the spin nature of the RP that is participating. The spin state of the hepatocyte changes according to the spin nature of the RP that is interacting. During the measurement process, the S and T<sup>0</sup> components do not change, but the hepatocytes change their spin states according to the RP spin character following the dynamical mapping [9, 79] ∥S > ∥0 >Hep ! β1∥S > ∥σ >Hep þ β2∥S > ∥0 >Hep, ∥T<sup>0</sup> > ∥0 >Hep

where η<sup>S</sup> and η<sup>T</sup><sup>0</sup> give us the strength of the interaction of the hepatocytes with the RP spin character, appearing in the new term, β3∥T<sup>0</sup> > ∥χ>Hep, without ∥P > states. In this case, the probabilities <sup>∥</sup>β1∥<sup>2</sup> or <sup>∥</sup>β3∥<sup>2</sup> express the appearance of <sup>∥</sup><sup>S</sup> <sup>&</sup>gt; or <sup>∥</sup>T<sup>0</sup> <sup>&</sup>gt; spin states, represented by

! <sup>r</sup>^<sup>n</sup> <sup>¼</sup> <sup>β</sup> <sup>∥</sup>η1∥<sup>2</sup>

! β3∥T<sup>0</sup> > ∥χ>Hep þ β4∥T<sup>0</sup> > ∥0>Hep, instituted by the interaction [9, 79]:

<sup>⨂</sup> <sup>∥</sup>χ>Hep <sup>&</sup>lt; <sup>0</sup><sup>∥</sup> <sup>þ</sup> <sup>∥</sup>0>Hep <sup>&</sup>lt; <sup>χ</sup><sup>∥</sup> � �

<sup>H</sup>^ in <sup>¼</sup> <sup>η</sup>S∥<sup>S</sup> >< <sup>S</sup><sup>∥</sup> � �<sup>⨂</sup> <sup>∥</sup>σ>Hep <sup>&</sup>lt; <sup>0</sup><sup>∥</sup> <sup>þ</sup> <sup>∥</sup>0>Hep <sup>&</sup>lt; <sup>σ</sup><sup>∥</sup> � � <sup>þ</sup> <sup>η</sup><sup>T</sup><sup>0</sup>

, <sup>r</sup><sup>T</sup>0<sup>S</sup>ðÞ¼ <sup>t</sup> <sup>r</sup><sup>T</sup>0<sup>S</sup>ð Þ<sup>0</sup> <sup>e</sup>�kS

2 t

P <sup>k</sup>tαμkt ∗

representation, involving all terms of the system [9] <sup>d</sup>r^<sup>c</sup> ð Þ<sup>t</sup>

†

, whose influence is characterized by bath spectral

http://dx.doi.org/10.5772/intechopen.78642

dt ¼ �<sup>i</sup> <sup>H</sup>^ <sup>T</sup> <sup>þ</sup> <sup>H</sup>^ <sup>P</sup> <sup>þ</sup> <sup>R</sup>^ <sup>S</sup>; <sup>r</sup>^cð Þ<sup>t</sup>

, <sup>r</sup>PPðÞ¼ <sup>t</sup> <sup>r</sup>SSð Þ<sup>0</sup> <sup>1</sup> � <sup>e</sup>�kSt � ).

h i, where

,

55

αν<sup>k</sup>δ ωð Þ � ωosc . In our approach such terms are expressed

<sup>N</sup>^ � <sup>þ</sup> <sup>α</sup><sup>∗</sup>^bN^ <sup>þ</sup>, representing the spin-selective recombination of

<sup>0</sup>, <sup>H</sup>^ <sup>0</sup> <sup>¼</sup> <sup>H</sup>^ <sup>T</sup> <sup>þ</sup> <sup>H</sup>^ <sup>P</sup>, and <sup>U</sup>^ <sup>0</sup>ð Þ¼ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>e</sup>�iH^ <sup>0</sup>ð Þ <sup>t</sup>�t<sup>0</sup> is the evolution operator of

2 t

<sup>∥</sup>AS∥<sup>2</sup> <sup>η</sup>2ASA<sup>∗</sup>

!

