Laplace deconvolution on the basis of time domain data and its application to dynamic contrast‐enhanced imaging

In the present paper we consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval which appears in many different contexts. We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over Laguerre functions basis (which acts as a surrogate eigenfunction basis) using regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. The number m of the terms in the expansion of the estimator is controlled via complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at T and provides an estimator with the risk within a logarithmic factor of m of the oracle risk under no assumptions on the model and within a constant factor of the oracle risk under mild assumptions. We emphasize that, in the present paper, we consider the true observational model with possibly non-equispaced observations which are available on a finite interval of length T and account for the bias associated with this model. The study is motivated by perfusion imaging using a short injection of contrast agent, a procedure which is widely applied for medical assessment of blood flows within tissues such as cancerous tumors. Presence of a tuning parameter a allows to choose the most advantageous time units, so that both the kernel and the unknown right hand side of the equation are well represented for the further deconvolution. In addition, absence of boundary effects allows for an accurate estimator of the tissue blood flow, parameter which is of great interest to medical doctors. The methodology is illustrated by an extensive simulation study and a real data example which confirms that the proposed technique is fast, efficient, accurate, usable from a practical point of view and competitive.

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