^{1}

^{2}

^{2}

^{2}

^{*}

In this paper we will show the possibility of studying physical properties and irreversible phenomena that occur in biological tissues by applying the dielectric Kluitenberg’s non-equilibrium thermodynamic theory. Namely, we shall use some recent extensions of this theory that allows deducing its main characteristic parameters from experimental measurements. We determine frequency spectrum for phenomenological and state coefficients of the non-equilibrium thermodynamic approach. Applying these results to the study of human liver tumor and normal liver we show, for comparison, that it is possible to determine the difference, in some detail, of the amount of single irreversible phenomena occurring inside them.

Liver cancer is the third most common cause of death from cancer worldwide. Hepatocellular carcinoma (HCC), a common cancer that typically occurs in the setting of cirrhosis and chronic hepatitis virus infections, is currently the fifth most common malignancy in men and the eighth in women worldwide [

The lack of distinctive pathological symptoms leads to a significant delay in the diagnosis of HCC; this often results in poor prognosis, with a median survival of a few months [

Multiple advances over the previous 30 years and the severe shortage of liver donors have expanded the indications for hepatic resection as the best treatment of this cancer, but unfortunately the vast majority of patients with liver tumors are not suitable for resection due to multifocal disease, anatomic limitations, inadequate functional liver reserve, extra-hepatic metastasis or medical co-morbidities [

In the last years alternative techniques have been developed, as ablation by radio frequencies, chemoembolization, radioembolization, magnetic chemotherapy and cryo-ablation, for local treatment of the liver tumour [

In particular by means of new microwave ablation (MWA) techniques good results have been obtained. This technique consists in the insertion of a microwave antenna that, by suitable frequencies of electromagnetic wave, causes cellular coagulation necrosis of the tumour.

The potential advantages of microwave technology include consistent production of high intra-tumoral temperatures, fast ablation times, large tumor ablation volumes and the use of multiple antennae; also it does not require the use of grounding pads, which decreases the time required for patient preparation [

However, few data are available on the extent of tumour destruction possible with microwave ablation, which are now used in conventional microwave ovens given optimal heating profiles [

Unfortunately, few studies have reported dielectric properties of tumour and normal liver tissues; the insufficient number of data is an handicap for the development of MWA techniques because physiological behaviour, as perfusion with temperature and water content change, affects dielectric properties of the biological tissue showing a substantial difference between physiological and pathological, and between living and non-living tissue.

Our goal is to deepen the study of dielectric properties of liver samples in the microwave frequency range investigating in the context of non-equilibrium thermodynamics (NET) [

In particular, we shall apply some recent theoretical methodologies in NET [

A central point of the formalism of NET is the definition of appropriate thermodynamic generalized forces (also called affinities) and fluxes in order to fully represent the entropy variation. Generalized forces and fluxes are connected by relationships, called phenomenological equations, which in many cases can be considered linear. The coefficients of those linear relationships are called phenomenological coefficients [

A more deep insight in the underlying phenomena is obtained by the Kluitenberg’s thermodynamic theory [

A strong limitation of the applicability of this theory is that an exact analytical computation of these coefficients is often impossible. However, recently approximate analytical expression of the dependence of the dielectric phenomenological and state coefficients on the frequency have been proposed [

The proposed approach constitutes an advancement with respect to traditional measurements of dielectric (or mechanical) properties. Indeed, standard dielectric (or mechanic) measurements of complex physical systems are only able to obtain complex dielectric constant (or complex Young module) [

We state that our point of view is in agreement with the assertion that physical principles are valid even for biological materials and that the greater difficulty to apply them is the identification of the variables that are to be considered as input in a contest of cause-effect principle. Since we are interested only to dielectric properties of human normal and malignant liver tissue and so to the difference between them we will consider the aforementioned coefficient as a function of the electric harmonic perturbation to which the tissue is subject. In particular as function of the frequency of harmonic perturbations.

The Kluitenberg’s theory [

Generally, the specific entropy “s” will be (for an elastic dielectric) function of the specific internal energy u, the strain tensor ε_{ik} and the specific polarization ^{(1)} and rewrote the entropy as

This lead to the following variables

where

and therefore to the introduction of the electric field E^{(1)} correlated to

By defining:

and the following state equations

where possible cross-effects among dielectric relaxation and other irreversible phenomena were neglected, and it is assumed that the mass density ρ is constant.

L^{(0,0)} and L^{(1,1)} are called phenomenological coefficients, whereas a^{(00)} and a^{(11)} state coefficients. We report the physical meaning to them associated:

a^{(00)} and a^{(11)} reciprocal dielectric constants

L^{(00)} resistance

L^{(11)} conductivity

We emphasize that L^{(0,0)} and L^{(1,1)} are not related to conduction current, but only to displacement one.

Combining Equations (6)-(9) it can be shown that the following differential equation for dielectric relaxation phenomena can be obtained:

where

In that which follows we refer only to one-dimension problem and so we consider only scalar quantity. In agreement with linear response theory [

it follows that E (as effect, intensive variable) vary as

were ϕ(ω) is the phase lag between P and E.

