# A New Methodology for Multiscale Myocardial Deformation and Strain Analysis Based on Tagging MRI

Authors: Luc Florack; Hans van Assen
Journal: International Journal of Biomedical Imaging
 (2010)

---

## Abstract

Myocardial deformation and strain can be investigated using suitably encoded
cine MRI that admits disambiguation of material motion. Practical limitations
currently restrict the analysis to in-plane motion in cross-sections of the heart
(2D + time), but the proposed method readily generalizes to 3D + time. We propose
a new, promising methodology, which departs from a multiscale algorithm that
exploits local scale selection so as to obtain a robust estimate for the velocity
gradient tensor field. Time evolution of the deformation tensor is governed by a
first-order ordinary differential equation, which is completely determined by this
velocity gradient tensor field. We solve this matrix-ODE analytically and present
results obtained from healthy volunteers as well as from patient data. The proposed
method requires only off-the-shelf algorithms and is readily applicable to planar or
volumetric tagging MRI sampled on arbitrary coordinate grids.


---

## Body

## 1. Introduction

Cine MRI, combined with (C)SPAMM ((C)SPAMM = (Complementary) SPAtial Modulation of Magnetization) encoding technology [1–4], admits disambiguation of local tissue motion, thus enabling the extraction of myocardial deformation and strain [5], which are known to correlate with cardiac pathologies. In particular, Götte et al. found that strain is more accurate than geometry in discriminating dysfunctional from functional myocardium [6]. Deformation and strain can be operationalized in various ways, either without explicit a priori regularization, or through exploitation of sparse constraints combined with interpolation and/or regularization [5, 7–33]. Possible encodings are (DENSE) Displacement ENcoding with Stimulated Echoes, cf. Aletras et al. [2), and (HARP) HARmonic Phase, cf. Osman et al. [20, 21]. Tagging-based methods using HARP technology form our point of departure, compare with Figure 1 for an illustration.Vertical (a) and horizontal (b) SPAMM encoding in a basal short-axis slice at approximately mid-systole for a patient.
(a)(b)Given a dense motion field within the myocardium, our aim is to devise an operational procedure for direct extraction of myocardial deformation and strain. By “direct” we mean that we seek to obviate sophisticated preprocessing steps, such as segmentation of, or interpolation between tag lines, and finite element methods explicitly coupled to the tagging pattern. Although such sophisticated “indirect” procedures exist and have been proven powerful, they require specific algorithmics that is neither trivially implemented nor readily available. Instead we aim for a multiscale, optimally conditioned, intrinsically parallelizable, linear algorithm for obtaining deformation (and thus strain) in analytically closed form. Optimal conditioning is achieved by exploitation of the scale degree of freedom in the definition of spatiotemporal differential image structure. In addition we aim to minimize the number of extrinsic control parameters. We believe that the parsimony of our method facilitates applicability and optimization, since only off-the-shelf algorithms (linear filtering and inversion of linear systems) are needed in our computation of myocardial deformation and strain.Our approach is as follows. To begin with, thevelocity gradient tensor field, which is the prerequisite for the model we present in this paper, is obtained using the multiscale motion extraction algorithm for scalar image sequences proposed by Florack et al. [12]. This algorithm is adapted to the situation of tagging MRI data [10, 13]. Slick encoding and/or linear filtering yields an n-tuple of independent scalar phase images of some fixed spatiotemporal region of interest, in which n is the dimension of space (n=2,3, as may be the case). The intrinsic, n-fold underdetermined “optic flow constraint equation” can, by construction, be applied to each of these n phase images, yielding an unambiguous system of equations for the underlying dense motion field to any desired differential order, and obviating Tikhonov regularization (which would bring in at least one additional control parameter) as a disambiguating prior (as is typically the case in optic flow applications due to the aperture problem). It has been verified that the first-order scheme indeed produces a plausible, dense, and robust velocity gradient tensor field within the myocardium. Details of its construction as well as the underlying automatic scale selection mechanism can be found in the cited literature (for an example of automatic scale selection, cf. also Niessen et al. [34]). In view of the cited work we will de-emphasize multiscale motion extraction and concentrate on subsequent deformation and strain analysis.In Section2 we outline in detail how to arrive at a closed-form analytical solution for the deformation tensor field (Section 2.1) and hence the strain tensor field (Section 2.2), given the velocity gradient tensor field. The novelty of this approach is that we circumvent numerical approximations in all intermediate steps, and introduce numerics only at the ultimate stage where we sample the resulting analytical tensor field expressions. Besides avoiding in this way numerical errors that may be difficult to quantify, this procedure has the advantage of being mathematically transparent, computationally trivial, and intrinsically parallellizable. (These properties, in fact, also hold for the multiscale motion extraction algorithm used to provide the input velocity gradient tensor field, loc. cit.) Some experimental results are given in Section 3 to demonstrate the feasibility of our approach.Section4 concludes our work with a summary and briefly sketches work in progress.Data acquisition details for the scans used in this paper are given in the appendix.