<sup>S</sup> ∥AT0∥<sup>2</sup>

T0

, <sup>β</sup> <sup>¼</sup> <sup>1</sup>

∥T<sup>0</sup> >< T0∥ n o

<sup>1</sup> � <sup>∥</sup>η1AS∥<sup>2</sup> : (4)

(5)

the bath operators F^� αμ <sup>¼</sup> <sup>P</sup> <sup>k</sup>tαμ<sup>k</sup> ^aα<sup>k</sup> <sup>¼</sup> <sup>F</sup>^αμ � �† , whose influence is characterized by bath spectral density functions Jαμνð Þ¼ ω π P <sup>k</sup>tαμkt ∗ αν<sup>k</sup>δ ωð Þ � ωosc . In our approach such terms are expressed by the reaction operator <sup>R</sup>^ <sup>S</sup> <sup>¼</sup> <sup>α</sup>^<sup>b</sup> † <sup>N</sup>^ � <sup>þ</sup> <sup>α</sup><sup>∗</sup>^bN^ <sup>þ</sup>, representing the spin-selective recombination of the singlet state. To evaluate the recombination process, which is responsible for the time evolution (see details in Ref. [9]), we express the Liouville equation (Eq. 2) in the quantum interaction representation, involving all terms of the system [9] <sup>d</sup>r^<sup>c</sup> ð Þ<sup>t</sup> dt ¼ �<sup>i</sup> <sup>H</sup>^ <sup>T</sup> <sup>þ</sup> <sup>H</sup>^ <sup>P</sup> <sup>þ</sup> <sup>R</sup>^ <sup>S</sup>; <sup>r</sup>^cð Þ<sup>t</sup> h i, where <sup>r</sup>^cðÞ¼ <sup>t</sup> <sup>U</sup>^ <sup>0</sup>ð Þ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>r</sup>^cð Þ<sup>t</sup> <sup>U</sup>^ † <sup>0</sup>, <sup>H</sup>^ <sup>0</sup> <sup>¼</sup> <sup>H</sup>^ <sup>T</sup> <sup>þ</sup> <sup>H</sup>^ <sup>P</sup>, and <sup>U</sup>^ <sup>0</sup>ð Þ¼ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>e</sup>�iH^ <sup>0</sup>ð Þ <sup>t</sup>�t<sup>0</sup> is the evolution operator of the system. Formally, we evaluate all operator quantities involved in the interaction representation. Once we calculate the thermal bath's degree of freedom contributions and apply the detailed balance principle, we arrive at the solution to the Haberkorn approach, <sup>r</sup>SSðÞ¼ <sup>t</sup> <sup>r</sup>SSð Þ<sup>0</sup> <sup>e</sup>�kSt , <sup>r</sup><sup>T</sup>0T<sup>0</sup> ðÞ¼ <sup>t</sup> <sup>r</sup><sup>T</sup>0T<sup>0</sup> ð Þ<sup>0</sup> , <sup>r</sup>ST<sup>0</sup> ðÞ¼ <sup>t</sup> <sup>r</sup>ST<sup>0</sup> ð Þ<sup>0</sup> <sup>e</sup>�kS 2 t , <sup>r</sup><sup>T</sup>0<sup>S</sup>ðÞ¼ <sup>t</sup> <sup>r</sup><sup>T</sup>0<sup>S</sup>ð Þ<sup>0</sup> <sup>e</sup>�kS 2 t , <sup>r</sup>PPðÞ¼ <sup>t</sup> <sup>r</sup>SSð Þ<sup>0</sup> <sup>1</sup> � <sup>e</sup>�kSt � ).

From the last equation, we can present the form in which products are formed. We suppose that in the initial process at t ¼ 0, rPPð Þ¼ 0 0, there are no damaged hepatocytes. The behavior of the generation of damaged hepatocytes will depend on the initial singlet spin population rSSð Þ0 , which means that the medium absorbs all that is produced by the enzymatic reaction rPPðÞ¼� t rSS ½ � ð Þ� t rSSð Þ0 [9].