By indicating with Г_{1} e Г_{2} storage and loss dielectric moduli respectively and remembering that they are related to no dissipative and dissipative phenomena respectively we have [

and the following expression for phenomenological and state coefficients as function of the frequency can be obtained:

where σ is the relaxation time and we have introduced a new quantity defined as:

By remembering the meaning of Г_{2} the last equation represents the difference between the total loss modulus and the loss associated to E^{(ir)} (E^{(ir)} and therefore L^{(00)} are not related to internal degree of freedom); in other words Equation (16) represents the dissipation associated to internal degree of freedom.

From 15_{1} and 15_{3} it follows that, if

And from (18) and (7) it follows

and from (4) we obtain:

Moreover from Equation (5) we deduce that E^{(1)} is not defined and therefore a^{(1,1)} is not defined. This means that no internal degree of freedom appears in agreement with physical meaning of

From Equations (8) and (9) we see that

This means that the difference between a^{(0,0)} and a^{(1,1)} is related to the difference between E^{(1)} and E^{(eq)}. In particular if a^{(0,0)} = a^{(1,1)} it follows

We observe that ^{(11)} = 0); this occurs for sufficiently low frequency where internal degree of freedom does not appear. So we can put

It is to note that, although no internal variables appear, we are not in an equilibrium state because Г_{2} = ωL^{(00)} is different from zero.

Now it is easy to see that if Г_{2} = 0 it follows

where T is the temperature (which we have assumed constant). In fact the Equation (30) is the sum of the only two dissipative terms:

1) dissipation related to E^{(ir)} (see Equation (6))

2) dissipation related to int. degree of freedom (see Equation (16))

If both are equal to zero, the system is in equilibrium state with zero entropy production.

In brief:

If

If

It is easy to consider the case in which L^{(00)} = 0; in this case

In this paper the results obtained by O’Rourke et al. [

In the contest of NET [

This new approach enables a more detailed investigation since it allows determining the amount of single phenomena which are hidden in the tissue by means of the relationships, among the coefficients mentioned above and the complex dielectric constant measured.

We indicate the function referred to pathological and normal tissues with the index “p” and “n” respectively. In this section we analyze experimental results for normal and pathological tissue taking into account the theoretical results described above. In particular we will investigate dissipative phenomena and therefore we start by analyzing the modulus G_{2}.

By observing the _{2p} < G_{2n}, which we call “low frequency region”, and the range of w for which it results G_{2p} > G_{2n}, which we call “high frequency region”. By remembering the meaning of the modulus G_{2}, the two curves show a greater dissipation of the pathologic tissue with respect to normal one in the region of high frequency (G_{2p} > G_{2n}); on the contrary a greater dissipation of the normal tissue with respect to pathologic one is shown in the region of low frequency (G_{2p} < G_{2n}).

It is easy to see that this change can be associated to the change of ^{(00)} is constant (see Equations (16) and (23)). In particular:

1)

2)

This means that for low frequencies, for which condition 2) is valid (

On the contrary, for high frequencies, for which condition 1) is valid (

So, in agreement with the entropy production (Equation (24)) we have two types of dissipation phenomena.

1) due to

2) due to ωL^{(00)}

The first is associated to the phenomenological L^{(1,1)} and to the state coefficient a^{(0,0)} (we do not consider a^{(1,1)} because it shows the reciprocal trend with respect to L^{(1,1)} (Equation (15)_{2,3}). The second to phenomenological coefficient L^{(0,0)}.

For that which concern L^{(1,1)} (

By physical meaning of L^{(1,1)} we deduce that dissipation due to conductivity in the region of high frequencies is greater in pathological tissue then normal one.

Moreover it is very important to observe that the increment of

Generally, in the differentiated form, the neoplastic cell have eosinophilic and cuboidal cytoplasm; vesicular nuclei with prominent nucleoli. Tumor cells show an increase in size with inverse nucleus/cytoplasm ratio [

This can be the motivation of the increase of the coefficient L^{(1,1)} and so of the conductivity shown in ^{(0,0)} (^{(00)} = G_{2R} = constant. Moreover

For that which concern a^{(0,0)} we see that _{1}).

It is important to observe that

The behavior of

In brief:

low frequency

high frequency

Finally we analyze the entropy production. Observing

We will conclude by specifying that the hope of this work is to suggest a new point of view based on application of no equilibrium thermodynamics with internal variables (internal degree of freedom) formulated by Kluitenberg and developed, for an experimental approach, by us (in previous papers) [

We are conscious of the difficulties to adapt the mathematical model of the aforementioned theory to the very complex phenomena that occur in biological tissues, but we are sure that the development of these ideas can lead to new results as it has been showed in this paper.

An analogous approach can be used if we investigate the tissues by mean of mechanical perturbation [

Francesco Farsaci,Annamaria Russo,Silvana Ficarra,Ester Tellone, (2015) Dielectric Properties of Human Normal and Malignant Liver Tissue: A Non-Equilibrium Thermodynamics Approach. Open Access Library Journal,02,1-12. doi: 10.4236/oalib.1101395

Hepatocellular carcinoma (HCC);

Microwave ablation (MWA);

Non-equilibrium thermodynamics (NET).