Metabolic reprogramming dynamics in tumor spheroids: Insights from a multicellular, multiscale model

Mathematical modeling provides the predictive ability to understand the metabolic reprogramming and complex pathways that mediate cancer cells’ proliferation. We present a mathematical model using a multiscale, multicellular approach to simulate avascular tumor growth, applied to pancreatic cancer. The model spans three distinct spatial and temporal scales. At the extracellular level, reaction diffusion equations describe nutrient concentrations over a span of seconds. At the cellular level, a lattice-based energy driven stochastic approach describes cellular phenomena including adhesion, proliferation, viability and cell state transitions, occurring on the timescale of hours. At the sub-cellular level, we incorporate a detailed kinetic model of intracellular metabolite dynamics on the timescale of minutes, which enables the cells to uptake and excrete metabolites and use the metabolites to generate energy and building blocks for cell growth. This is a particularly novel aspect of the model. Certain defined criteria for the concentrations of intracellular metabolites lead to cancer cell growth, proliferation or death. Overall, we model the evolution of the tumor in both time and space. Starting with a cluster of tumor cells, the model produces an avascular tumor that quantitatively and qualitatively mimics experimental measurements of multicellular tumor spheroids. Through our model simulations, we can investigate the response of individual intracellular species under a metabolic perturbation and investigate how that response contributes to the response of the tumor as a whole. The predicted response of intracellular metabolites under various targeted strategies are difficult to resolve with experimental techniques. Thus, the model can give novel predictions as to the response of the tumor as a whole, identifies potential therapies to impede tumor growth, and predicts the effects of those therapeutic strategies. In particular, the model provides quantitative insight into the dynamic reprogramming of tumor cells at the intracellular level in response to specific metabolic perturbations. Overall, the model is a useful framework to study targeted metabolic strategies for inhibiting tumor growth.

Model-simulated tumor growth profile using the optimized parameter set, initiating with the five different cell cluster configurations. Ten iterations for each case were simulated. The black solid line is the mean of all 50 simulations, and the shaded grey area represents the standard deviation of the simulations. Squares represent published experimental tumor spheroid data obtained from a prostate cancer cell line (PC3) [5] and two different glioma cell lines (AN1 and U-87) [6].  Description of parameters given in Table 1 of the main text 1. incvol : The rate at which cells increase their volume to proliferate, a property of only proliferating cells (PCancer and PStem). The higher the rate, the greater the increase in the cell's volume, leading to a higher rate of proliferation.
2. decvol : The rate at which necrotic cells decrease their volume, mimicking apoptosis.
3. PGrThr : The threshold value for the total concentration of ATP, intracellular glucose and glutamine needed for a proliferating cell to increase its volume.
4. SGrThr : The threshold value for the total concentration of ATP, intracellular glucose and glutamine needed for a stem cell to increase its volume.

StressIncrement:
The value by which the Stress attribute is increased when the required number of neighbors (N ) surround a cell and exert compressive stress.
6. StressThr : The threshold value of the Stress attribute that the cells have to cross to transition into a necrotic cell.
7. N : The number of neighbors that a cell has, which influences the accumulation of Stress.
8. V atpmax : A parameter influencing how cell attributes depend on ATP.
9. atpD: Threshold value below which the Starvation attribute is incremented for the cell, leading to necrosis and above which Health is acquired, leading to the conversion of a quiescent cell to a proliferating cell.
10. P N eT hr: This is the threshold for a proliferating cell to transition into necrotic state due to a lack of ATP.
11. P N eT hr acidosis : This is the threshold for a proliferating cell to transition into necrotic state due to excessive exposure to an acidic environment.
12. T otal time : The total time for which the cells have to be exposed to a low (N eg concAT P ) or high (P os concAT P ) concentration of ATP in order to transition to a necrotic or proliferating cell.
13. N eg concAT P : The low concentration of ATP used in the calculation of PNeThr. A low value of N eg concAT P leads to low value of PNeThr and an easier transition of a proliferating cell to a necrotic cell. Alternatively, a high value of N eg concAT P leads to high value of PNeThr respectively and an harder transition of a proliferating cell to a necrotic cell.
14. P os concAT P : Total amount of intracellular ATP, glucose and glutamine used in the calculation of QCPThr and QSSThr, which affects the transition from a quiescent cell to a proliferating cell.
15. V lacmax : A parameter for attribute factor calculation dependent on lactate.
16. LacDeath: Threshold value of lactate concentration, above which the Starvation attribute is incremented for the cell, leading to necrosis.
17. P os concLac : The high concentration of lactate used in the calculation of P N eT hr acidosis .
May 19, 2019 7/8 18. T otal timelac : This is the total time for which the cells have to exposed to high (P os concLac ) concentration of lactate to transition to necrotic state due to acidosis.
19. maxdiv : Maximum number of divisions the cell can have before senescence sets in and the cell transitions to a necrotic cell.

21.
C : This measures the amount of contribution of nutrients towards the accumulation of Health. Higher C values leads to faster accumulation of Health and an easier transition from a quiescent cell to a proliferating cell.
22. a: The Hill coefficient representing the cooperativity between nutrients responsible for the growth and transition of a cell.