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Cartan Frames for Heart Wall Fiber Motion

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Cartan Frames for Heart Wall Fiber Motion Cartan Frames for Heart Wall Fiber Motion Babak Samari1, Tristan Aumentado-Armstrong1, Gustav Strijkers2, B Martijn Froeling3, and Kaleem Siddiqi1( ) 1 School of Computer Science and Centre for Intelligent Machines, McGill University, Montreal, Canada [email protected] 2 Academic Medical Center, University of Amsterdam, Amsterdam, The Netherlands 3 University Medical Center, Utrecht, Utrecht, The Netherlands Abstract. Current understanding of heart wall fiber geometry is based on ex vivo static data obtained through diffusion imaging or histology. Thus, little is known about the manner in which fibers rotate as the heart beats. Yet, the geometric organization of moving fibers in the heart wall is key to its mechanical function and to the distribution of forces in it to effect efficient, repetitive pumping. We develop a moving frame method to address this problem, with a spatio-temporal formulation of the asso- ciated Cartan matrix. We apply our construction to simulated (canine) data obtained from a left ventricular mechanics challenge, and to in vivo human left ventricular data. The method shows promise in providing Cartan connection parameters to describe spatio-temporal rotations of fibers, which in turn could benefit subsequent analyses or be used for diagnostic purposes. 1 Introduction Mammalian heart wall muscle is comprised of densely packed elongated myocytes in an extra-cellular matrix [4,11]. The geometry of this packing facilitates effi- cient pumping to optimize ejection fraction while also providing strength. The precise manner in which fibers move and rotate with the material medium in which they are placed during the heart beat cycle is as yet not known. Current models of heart wall fiber geometry are derived from diffusion imaging of static ex vivo hearts and from histology. Models do exist for such data, for example the rule-based model of [1], and the techniques in [7]. However, such methods lack a temporal dimension to capture fiber rotation as the heart beats. Whereas there is on-going progress in our community towards in vivo cardiac diffusion imaging, with the possibility to now acquire full heart fiber orientation data in beating hearts [6,17], the mathematical tools for analysis lag behind. The key contribution of the current article is the use of Cartan connec- tion forms [12,16] to model both spatial (within a time sample) and temporal (between time samples) rotations of frame fields attached to heart wall fibers. This allows one to capture spatial and temporal geometric signatures within a c Springer International Publishing AG 2017 M. Pop and G.A. Wright (Eds.): FIMH 2017, LNCS 10263, pp. 32–41, 2017. DOI: 10.1007/978-3-319-59448-4 4 Cartan Frames for Heart Wall Fiber Motion 33 single consistent framework. We begin by reviewing Cartan connection forms and their use in moving frame methods. Extending a method for the case of sta- tic fibers in [13], we add a temporal dimension to the Cartan matrix. We then specialize the model by adopting frames fit to heart wall myofiber orientations recovered from diffusion data. We demonstrate the promise of this method for the recovery of geometric curvature type spatio-temporal signatures for moving myofibers during a heart beat cycle, using both finite element simulation on canine data [2] and human in vivo cardiac diffusion imaging. 2 Materials and Methods 2.1 Cartan Connection Forms We begin by reviewing the mathematics of Cartan connection forms for the case T 3 of an orthonormal unit frame field F =[f1,f2,f3] defined in R , which is the construction used in [13] to describe the geometry of myofibers in a static heart wall. A covariant derivative of this frame field with respect to a vector v at point p is given by ∇vfi = ωi1(v)f1(p)+ωi2(v)f2(p)+ωi3(v)f3(p), (1) where i ∈{1, 2, 3},andωij(v)=∇vfi · fj.