Quantification of cerebral blood flow (CBF) and the tissue residue function (R) using bolus-tracking MRI requires deconvolution of the arterial input function (AIF). Currently, the most commonly used deconvolution method is singular value decomposition (SVD), which has been shown to produce accurate estimations of CBF. However, this method introduces unwanted oscillations in the time course of R, and there are situations in which the actual shape is of interest (e.g., in calculating flow heterogeneity and assessing bolus dispersion). In such cases, the conventional SVD method may no longer be suitable, and an alternative approach may be required. This work describes the implementation of Tikhonov regularization with the L-curve criterion to quantify CBF and obtain a better characterization of R. The methodology is tested on simulated and patient data, and the results are compared to those found using the conventional SVD approach. Although both methods produce similar CBF values, the deconvolved R shape obtained using SVD is dominated by oscillations and fails to characterize the shape in the presence of dispersion. On the other hand, the use of the proposed regularization method improves the characterization of the tissue residue function. Copyright 2003 Wiley-Liss, Inc.