Toward a gibbsian approach to the problems of growth and cancer

Summary

Certain sections ofJosiah Willard Gibbs's thermodynamics papers might be applicable to biological equilibrium and growth, normal or abnormal.Gibbs added terms⌆ Μ _i dm _i to the differential of the internal energy dε=tdη−pdΝ, (t=temperature,p=pressure,η=entropy,Ν=volume) where$\mu _i = \frac{{\delta \varepsilon }}{{\delta m_i }}$ is the potential of substancem _i , to provide for chemical as well as thermal and mechanical equilibrium. In this article a further generalization is suggested, to include biological equilibrium by adding to de terms of the form GdN, the variableN being the number of cells, where$G = \frac{{\delta \varepsilon }}{{\delta N}}$ is a “growth potential” that measures exactly the resistance toward spontaneous growth. The functionG, likeΜ _i is intensive in nature (i.e. depends on intensive variables only) except for a conversion factor$\frac{{dM}}{{dN}}$,M=⌆m _i , affording possible insight into why incipient abnormal growth is often independent of the number of cells. Useful applications might follow from identities between$\frac{{\delta G}}{{\delta \eta }},\frac{{\delta G}}{{\delta v}}$, or$\frac{{\delta G}}{{\delta m_i }}$ and$\frac{{\delta t}}{{\delta N}}, - \frac{{\delta p}}{{\delta N}}$ or$\frac{{\delta \mu _i }}{{\delta N}}$ respectively. The following new function is studied,$\bar \zeta = \zeta - GN$, a natural generalization of theGibbs free energy function ζ, the possibility of measuring it electrically, and comparison of its role with that of ζ for the possible experimental determination ofG. Gibbs's necessary and sufficient conditions for heterogeneous equilibrium ofn components inm phases are generalized and also modified to include broader restraining conditions like$\mathop \sum \limits_{i = 1}^m \delta N_j (i) \geqslant o$,j=1,f,n, the > being characteristic of only living cellular phases. Careful appraisal of the term “biological stability” is followed by new criteria for stability, instability, and limits of stability, (neutral equilibrium) in terms of derivatives ofG, with possible medical applications. Three different sections of Gibbs's works tend to indicate that, for a biological phase, lower pressure usually increases its stability. The equation$p'' - p' = \sigma \left( {\frac{I}{r} + \frac{I}{{r'}}} \right)$, where σ=surface tension,p′, p′ = pressures,r, r′=radii of curvature, is applied to possible control of tissue growth at interfaces. Methods of altering the equilibrum between three phasesA, B, C by varying the interfacial tensionsσ _AB ,σ _BC ,σ _AC , using relations like _AB <σ _AC + _BC for stability of theA, B interface, suggest different means for shifting biological equilibrium between normal and abnormal cells through the introduction of new third phases at the interface. Various devices are mentioned for possible control of growth through proper channeling of surface or other equivalent forms of energy.