#### 4.2.3. Results on hepatocytes

the exponential approach, the Φ<sup>S</sup> represents the effect of all reencounter times for the reencounter probability of a diffusive geminate RP when it describes the time evolution after their formation, and <sup>τ</sup>�<sup>1</sup> <sup>¼</sup> kS is the average reencounter time when kS <sup>¼</sup> kT [9]. We use as an initial condition the fact that the population is born in a singlet state ∥Ψð Þ t ¼ 0 >¼ ∥S >. The

this sense, the quantum measurements [92] give us the formation of products and then the effect of singlets in the hepatocytes with an intensity <sup>∥</sup>ASð Þ<sup>t</sup> <sup>∥</sup><sup>2</sup> [9, 79]. With use of the spinbased quantum mechanical model, we perform the calculation of singlet spin population and determine the MFE, obtaining a result of 61% compared with the experimental findings of 56% and 58%. Evaluating the quantum yield for the RP intermediaries in the substrate-product system of the CYP450, it is interesting to illustrate the cytoprotective mechanism, which consists of the diminution of the singlet population, responsible for diminishing the number of initiated cells, and, therefore, the preneoplastic lesion formation. To study the spin population behavior of the system, we diagonalize the interaction spin Hamiltonian in the superstate representation, applying the Lanczos method [9]. Beyond the mathematical model, the biology of the problem concerns the action of the EMF on RP affecting the hepatocytes during the enzymatic procarcinogen activation of the CYP450, precisely modulating the charges that are in migration during the electron transfer reactions generated by the interaction of the CYP450 in their substrate-producing electrophilic species and ROS. The intermediaries generated during this process are the source of the first insult to hepatocytes on their way to becoming preneoplastic lesions in the ChemIndHep protocol. We used three assumptions: (1) the ChemReac are spin selective, (2) ChemReac are nuclear spin selective, and (3) ChemReac are selective with the spin of the electron. Under such circumstances, only the reactions with singlets favored the formation of standard molecules. The reactions with triplet RP are forbidden. Under this outline, the spin of the electron controls the generation of the magnetic spin effects [9]. One of the keys of the model is to consider the role of the enzymatic protein as a molecular motor, catalyzing electrons to the reaction, where the RP is generated into the substrate of CYP450 when it is metabolizing the xenobiotics used in the MHRM. The hepatocytes are in contact with the RP in the liver as in a thermal bath, and we assume a Gibbsian distribution, interacting harmonically with them. This strategy includes the very tough dissipation problem, whose quantization process involves some difficulties. To do this, we employed the Caldera–Legget model [93, 94], which explicitly includes it. We use the path integral method in the Feynman–Vernon functional approach to describe the time evolution of the spin population of the system. Also, we employ the influence-functional technique to incorporate the Brownian motion at any temperature. Thus, our system can be considered close and we can apply the traditional quantization method [93]. We consider conservation of energy, and to give an exact treatment of the quantum dissipation dynamics, we use the hierarchical equation of motion, which is more tractable from a numerical point of view. We use for

^b modeling the ChemRec and consider the thermal bath as a set of

, representing ^a<sup>þ</sup>

<sup>μ</sup> � ^a†

<sup>μ</sup> ^a� <sup>μ</sup> � ^a<sup>μ</sup> � �

, the

αμ ^a� μ

harmonic oscillators with equally spaced energy levels at the frequency ωosc. This expresses the

creation (annihilation) operator of the electron in some specified spin-orbit state. Moreover,

� �

αμ ^a<sup>þ</sup> <sup>μ</sup> <sup>F</sup>^� αμ <sup>þ</sup> <sup>F</sup>^<sup>þ</sup>

<sup>n</sup> APn ð Þt ∥Pn > þ ASð Þt ∥S > þA<sup>T</sup><sup>0</sup> ð Þt ∥T<sup>0</sup> >. In

evolution time wave function reads <sup>∥</sup>Ψð Þ<sup>t</sup> <sup>&</sup>gt;<sup>¼</sup> <sup>P</sup>