## 2. Theory

### 2.1. Deformation

The velocity gradient tensor, with componentsLβα relative to a coordinate frame, relates the rate of change of a momentary infinitesimal line element dẋα to the line element dxβ itself. (One can either view dxα as the α-component of an “infinitesimal vector” attached to a fiducial material point in Euclidean space and representing a directed material line element, that is, an “infinitesimal” number—or as the αth element of a local covector frame, depending on engineering or mathematical mind set.) From dẋα=dvα it follows, using the chain rule, that(1)dẋα=LβαdxβwithLβα=∂vα∂xβ(α,β=1,…,n).
(The Einstein summation convention applied here will be used henceforth.) The numbers Lβα constitute a matrix L with row and column index α,β. To get an estimate of this tensor field we have applied the algorithm of Florack et al. and van Assen et al., solving a linear system of algebraic equations enforcing optimal conditioning through scale selection. The reader is referred to the literature for details [10, 12, 13].Tissue deformation can be described by a mapf:M→N:x(X,t0)↦x(X,t), in which x(X,t) denotes the spatial position of a material point at time t, with reference position X=x(X,t0) at time t0 (Lagrange picture). Domain M and codomain N are copies of the deformable tissue medium (a subset of ℝn) at times t0, respectively, t≥t0. We fix t0 so as to correspond to end-diastole. (During the diastolic phase the heart relaxes and is refilled with blood. It is followed by the systolic phase, in which the heart contracts. It is this phase that traditionally receives most attention by cardiologists.)The associated differential map,(2)f*≡df:TMX→TNx:v↦f*(v)=viFiαfα,
relative to a basis {fα} of TNx, provides a local linearization of tissue deformation, that is, the deformation tensor. Here TMX is the tangent space of M at the fiducial point X=x(X,t0), and TNx that of N at the f-mapped point x=x(X,t). The matrix F(t,t0) of this linear map relative to the local coordinate charts {xα,N} and {Xi,M} is given by the Jacobian(3)Fiα=∂xα∂Xi.
By virtue of the chain rule, the relation between deformation and velocity gradient tensors, (1) and (3) is given by the first-order ODE [35](4)Ḟ=LF,
subject to an initial condition, namely, F(t=t0,t0)=I. Equivalently,(5)F(t,t0)=I+∫t0tL(s)F(s,t0)ds.
Only for stationary flows, that is, L(t)=L0 pointwise constant, this initial value problem admits a trivial solution F(t,t0)=exp((t-t0)L0). (For the sake of simplicity the spatial dependency of all field entities is suppressed in the notation.) However, we do not have stationarity, and so we must proceed more carefully.Equation (5) induces an expansion known as the matricant, compare with Gantmacher [35]:(6)F(t,t0)=I+∫t0tL(τ)dτ+∫t0tL(τ)∫t0τL(σ)dσdτ+⋯.
The matricant has the following property:(7)F(t,t0)=F(t,t1)F(t1,t0)(t0<t1<t).
It follows that, if we split the interval [t0,t] into n parts by using intermediate points t1,…,tn-1 separated by Δtk=tk-tk-1 (k=1,…,n, with tn=t),(8)F(t,t0)=F(t,tn-1)⋯F(t1,t0),witht0<t1<⋯<tn-1<tn=t.
For an infinitesimally narrow time interval [tk-1,tk] we have by approximation(9)F(tk,tk-1)=I+L(tk*)Δtk+higherordertermsinΔtk,
in which tk* is any point satisfying tk-1≤tk*≤tk. Equations (8) and (9) give rise to a representation in terms of a so-called multiplicative integral [35]:(10)F(t,t0)=~∫t0t(I+L(τ)dτ)=deflimΔtk→0(I+L(tn*)Δtn)⋯(I+L(t1*)Δt1).
One recognizes the multiplicative counterpart of the Riemann sum approximation for ordinary (additive) integrals. One can show that this is identical to(11)F(t,t0)=[35]~∫t0texp(L(τ)dτ)=deflimΔtk→0exp(L(tn*)Δtn)⋯exp(L(t1*)Δt1).
Several properties of the deformation tensor are manifested in this representation. For instance, for square matrices A, B, one has(i)
detAB=detAdetB,(ii)
det(I+ϵA)=1+ϵtrA+𝒪(ϵ2),(iii)
detexpA=exptrA.Consequently,(12)detF(t,t0)=~∫t0t(1+trL(τ)dτ)=~∫t0texp(trL(τ)dτ).
In particular, a divergence free velocity field (divv=trL=0) preserves volumes: detF(t,t0)=1. Furthermore, expAexpB=exp(A+B) if [A,B]=0, whence for a stationary velocity field one obtains F(t,t0)=exp((t-t0)L0), as we already noticed. Finally, the multiplicative integral suggests a straightforward numerical approximation, namely by using either  (10) or (11) without limiting procedure. In this case the two representations are of course no longer identical. For computational efficiency we have chosen the former, with Δtk=Δt corresponding to the (constant) frame interval of our tagging MRI sequence, and tk*=kΔt.


