Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.

In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a first-order Taylor series expansion; Newton-based methods, which use a second-order Taylor series expansion; and the Gauss-Newton method, which iteratively uses multiple linear regressions via a first-order Taylor series expansion. The Levenberg-Marquardt method modifies the Gauss-Newton method, while the Nelder-Mead simplex approach does not involve linearization procedures but instead examines the response surface using a series of moving and contracting or expanding polyhedra.

Algorithms used within NMEM include the FO (First-Order), FOCE (first-order conditional estimation), SAEM (stochastic approximation expectation-maximization), and MLEM (maximum likelihood estimation methods). In the context of NONMEM, FO and FOCE algorithms seek to minimize the objective function through linearization using first-order Taylor series expansions of the error model. Notably, the FOCE algorithm estimates interindividual variability simultaneously with the population mean and variance, unlike the FO algorithm, which does so in a post hoc step. The Laplacian FOCE method within NONMEM also utilizes a second-order Taylor series instead of the first-order expansion. In contrast, the MLEM algorithm maximizes a likelihood function through an iterative series of E-steps and M-steps without relying on linearization techniques. This involves computing conditional means and covariances in the E-step and updating population mean, covariance, and error variance parameters in the M-step to maximize the likelihood from the previous step. These algorithms demonstrate different approaches to numerical problem-solving, each tailored to specific applications and methodologies.