A deconvolution approach for dynamic contrast enhanced magnetic resonance imaging using an approximation basis of exponential functions constrained to be non-negative and non-increasing is developed and compared with widely used methods. Monotonicity in an exponential basis is implemented in terms of a newly derived condition which is a considerable generalization over a previous condition that implies complete monotonicity. Since the constraints imply a bound on the total variation, a well known staircasing effect may result with other approximation bases, but an exponential basis is shown to resist staircasing. In addition to the choice of approximation basis, further regularization is implemented in terms of the number of basis functions and the distribution of their parameters. The exponential approach is applied to dynamic contrast enhanced magnetic resonance imaging data to determine physiological parameters pixelwise to visualize a cerebral tumor, and the results are compared favorably with those of the standard truncated singular value decomposition approach. In particular, kernels estimated with constrained exponentials are free of oscillations and staircasing, and the images of estimated kernel parameters are sharper than those obtained by truncated singular value decomposition.