54 Vitamin E in Health and Disease

simplicity, Hp <sup>¼</sup> <sup>ω</sup>osc^<sup>b</sup>

†

complete system to study as <sup>H</sup>^ Sb <sup>¼</sup> <sup>P</sup>

We described the effect through the following dynamical mapping [9, 79] ∥S> ∥0 >Hep ! η1∥P > ∥ σ >Hep þ η2∥S > ∥0 >Hep, ∥T<sup>0</sup> > ∥0 >Hep ! ∥T<sup>0</sup> > ∥0 >Hep: When we normalize it, measurement of the generation of some reaction product on the hepatocytes ∥P > is represented according to [9, 79]:

$$\hat{\rho}(0) = \begin{pmatrix} \|A\_{\mathcal{S}}\|^2 & A\_{\mathcal{S}}A\_{T\_0}^\* \\ A\_{T\_0}A\_{\mathcal{S}}^\* & \|A\_{T\_0}\|^2 \end{pmatrix} \rightarrow \hat{\rho}\_n = \beta \begin{pmatrix} \|\eta\_1\|^2 \|A\_{\mathcal{S}}\|^2 & \eta\_2 A\_{\mathcal{S}} A\_{T\_0}^\* \\ \eta\_2^\* A\_{T\_0} A\_{\mathcal{S}}^\* & \|A\_{T\_0}\|^2 \end{pmatrix}, \beta = \frac{1}{1 - \|\eta\_1 A\_{\mathcal{S}}\|^2}. \tag{4}$$

This fact means that the hepatocytes measure the spin nature of the RP that is participating. The spin state of the hepatocyte changes according to the spin nature of the RP that is interacting. During the measurement process, the S and T<sup>0</sup> components do not change, but the hepatocytes change their spin states according to the RP spin character following the dynamical mapping [9, 79] ∥S > ∥0 >Hep ! β1∥S > ∥σ >Hep þ β2∥S > ∥0 >Hep, ∥T<sup>0</sup> > ∥0 >Hep ! β3∥T<sup>0</sup> > ∥χ>Hep þ β4∥T<sup>0</sup> > ∥0>Hep, instituted by the interaction [9, 79]:

$$\begin{split} \hat{H}\_{\text{in}} &= \left\{ \eta\_{\text{S}} \| \mathbb{S} > < \mathbb{S} \| \right\} \otimes \left[ \| \sigma >\_{\text{HP}} < 0 \| + \| 0 >\_{\text{HP}} < \sigma \| \right] + \left\{ \eta\_{T\_0} \| T\_0 > < T\_0 \| \right\} \\ &\quad \otimes \left[ \| \chi >\_{\text{HP}} < 0 \| + \| 0 >\_{\text{HP}} < \chi \| \right] \end{split} \tag{5}$$

where η<sup>S</sup> and η<sup>T</sup><sup>0</sup> give us the strength of the interaction of the hepatocytes with the RP spin character, appearing in the new term, β3∥T<sup>0</sup> > ∥χ>Hep, without ∥P > states. In this case, the probabilities <sup>∥</sup>β1∥<sup>2</sup> or <sup>∥</sup>β3∥<sup>2</sup> express the appearance of <sup>∥</sup><sup>S</sup> <sup>&</sup>gt; or <sup>∥</sup>T<sup>0</sup> <sup>&</sup>gt; spin states, represented by

∥σ > and ∥χ > [9]. They measure the fraction of singlets transformed into a reaction product or transformed cell. It is precisely this kind of product formation that changes the electronic configuration of the hepatocytes, as we claim. The result is evident from Figure 2, where we note that by increasing the hyperfine coupling constant, A, the singlet population is diminished. Thus, once the hepatocytes interact during δt with the CYP450, they change their spin states through the application of the selective spin operator. For each time interval, δt will apply some dynamical mapping to each new healthy hepatocyte in the tissue ∥0 >en, incorporating them into the enzymatic reaction that provides catalyzing electrons during the metabolization of xenobiotics [9], <sup>∥</sup>Ψð Þ <sup>δ</sup><sup>t</sup> <sup>&</sup>gt;s en <sup>¼</sup> <sup>U</sup>ð Þ <sup>δ</sup><sup>t</sup> <sup>∥</sup>Ψð Þ <sup>δ</sup><sup>t</sup> <sup>&</sup>gt;<sup>0</sup> <sup>¼</sup> <sup>e</sup>�iH^ in <sup>δ</sup>t∥ϕ0>s∥<sup>0</sup> <sup>&</sup>gt;en: Also, neglecting the memory effects due to the previous results with other singlets, which are diminishing as is evidenced by the quantum measurement <sup>∥</sup>β1∥<sup>2</sup> <sup>¼</sup> kSδt, changes the electronic configuration of the hepatocytes when the RP is converted in a reaction product by the recombination kinetics [9, 79].