[12] shows that these ωijs satisfy the 3 3 definition of 1-forms in R . For any vector field v in R , ∇vfi = j ωij (v)fj. Using the alternating property of 1-forms, ω11 = ω22 = ω33 =0andω21 = −ω12,ω31 = −ω13,ω32 = −ω23. Let us assume that the frame field F has the following parametrization in the universal Cartesian coordinate system E = T [e1,e2,e3] : fi = αi1e1 + αi2e2 + αi3e3, (2) where i ∈{1, 2, 3}, and each αij = fi·ej is a real valued function. The matrix A = [αij ] is called the attitude matrix, and dA =[dαij ] is a matrix whose elements are 1-forms. With ω =[ωij ] it can be shown that ωij = k(dαik)αkj [12], so ω = dAAT . Connection forms describe the rate of change of a frame field T [f1,f2,f3] in the direction of an arbitrary vector v. The dual 1-form of the T frame field [f1,f2,f3] is obtained when it is itself parametrized via 1-forms, i.e., for each vector v at p, ψi(v)=v · fi(p). For the sake of simplicity we will use the same notation for the frame and its dual representation, i.e., fi(v)=ψi(v)=v · fi(p). (3) From the definition of the 1-form, ∂x dx v v i p v . i( )= j ∂x ( )= i (4) j j Combining Eqs. (3), (4) results in fi(ej)=fi · ej =( αikek)ej = αij . (5) k 34 B. Samari et al. T From Eq. (4), (5)wehavefi = j αijdxj which, with F =[fi] , can be written T in the dual 1-form representation as F = A[dx1 dx2 dx3] . The Cartan structural equation is then given by [12]: df i = j ωij ∧ fj, i.e., dF = ωF, dωij = k ωik ∧ ωkj, i.e., dω = ωω. (6) Since there are three unique connection forms ω12(v),ω13(v),ω23(v), feeding the frame field’s unit vectors to them produces nine different connection parameters cijk = ωij(fk). Here, for the vector v at a point p, ωij (v) represents the amount of fi(p)’s turn toward fj(p) when p moves in the direction of v. The estimation of these connection parameters for frame fields attached to diffusion MRI data of ex vivo hearts was the strategy proposed in [13] to parametrize the static geometry of heart wall myofibers. 2.2 Cartan Forms for Moving Fibers We assume that for any query time t ∈ R we have a stationary state of an orthonormal moving frame field F t attached to heart wall fibers as in [13], with t T the corresponding universal coordinate E =[e1,e2,e3] : f1 is aligned with the fiber orientation, f3 is given by the component of the normal to the heart wall that is orthogonal to f1 and f2 is their cross-product. Our goal is to parametrize the local rotation of F t in both space and time so we extend the existing 3D coordinate system to add a 4th dimension to represent the time axis. Let E = T 4 [e1,e2,e3,e4] represent our extended universal coordinate system, R , in-which T ei · ej = δi,j where δi,j is the Kronecker delta, and e4 =[0, 0, 0, 1] is a basis vector for the time axis in our 4D representation. Thus a query < x,y,z,t >, where x, y, z, t ∈ R, represents a query < x,y,z > in Et.Weextendthelocal frames F t to 4D as follows: ⎡ ⎤ f t f t f t 1,x 1,y 1,z F t f ,f ,f T ⎣f t f t f t ⎦ Definition 1. Let =[ 1 2 3] = 2,x 2,y 2,z describe the frame field f t f t f t 3,x 3,y 3,z in R3 at time t, then: ⎡ ⎤ f t f t f t 1,x 1,y 1,z 0 ⎢f t f t f t ⎥ ⎢ 2,x 2,y 2,z 0⎥ ⎣f t f t f t ⎦ (7) 3,x 3,y 3,z 0 0001 is the 4D extension of the frame fields F t. t T Since we know that F =[f1,f2,f3] is an orthogonal matrix, it is not hard to T show that F =[f1,f2,f3,f4] is also orthogonal, i.e., ∀i = j, fi · fj =0. The next step is to extend the existing connection forms from 3D to 4D. Let T 4 F =[f1,f2,f3,f4] be an arbitrary frame field on R ,which,fori ∈{1, 2, 3, 4}, has the following parametrization in the extended universal coordinate system T E =[e1,e2,e3,e4] : fi = αi1e1 + αi2e2 + αi3e3 + αi4e4. (8) Cartan Frames for Heart Wall Fiber Motion 35 A covariant derivative of this frame field with respect to a vector v at point p is given by ∇vfi = ωi1(v)f1(p)+ωi2(v)f2(p)+ωi3(v)f3(p)+ωi4(v)f4(p), (9) 4 where [αij ]=[fi · ej] is the attitude matrix in R and ωij = k(dαik)αkj is a 1-form. Definition 2. Let i<jand i, j, k ∈{1, 2, 3, 4}. By feeding the frame field’s unit vectors (fis) to ωij (v) we have cijk = ωij (fk). (10) Remark: For j =4,cijk = ωij(fk) = 0. For a point p and with i<j∈ {1, 2, 3},ωij (fk) represents the amount of fi(p)’s turn toward fj(p) when taking a step towards fk(p). Given the skew-symmetry property of the 4 × 4 connection form matrix we build 3×4 = 12 different non-zero and unique cijks. We can use these coefficients to estimate the motion of the frame field using first order approximation of the Taylor expansion of the frame field F in the direction of the vector v at point p fi(v) fi(p)+df i(v).
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