Elie Bretin

Imen Mekkaoui

Jerome Pousin

Assessment of the effect of tissue motion in diffusion MRI: Derivation of new apparent diffusion coefficient formula

Keywords: Diffusion magnetic resonance imaging, Bloch-Torrey equation, cardiac diffusion imaging, diffusion encoding gradient, apparent diffusion coefficient, asymptotic analysis

We investigate in this paper the diffusion magnetic resonance imaging (MRI) in deformable organs such as the living heart. The difficulty comes from the hight sensitivity of diffusion measurement to tissue motion. Commonly in literature, the diffusion MRI signal is given by the complex magnetization of water molecules described by the Bloch-Torrey equation. When dealing with deformable organs, the Bloch-Torrey equation is no longer valid. Our main contribution is then to introduce a new mathematical description of the Bloch-Torrey equation in deforming media. In particular, some numerical simulations are presented to quantify the influence of cardiac motion on the estimation of diffusion. Moreover, based on a scaling argument and on an asymptotic model for the complex magnetization, we derive a new apparent diffusion coefficient formula. Finally, some numerical experiments illustrate the potential of this new version which gives a better reconstruction of the diffusion than using the classical one.

Introduction

Diffusion magnetic resonance imaging (diffusion MRI) is an imaging technique that is capable of providing MRI images with a contrast sensitive to the diffusion motion of water molecules [START_REF] Le Bihan | Imagerie de diffusion in vivo par résonance magnétique nucléaire[END_REF]. Diffusion MRI plays a very important role in studying the microscopic structure of biological tissues through the measure of the diffusion coefficient of water molecules in the tissues. This technique was successfully applied to static organs such as in brain imaging ( [START_REF] Horsfield | Applications of diffusion-weighted and diffusion tensor MRI to white matter diseases[END_REF], [START_REF] Warach | Fast magnetic resonance diffusion-weighted imaging of acute human stroke[END_REF], [START_REF] Lazar | Mapping brain anatomical connectivity using white matter tractography[END_REF]). However, its implementation on moving organs like the beating heart is very difficult because of the cardiac motion during acquisition. The heart has a pseudo-periodic and fast motion that can be tracked with dynamic series of images ( [START_REF] Spottiswoode | Tracking myocardial motion from cine DENSE images using spatiotemporal phase unwrapping and temporal fitting[END_REF], [START_REF] Wen | Adaptive post-processing techniques for myocardial tissue tracking with displacementencoded MR imaging[END_REF]). The tissue motion induces an attenuation of the signal measured in diffusion MRI which can considerably degrade the quality of estimation of the diffusion coefficient. Because of the sensitivity to motion, it is difficult to assess to what extent the diffusion characteristics obtained from diffusion MRI reflect the real properties of the cardiac tissues. Some experimental studies have investigated the influence of cardiac motion on the diffusion measurements ( [START_REF] Reese | Imaging myocardial fiber architecture in vivo with magnetic resonance[END_REF]- [START_REF] Rapacchi | Low b-Value Diffusion-Weighted Cardiac Magnetic Resonance Imaging[END_REF]).

The signal measured in diffusion MRI is described by the magnetization density of the water molecules which can be modeled by the complex valued Bloch-Torrey partial differential equation (PDE) [START_REF] Torrey | Bloch equation with diffusion terms[END_REF]. This equation provides a mathematical description of the diffusion phenomenon of water molecules in the field of magnetic resonance imaging. In static organs, such as the human brain, several approaches have been proposed in the literature to model the diffusion MRI signal from the Bloch-Torrey equation. These different approaches offer the possibility of understanding the link between the diffusion of water molecules and the geometric structure of biological tissues. For example, analytical ( [START_REF] Stejskal | Spin diffusion measurements: spin echoes in the presence of a timedependent field gradient[END_REF]- [START_REF] Pfeuffer | Restricted diffusion and exchange of intracellular water: theoretical modelling and diffusion time dependence of 1H NMR measurements on perfused glial cells[END_REF]) and numerical models ( [START_REF] Xu | Numerical study of water diffusion in biological tissues using an improved finite difference method[END_REF]- [START_REF] Moroney | Numerical analysis of NMR diffusion measurements in the short gradient pulse limit[END_REF]) of the solution of the Bloch-Torrey equation have been introduced under simplified assumptions on the geometry of spatial domains in which the diffusion process takes place.

Recently, a multi-compartment model for the Bloch-Torrey equation has been developed in [START_REF] Van | A finite element methods to solve the Boch-Torrey equation applied to diffusion magnetic resonance imaging[END_REF] within a more complex geometry and under more realistic experimental conditions. However, very few models have been proposed to study the influence of physiological motion on the signal measured in diffusion MRI. For example, in [START_REF] Rohmer | A Bloch-Torrey equation for diffusion in a deforming media[END_REF] the Bloch-Torrey equation was expressed in generalized curvilinear coordinates to describe the behavior of magnetization in the heart during its deformation over the cardiac cycle and a change of basis formulas were used in order to take into account the effect of motion on diffusion.

The focus of this work is firstly presenting a mathematical model based on the Bloch-Torrey equation, which is capable of taking into account the physiological motion of the heart, and allowing the quantification of tissue deformation on the complex magnetization. This model is obtained by writing the Bloch-Torrey PDE in a domain that deforms over time according to the laws of continuum mechanics. Then from the resulting model, the diffusion MRI images are simulated at different moments in the cardiac cycle by using an analytical motion field mimicking a realistic deformation of the heart, and this to validate the model by comparing the results of numerical simulations with the existing experimental results. Secondly, we focus on the mitigation issue of the effect of cardiac motion on the diffusion MRI images. For this purpose, we propose a method for correcting the diffusion MRI images which were reconstructed in the presence of cardiac motion. This method is derived from a singular perturbative model based on an asymptotic analysis of the Bloch-Torrey equation with motion. The resulting model provides an asymptotic model of ordinary differential equation (ODE) that allows to derive a new formula of apparent diffusion coefficient for the diffusion correction and reconstruction of diffusion MRI images free form motion effect. This paper is organized as follows, in Section 2 the Bloch-Torrey equation in static medium is recalled and diffusion measurement technique used in practice is described. The Bloch-Torrey equation established in the presence of physiological motion is introduced in Section 3. Section 4 is dedicated to numerical simulation of the diffusion MRI images during the cardiac cycle. These images are reconstructed by means of the complex magnetization obtained by solving the Bloch-Torrey equation with motion by a finite element method. In section 5, we present a singular perturbation method for correcting the diffusion MRI images that are affected by heart motion during the cardiac cycle. This method provides an asymptotic ODE model that allows to derive a relationship between the complex magnetization influenced by motion and the exact diffusion. This relationship is then used in numerical simulations to correct the diffusion images reconstructed in the presence of motion.

Bloch-Torrey equation and diffusion measurement

The signal measured by diffusion MRI technique can be modeled by the so-called Bloch-Torrey equation, that gives a mathematical description of the complex magnetization M : Ω ⊂ R 3 → C of water molecules including the effect of molecular diffusion. This equation is given in [START_REF] Torrey | Bloch equation with diffusion terms[END_REF] for all x ∈ Ω by:

∂ t M (x, t) -div D(x)∇M (x, t) + 1 T 2 M (x, t) = 0 (1)
where x = (x, y, z) t is the spatial position, T 2 is a physical parameter called transverse relaxation time, and D(x) represents a second order diffusion tensor which is a 3x3 symmetric positive definite matrix:

D =   D xx D xy D xz D yx D yy D yz D zx D zy D zz  
where D xx is the diffusion coefficient of water molecules in the x-direction and D xy is the correlation between the diffusion coefficients in the x and y-directions, etc. In diffusion MRI, the water molecules are subjected to a time dependent magnetic field gradient (called diffusion encoding gradient):

G(t) = (G x (t), G y (t), G z (t)) t ,
in order to encode the diffusion motion of water molecules in the different directions of space. The Bloch-Torrey equation reads then as:

