Mathematical Models of Tumour Invasion Mediated by Transformation‐Induced Alteration of Microenvironmental pH

The invasive phenotype, characteristic of malignancy, is found in multiple transformed populations with widely varying genotypes. The authors hypothesize that this genetic heterogeneity and instability precludes mechanisms of tumour invasion requiring consistent, coordinated production of one or more proteins or other macromolecules. Instead, this model assumes that the genetically disordered, unstable populations found in tumours must employ a simple mechanism of invasion that arises from one common trait of all transformed cells--reversion to a phenotype more primitive (less differentiated) than the tissue of origin. Specifically, this approach focuses on primitive metabolic pathways with preferential use of glycolysis for energy production. Mathematical models of invasive cancer based on tumour-induced acidification of the microenvironment are presented. Using population ecology and diffusion-reaction models, it is shown that normal tissue adjacent to the tumour edge is subjected to an extracellular pH that is significantly lower than normal. This leads to degradation of the interstitial matrix, loss of intercellular gap junctions and cell necrosis. Tumour cells have an optimal extracellular pH less than that of normal cells and are thus able to thrive in the acidic microenvironment expanding into the space of the dying normal tissue. The model is consistent with extant data on the tumour microenvironment as well as clinical data relating increasing tumour invasiveness with elevated glucose utilization and lactate content. It predicts well-established phenomena in tumorigenesis such as the adenoma-carcinoma sequence and the critical role of angiogenesis in the invasive phenotype. An acellular gap at the tumour host interface is also predicted and can be demonstrated in pathologic specimens. Novel treatment approaches based on this model are discussed.