Cytoprotective Effect of 120 Hz Electromagnetic Fields on Early Hepatocarcinogenesis: Experimental…

http://dx.doi.org/10.5772/intechopen.78642

57

We studied the ChemIndHep experimentally through the use of MRHM. We found that the periodical application with ELF-EMF (4.5 mT (120 HZ)) inhibits by more than 50% the number and area of GST-P liver foci and preneoplastic liver lesions in rats. Through the reduction of cell proliferation and without alteration of the apoptosis process, ELF-EMF interferes with the altered cell cycle continuity and DNA synthesis induced by ChemIndHep. Theoretically, we found that with the use of a 120 Hz ELF-EMF it was possible to achieve modulation of the magnetic-sensitive short-lived RP intermediaries produced by the OS, generated when xenobiotics are metabolized during the catalytic cycle by the CYP450 in the early ChemIndHep. The process of modulation uses the competitive kinetics of the RP selective reactions [71], giving the role of a molecular motor to the enzymatic protein [9]. To achieve this, we studied a quantum mechanics model to describe the interaction of RP/hepatocytes following the typical Haberkorn approach studying the spin dynamics and employing the path integral method and second quantization. The idea of this chapter was to obtain an explanation of the way in which hepatocytes can modify their electronic structure when interacting with the RP, which comes from the enzymatic reaction. Although it is an in-depth mathematical model, the results of our research provide us with further details of the MF's control on BS, specifically in the ChemReac that modulates the electrons in OS (cytoprotective effect) generated in the reactive hepatocytes-

RP system in the liver, with the outcome of understanding hepatocarcinogenesis.

One of the authors (JJGN) would like to acknowledge the financial support of the Brazilian Agency PNPS/CAPES by the Programa Nacional de Pós-Doutorado in UESC-BA-Brazil.

5. Conclusions

Acknowledgements

Figure 2. (a) Singlet population normalized with r<sup>4</sup> for A (=0.1): With the range of values of (γ1,J) (a1) [(10,10), (0,3)], (a2) [(1,1), (0,1)], (a3) [(0.5,0.5), (5,5)]. (b) Singlet population normalized with r<sup>4</sup> for A (=1): With the range of values of (γ1,J ) (b1) [(10,10), (0,3)], (b2) [(1,1), (0,1)], (b3) [(0.5,0.5), (0.1,0.1)], (b4) [(0,5,0.5), (0.01,0.01)], (b5) [(0.5,0.5), (0.001,0.001)]. (c) Singlet population normalized with r<sup>4</sup> for A (=10): With the range of values of (γ1,J ) (c1) [(10,10), (0,3)], (c2) [(5,5), (0,1)], (c3) [(2,2), (0,1)] (from reference [9]).

∥σ > and ∥χ > [9]. They measure the fraction of singlets transformed into a reaction product or transformed cell. It is precisely this kind of product formation that changes the electronic configuration of the hepatocytes, as we claim. The result is evident from Figure 2, where we note that by increasing the hyperfine coupling constant, A, the singlet population is diminished. Thus, once the hepatocytes interact during δt with the CYP450, they change their spin states through the application of the selective spin operator. For each time interval, δt will apply some dynamical mapping to each new healthy hepatocyte in the tissue ∥0 >en, incorporating them into the enzymatic reaction that provides catalyzing electrons during the metabolization of xenobiotics [9], <sup>∥</sup>Ψð Þ <sup>δ</sup><sup>t</sup> <sup>&</sup>gt;s en <sup>¼</sup> <sup>U</sup>ð Þ <sup>δ</sup><sup>t</sup> <sup>∥</sup>Ψð Þ <sup>δ</sup><sup>t</sup> <sup>&</sup>gt;<sup>0</sup> <sup>¼</sup> <sup>e</sup>�iH^ in <sup>δ</sup>t∥ϕ0>s∥<sup>0</sup> <sup>&</sup>gt;en: Also, neglecting the memory effects due to the previous results with other singlets, which are diminishing as is evidenced by the quantum measurement <sup>∥</sup>β1∥<sup>2</sup> <sup>¼</sup> kSδt, changes the electronic configuration of the hepatocytes when the RP is converted in a reaction product by the recombination kinetics [9, 79].