### 2.2. Strain

On the basis of the differential mapf*,  (2), and its transpose f*T, one defines an intrinsic mapping on TMX, known as the Lagrangian strain tensor [36],(13)E≡12(f*T∘f*-idTMX):TMX→TMX:v↦E(v)=viEijej,
relative to a basis {ej} of TMX, with mixed tensor components(14)Eij=12(gℓjFℓαhαβFiβ-δij).
Here gij are the components of the dual Euclidean metric tensor in domain coordinates {Xi,M}, and hαβ are those pertaining to codomain coordinates {xα,N}. E vanishes identically if f* is an isometry, thus E captures genuinely nonrigid deformations.The general coordinate convention, see (14), which admits different bases for deformed and undeformed configurations, is instructive. Although one would normally prefer identical bases, let us consider the case in which the metric components hαβ are those induced from gij by the deformation map itself (carry-along). That is, we assume that the coordinate frame for the deformed configuration is just the deformed coordinate frame of the reference configuration. One then expects the Lagrangian strain tensor to be nullified, because everything, including the local reference frames, is intrinsically deformed in a consistent manner. Indeed, if we write the infinitesimal material line element as(15)ds2=gijdXidXj=gij∂Xi∂xα∂Xj∂xβdxαdxβ≡hαβdxαdxβ,
we recognize the deformation tensor, see (3), and it follows that(16)gij=hαβFiαFjβ.
As anticipated for the coordinate frames employed,  (14) indeed reduces to(17)Eij=12(gℓjhαβFℓαFiβ︸(16)-δij)=12(gℓjgℓi-δij)=0.
In practice one of course carries out computations relative to fixed (deformation independent) coordinate frames for undeformed and deformed configurations, but there is no reason to assume input and output frames to coincide a priori, nor to restrict oneself to Cartesian coordinate frames. Equation (14) permits us to use whatever bases we may consider convenient. This may be beneficial in certain situations, for example, those that suggest spherical, cylindrical or other coordinate systems depending on the configuration of acquisition data, compare with the phantom study by Young et al. [37], and the desired output representation.Below we consider a single Cartesian coordinate system for both domain and codomain. (In such a system the representations of the various metric tensors,gij and hαβ and their duals, gij and hαβ, all simplify to identity matrices.) Our analysis is confined to a single short-axis plane, whence n=2, that is, we only account for in-plane motion components. The theory trivially generalizes to n=3 and non-Cartesian coordinate grids.


