Euro. Jn! of Applied Mathematics (1993), vol. 4, pp. 241-270. © 1993 Cambridge University Press 24 l Cardiac magnetic resonance imaging by retrospective gating: mathematical modelling and reconstruction algorithms J.B. T. M. ROERDINK and M. ZWAAN Centre for Mathematics and Computer Science, PO Box 4079, 1009 AB Amsterdam, The Netherlands (Received in revised form 12 June 1992) This paper is concerned with some mathematical aspects of magnetic resonance imaging (MRI) of the beating human heart. In particular, we investigate the so-called retrospective gating technique which is a non-triggered technique for data acquisition and reconstruction of (approximately) periodically changing organs like the heart. We formulate the reconstruction problem as a moment problem in a Hilbert space and give the solution method. The stability of the solution is investigated and various error estimates are given. The reconstruction method consists of temporal interpolation followed by spatial Fourier inversion. Different choices for the Hilbert space .llf of interpolating functions are possible. In particular, we study the case where .llf is (i) the space of bandlimited functions, or (ii) the space of spline functions of odd degree. The theory is applied to reconstructions from synthetic data as well as real MRI data. 1 Introduction Magnetic resonance imaging (MRI) is a diagnostic technique to measure and display cross- sections of human organs. In this paper we consider the reconstruction of a cross-section of the beating heart. The general problem in dynamic MRI is that because of physical limitations, the standard measurement technique is not fast enough to acquire all the data necessary for the reconstruction of a single heart phase, in a time which is short enough that the motion of the heart is negligible. 1 In the case of the beating heart, one can make use of the (approximate) periodicity of the motion. That is, data corresponding to the same relative heart phase may be recorded in different heartbeats. This presupposes exact reproducibility of the heart motion in successive cycles, a condition which will be violated in practice. There have been various ways to deal with this problem. McKinnon & Bates [l], who considered cardiac imaging in the context of computerized tomography (CT), assumed the number of cycles to be sufficiently small so that the heart motion during these cycles can be assumed to be 'quasi- stationary '. Another alternative, which will be pursued in this paper, is to assume that there is a simple rule to map heart intervals of different duration to a standard heart interval of unit length in such a way that data are assigned to the correct heart phase. To perform this synchronization of the data, the electrocardiogram (ECG) is simultaneously recorded and 1 There are reports of attempts to do real time imaging (see e.g. Mansfield & Morris [5]).