∂ t M xy (x, t) -div D(x)∇M xy (x, t) + iγx • G(t) + 1 T 2 M xy (x, t) = 0 (2)
where γ is the gyromagnetic ratio of the hydrogen atom such that γ 2π = 42.58MHz/Tesla. To measure the diffusion motion of water molecules with magnetic resonance imaging, a special shape of diffusion encoding gradients is performed. This shape is usually the spin echo diffusion gradient sequence introduced by Stejskal and Tanner in [START_REF] Stejskal | Spin diffusion measurements: spin echoes in the presence of a timedependent field gradient[END_REF]. The general idea of this sequence is to excite the water molecules with a 90 o radio-frequency (RF) pulse [START_REF] Nishimura | Principles of Magnetic Resonance Imaging[END_REF], dephase them and encode their positions in a specific spatial direction with a time-constant magnetic field gradient of intensity G and duration δ. After that, a 180 o RF pulse [START_REF] Nishimura | Principles of Magnetic Resonance Imaging[END_REF] is applied to invert the phase of magnetization of water molecules. A second diffusion gradient with equal intensity and duration to those of the first gradient is applied at a time ∆ after the first gradient to rephase the water molecules (implying an effect of -G with the 180 o RF pulse). The signal is finally acquired at a time called echo time T E (Fig. 1). Noting that to produce a signal at the time t = T E, we have the condition that If the water molecules move due to diffusion during the time ∆, then the effect of the second gradient will not be exactly opposite to the first gradient. This leads to a phase incoherence of water molecules resulting in a reduction of the magnitude of the complex magnetization M xy , and consequently, the signal will be smaller relative to the case in which there was no diffusion encoding gradients. The decrease in the acquired signal will reflect the amount of diffusion which occurred during the application of the diffusion encoding gradients. Hence, measuring the signal decay makes possible the measure of the diffusion coefficient in the direction of application of the diffusion gradient sequence. In order to estimate the diffusion coefficient, two MRI images are acquired, one with diffusion encoding gradient and the another without diffusion encoding gradient. These two images correspond to the MRI signals calculated over each voxel V at the echo time T E with diffusion gradient as well as with no diffusion gradient:

T E 0 G(t)dt = 0.
S = V M xy (x, T E)dx ; S 0 = V M xy (x, T E)| G=0 dx.
Then, the diffusion coefficient can be estimated for each point of the MRI image through the so-called Apparent Diffusion Coefficient (ADC) ( [START_REF] Stejskal | Spin diffusion measurements: spin echoes in the presence of a timedependent field gradient[END_REF], [START_REF] Le Bihan | MR imaging of intra-voxel incoherent motions: application to diffusion and perfusion in neurologic disorders[END_REF]) as:

ADC := - 1 b ln S S 0 , (3)
with the diffusion weighting factor (or b-value):

b = γ 2 T E 0 t 0 G(s)ds 2 dt .
In the case of anisotropic diffusion, the apparent diffusion tensor D A can be estimated according to the relation [START_REF] Mattiello | Analytical Expressions for the b Matrix in NMR Diffusion Imaging and Spectroscopy[END_REF]:

ln S S 0 := -B : D A ,
where the diffusion weighting B-matrix is given by:

B = γ 2 T E 0 t 0 G(s)ds ⊗ t 0 G(s)ds dt .
The apparent diffusion tensor D A is a 3x3 symmetric positive definite matrix. This implies that there are six apparent diffusion coefficients to be estimated. To do so, a set of diffusion MRI signals is acquired by the application of the diffusion encoding gradients in at least six non-coplanar spatial directions to get a linear system of at least six equations that can be resolved to find the coefficients of the tensor D A .

3 Bloch-Torrey equations in deforming media

Mathematical model

In this section we consider an evolving set Ω(t) ⊂ R d (d=2 or 3) for all t ∈ [0, T ] with T > 0. Let us introduce the geometric transformation ϕ which is a regular time-space dependent function:

ϕ : (0, T ) × Ω(0) → Ω(t) (t, x) → ϕ(t, x) = X
and assume that for any point x, the curve t → (t, x) satisfies the system:

∂ t ϕ(t, x) = v(ϕ(t, x)) ϕ(0, x) = x
for a given velocity field v : R d → R d . We assume that the evolving set {Ω(t)} t∈[0,T ] is defined from an initial domain Ω(0) ⊂ R d by:

Ω(t) = {X = ϕ(t, x), x ∈ Ω(0), t ∈ (0, T )}.
The following notations will be used in this section: If for some time T > 0, a function g is defined in Ω(t) as:

g : ∪ t∈(0,T ) Ω(t) × {t} → R (X, t) → g(X, t)
then it can be defined in Ω(0) as:

ḡ : Ω(0) × (0, T ) → R ḡ(x, t) → g(ϕ(t, x), t).
We now give a description of the Bloch-Torrey equation in the moving set Ω(t). The idea consists to analyze the magnetic flux of M in a elementary volume V (t) ⊂ R d which evolves by the geometric transformation ϕ from an elementary volume V (0) at time t 0 = 0. We denote by M (X, t) the magnetization at X ∈ V (t) at time t. In order to establish the conservation law for the magnetization density we have to specify the change of magnetization in the volume V (t) which is given by d dt V (t) M (X, t)dX. We write the equation of conservation taking into account losses J due to the flux through the boundary of V (t), which is assumed to be:

J(X, t) = -D(X)∇ X M (X, t),
where D(X) is the diffusion tensor at X. We get:

d dt V (t) M (X, t)dX = - ∂V (t) J(X, t) • n X dS(X) = ∂V (t) D(X)∇ X M (X, t) • n X dS(X)
where n X is the normal unit vector to the surface dS at point X. By using the divergence theorem we get:

d dt V (t) M (X, t)dX = V (t) div X (D(X)∇ X M (X, t))dX. (4)
Proposition 3.1. If the magnetization M satisfies Equation (4) in the domain V (t), then it satisfies the following equivalent equations that describe M in the deformed and initial configurations:

∂ t M (X, t) -div X (D(X)∇ X M (X, t)) + div X (M (X, t)v(X)) = 0, X ∈ Ω(t), t > 0 (5)
∂ t M (x, t) -div F -1 (t, x)D(x)F -t (t, x)∇M (x, t) + div X (v)(x, t)M (x, t) -F -1 (t, x)D(x)F -t (t, x)∇ t 0 div X (v)(x, s)ds • ∇M (x, t) = 0, x ∈ Ω(0), t > 0, (6)
where F(t, x) := Dϕ(t, x) is the Jacobian matrix of ϕ and F -t the transpose of F -1 .

Proof. By Reynolds transport theorem we have:

d dt V (t) M (X, t)dX = V (t) (∂ t M (X, t) + div X (M (X, t)v(X)))dX.
By identification with Eq. ( 4) we find:

V (t) (∂ t M (X, t) + div X (M (X, t)v(X)))dX = V (t) div X (D(X)∇ X M (X, t))dX.
Hence equation ( 5) is obtained for every volume V (t). Now to get Eq. ( 6), Eq. ( 4) is expressed in the domain V (0). We use the change of variable formulas for the gradient and divergence [START_REF] Duvaut | Mécanique des milieux continus[END_REF]:

∇ X g(X, t) = F -t (t, x)∇g(x, t) ; div X (g(X, t)) = 1 |F(t, x)| div(|F(t, x)|F -1 (t, x)g(x, t)),
and the Liouville-Jacobi formula for the derivative of the determinant:

∂ t |F(t, x)| = tr(Dv)(ϕ(t, x))|F(t, x)| = div(v)(ϕ(t, x))|F(t, x)|,
which gives:

|F(t, x)| = exp t 0 div(v)(ϕ(s, x))ds .
Then the second member of Eq. ( 4) can be written as:

V (t) div X (D(X)∇ X M (X, t))dX = V (0) 1 |F| div |F|F -1 D(x)F -t ∇M (x, t) |F|dx = V (0) |F|div F -1 D(x)F -t ∇M (x, t) dx + V (0) [∇|F|] • [F -1 D(x)F -t ∇M (x, t)]dx = V (0) div F -1 D(x)F -t ∇M (x, t) |F|dx+ V (0) ∇ t 0 div(v)(x, s)ds • F -1 D(x)F -t ∇M (x, t) |F|dx. (7)
Now, by applying the change of variable formula for the integrals and the Liouville-Jacobi formula, the first member of Eq. ( 4) gives:

d dt V (t) M (X, t)dX = d dt V (0) M (x, t)|F(t, x)|dx = V (0) |F(t, x)|∂ t M (x, t)dx + V (0) M (x, t)∂ t |F(t, x)|dx = V (0) ∂ t M (x, t)|F(t, x)|dx + V (0) M (x, t)div(v)(x, t)|F(t, x)|dx. (8)
By combining Eqs. ( 7) and ( 8), we get:

V (0) ∂ t M (x, t)+div(v)(x, t)M (x, t) |F(t, x)|dx = V (0) div F -1 (t, x)D(x)F -t (t, x)∇M (x, t) |F(t, x)|dx + V (0) F -1 (t, x)D(x)F -t (t, x)∇ t 0 div(v)(x, s)ds • ∇M (x, t)|F(t, x)|dx.
Which is true for every V (0):

∂ t M (x, t) -div F -1 (t, x)D(x)F -t (t, x)∇M (x, t) + h(x, t) • ∇M (x, t) + div(v)(x, t)M (x, t) = 0, (9)
with

h(x, t) = -F -1 (t, x)D(x)F -t (t, x)∇ t 0 div(v)(x, s)ds .
Equation ( 9) allows to take into account the effect of motion in the Bloch-Torrey equation ( 2). The frequency term γx • G(t) in Eq. ( 2) is transformed into γϕ(t, x) • G(t) according to the motion. Then the Bloch-Torrey equation with motion effect is written as follows:

