Constructal approach to cell membranes transport: Amending the 'Norton-Simon' hypothesis for cancer treatment

Abstract

To investigate biosystems, we propose a new thermodynamic concept that analyses ion, mass and energy flows across the cell membrane. This paradigm-shifting approach has a wide applicability to medically relevant topics including advancing cancer treatment. To support this claim, we revisit 'Norton-Simon' and evolving it from an already important anti-cancer hypothesis to a thermodynamic theorem in medicine. We confirm that an increase in proliferation and a reduction in apoptosis trigger a maximum of ATP consumption by the tumor cell. Moreover, we find that positive, membrane-crossing ions lead to a decrease in the energy used by the tumor, supporting the notion of their growth inhibitory effect while negative ions apparently increase the cancer's consumption of energy hence reflecting a growth promoting impact. Our results not only represent a thermodynamic proof of the original Norton-Simon hypothesis but, more concretely, they also advance the clinically intriguing and experimentally testable, diagnostic hypothesis that observing an increase in negative ions inside a cell in vitro, and inside a diseased tissue in vivo, may indicate growth or recurrence of a tumor. We conclude with providing theoretical evidence that applying electromagnetic field therapy early on in the treatment cycle may maximize its anti-cancer efficacy.

Figure 1. This figure represents, during mitosis, the ATP flow [mol s⁻¹] for a tumor cell vs. the ATP flow [mol s⁻¹] for a normal, non-cancerous cell as a function of the cell reproduction and death rates , weighted on their maximum values.

These last quantities are non-dimensional values representing the growth and apoptosis rate of cells within a range [0,1]. It can be seen that both, an increase in proliferation and a reduction in apoptosis trigger a maximum of ATP consumption by the tumorous cell.

Figure 2. This figure represents the ratio of energy consumption vs. the volumetric growth of cancer.

The y-axis is the ratio between the energy used by a cancer cell, Q, related to the energy used by a normal, non-cancerous cell, Q₀. The x-axis is the ratio between the volume of cancer V related to its initial value V₀ at time 0. This result has been obtained by using Eq. (29). Note that a cancer cell and therefore tumor tissue consumes much more energy than normal tissue; moreover, the energy required by the cancer is proportional to the exponential tumor growth rate.

Figure 3. The figure represents the volumetric growth of cancer tissue under a magnetic field (50 μT, 40 Hz) in relation to the energy used by the cancer with regards to the ion involved.

The y-axis represents growth, i.e. the ratio between the volume of cancer V related to its initial value V₀. The x-axis displays the ratio (in percentage) in energy consumption used for growth between a cancer cell and a normal, non-cancerous cell; here, Q is the energy used by the cancer cell while Q₀ is the energy consumed by the normal cell; minus (−) and plus (+) are related to the motion of the ions in relation to the membrane’s bioelectric field; consequently, in the case of Cl⁻ inflow the cell gains energy, which in turn can be used for continued growth, while, to process Ca²⁺, K⁺ and Na⁺ inflow, the cell loses energy and thus growth potential as it works against the membrane potential to sustain the influx of these ions. As the tumor grows, all ion fluxes change but their percentage changes are much larger when the tumor is exposed to the magnetic field in (a), reflecting a much more pronounced consumption of ATP to process these fluxes. The inhibitory effect of the magnetic field is evident as the tumor, to achieve the same volumetric growth, is forced to consume much more energy in (a) than in (b) when it is not exposed to a magnetic field. Still, since Cl⁻ influx increases also in (a), particularly at later growth stages, this further supports the argument that magnetic field therapy may lose efficacy eventually (which is being explored in more detail in Fig. 4).

Figure 4. The figure represents the ratio between the energy consumption of a cancer cell inside the magnetic field (50 μT, 40 Hz) and outside of it; the y-axis is the quotient of the x-axis of Fig. 3(a,b), with the x-axis again depicting the volumetric increase of the resulting tumor.

It is evident that tumor tissue consumes much more energy at the early growth stages, which supports the temporary effectiveness of the magnetic field therapy that rapidly diminishes at later, more advanced tumor growth stages, hence advocating for applying this therapeutic modality early on in the treatment cycle (see also Fig. 3). While this treatment response is reflected in the flux patterns of all four ions, it is conceivable to pinpoint the maximum benefit of the magnetic field to the largest ratio between the Na⁺ and Ca²⁺ fluxes which may have diagnostic value (note that the Na⁺ and Cl⁻ fluxes are superimposed in the figure). As to the tumor, based on Eq. (16) and Fig. 1, we argue that rapid cancer cell proliferation dominates early on whereas marked apoptosis impacts the result at later stages. (Note: due to the current setup, we are unable to assign a distinct weight to the relative contributions to [total] energy consumption of intrinsic tumor growth vs. therapeutic impact of the magnetic field, an area that we would like to explore in future works).