A cellular automaton model examining the effects of oxygen, hydrogen ions and lactate on early tumour growth.

Some tumours are known to exhibit an extracellular pH that is more acidic than the intracellular, creating a 'reversed pH gradient' across the cell membrane and this has been shown to affect their invasive and metastatic potential. Tumour hypoxia also plays an important role in tumour development and has been directly linked to both tumour morphology and aggressiveness. In this paper, we present a hybrid mathematical model of intracellular pH regulation that examines the effect of oxygen and pH on tumour growth and morphology. In particular, we investigate the impact of pH regulatory mechanisms on the cellular pH gradient and tumour morphology. Analysis of the model shows that: low activity of the Na+/H+ exchanger or a high rate of anaerobic glycolysis can give rise to a "fingering" tumour morphology; and a high activity of the lactate/H+ symporter can result in a reversed transmembrane pH gradient across a large portion of the tumour mass. Also, the reversed pH gradient is spatially heterogeneous within the tumour, with a normal pH gradient observed within an intermediate growth layer within the spheroid. We also include a fractal dimension analysis of the simulated tumour contours, in which we compare the fractal dimensions of the simulated tumour surfaces with those found experimentally via photomicrographs.

Theoretical predictions of lactate and hydrogen ion distributions in tumours.

High levels of lactate and H(+)-ions play an important role in the invasive and metastatic cascade of some tumours. We develop a mathematical model of cellular pH regulation focusing on the activity of the Na(+)/H(+) exchanger (NHE) and the lactate/H(+) symporter (MCT) to investigate the spatial correlations of extracellular lactate and H(+)-ions. We highlight a crucial role for blood vessel perfusion rates in determining the spatial correlation between these two cations. We also predict critical roles for blood lactate, the activity of the MCTs and NHEs on the direction of the cellular pH gradient in the tumour. We also incorporate experimentally determined heterogeneous distributions of the NHE and MCT transporters. We show that this can give rise to a higher intracellular pH and a lower intracellular lactate but does not affect the direction of the reversed cellular pH gradient or redistribution of protons away from the glycolytic source. On the other hand, including intercellular gap junction communication in our model can give rise to a reversed cellular pH gradient and can influence the levels of pH.

Acid-mediated tumour cell invasion: a discrete modelling approach using the extended Potts model.

Acidic extracellular pH has been shown to play a crucial part in the invasive and metastatic cascade of some tumours. In this study, we examine the effect of extracellular acidity on tumour invasion focusing, in particular, on cellular adhesion, proteolytic enzyme activity and cellular proliferation. Our numerical simulations using a cellular Potts model show that, under acidic extracellular pH, changes in cell-matrix adhesion strength has a comparable effect on tumour invasiveness as the increase in proteolytic enzyme activity. We also show that tumour cells cultured under physiological pH tend to be large and the tumours develop a "diffuse" morphology compared to those cultured at acidic pH, which display protruding "fingers" at the advancing front. A key model prediction is the observation that the main effect on invasion from culturing cells at low extracellular pH stems from changes in the intercellular and cell-matrix adhesion strengths and proteolytic enzyme secretion rate. However, we show that the effects of proteolysis needs to be significant as low to moderate changes only has nominal effects on cell invasiveness. We find that the low pH e effects on cell size and proliferation rate have much lower influence on cell invasiveness.

Regulation of tumour intracellular pH: a mathematical model examining the interplay between H+ and lactate.

Non-invasive measurements of pH have shown that both tumour and normal cells have intracellular pH (pHi) that lies on the alkaline side of neutrality (7.1-7.2). However, extracellular pH (pHe) is reported to be more acidic in some tumours compared to normal tissues. Many cellular processes and therapeutic agents are known to be tightly pH dependent which makes the study of intracellular pH regulation of paramount importance. We develop a mathematical model that examines the role of various membrane-based ion transporters in tumour pH regulation, in particular, with a focus on the interplay between lactate and H(+) ions and whether the lactate/H(+) symporter activity is sufficient to give rise to the observed reversed pH gradient that is seen is some tumours. Using linear stability analysis and numerical methods, we are able to gain a clear understanding of the relationship between lactate and H(+) ions. We extend this analysis using perturbation techniques to specifically examine a rapid change in H(+)-ion concentrations relative to variations in lactate. We then perform a parameter sensitivity analysis to explore solution robustness to parameter variations. An important result from our study is that a reversed pH gradient is possible in our system but for unrealistic parameter estimates-pointing to the possible involvement of other mechanisms in cellular pH gradient reversal, for example acidic vesicles, lysosomes, golgi and endosomes.