∂ t M xy -div(F -1 DF -t ∇M xy ) + h • ∇M xy + (iγϕ • G + 1 T 2 + div(v))M xy = 0 for x ∈ Ω(0). ( 10
)
When the frequency term γϕ • G is high, the magnetization M xy becomes highly oscillating and this limits the time step that can be used in the numerical simulations. To address this limitation, Eq. ( 10) is demodulated by writing the magnetization M xy as follows:

M xy (x, t) = m(x, t) exp -iγ t 0 ϕ(x, t ) • G(t )dt . (11)
Substituting [START_REF] Torrey | Bloch equation with diffusion terms[END_REF] into Eq. ( 10) and multiplying the resulting equation by exp iγ

t 0 ϕ(x, t ) • G(t )dt ,
we get:

∂ t m -exp(iΦ)div K∇(m exp(-iΦ) + exp(iΦ)h • ∇(m exp(-iΦ)) + 1 T 2 + div(v) m = 0, (12)
where we noted the new diffusion tensor by:

K = F -1 DF -t ,
and the phase term by:

Φ(x, t) = γ t 0 ϕ(x, t ) • G(t )dt ,
We have:

exp(iΦ)h • ∇(m exp(-iΦ)) = h • ∇m -imh • ∇Φ, (13)
and

exp(iΦ)div(K∇(m exp(-iΦ)) = div(K∇m) -K∇Φ • ∇Φm -2iK∇Φ • ∇m -iK : (D∇Φ)m -i(∇ t K)∇Φm ( 14
)
where D∇Φ is the Jacobian matrix of ∇Φ(x, t), and K : (D∇Φ) is the double dot product of K and D∇Φ. By replacing Eqs. ( 13) and ( 14) in Eq. ( 12), we get the following equation for the magnetization m:

∂ t m -div(K∇m) + c 1 • ∇m + c 2 m = 0, (15)
with

c 1 (x, t) = h(x, t) + 2iK(x, t)∇Φ(x, t),
and c 2 (x, t) = 1 T 2 + div(v) + K∇Φ • ∇Φ + i K : (D∇Φ) -h • ∇Φ + (∇ t K)∇Φ .
Noting that to get the solution of Eq. ( 10), we solve Eq. ( 15) to obtain m and by using the expression [START_REF] Torrey | Bloch equation with diffusion terms[END_REF] we deduce the solution of Eq. (10).

Existence and uniqueness

Assume that Ω(0) be a bounded Lipschitz subset of R d , and Q = Ω(0) × [0, T ] for T > 0. We associate to the Bloch-Torrey equation [START_REF] Rapacchi | Low b-Value Diffusion-Weighted Cardiac Magnetic Resonance Imaging[END_REF] an initial condition and homogenous Dirichlet boundary condition. In this case the magnetizations M xy and m have the same initial and boundary values. Then the same initial and boundary conditions can be associated to Eq. ( 15):

   ∂ t m -div(K∇m) + c 1 • ∇m + c 2 m = 0 in Q, m = 0 on ∂Ω(0) × [0, T ], m = m 0 on Ω(0) × {0}. (16)
For each t ∈ [0, T ], we denote by L the second order differential operator given by:

Lm = -div(K(x, t)∇m) + c 1 (x, t) • ∇m + c 2 (x, t)m, (17)
with

K(x, t) = (K ij (x, t)) 1≤i,j≤d and c 1 (x, t) = (c 1i (x, t)) 1≤i≤d . The deformation function ϕ is sup- posed in C 2 (Q, R) d .
We recall that the tensor K is given by F -1 DF -t , where F is the Jacobian of the deformation ϕ that can be written in function of the displacement u. We get then F := I + Du.

In order to guarantee the definite positivity of K we suppose that Du(x, t) is small compared to 1 for each x ∈ Ω(0) and t ∈ [0, T ]. This allows to satisfy the following ellipticity hypothesis for the operator L:

∃δ > 0 such that ∀ξ ∈ C d , Re K(x, t)ξ • ξ ≥ δ|ξ| 2 a.e. , x ∈ Ω, t ∈ [0, T ].
The coefficients c 1i for i = 1, ..., d, and c 2 are supposed in L ∞ (Q, C). In order to de formulate problem [START_REF] Xu | Numerical study of water diffusion in biological tissues using an improved finite difference method[END_REF] in a weak sense, let v be a test function in V = H 1 0 (Ω(0), C) equipped with the scalar product and the induced norm:

(u, v) V = Ω(0) (∇u • ∇v + uv)dx ; u 2 V = Ω(0) (|∇u| 2 + |u| 2 )dx.
By multiplying ( 16) by v, by integrating by parts on Ω(0), and by using the Green's formula, a weak formulation of problem ( 16) is obtained:

   Find m ∈ L 2 ([0, T ], H 1 0 (Ω(0), C)) ∩ H 1 ([0, T ], H -1 (Ω(0), C)) such that ∂ t m, v + a(t; m, v) = 0, ∀v ∈ V, a.e. t ∈ [0, T ], m(x, 0) = m 0 ,
where m 0 ∈ H = L 2 (Ω(0), C) and the sesquilinear form a is given by:

a(t; m, v) = Ω(0) (K(x, t)∇m • ∇v + c 1 (x, t) • (∇m)v + c 2 (x, t)v)dx, ∀t ∈ [0, T ].
The existence and unicity of this problem can be obtained as a consequence of the following theorem. Theorem 3.1. (J.L. Lions) [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] Let V ⊂ H two separable Hilbert spaces such that the injection of V in H is continuous and V is dense in H. We identify H to its dual H , and we note V the dual space of V , then H can be identified to a sub-space of

V : V ⊂ H ⊂ V . If f ∈ V and v ∈ V , f, v is the pairing of V and V . For t ∈ [0, T ], let a(t; u, v) : V × V → C be a sesquilinear form defined by: a(t; u, v) = L(t)u, v ; L(t)u ∈ V ; L(t) ∈ L(V, V )
and satisfies the properties:

(i) ∀u, v ∈ V , the function t → a(t; u, v) is measurable,
(ii) (coercivity) there exists a constants α > 0 and σ ∈ R such that:

α v 2 V ≤ Re a(t; v, v) + σ v 2 H , ∀v ∈ V, ∀t ∈ [0, T ].
(iii) (continuity) there exists a constant c > 0 such that:

|a(t; u, v)| ≤ c u V v V , ∀t ∈ [0, T ], ∀u, v ∈ V.
Then, for all u 0 ∈ H, the following problem

∂ t u, v + a(t; u, v) = f, v a.e. t ∈ [0, T ], ∀v ∈ V, u(0) = u 0 , admits a unique solution u in W (V ) = {v : [0, T ] → V ; v ∈ L 2 ([0, T ]; V ); ∂ t v ∈ L 2 ([0, T ]; V )}.

Finite element approximation

We note that to calculate the magnetization M xy , we solve numerically Eq. ( 16) for the magnetization m and by using the relation [START_REF] Torrey | Bloch equation with diffusion terms[END_REF] we deduce M xy . We use two-dimensional Lagrangian finite element method for the resolution of Eq. [START_REF] Xu | Numerical study of water diffusion in biological tissues using an improved finite difference method[END_REF]. A weak formulation for this problem is written as follows:

d dt Ω(0) mvdx + Ω(0) K∇m • ∇vdx + Ω(0) c 1 • (∇m)vdx + Ω(0) c 2 mvdx = 0.
where v ∈ V is a real test function. We define a mesh for the domain Ω(0), that enables to introduce an approximation space V h which is a finite dimension sub-space of V . To discretize Ω(0), we choose a triangular mesh T h (Fig. 2), where h is the maximum mesh size. The space of approximation V h is the set of continuous functions on Ω(0) and which are polynomials on each triangle K of the mesh. By choosing an approximation by polynomials of degree less than or equal to 1, we have:

V h = {v continuous on Ω(0), v| K ∈ P 1 for all K ∈ T h }.
This space admits a base functions (ϕ j ) j that take the value 1 at node x j and 0 at the other nodes x k of the mesh. We get then the following approximate problem:

               Find m h ∈ C 1 ([0, T ], V h ) such that : d dt Ω(0) m h v h dx + Ω(0) K∇m h • ∇v h dx + Ω(0) c 1 • (∇m h )v h dx + Ω(0) c 2 m h v h dx = 0 a.e. t ∈ [0, T ], ∀v h ∈ V h m h (0) = m 0h (18)
where m 0h ∈ V h is an approximation of the initial condition m 0 . The unknown function m h is discretized as follows:

m h (x, t) = N j=1 m j (t)ϕ j (x), t > 0,
where m j is the value of the solution m h at the node x j , and N is the total number of nodes. We set v h = ϕ i , the weak formulation in [START_REF] Moroney | Numerical analysis of NMR diffusion measurements in the short gradient pulse limit[END_REF] gives:

N j=1 d t m j (t) Ω(0) ϕ j ϕ i dx + m j (t) Ω(0) K∇ϕ j • ∇ϕ i dx + Ω(0) c 1 • (∇ϕ j )ϕ i dx + Ω(0) c 2 ϕ j ϕ i dx = 0, (19)
for i = 1, . . . , N . We Consider m(t) = (m i (t)) 1≤i≤N . For t ∈ [0, T ], we introduce the matrices:

M ij = Ω(0) ϕ j ϕ i dx ; S ij (t) = Ω(0) K(x, t)∇ϕ j • ∇ϕ i dx, K ij (t) = Ω(0) c 1 (x, t) • (∇ϕ j )ϕ i dx ; H ij (t) = Ω(0) c 2 (x, t)ϕ j ϕ i dx,
for i, j = 1, . . . , N . The weak formulation [START_REF] Van | A finite element methods to solve the Boch-Torrey equation applied to diffusion magnetic resonance imaging[END_REF] can be written in matrix form:

M d dt m(t) + A(t)m(t) = 0, t > 0, (20)
with

A(t) = S(t) + K(t) + H(t),
and the initial condition m(0) is given. For the temporal discretization of ODE [START_REF] Rohmer | A Bloch-Torrey equation for diffusion in a deforming media[END_REF] an Euler implicit scheme is used. The interval [0, T ] is subdivided into n T ∈ N sub-intervals of equal length ∆t := T /n T . We therefore have the sequence (m n ) 0≤n≤n T satisfying:

1 ∆t M(m n+1 -m n ) + A n+1 m n+1 = 0, with A n = A(n∆t) and m 0 = m(0) is given.
4 Application to cardiac diffusion MRI

Analytical 2D short-axis cardiac motion model

To study the effect of cardiac motion on diffusion MRI images, we implement a 2D cardiac image simulator introduced in [START_REF] Clarysse | 2D spatial and temporal displacement field fitting from cardiac MR tagging[END_REF]. This simulator is based on a spatial-temporal analytical cardiac motion model mimicking a realistic deformation of the heart. It can be applied to a real short-axis image of the left ventricle to generate from it a synthetic sequence of MRI images of the beating heart. The resulting images resemble to a real MRI sequence and can be used to control the dynamics of the heart (Fig. 2).

The initial form of a short-axis slice of the left ventricle can be modeled by a ring delimited by two circles of internal R int and external R ext radii, respectively (Fig. 2). The heart deformation is defined analytically during the cardiac cycle and controlled by the temporal function S given by:

S(t) = 1 2 (1 -cos( πt Ts )) if 0 ≤ t ≤ T s 1 2 (1 -cos( π(t-Ts-T d ) T d )) if T s < t < T s + T d ,
where the time T s is the cardiac contraction duration that is called systolic phase and T d is the cardiac expansion duration known as diastolic phase. The duration of one cardiac cycle in human heart is about 1000ms. The systolic duration covers one third of the cardiac cycle and the diastolic duration covers two thirds (Fig. 3). We denote by (R 0 , θ 0 ) the polar coordinates at an initial time t 0 of a material point c 0 in the ring

Ω(0) = [R int , R ext ]×[0, 2π],
and by (R(t), θ(t)) its coordinates during deformation. The heart deformation ϕ is defined in polar coordinates by [START_REF] Clarysse | 2D spatial and temporal displacement field fitting from cardiac MR tagging[END_REF]:

ϕ : [0, T ] × Ω(0) → Ω(t) (t, c 0 ) → ϕ(t, c 0 ) = c t with c t = ϕ(t, (R 0 , θ 0 )) = R(t) = R 2 0 -R 2 int λ(t) g(θ 0 ) + R int (θ 0 , t) 2 1/2 θ(t) = θ 0 + ψ(t, R 0 ) + χ(t).
The variation of thickening of the myocardial wall over time is given by λ(t) = 1 -0.2S(t). The radial contraction is:

R int (θ 0 , t) = R int -(R int -R min )g(θ 0 )S(t),
with g(θ 0 ) = 1 + 0.1(cos(θ 0 + 3π/4) + 1),

R min = R int -5(R ext -R int )/6
is the minimal limit value of the internal radius R int during systole, and

ψ(t, R 0 ) = -(0.2π/180)S(t)R 0 ; χ(t) = (10π/180)S(t).
Then, the heart deformation field in Cartesian coordinates is: ϕ(t, (x, y)) = R(t) cos θ(t), R(t) sin θ(t) .

Diffusion measurements in cardiac diffusion MRI

There are several forms of diffusion encoding sequences proposed in literature for diffusion measurement in the beating heart. In this paper we choose a diffusion encoding sequence called STEAM (see [START_REF] Reese | Measuring diffusion in the presence of material strain[END_REF], [START_REF] Tseng | Cardiac diffusion tensor MRI in vivo without strain correction[END_REF]) for the numerical simulations of the diffusion MRI images. The idea of this sequence is to divide the spin echo diffusion encoding sequence (Fig. 1) into two parts and replace the 180 o RF pulse by two RF pulses of 90 o which will be applied in two successive cardiac cycles (Fig. 4). To use this diffusion encoding sequence, the heartbeat should be regular and the motion is periodic during two consecutive heart beats. The diffusion encoding gradients must be applied exactly at the same time in two successive cardiac cycles in order to fix the cardiac position and compensate the two phase terms that are equal in amplitude and opposite in sign. A duration equal to T E/2 separates between the first and second RF pulse 90 o , and between the third RF pulse and the acquisition of diffusion MRI signal. The time ∆ is equal to the duration of one cardiac cycle, and the two parts of the diffusion encoding sequence are separated by a period called mixing time T M := ∆ -T E/2. In cardiac diffusion imaging, the diffusion encoding sequences are synchronized with the electrocardiogram (ECG). This serves to trigger acquisitions at different times of the cardiac cycle by varying the trigger delay (TD) that corresponds to the time between the beginning of the cardiac cycle and the beginning of the sequence of diffusion encoding magnetic field gradients. Synchronization makes it possible to define an optimal acquisition windows during which the influence of cardiac motion on the diffusion MRI images is reduced.

Numerical simulations

In this section we study numerically the influence of cardiac motion on diffusion MRI images at different moments of the cardiac cycle. A STEAM diffusion encoding gradients (Fig. 4) is applied and for the cardiac motion, the analytical cardiac deformation field introduced in Section 4.1 is used. The magnetization M xy is calculated by solving the Bloch-Torrey equation with motion by finite element method as described in Section 3. The relative error in diffusion is calculated on the whole domain Ω(0), over the cardiac cycle, and presented on Figure 7(a). This result confirms that the diffusion is unaffected by the cardiac deformation at two points in the cardiac cycle called sweet spots that correspond to times when the cardiac deformation approximates its time average during the cardiac cycle. These time points are identified at mid-systole and mid-diastole (Fig. 7(b)).

These numerical simulations were already published in [START_REF] Mekkaoui | Quantifying the Effect of Tissue Deformation on Diffusion-Weighted MRI: A Mathematical Model and an Efficient Simulation Framework applied to Cardiac Diffusion Imaging[END_REF] in a comparative framework for evaluating different forms of diffusion encoding sequences used in cardiac diffusion MRI. These results are in agreement with the previous experimental studies (see [START_REF] Tseng | Cardiac diffusion tensor MRI in vivo without strain correction[END_REF], [START_REF] Dou | Cardiac diffusion MRI without motion effects[END_REF], [START_REF] Stoeck | Dual-Phase Cardiac Diffusion Tensor Imaging with Strain Correction[END_REF] for example). In literature, it was proposed to perform diffusion measurements at sweet spots when the deformation effect is minimized. However, this prevents acquisitions at the other time points of cardiac cycle.

Generalization of ADC formula

As we have seen in the previous section, the classical ADC formula does not give a sufficiently good reconstruction of the diffusion in the case of moving organ. To address this limitation we propose a new motion correction method of the diffusion MRI images which were reconstructed in the presence of motion. This method is established from an asymptotic model based on a particular scale of coefficients of the Bloch-Torrey equation with motion for the demodulated magnetization m (Eq. ( 16)).

