Reconstruction of the space-dependent perfusion coefficient from final time or time-average temperature measurements

Abstract

Knowledge of the blood perfusion in biomedicine is of crucial importance in applications related to hypothermia. In this paper, we consider the inverse bio-heat transfer nonlinear problem to determine the space-dependent perfusion coefficient from final time or time-average temperature measurements, which are themselves space-dependent quantities. In other applications this coefficient multiplying the temperature function represents a reaction rate. Uniqueness of solution holds but continuous dependence on the input data is violated. The problem is reformulated as a least-squares minimization whose gradient is obtained by solving the sensitivity and adjoint problems. The newly obtained gradient formula is used in the conjugate gradient method (CGM). This is the first time that the CGM is applied to solve the inverse problems under investigation. For exact data, we investigate the convergence of the iterative CGM. We also test that the iterative algorithm is semi-convergent under noisy data by stopping the iteration using the discrepancy principle criterion to produce a stable solution. Furthermore, because the search step size is computed using an optimization scheme at each iteration the CGM is very efficient. Three examples are investigated to verify the accuracy and stability of the numerical method.