Solvers

Overview

Method	Function	Gives	Assumptions
Bloch-Torrey equation	solve_btpde	magnetization, signal	none (reference)
Homogenized ADC	solve_hadc, solve_hadc_tensors	ADC, apparent diffusion tensor	impermeable membranes, low b
Matrix formalism	solve_laplace, solve_mf	magnetization, signal	truncated eigenfunction basis
Matrix formalism, effective tensor	compute_mf_diffusion_tensor	apparent diffusion tensor	low b
Kärger model	solve_karger	signal	well mixed compartments
Short diffusion time approximation	compute_adc_sta	ADC	short diffusion times
Analytical matrix formalism	solve_analytical_mf	signal	layered plates, cylinders, spheres

The ADC follows from the signals of several b-values with fit_adc, and apparent diffusion tensors from ADCs in several directions with fit_tensors. See Apparent diffusion coefficients for a comparison.

Time stepping

solve_btpde uses a ThetaSolver by default (Crank-Nicolson): one sparse factorization per interval for PGSE-like sequences, and GMRES with a preconditioner for other gradients. The time step is a thousandth of the echo time by default:

u = solve_btpde(model, matrices, gradient, ThetaSolver(; timestep = 50.0))
signal = compute_signal(matrices.M, u)

The adaptive ODE solvers of OrdinaryDiffEq are also available, after loading one of its solver packages, for example using OrdinaryDiffEqBDF:

u = solve_btpde(model, matrices, gradient, QNDF(); abstol = 1e-8, reltol = 1e-6)

Periodic boundary conditions

The meshes of the samples of periodic tissue (PeriodicBoxECS, VoronoiSetup) are periodic. With

matrices = assemble_matrices(model; periodic = true)

the points on opposite faces of the box are identified, and solve_btpde (with ThetaSolver) and solve_hadc use periodic boundary conditions instead of impermeable box faces (see Periodic boundary conditions). The signals and ADCs then do not depend on the position of the cells in the box, and the water of the ECS is not trapped by the box. See Dense tissue with periodic boundary conditions and White matter.

Callbacks

solve_btpde calls the callbacks after each time step: Printer prints information, VTKWriter saves the magnetization for ParaView, and Plotter plots it (see Plotting and output).