The derivation of the asymptotic model for the complex magnetization M xy is based on an analysis of order of magnitude of the diffusion D and the gradient of phase ∇Φ which are present in the coefficients of problem [START_REF] Xu | Numerical study of water diffusion in biological tissues using an improved finite difference method[END_REF] for the magnetization m. In the biological tissues, the diffusion D is usually in the order of 10 -8 cm 2 /ms [START_REF] Price | Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: Part 1. Basic theory[END_REF]. The phase term Φ in the case of tissue deformation is given by:

Φ(x, t) = γ t 0 ϕ(x, t ) • G(t )dt ,
where G is the diffusion encoding gradient and ϕ is the cardiac deformation. The constant γ is the gyromagnetic ratio of the hydrogen atom that satisfies γ 2π = 42.58MHz/Tesla, and thus γ = 2.67513 × 10 5 rad ms -1 Tesla -1 . We calculate the evolution of the squared norm of ∇Φ(x, t) at different times of the cardiac cycle. The results are shown in Figure 8. As we can see, the quantity ∇Φ(x, t) 2 2 is in the order of 10 6 . Thus we can choose a strictly positive small parameter ε that is in the order of 10 -4 , and consider D = 2 D and ∇Φ = 1 ε ∇Φ. Then problem ( 16) can be transformed into the following problem, where its solution will be denoted m ε :

   ∂ t m ε -div(ε 2 K∇m ε ) + εc 1 • ∇m ε + (ε C2 + C 3 )m ε = 0 in Q, m ε = 0 on ∂Ω(0) × [0, T ], m ε = m 0 on Ω(0) × {0}, (21)
with K = ε 2 K and the coefficients c1 , C2 , and C 3 are given by:

c1 = ε h + 2i K ∇Φ ; h = K∇ t 0 div(v)(x, s)ds , C2 = i( K : (D ∇Φ) -h • ∇Φ + ∇ t K ∇Φ), C 3 = 1 T 2 + div(v) + K ∇Φ • ∇Φ = 1 T 2 + div(v) + K∇Φ • ∇Φ.
Now we study the asymptotic behavior of the solution m ε of problem ( 21) when ε tends to 0. We write this solution as an asymptotic series with ε:

m ε = m 0 + εm 1 + • • •
We substitute this approximation in [START_REF] Nishimura | Principles of Magnetic Resonance Imaging[END_REF], we find that m ε tends formally to m 0 when ε → 0 where m 0 satisfies:

∂ t m 0 (x, t) + C 3 t)m 0 (x, t) = 0 in Q with m 0 (x, 0) = m 0 (x) on Ω(0). (22)

Existence and unicity of solution m

For each time t ∈ [0, T ], we denote by L ε the differential operator given by:

L ε u = -div(ε 2 K∇u) + εc 1 • ∇u + (ε C2 + C 3 )u
where the components of c1 , and the coefficients Kij , C2 , C 3 are supposed in L ∞ (Q). Under the assumption of small deformations, as in Section 3.2 the operator L ε , ε > 0 is elliptic in the sense:

∃δ(ε) > 0, δ(ε) → 0 with ε, such that ∀ξ ∈ C d , Re ε 2 Kξ • ξ ≥ δ(ε)|ξ| 2 p.p., x ∈ Ω(0), t ∈ [0, T ].
A weak formulation for problem [START_REF] Nishimura | Principles of Magnetic Resonance Imaging[END_REF] can be written as follows:

   Find m ε ∈ L 2 ([0, T ], H 1 0 (Ω(0), C)) ∩ H 1 ([0, T ], H -1 (Ω(0), C)) such that ∂ t m ε , v + a(t; ε; m , v) = 0, ∀v ∈ V, a.e. t ∈ [0, T ], m ε (x, 0) = m 0 , (23)
where a(t; ε; m ε , v) : V × V → C, ε > 0, is a family of sesquilinear forms given by:

a(t; ε; m ε , v) = L ε (t)m ε , v = Ω(0) (ε 2 K∇m ε • ∇v + εc 1 • ∇m ε v + (ε C2 + C 3 )m ε v)dx.
Then as previously, the unique existence of solution u of the problem (23) can be obtained as a consequence of theorem (3.1). notice that the coercivity of a can then be obtained by the following lemma.

Lemma 5.1. The form a associated to the operator L ε is coercive if there exists α(ε) > 0, σ(ε) ∈ R, and

α(ε) → 0, σ(ε) → σ, when ε → 0, satisfying: α(ε) v 2 V ≤ Rea(t; ε; v, v) + σ(ε) v 2 H , ∀t ∈ [0, T ], ∀u, v ∈ V.
Proof. By using the ellipticity of L ε , we have:

δ(ε) Ω(0) |∇v| 2 dx ≤ Re a(t; ε, v, v) -Re Ω(0) (εc 1 • ∇vv + (ε C2 + C 3 )|v| 2 )dx ≤ Re a(t; ε; v, v) + ε |c 1 | L ∞ (Q) Ω(0) |∇v||v|dx + (ε C2 L ∞ (Q) + C 3 L ∞ (Q) ) Ω(0) |v| 2 dx ≤ Re a(t; ε; v, v) + ε |c 1 | L ∞ (Q) Ω(0) |∇v||v|dx + (ε C2 L ∞ (Q) + C 3 L ∞ (Q) ) Ω(0) |v| 2 dx.
By using the Young's inequality with > 0, we have:

(δ(ε) -ε |c 1 | L ∞ ) Ω(0) |∇v| 2 dx ≤ Re a(t; ε; v, v) + ( ε 4 |c 1 | L ∞ + ε C2 L ∞ + C 3 L ∞ ) Ω(0) |v| 2 dx. We add (δ(ε) -ε |c 1 | L ∞ ) Ω(0) |v| 2 dx
to each member of the inequality, it yields:

α(ε) v 2 V ≤ Re a(t; ε; v, v) + σ(ε) v 2 H with σ(ε) = δ(ε) + ( ε 4 -ε ) |c 1 | L ∞ + ε C2 L ∞ + C 3 L ∞ (24)
and

α(ε) = δ(ε) -ε |c 1 | L ∞
where is chosen such that ∃α > 0 satisfying 0 < ε 2 α < α(ε). We have α(ε) tends to 0 and σ(ε) tends to σ = C 3 L ∞ (Q) when ε tends to 0.

Convergence of m to m 0

A convergence result of m ε to m 0 when ε tends to zero is given by the following theorem:

Theorem 5.1. Under the ellipticity assumption of L ε , ε > 0, and [START_REF] Le Bihan | MR imaging of intra-voxel incoherent motions: application to diffusion and perfusion in neurologic disorders[END_REF]. In addition, α(ε)m ε converges weakly to 0 in L 2 ([0, T ], H), and ∂ t m ε converges weakly in L 2 ((0, T ), V ) to ∂ t m 0 , when ε tends to 0.

c1 ∈ C 1 (Q, C) d , the solution m ε of problem (23) converges weakly in L 2 ([0, T ], H) to m 0 solution of
Proof. We have:

∂ t m ε (t), v + a(ε; t; m ε (t), v) = 0, for v ∈ V.
We take v = m , we get:

1 2 d dt m ε 2 H + Rea(ε; t; m ε (t), m ε (t)) = 0.
According to Lemma 5.1, the form a is coercive. Then:

d dt m ε (t) 2 H + 2α(ε) m ε (t) 2 V ≤ 2σ(ε) m ε (t) 2 H . (25)
This implies:

m ε (t) H ≤ exp(σ(ε)t) m 0 H . (26)
According to [START_REF] Duvaut | Mécanique des milieux continus[END_REF] we have:

∃ε 0 such that ∀ε < ε 0 , σ(ε) ≤ 2 C 3 L ∞ (Q)
. By integrating (26) between 0 and T , we get the estimation: m ε L 2 ([0,T ],H) ≤ C m 0 H , with C a constant independant of ε. We deduce that the function m ε est bounded in L 2 ((0, T ), H) independently of ε. By substituting the inequality [START_REF] Clarysse | 2D spatial and temporal displacement field fitting from cardiac MR tagging[END_REF] in [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] and integrating between 0 and T , we find:

m ε (T ) 2 H + 2α(ε) m ε 2 L 2 ([0,T ],V ) ≤ exp(2σ(ε)T ) m 0 2 H .
Then:

α(ε)m ε L 2 ([0,T ];V ) ≤ C m 0 H , (27)
with C a constant independent of ε. Then α(ε)m ε is bounded in L 2 ([0, T ]; V ), and likewise for the function εm ε since we have ε 2 α < α(ε).

We show now that ∂ t m ε is bounded in L 2 ([0, T ], V ). We have:

| ∂ t m ε (t), v | ≤ |a(t; ε; m ε , v)| ≤ ε 2 K 2 L ∞ (Q) Ω(0) |∇m ε ||∇v|dx + ε |c 1 | L ∞ (Q) Ω(0) |∇m ε ||v|dx + (ε C2 L ∞ (Q) + C 3 L ∞ (Q) ) Ω(0) |m ε ||v|dx ≤ ε K 2 L ∞ εm ε V + |c 1 | L ∞ εm ε V + C2 L ∞ εm ε V + C 3 L ∞ m ε H v V .
Since 0 < ε ≤ 1, then:

∂ t m ε V := sup v∈V, v ≤1 | ∂ t m ε (t), v | ≤ K 2 L ∞ + |c 1 | L ∞ + C2 L ∞ εm ε V + C 3 L ∞ m ε H .
By integrating between 0 and T , and using the fact that εm ε L 2 ([0,T ],V ) and m ε L 2 ([0,T ],H) are bounded, we have:

