Cardiac motion presents a major challenge in diffusion weighted MRI, often leading to large signal loss that necessitates specific acquisition strategies. The diffusion process in the myocardium is difficult to investigate because of the unquantified sensitivity of diffusion measurements to cardiac motion. We introduce a rigorous mathematical formalism based on the Bloch-Torrey equation to quantify the effect of tissue motion on the measurement of apparent diffusion coefficient (ADC) for different types of diffusion encoding sequences.

The presented mathematical model is based on the Bloch-Torrey equation, and takes into account deformations according to the laws of continuum mechanics. The Bloch-Torrey equation is usually used to describe the magnetization in immobile tissues. In deformed tissues, the Bloch-Torrey equation is modified in the following way: the diffusion tensor is affected by the deformation, advection term involving the velocity of the displacement is added and the zero order terms are increased by the divergence of the velocity. The phase term of the modified Bloch-Torrey equation is also changed according to the motion:

$$\partial_t M_{xy}(\mathbf{x},t)-\text{div}(\mathbf{F}^{-1}(\mathbf{x},t) \mathbf{D}(\mathbf{x}) (\mathbf{F}^{-1}(\mathbf{x},t))^t \nabla M_{xy}(\mathbf{x},t))+\mathbf{v}(\mathbf{x},t)\cdot \nabla M_{xy}(\mathbf{x},t)\\+(i\gamma \varphi(\mathbf{x},t)\cdot \mathbf{G}(t)+\frac{1}{T_2}+\text{div}(\mathbf{v}(\mathbf{x},t)))M_{xy}(\mathbf{x},t)=0 \text{ in } \Omega\times[0,T]$$

$$M_{xy}(\textbf{x},t)=0 \text{ on } \partial\Omega\times[0,T], \quad M_{xy}(\textbf{x},0)=M_{xy}^0 \text{ on } \Omega\times\{0\}.$$

The initial condition $$$M_{xy}^0(\mathbf{x})$$$ is the thermal equilibrium magnetization, $$$\varphi(\mathbf{x},t)$$$ is the deformation field which is given, in our case, by an analytical 2D cardiac deformation model [1], $$$\mathbf{F}(\mathbf{x,t})=D\varphi(\mathbf{x},t)$$$ is the Jacobian matrix of $$$\varphi$$$ and $$$\mathbf{v}(\mathbf{x},t)$$$ is the velocity field of the displacement. Dirichlet homogeneous boundary conditions were imposed on the computational domain $$$\Omega$$$ which is represented by a ring mimicking the shape of 2D-short axis of the left ventricle. Approximating this mathematical model by coupling a finite element discretization in space with Euler implicit discretization in time, the aim of this study is to propose a numerical simulations to predict the sensitivity of the diffusion signal to cardiac motion under the influence of diffusion gradients. Different diffusion encoding schemes are considered: unipolar Spin Echo, bipolar, and asymmetric bipolar Spin-Echo sequences [2], Twice-Refocused Spin-Echo sequence [3], Stimulated-Echo sequence with unipolar and bipolar diffusion encoding gradients [4,5], and acceleration motion compensation (AMC) using spin echoes [6]. Magnitude intensity images are simulated for different phases of the cardiac cycle, and ADC error is computed in each case.

Fig.2 shows the magnitude images along the cardiac cycle for the simulated cardiac deformation given in Fig1. Corresponding simulated strain and strain rate are given in Fig.3. The resulting global signal intensity and relative error on the ADC estimates are presented in Fig.4 and Fig.5, respectively. Our numerical simulations show that the motion sensitivity of the diffusion sequence is reduced by using either spin echo AMC diffusion gradients or unipolar stimulated acquisition mode, the latter being insensitive to bulk motion, while the first one is unaffected by strain. Our numerical model identifies the existence of time points called "sweet spots" of the cardiac cycle, located at end-systole and end-diastole for unipolar stimulated se- quence and at the time points of null strain-rate for the spin echo sequences, on which the diffusion is unaffected by the strain or cardiac deformation.Seven popular diffusion preparation techniques were compared side-by-side through numerical simulation framework using a simulated macroscopic deformation of the left ventricle wall. This formalism allowed to quantify the effect of a given model of cardiac motion on the diffusion weighted MRI signal for each available diffusion preparation scheme. This study also demonstrates that the sweet spots depend on the type of diffusion encoding scheme, which has never been clearly stated in the literature. Our numerical results are based on the Bloch equation, which is used in Magnetic Resonance Fingerprinting (MRF) [7] to simulate the signal evolution and predict the tissue parameters of interest ($$$T_1$$$, $$$T_2$$$, ADC,...), without any requirement on the shape of the signal evolution curve. Then this technique can be used in cardiac diffusion imaging to obtain new quantitative information of many MR parameters simultaneously and without the high sensitivity to measurement errors found in many other fast methods.