Vol.:(0123456789) 1 3 European Biophysics Journal https://doi.org/10.1007/s00249-018-1275-5 BIOPHYSICS LETTER A kinetic view of acid‑mediated tumor invasion Ahmed M. Fouad 1 Received: 17 October 2017 / Revised: 28 December 2017 / Accepted: 2 January 2018 © European Biophysical Societies’ Association 2018 Abstract According to the acid-mediated tumor invasion hypothesis, tumor-induced alteration of microenvironmental pH may provide a simple, yet complete mechanism for tumor invasion. The acid-mediation hypothesis analyzes the tumor growth and invasion process from a reaction–difusion system perspective, where it incorporates the H + ion concentration as a reaction factor and adds density-dependent difusion parameters to the reaction terms, yielding independent reaction–difusion equations for the normal, tumor, and acid populations. In this article, we apply the dynamical stability theory to the acid-mediation hypothesis. For reasonable biological parameters, we study the fxed points central to the model and their stability by calcu- lating the eigenvalues of the Jacobian matrix of the partial diferential equations that represent how these three populations evolve with time. For the case where a malignant behavior has not already taken place yet (the time rates of change of the densities of the three populations are equal to zero), our numerical results convey two diferent, yet possible confgurations in three-dimensional space: stable and unstable dynamical equilibriums, and we discuss possible prospective trajectories for the normal and tumor populations starting from each confguration. Moreover, we discuss potential applications of our approach. Keywords Cancer biology · Dynamical stability theory · Fixed points · Reaction–difusion systems · Acid-mediated tumor invasion Introduction Malignant cells are remarkably heterogeneous and character- istically exhibit altered metabolic patterns due to the critical role of the mutator phenotype during carcinogenesis, where hundreds, thousands, and often hundreds of thousands of genetic mutations are typically exhibited by the transformed genomes (Nowell 1976; Lengauer et al. 1998). Through stud- ies of breast and renal cancers, it has been found that every tumor cell exhibited a novel genotype (Jiang et al. 2000), meaning no prototypical cancer cell exists, and each malig- nant genotype seems to be the result of a unique genetic pathway exhibited during carcinogenesis. The progressive invasion and destruction of normal tissue is a unique growth pattern exhibited by virtually all cancers, leading to death of the host. This paradox of common clinical behavior with its marked genotypic diversity has been examined (Gatenby 1991; Gatenby and Gawlinski 1996), where it is proposed that the similarity of invasive behavior suggests a common underlying mechanism that must be fundamentally related to the few phenotypic traits exhibited by virtually all tumors. Warburg was the frst to establish that tumors consist- ently rely on anaerobic pathways to convert glucose to ATP, even in the presence of abundant oxygen (Warburg 1930). Tumor cells maintain ATP production by increasing the glucose fux, where the anaerobic metabolism of glucose to lactic acid is substantially less efcient than oxidation to CO 2 and H 2 O. The latter forms the basis for tumor imaging with FDG-PET (Haberkorn et al. 1991; Patz et al. 1994). In addition, PET imaging has also demonstrated a direct correlation between tumor aggressiveness (and progno- sis) and the rate of glucose consumption (Haberkorn et al. 1991). The pioneering work of Warburg and others showed that the increased reliance on glycolysis to produce energy occurs even in the presence of abundant oxygen (Warburg et al. 1927; Vaupel et al. 1989) for oxidative phosphoryla- tion, indicating that the metabolic shift from oxidative phos- phorylation to glycolysis (now referred to as the Warburg efect or glycolytic phenotype) is a crucial transformation in tumor metabolism. The underlying causation of the War- burg efect remains mysterious; however, it yields signifcant * Ahmed M. Fouad ahmed.fouad@temple.edu 1 Department of Physics, Temple University, Philadelphia, PA, USA