∂ t m ε L 2 ([0,T ],V ) ≤ C m 0 H , where C is a constant independent of ε. Then we deduce that the function ∂ t m ε is bounded in L 2 ([0, T ], V ).
We can then extract a subsequence of m ε that converges weakly in L 2 ([0, T ], H) to a function u 1 ∈ L 2 ([0, T ], H), and a subsequence of α(ε)m ε weakly converging to a function u 2 in L 2 ([0, T ], V ). By using the result of compactness of the canonical injection of V in H (Theorem Rellich-Kondrashov), the convergence of α(ε)m ε to u 2 is in L 2 ([0, T ], H). As the convergence of m ε to u 1 is in L 2 ([0, T ], H), and α(ε) tends to 0 with ε, we have that α(ε)m ε tends to 0 in L 2 ([0, T ], H). By uniqueness of the limit, it comes u 2 = 0. Similarly, we have the weak convergence of ∂ t m ε in L 2 ([0, T ], V ) to a function u 3 . By the continuity of derivation in the space distributions D (Q), ∂ t m ε converges weakly to ∂ t u 1 in L 2 ([0, T ], V ). We use these weak convergences to go to the limit as ε tends to 0 in the weak formulation [START_REF] Mattiello | Analytical Expressions for the b Matrix in NMR Diffusion Imaging and Spectroscopy[END_REF]:

∂ t m ε , v + Ω(0) (ε 2 K∇m ε • ∇v + εc 1 • ∇m ε v + (ε C2 + C 3 )m ε v)dx = 0,
and to show that the second term tends to zero we use the fact that ε∇m ε is bounded in L 2 ([0, T ], H). For the third term we integrate by parts which is possible because c1 ∈ C 1 (Q, C) d , then we use the fact that m ε is bounded in L 2 ([0, T ], H) to show that it tends to zero. Finally, we get the problem:

∂ t u 1 , v + Ω(0) C 3 u 1 vdx = 0, ∀v ∈ V p.p. t ∈ [0, T ], u 1 (x, 0) = m 0 ,
where u 1 (•, 0) has a sense because we have u 1 , ∂ t u 1 ∈ L 2 ([0, T ], V ), then u 1 ∈ C 0 ([0, T ], V ). By comparing this problem to Eq. ( 22), we get u 1 = m 0 . Hence the weak convergences of m ε to m 0 in L 2 ([0, T ], H), and of ∂ t m ε to ∂ t m 0 in L 2 ([0, T ], V ).

Analysis of the ADC formula and correction

We recall that in the case static domain, the ADC formula reads as:

ln S(x) S 0 (x) ln M xy (x, T E) M xy (x, T E)| G=0 -B : D(x),
where the diffusion weighting B-matrix is given by:

B = γ 2 T E 0 t 0 G(s)ds ⊗ t 0 G(s)ds dt .
We have seen in the previous section that this formula is not valid in the case of moving media. By using the explicit expression of m 0 and the convergence result of m to m 0 , we can derive the following generalized ADC formula:

ln S(x) S 0 (x) = ln M xy (x, T ) M xy (x, T )| G=0 -B ϕ (x) : D(x), (28)
where T = T E + T M , and B ϕ represents a B-matrix modified according to the deformation ϕ:

B ϕ (x) = γ 2 T 0 w(x, t ) ⊗ w(x, t )dt , with w(x, t) = F -t (x, t)∇Φ(x, t) ; Φ(x, t) = t 0 ϕ(x, t ) • G(t )dt .
Indeed, remark that using the asymptotic ODE model [START_REF] Le Bihan | MR imaging of intra-voxel incoherent motions: application to diffusion and perfusion in neurologic disorders[END_REF], the magnetization m 0 satisfies:

m 0 (x, t) = m 0 (x) exp(-t/T 2 ) exp - t 0 ∇Φ(x, t ) t K(x, t )∇Φ(x, t )dt exp - t 0 div(v)(x, t ) ,
and then we have the magnetization:

M xy (x, t) m 0 (x) exp(-t/T 2 ) exp t 0 div(v)(x, t ) exp(-iΦ(x, t )dt ) × exp - t 0 ∇Φ(x, t ) t K(x, t )∇Φ(x, t )dt .
From a practical point of view the deformation field ϕ is supposed to be known at each time interval during which the diffusion encoding sequence is applied in order to be able to calculate the modified diffusion weighting matrix B ϕ for the estimation of the diffusion tensor D.

Numerical simulations

In Section 4.3 we have seen that the diffusion images (see Fig. 6) are influenced by cardiac motion during the cardiac cycle. This results show the need to correct the diffusion images by comparing them with the image of exact diffusion. In this section the diffusion MRI images will be corrected according to Eq. ( 28) to reconstruct new diffusion images without motion effect. On figures 9 and 10 we present the diffusion images reconstructed at different times in the cardiac cycle before and after motion correction. These results are shown in the deforming configuration because the image reconstruction is done when the deformation takes place. The images of the error between the exact diffusion and the diffusion reconstructed without motion effect are also shown. We observe that the corrected diffusion is close to the exact diffusion shown on Figure 6. A case of non-homogeneous isotropic diffusion in also tested. On Figure 11 the exact diffusion coefficient is shown. Figures 12 and13 present the reconstruction results of the diffusion MRI images before and after motion correction in systole and diastole, at TD=250ms and TD=850ms, respectively. As we can see on the figures, the corrected diffusion obtained in systole and diastole is close to the exact diffusion coefficient.

Conclusion

In this paper, we introduced modified model of the Bloch-Torrey equation that describes the behavior of the complex magnetization and the diffusion MRI signal in the presence of physiological motion. Then, we quantified, by using numerical simulations, the effect of cardiac motion on the diffusion MRI images reconstructed at different moments of the cardiac cycle. Two time points called sweet spots have been identified in the cardiac cycle, wherein the diffusion images are not affected by cardiac deformation. However, when the diffusion encoding sequence is performed at time points of the cardiac cycle other than sweet spots, a motion correction is necessary after the reconstruction of the diffusion MRI images. For this purpose, we have proposed a motion correction method by means of a new apparent diffusion coefficient formula which is based on an asymptotic ODE model for the complex magnetization. This model has been obtained by an asymptotic analysis of the Bloch-Torrey equation with motion. This motion correction method has been used to correct the simulated diffusion MRI images which were reconstructed at different moments in the cardiac cycle. This method is valid at any time of the cardiac cycle, since it is based on a mathematical expression that shows explicitly the influence of cardiac deformation on the diffusion.

Figure 1 :

 1 Figure 1: Spin echo diffusion encoding sequence. Two identical gradients are applied around the 180 o RF pulse. G is the gradient intensity, δ the gradient duration and ∆ the gradient spacing.

Figure 2 :

 2 Figure 2: (Left) Cardiac MRI images generated by the simulator introduced in [26]. The region of interest (the left ventricle zone) is shown inside the yellow squares. (Right) A domain Ω(0) in the form of a ring is chosen for representing the left ventricle zone.

Figure 3 :

 3 Figure 3: Behavior of the function S over one cardiac cycle. Ts = 333ms, T d = 667ms.

Figure 4 :

 4 Figure 4: STEAM diffusion encoding sequence.

3 .

 3 The initial condition is m(x, 0) = 1. The transverse relaxation time is T 2 = 60ms. The diffusion is supposed homogeneous and isotropic D = 2.5 × 10 -3 mm 2 /s. The diffusion encoding gradient is applied in the x-direction. The parameters of the diffusion encoding sequence are ∆ = 1000ms, δ = 1ms, T E = 6ms, G = 79.5mTesla/m. The duration of the simulated cardiac cycle is 1000ms, the trigger delay TD varies between 0 and 950ms, and the position of the diffusion encoding sequence in the cardiac cycle varies according to TD. On figure5we show the norm of Du at different values of TD where u is the displacement during the application of the diffusion encoding gradients. As we can see the assumption Du <1 made in Section 3.2 is satisfied.

Figure 5 :

 5 Figure 5: Du calculated during the application of the diffusion encoding gradients for different values of TD.

Figure 6 :

 6 Figure 6: (Top) Diffusion MRI images at different moments of cardiac cycle. (Bottom) Exact diffusion coefficient.

Figure 7 :

 7 Figure 7: (a) Relative error in diffusion coefficient. (b) Localization of the sweet spots when the cardiac deformation is approximately equal to its temporal mean during the cardiac cycle.

Figure 8 :

 8 Figure 8: The squared norm of ∇Φ(x, t) calculated at different moments of the cardiac cycle: (a) (b) TD=200ms, (c) TD=350ms, (d) TD=600ms, (e) TD=900ms.

Figure 9 :Figure 10 :

 910 Figure 9: Diffusion images reconstructed in systole. 1 st column: Before correction at: TD=0ms, TD=100ms, TD=350ms. 2 nd column: After correction. 3 rd column: Absolute error between the exact diffusion and the corrected diffusion images.

Figure 11 :Figure 12 :

 1112 Figure 11: Exact diffusion.

Figure 13 :

 13 Figure 13: Images constructed at TD=850ms. 1 st row: Diffusion encoding gradient applied in x-direction: (a) Diffusion before correction. (b) Diffusion after correction. (c) Absolute error between the exact diffusion and the corrected diffusion images. 2 nd row: Diffusion encoding gradient applied in y-direction: (d) Diffusion before correction. (e) Diffusion after correction. (f) Absolute error between the exact diffusion and the corrected diffusion images.


