An algorithm to calculate the NMR signal of a multi spin-echo sequence with relaxation and spin-diffusion

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Abstract

An algorithm to calculate NMR signals of a multi-echo pulse sequence with arbitrary position dependent B0 and B1 fields taking into account relaxation and spin-diffusion is presented. The multi-echo pulse sequence consists of an initial RF pulse (“90°” RF pulse) and a series of L refocusing RF pulses with arbitrary phases and flip-angles. The calculation is exact and takes into account all the magnetization pathways that contribute to the signal on a predefined spatial grid. The theoretical prediction is verified experimentally using a high field NMR microscopy system. The algorithm was implemented in a simulation program in order to optimize the design of an inside-out MR intra-vascular catheter that is used for characterization of vessel wall tissue. Measured data obtained with the catheter are in good agreement with the theoretical prediction of the simulation.

Introduction

In this paper we present an algorithm to calculate the magnetization M and signal s of a multi-echo pulse sequence vs. time on a grid of spatial points r = {xi, yi, zi}. M and s are calculated for a known position dependent static magnetic field B0 (r) and RF field B1 (r) by solving the Bloch Equations taking into account relaxation and spin-diffusion. The algorithm is utilized to optimize the design of a new MR intra-vascular catheter that is used for tissue characterization at the vessel wall.

Based on reference [18] we assume throughout the paper that any RF pulse can be considered an instantaneous operator applied at the RF pulse center and the evolution of the magnetization is calculated by applying free rotation, relaxation and spin-diffusion between adjacent RF pulse centers. The evolution of the magnetization along the train is analyzed using phase diagrams.

During the past few years, new NMR applications that operate in inhomogeneous fields where B1 and B0 vary by more than an order of magnitude over the volume of interest were developed. These applications include stray field NMR [1], oil well logging [2], [3], material testing [4], [5], and other applications [6] where the magnetic fields are generated from outside the sensitive volume. Such MR systems are referred to as inside-out systems [5]. The algorithm presented in this work is suitable for systems with very inhomogeneous magnetic fields such as inside-out systems and also for conventional MRI scanners.

Kiselev [7] calculated the evolution of M in a sequence with many RF pulses taking into account spin-diffusion and relaxation. In this paper we use a different although equivalent approach (see Discussion and conclusion) to solve for M and s by writing it as a finite sum of coherence pathways [8].

An elegant analysis of the NMR signal with and without spin-diffusion using coherence pathways for a train of RF pulses with a constant inter pulse delay was given by Kaiser et al. [8]. They showed that in the presence of spin-diffusion the decay rate due to diffusion varies for different pathways. Since the number of pathways increases with the number of RF pulses an analytical calculation of the echo signal in terms of coherence pathways is not feasible. Zur et al. [9] extended this analysis to time-varying gradients with an arbitrary waveform in a steady-state free precession (SSFP) pulse sequence. Hennig [10], [11] and Zur et al. [12] used phase diagrams to calculate signals in a multi-echo pulse sequence using the pioneering work of Woessner [13]. Other investigators [14], [15], [16], [17] analyzed the signal of a multi-echo sequence based on coherence pathways and the effect of spin-diffusion on these pathways.

In the first part of the paper we shall calculate the magnetization and the signal on the grid r where B1 (r), B0 (r), the diffusion coefficient D (r) and the relaxation times depend on r. Then we shall present an experimental verification of the algorithm using a high field MR microscopy system and an inside-out intra-vascular MR catheter.

Section snippets

Theory

A multi spin-echo pulse sequence is shown in Fig. 1. It consists of a train of L refocusing RF pulses with flip angles θj and phases ϕj (j = 1 to L), preceded by an initial RF pulse of flip angle θ0 and phase ϕ0. The j refocusing RF pulse operator is denoted Pj, and the initial RF pulse operator is denoted P0. The time between P0 and P1 is τ, and between adjacent refocusing RF pulses 2τ. The center of P0 is defined as time t = 0, and the center of Pj is at t = (2j  1) · τ. There are L echoes and echo j

Measurements with a high field NMR microscopy system

To verify the theory we measured the decay time constants of water with known diffusion coefficient and relaxation times at various RF flip angles and gradient amplitudes and compared it to the decay rates calculated by the theory. The measurements were performed on a high field NMR system equipped with magnetic field gradients operating at 400 MHz (Bruker Avance 400 WB, Bruker. Karlsruhe Germany). We used a sample of doped water at 20 °C with diffusion coefficient D = 2.0 × 10−9 m2/s. The RF flip

Discussion and conclusion

The algorithm presented in this work computes the time evolution of the magnetization M and signal s on a predefined spatial grid taking into account relaxation and spin-diffusion.

The accuracy and efficiency of the calculation stems from the fact that Mxy after RF pulse j is a polynomial of exp(iΦ) with N = 4j  1 coefficients am Eqs. (5a), (5b):Mxy(Φ)=m=-(2j-1)2j-1am·exp(imΦ).It can be shown [24] that am can be derived from Mxy (Φ) evaluated (using Eq. (24)) at N discrete values of Φ around the

Acknowledgments

The author gratefully acknowledges Lev Muchnik for writing a major part of the simulation program code, Dr. Peter Bendel from the Weizmann Institute of Science in Rehovot, Israel for running the high field NMR system, and Assaf Weiss for running the measurements of the MR Intra-vascular catheter.

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