A mathematical model represents selected features of a system with mathematical objects and relations so that its behavior can be analyzed or predicted. Choosing the representation is part of the work: specify the quantities tracked, the equations or probability distributions connecting them, and the conditions under which those connections are expected to apply. A mathematical proof can establish a consequence of the model; comparing that consequence with observations tests its suitability for the intended use.
For a simple pendulum, the angle and gravitational acceleration give the ideal equation . Its assumptions include a point mass, fixed length , uniform gravitational acceleration, and no damping. The small-angle approximation gives the simpler harmonic oscillator equation and oscillation period . A larger amplitude changes the full model's period; drag requires a damped pendulum. The simpler model remains useful when its omitted effects are small for the question being asked.
A model need not track every detail. A Bernoulli process models repeated success/failure observations using one probability and an independence assumption; a Gaussian process specifies a mean and a covariance function to model related measurements. Each representation enables calculations by retaining some distinctions and omitting others. Useful simplification therefore comes with a stated domain of use and quantities against which its predictions can be checked.
A parameter specifies a member of a family of mathematical models and is held fixed while a particular solution evolves. Varying it can change equilibrium existence, stability or long-term dynamics, producing a bifurcation. A model parameter differs from its evolving state variables and from an initial condition, even though all may be numerical inputs to a computation.
A mathematical model parameter is a quantity held fixed while solving a particular instance of a model, but allowed to vary between instances. Changing a parameter can change equilibria, their stability and the qualitative behavior of trajectories. Parameter constancy does not mean probabilistic independence.
A box model represents a system by finitely many internally uniform compartments. Balances of transported material and local sources give coupled ordinary differential equations. For a constituent's concentration in fixed box volume , the stored amount is ; changing volumes requires the balance of that product. A transport-only closed box model preserves total inventory, even when its transport matrix is nonsymmetric.
Computation of the behavior implied by a specified mathematical model. Agreement between a simulation and its equations is distinct from agreement between those equations and the physical system.
Modeling a system through interacting individual entities, such as cells, with rules for their states, behavior and environment.
PhysiCell is an open-source agent-based modeling framework for simulating interacting cells in a spatial microenvironment. Individual cell agents carry phenotypes and interact mechanically, while diffusing substrates are represented by continuum transport fields. Linking these descriptions permits cell behavior to depend on the local environment and to modify that environment in turn.
The original publication demonstrates multicellular tumor simulations and describes cell cycling, death, volume, mechanics and motility models. The scientific value is the coupling of scales; a simulated pattern remains conditional on the specified rules and parameters, rather than establishing biological causation by itself.[ref]
Representing and propagating uncertainty in measurements, parameters, initial conditions and model structure into predictions.
