Mathematical Modeling Reveals the Role of Hypoxia in the Promotion of Human Mesenchymal Stem Cell Long-Term Expansion

Figure 3

Workflow of parameter fitting via nonlinear regression. First, given initial conditions, we can solve the time-variant ODEs in our model (3) by numerical ODE solvers to get the model predications for time points in interest. Then, the constrained optimization problem (5) for parameter fitting can be tackled with iterative nonlinear optimization algorithms, for instance, the Nelder-Mead simplex search approach (like the fminsearch, fmincon, or lsqcurvefit functions in MATLAB). Here, it should be noted that to avoid the possible bad local minima associated with nonlinear, nonconvex optimization problems, we may need to try multiple initial guesses of the parameter vector . We use a systematic approach based on grid search to coordinate multiple initial value trials [44], thanks to the low dimension and small dataset size in this study. This approach can increase the probability that we find the global minimum or at least a good local minimum close to the global one. A detailed description of the fitting procedures is provided in Supplementary Materials (available here).