# Non-Equilibrium Thermodynamic Approach to Ca\({}^{2+}\)-Fluxes in Cancer
###### Abstract  
Living systems waste heat in their environment. This is the measurable effect of the irreversibility of the biophysical and biochemical processes fundamental to their life. Non-equilibrium thermodynamics allows us to analyse the ion fluxes through the cell membrane, and to relate them to the membrane electric potential, in order to link this to the biochemical and biophysical behaviour of the living cells. This is particularly interesting in relation to cancer, because it could represent a new viewpoint, in order to develop new possible anticancer therapies, based on the thermoelectric behaviour of cancer itself. Here, we use a new approach, recently introduced in thermodynamics, in order to develop the analysis of the ion fluxes, and to point out consequences related to the membrane electric potential, from a thermodynamic viewpoint. We show how any increase in the cell temperature could generate a decrease in the membrane electric potential, with a direct relation between cancer and inflammation. Moreover, a thermal threshold, for the cell membrane electric potential gradient, has been obtained, and related to the mitotic activity. Finally,we obtained the external surface growth of the cancer results related (i) to the Ca\({}^{2+}\)-fluxes, (ii) to the temperature difference between the the system and its environment, and (iii) to the chemical potential of the ion species.  
biophysics; cancer; non-equilibrium thermodynamics; heat and ions fluxes; transport theory; thermodynamics of biosystems +
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a fundamental second messenger, involved in a variety of cellular processes, such as proliferation and apoptosis.  
Considering the experimental evidence [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31], tumour progression has also been related to the accumulation of some alterations in the Ca\({}^{2+}\) signal, which inhibits its cytotoxic activity [32].  
Calcium signal modulation can change cells' sensitivity to signals [15; 32]. High levels of mitochondrial Ca\({}^{2+}\) concentration for a long time, have been shown to induce the mitochondrial permeability transition pore, a pathological and physiological phenomenon, discovered over 40 years ago, and still not completely understood [15; 33]. The mitochondrial permeability transition pore causes the formation of a non-specific channel within the inner mitochondrial membrane, useful for Ca\({}^{2+}\) release and metabolite exchange, between the mitochondrial matrix and cytosol. However, a prolonged mitochondrial permeability transition pore causes changes in the inner mitochondrial membrane potential, cessation of ATP synthesis, bioenergetic crisis, and apoptotic or necrotic cell death [14; 16; 33].  
All this experimental evidence moves the research interest towards the analysis of the Ca\({}^{2+}\)-fluxes. At present, most of the mechanisms related to intracellular Ca\({}^{2+}\) responses have been understood by developing in vitro experiments, but comprehension of the physiological role of these processes, in relation to tumour environment, remains an problem [32].  
In this paper, we wish to develop a new viewpoint in the analysis of ion fluxes, recently published in [34], based on thermodynamics, with particular regards to the non-equilibrium thermodynamics. Our aim is to suggest an approach which takes into account the ion fluxes, in relation to the membrane electric potential gradient, in order to analytically describe the link between ion fluxes and membrane potential, in relation to cancer behaviour. We will focus our analysis on Ca\({}^{2+}\) fluxes, due to the fundamental role of this ion in the regulation of a great number of cell functions.