Patrick Goujon

PRISMES Tiffane Levick

Translating homophonic wordplay in

Keywords: wordplay, rap, compensation, creativity, compromise 1 Bradley, Adam

This paper presents a discussion of the issues raised in the translation of homophonic wordplay from French into English. To this end, it takes as its point of departure examples from the translation into English of a contemporary French novel, a novel entitled Moi non, written by Patrick Goujon and published by Gallimard in 2003. These examples are rife with the manifestations of the author's stylistic game which harnesses the techniques of rap in an exploration of the implications of sound and sense in poetic prose. When faced with the intricacy of the expression in the original text, the translator must strive to identify solutions that respect simultaneously the playful and the connotative functions of the words used. This process beckons questions relative to the artistic licence afforded to the translator who is bound to an assessment of the importance of sound versus that of semantics, seeking a creative compromise between meaning and wordplay.

Homophonic translation, or the translation of homophones? To the untrained ear, the two options prove a source of confusion, beckoning the question of whether these two types of translation are synonymous. Alas, they are not; or, at least, not entirely. Homophones, homophonic; the two are not one and the same. The sounds are similar, and indeed so are the meanings, and though the implications courting these two processes of translation share common concerns in the transfer of sound and of sense, they stand simultaneously one to the other in stark contrast. On one end of the spectrum sits homophonic translation, adopting as its agenda the creation of new semantic networks in the translated text. The enterprise of the translation of homophones sits towards the other end of this spectrum, focussing on the creation of new sound networks in the translated text and undertaken within a more orthodox overall strategy of translation. Homophonic translation, then, is akin to a recreation of sense, whereas the translation of homophones involves a recreation of sound. It is apparent that the interaction of sound and sense in the two types of translation is what sets these two processes apart. This paper offers therefore not an investigation of homophonic translation as such, but a case study of the translation of homophonic wordplay in a novel containing a large amount of rap and in which the techniques of rap surface within the prose itself. This novel is entitled Moi non, and it was written by Patrick Goujon and published by Gallimard in 2003. Though the translation of this novel provided me with the opportunity to test certain hypotheses, since I undertook the translation within the context of my PhD. my overall approach to the translation was fundamentally conventional insofar as my primary concern was that of producing a text in English which could be readily compared with the original French both in terms of style and of meaning. The novel translated relates the thoughts and experiences of the two young men, the main narrators Hoch and Flex, and of their friends and family. The characters all live on the outskirts of Paris and the book is set in the early 2000s, though this information is not directly divulged to the readers. Making use of highly informal language characterised by slang and musicality, Goujon forges in Moi non a distinct style of writing combining the colloquial and the lyrical, and homophones provide for him the ideal platform to dabble in a juggling of sound and of polysemy. As such, the novel is sewn together with strings of sounds and semantic fields, filled with the homophones that he sows in his lines and verses and paragraphs.

The style and effect of the game that Goujon plays in Moi non can be likened to the "flow" of a piece of rap, or the rhythm and the rhyme of the words, and the way in which these elements interact. In The Book of Rhymes: The Poetics of Hip Hop, Adam Bradley defines flow as "an MC's distinctive lyrical cadence, usually in relation to a beat. It is rhythm over time." 1 Goujon's focus on flow renders his text lively and creative, and the webs of linguistic complexity weaved into his lines provided ample matter for reflection in the attempt to translate this wordplay into English. My concerns in translation orbited around the ideal of identifying creative solutions that respected the playful acoustics of the original text without straying too far from its structure and content. Determining to what extent the flow of the original needed to be respected, and in what ways, and what liberties I could therefore take in the manipulation of the structure of the original text and of the specific images it conveys, was by no means an objective process: I was acutely aware of my agency as a translator, and of the subjectivity of the reasoning behind my own decisions, with which not all translators, readers or critics would agree.

In their introduction to The Practices of Literary Translation, Constraints and Creativity, Jean Boase-Beier and Michael Holman point out that "There is no agreement on the level of intervention permitted to the translator." 2 To cite two other theorists writing about the role of creativity in translation, Manuela Perteghella and Eugenia Loffredo, we note that "The source text offers the starting point for a journey and becomes the space 'into' and 'through' which the translator is given the opportunity to explore creatively and perform his/her subjectivity." 3 It therefore seems reasonable to assert that what links homophonic translation with my own approach to translation, and, indeed, with all types of translation, is the steps all translators must necessarily take each time they undertake a translation: namely, those of interpretation and of expression, or of re-expression. When reading a text to be translated homophonically, the translator must privilege certain sounds or combinations of sounds of this original, and identify within the abundance of possibilities available to translate each sound unit or combination of sound units in the language of translation the solutions they find the most fitting. In the same vein, the translator undertaking a "standard' translation must identify, consciously or unconsciously, the elements of the original they find the most relevant and then sift through the abundance of possibilities available to translate the series of words from the source language into the target language. And the choices made by each translator will necessarily be different.

When translating the passages of Moi non particularly marked by creativity and inventiveness, it frequently occurred to me that I was writing a version of these passages of the original text in English that constituted my own version of the original. As Susan Bassnett would put it, I had become a recreator of the text in English,4 and while this text in English was the product of my own imagination, it was also bound by the limits of the general form and content of the original text. I had become a co-author of sorts, and I was conscious that my version of the text would be vastly different from that which might be produced by another translator, with a set of interpretive and imaginative capacities necessarily different from my own. This much can be said for any translation, of course: no two translators will produce an identical version of the same text in the target language, but it seems to be especially true of translations of heavily-stylised texts, where one notices a distinct effort to play with literary devices, inviting, and indeed obliging, the translator to unearth particularly creative solutions in the target language.

Having now laid the foundations of the concerns with which the translator must contend when dealing with homophonic wordplay, we are ready to explore more concretely the implications of such concerns in the work chosen for discussion in this paper. The following seven examples from the translation of Moi non are organised in terms of the types of homophones used in the original and in the translation.

Translation through exact correspondence

Of the numerous techniques adopted to translate homophonic wordplay in Moi non, the first technique, that of exact correspondence in both languages, is the least commonly exploited. The following example is in fact the sole case in which I deemed the technique both possible and appropriate: Tu connais la presse ? Do you know the press?

Non.

J'ai pensé Presse, presse comme journaux.

Parce que tu es à la presse. Le préposé est en congé maladie et tu le remplaces. À la presse à mannequins.

(p. 28)

No.

I thought Press, press as in newspapers.

Because you're in the press. The attendant is on sick leave and you're replacing him. In the mannequin press.

Since the nouns la presse and "the press" share the same double meaning in both languages, indicating both the machine and the journalism industry, we are afforded a swift and straightforward transfer between the two texts. A more linguistically nimble translator may have found appropriate solutions to employ this technique elsewhere in the translation but the inclusion of homophones often appeared incongruous, prodding me in the direction of a more liberal approach, as outlined in the remaining examples.

Compensation through the use of added homophones

In order to compensate for the several instances in which I was unable to use homophones in the translation in the same passage as which they appeared in the original, I endeavoured to identify opportunities to add to the translation a number of homophones at different places where there was no corresponding homophone in the same passage in the original.

je suis le rêve de l'arbre sur des milliers d'années / l'écorce patiente qui sent la main du nouveau-né / je suis les initiales gravées sur le torse, le bois, le flanc / je suis l'un, je suis l'autre, ceci est ma force. (p. 204) I'm the dream of the tree that thousands of years have borne / the patient bark feeling the hand of the baby just born / I'm the initials engraved in the hide, the wood, the side / I'm one, I'm the other, this is my pride.

In this example, I seized the opportunity to play on the sound "born," by using two of the three past participles of the verb "to bear:" so, "borne," with an e, and "born," without an e. This choice enabled both the use of a rhyme, essential to the two bars of rap, and of a homophone. I fleshed out the sur of the original and transposed it into a verb: the tree had carried, held, or borne a dream for thousands of years, while the baby had just come into the world, or been born.

3.

Change in part of speech

When I was able to keep a homophone in the same place in the translation as in the original, where the words in both languages could convey different meanings in similar ways, it was, as previously established, rarely possible to use the homophone in the exact same way, that is to say, to keep the same part of speech. This passage again presents an example of rap lyrics, written by Flex.

Je veux pas me sentir enfermé, l'coeur en ciment en si peu d'temps / m'retrouver dans une peau que je kifferais pas, du vent / sur mes couplets, dans les poches de mes vêtements / le regard fatigué et froid de ceux qui squattent les bancs, mais bon / c'est comme ça nos vies, on pense qu'à se plaindre et à se tirer d'ici / c'qu'est toujours mieux que se tirer dessus / J'ai vu tellement de rêves échouer dans le décor / emboutir des hommes et des femmes sans même leur donner la mort / les r'garder droit dans les yeux, sans même les plisser / leur dire « à compter d'aujourd'hui ton seul bonheur s'ra d'aller pisser ». (p.

