Adjoint equation-based estimation of porous media permeability

Contrast enhanced myocardial perfusion imaging uses an injected contrast agent (CA) as a proxy for blood flow in cardiac tissue. This imaging modality has previously been modelled as the advection-diffusion of the CA through a porous medium. Here, an adjoint equation approach is developed for estimating the parameters of this porous medium model, as a route to more accurate and detailed quantitative diagnoses. Several test cases – of varying porous parameters, Peclet number (Pe), source terms, and boundary conditions – were used to generate “observed” data, on which to test the parameter estimation. Forward models were solved using the finite element method, implemented in FEniCS, with adjoints computed via the dolfin-adjoint module. These models were used to estimate the permeability tensor, by minimising an objective function containing two penalty terms: (1) the error between estimated concentration and observed concentration, and (2) the gradient of the permeability tensor. An iterative approach to adjusting the weighting of the penalty terms was developed, which allows the optimiser to converge to a low overall error while resolving local inhomogeneities in the porous medium parameters. For an homogeneous 2D test case with Pe≤0.1, the estimated permeability and concentration matched well with the observed data (with an L2 error norm < 10^(-3) for all cases). A test case with a homogeneous background permeability containing a localised area of lowered permeability (between 0% - 70% of the background value), showed that the method could successfully determine the location and extent of the low permeability region. This method was found to be compatible with features such as noise filtering and spatio-temporal down-sampling. In addition to its potential for analysing perfusion images, this method could prove useful for other applications where experimental determination of porous medium parameters is essential. Future work will test and develop the method on real data.