203)

I don't want to feel trapped in this urban sprawl, heart like a stone wall in no time at all / end up with a skin I don't want to be in, scrawl / lyrics as thin as the clothes beneath my chin, crawl / tired and cold, like drifters idling in a hall, but, after all / that's our life, we think only of complaining and of escaping on the next bullet train / which is always better than putting a bullet in the brain / I've seen so many dreams drown with the fish / crush men and women, not even grant their death wish / look them right in the eye and not even blink / tell them "from now on in your only pleasure will be smelling your shit stink."

Arriving at and settling on this translation was by no means a fast or facile process, but I departed from the two meanings of the verb se tirer evoked here, that of leaving and that of shooting, to brainstorm a list of words and expressions with similar meanings. Somewhere in this mess of notes was the word "bullet," noted to correspond with the idea of shooting oneself. When fleshed out into an expression, bullet could become "bullet in the head" or, indeed, "bullet in the brain," which conveniently rhymes well with the idea of running away on a bullet train. While they adjust the exact images projected by the original text, these two new rhyming images in English reflect the overall idea elicited by the French lines. The two expressions also function in terms of flow and establish a consistent and mirrored rhythm over the two bars.

Displacement of the homophone

In cases where it did not appear fitting to translate a homophone by another homophone, using the same or a different part of speech, it was still often possible to use a homophone in another part of the corresponding passage of the translation.

Je demande pas où sont les ceintures à l'arrière, vu que c'est la seule option absente dans la bagnole, et puis, avec le caisson de basses dans le coffre, faudrait que j'aie plus de coffre encore pour gueuler ma question. (p. 74) I don't ask where the seat belts are in the back seat since that's the only "extra" missing in the car, and, plus, with the subwoofer thumping behind us, it's not like my question'd be loud enough to beat the beat of the bass.

Another constraint framing my approach to the translation here necessitated the restructuring of the sentence; I generally avoided using vocabulary specific to a particular part of the Anglosphere in the translation and so chose not to translate the homophone coffre as "boot" or "trunk." This choice lead subsequently to the identification of an alternative solution, involving the use of a homophone, as I translated the location coffre more generally through an adverb of place ("behind") and displaced the use of a homophone to another part of the sentence. The noun homophone coffre in the original was translated by the homophone "beat" in English, used as a verb and then as a noun. This choice allowed the same sort of wordplay to be maintained, but inexactly, since the sentence is rearranged to allow the homophonic wordplay to be transferred to another word.

Partial homophones

Partial homophones are used extensively in the original text, especially within the passages of rap lyrics. These partial homophones are the product of assonance and consonance, featuring prominently in rap lyrics and often used to make half rhymes, or slant rhymes. The term partial homophones here refers not only to the use of words with similar sounds, but to the repetition of entire syllabic sounds through the use of the same base word or sound with different endings. Below are two examples of partial homophones, used in the original text and in the translation.

On est paumés, on compte pour des pommes et on tombe comme des mouches, pas de mouchoirs quand on meurt, les langues de putes se touchent et nous gomment, la vie est tellement conne parfois qu'on peut dégommer son voisin et rester sans voix, vis ce que je vis vois ce que je vois, j'ai la rage, la gorge et le poing serrés, elle pour le lancer lui et lui pour me rassurer... (p. 192) We're spaced out, a waste of space, and we're dropping like flies, not a tear will drop at our death, the catty bitches come out and drown us out, life's so dim sometimes you can make your bro your foe, drown him, take him out, and see the cat get your tongue, live what I live, see what I see, I'm riled up, throat tight, fist out to fight, the throat to thrust the fist and the fist to ease my mind… Paumés and pommes are translated as the slightly different sounding "space" and "spaced," in "spaced out" and "a waste of space," and mouche and mouchoir through two forms of the verb "to drop": "dropping" and "drop." I used the same technique of building on base words by changing the ending further down in the passage where I added a partial homophone: the noun "cat" and the adjective "catty." One might argue that this addition of a partial homophone compensates for the lack of homophone in the translation corresponding to the different uses of vois/voix. This example in particular demonstrates the significance of structural compromises through syntactical rearrangement in the translation to allow the sounds and images to reflect what I perceived to be the essence of the ideas conveyed in the original lines. The passage in English is markedly different from its French counterpart, but both are comparable in their use of rhythm and sound to communicate very similar messages.

The next example is taken from the narration in the first half of the novel and shows again the use of a partial homophone in the translation of a passage containing a pure homophone:

On se raconte nos petites histoires, enfin je t'ai déjà dit, les taffes soporifiques et les tafs en sous-marin, les coups de vice à la scred '. (p. 31) We trade stories, like I said, about trading under the table. About smoking and joking, puffing to relax, about wheeling and dealing, covering our tracks.

The double meaning of "trade," used in two forms ("trade" and "trading") is used to emulate the use of homophones in the original. In this example as in the previous, it is clear that I departed considerably from the syntactic arrangement of the original text. I took the first use of a gerundive and built it on in the second part of the passage, attempting to achieve a robust flow. This flow is different from in the original, but has more impact on an acoustic level than what I might have settled on had I followed the syntax of the original more closely. This focus on flow is what most influenced my translation strategies, and the final example to be studied reveals the extent to which considerations of rhythm and rhyme dominated my translation choices.

6.

Rhyme and rhythm

When it was neither possible nor apt to include homophones, partial or full, in the translation, the primordial concern still remained rooted in sound. I systematically strove to achieve a strong rhythm and rhyme in the English text by privileging words with complementary sounds arranged to enhance stress sonorous echoes.

Mon ami est mon paravent et d'ici que le type calcule, je prends ma part avant. (p. 20)

My friend's my shield and, by the time the guy's put two and two together, I've already sealed the deal.

In the above example, the homophonic wordplay of the French is not reproduced in the English. I opted instead for a rhythmic translation that channelled assonance and consonance, using the expression "to put two and two together" for calculer, and "to seal the deal," for prendre ma part avant. Though establishing homophonic wordplay in the English could have been a possibility, through playing with the word "shield," for example, I preferred the final solution without homophones in English as the flow of the sentences was particularly forceful, therefore placing the emphasis on rhythm.

The examples presented for analysis in this paper have explored indirectly the role of constraints in sparking and structuring creativity. When writing and translating rap, the writer and translator find themselves preoccupied by the conjuring of rhymes as well as of a certain rhythm that corresponds to the beats available in each bar. The writer of rap lyrics is faced with the arduous task of exploring the linguistic and stylistic possibilities present in their language that respect the requirements of rap so that the lyrics produced are pleasing to both the ear and to the mind. As we have established, even outside of the numerous passages of rap lyrics that punctuate the narration and the dialogue of Moi non, the prose remains heavily influenced by the concerns of flow, the hallmark of rap music. As such, the stylistic devices so common in rap lyrics frequently make a noticeable appearance in the narration and the rhythm that imbues each line enhances the overall force of each passage. Homophonic wordplay is perhaps the most salient of these devices.

The rap-related constraints placed on the writer of the original text are compounded in translation, where the translator must attempt to respect both the style and the content of the text as far as possible while making use of the possibilities available to them in the language of translation. In both the original and the translation, however, the constraints imposed on the writer and the translator by the features of rap music, by linguistic (im)possibilities, and by a range of other requirements both in and out of the text, can, and indeed do, provide a source of creativity. Boase-Beier and Holman, cited earlier, dedicated a book to case studies exploring the role of constraints in sparking creativity in literary translation, and in their introduction, they assert that "… creativity is often intimately tied to constraint; it is a response to it, it is enhanced by it."5 Framed within the deliberations of the articles in this collection, the resonance of such a statement hardly requires reinforcing.

In his book Translation, Rewriting and the Manipulation of Literary Fame, André Lefevere reminds us of the tendency to assume that a translator must privilege either the sound or the sense of the text being translated; that is impossible to render both in translation, to be a servant of two masters. "Translators, we are likely to be told, will be able to render the sense of the original only at the expense of the sound, and often also of the morphosyntactic features organising the original."6 I hope that in my translation of these passages containing homophonic wordplay, I have rendered the essence of the sense of the original text, sacrificing the morphosyntactic features of the original, which incidentally I felt was inevitable or at the very least indispensable, but not the sound. In taking liberties to perform syntactic reshuffling and modify specific images to suit the sound patterns that I, personally, felt had the most force in English. I have created a new text in my own language and my own words, which is able to stand alone, in its own right, but which is at the same time reflective of and clearly comparable to the original text in French.

Bassnett, Susan, and Peter Bush, The Translator as Writer, London, New York, Continuum, 2006

Boase-Beier, Joan, and Michael Holman, The Practices of Literary Translation: Constraints and Creativity, Manchester, St Jerome Publishing, 1999, p.

Lefevere, André, Translation, Rewriting and the Manipulation of Literary Fame, London, New York, Routledge, 1992, p. 100