## 2.1. Deformation

The velocity gradient tensor, with componentsLβα relative to a coordinate frame, relates the rate of change of a momentary infinitesimal line element dẋα to the line element dxβ itself. (One can either view dxα as the α-component of an “infinitesimal vector” attached to a fiducial material point in Euclidean space and representing a directed material line element, that is, an “infinitesimal” number—or as the αth element of a local covector frame, depending on engineering or mathematical mind set.) From dẋα=dvα it follows, using the chain rule, that(1)dẋα=LβαdxβwithLβα=∂vα∂xβ(α,β=1,…,n).
(The Einstein summation convention applied here will be used henceforth.) The numbers Lβα constitute a matrix L with row and column index α,β. To get an estimate of this tensor field we have applied the algorithm of Florack et al. and van Assen et al., solving a linear system of algebraic equations enforcing optimal conditioning through scale selection. The reader is referred to the literature for details [10, 12, 13].Tissue deformation can be described by a mapf:M→N:x(X,t0)↦x(X,t), in which x(X,t) denotes the spatial position of a material point at time t, with reference position X=x(X,t0) at time t0 (Lagrange picture). Domain M and codomain N are copies of the deformable tissue medium (a subset of ℝn) at times t0, respectively, t≥t0. We fix t0 so as to correspond to end-diastole. (During the diastolic phase the heart relaxes and is refilled with blood. It is followed by the systolic phase, in which the heart contracts. It is this phase that traditionally receives most attention by cardiologists.)The associated differential map,(2)f*≡df:TMX→TNx:v↦f*(v)=viFiαfα,
relative to a basis {fα} of TNx, provides a local linearization of tissue deformation, that is, the deformation tensor. Here TMX is the tangent space of M at the fiducial point X=x(X,t0), and TNx that of N at the f-mapped point x=x(X,t). The matrix F(t,t0) of this linear map relative to the local coordinate charts {xα,N} and {Xi,M} is given by the Jacobian(3)Fiα=∂xα∂Xi.
By virtue of the chain rule, the relation between deformation and velocity gradient tensors, (1) and (3) is given by the first-order ODE [35](4)Ḟ=LF,
subject to an initial condition, namely, F(t=t0,t0)=I. Equivalently,(5)F(t,t0)=I+∫t0tL(s)F(s,t0)ds.
Only for stationary flows, that is, L(t)=L0 pointwise constant, this initial value problem admits a trivial solution F(t,t0)=exp((t-t0)L0). (For the sake of simplicity the spatial dependency of all field entities is suppressed in the notation.) However, we do not have stationarity, and so we must proceed more carefully.Equation (5) induces an expansion known as the matricant, compare with Gantmacher [35]:(6)F(t,t0)=I+∫t0tL(τ)dτ+∫t0tL(τ)∫t0τL(σ)dσdτ+⋯.
The matricant has the following property:(7)F(t,t0)=F(t,t1)F(t1,t0)(t0<t1<t).
It follows that, if we split the interval [t0,t] into n parts by using intermediate points t1,…,tn-1 separated by Δtk=tk-tk-1 (k=1,…,n, with tn=t),(8)F(t,t0)=F(t,tn-1)⋯F(t1,t0),witht0<t1<⋯<tn-1<tn=t.
For an infinitesimally narrow time interval [tk-1,tk] we have by approximation(9)F(tk,tk-1)=I+L(tk*)Δtk+higherordertermsinΔtk,
in which tk* is any point satisfying tk-1≤tk*≤tk. Equations (8) and (9) give rise to a representation in terms of a so-called multiplicative integral [35]:(10)F(t,t0)=~∫t0t(I+L(τ)dτ)=deflimΔtk→0(I+L(tn*)Δtn)⋯(I+L(t1*)Δt1).
One recognizes the multiplicative counterpart of the Riemann sum approximation for ordinary (additive) integrals. One can show that this is identical to(11)F(t,t0)=[35]~∫t0texp(L(τ)dτ)=deflimΔtk→0exp(L(tn*)Δtn)⋯exp(L(t1*)Δt1).
Several properties of the deformation tensor are manifested in this representation. For instance, for square matrices A, B, one has(i)
detAB=detAdetB,(ii)
det(I+ϵA)=1+ϵtrA+𝒪(ϵ2),(iii)
detexpA=exptrA.Consequently,(12)detF(t,t0)=~∫t0t(1+trL(τ)dτ)=~∫t0texp(trL(τ)dτ).
In particular, a divergence free velocity field (divv=trL=0) preserves volumes: detF(t,t0)=1. Furthermore, expAexpB=exp(A+B) if [A,B]=0, whence for a stationary velocity field one obtains F(t,t0)=exp((t-t0)L0), as we already noticed. Finally, the multiplicative integral suggests a straightforward numerical approximation, namely by using either  (10) or (11) without limiting procedure. In this case the two representations are of course no longer identical. For computational efficiency we have chosen the former, with Δtk=Δt corresponding to the (constant) frame interval of our tagging MRI sequence, and tk*=kΔt.


## 2.2. Strain

On the basis of the differential mapf*,  (2), and its transpose f*T, one defines an intrinsic mapping on TMX, known as the Lagrangian strain tensor [36],(13)E≡12(f*T∘f*-idTMX):TMX→TMX:v↦E(v)=viEijej,
relative to a basis {ej} of TMX, with mixed tensor components(14)Eij=12(gℓjFℓαhαβFiβ-δij).
Here gij are the components of the dual Euclidean metric tensor in domain coordinates {Xi,M}, and hαβ are those pertaining to codomain coordinates {xα,N}. E vanishes identically if f* is an isometry, thus E captures genuinely nonrigid deformations.The general coordinate convention, see (14), which admits different bases for deformed and undeformed configurations, is instructive. Although one would normally prefer identical bases, let us consider the case in which the metric components hαβ are those induced from gij by the deformation map itself (carry-along). That is, we assume that the coordinate frame for the deformed configuration is just the deformed coordinate frame of the reference configuration. One then expects the Lagrangian strain tensor to be nullified, because everything, including the local reference frames, is intrinsically deformed in a consistent manner. Indeed, if we write the infinitesimal material line element as(15)ds2=gijdXidXj=gij∂Xi∂xα∂Xj∂xβdxαdxβ≡hαβdxαdxβ,
we recognize the deformation tensor, see (3), and it follows that(16)gij=hαβFiαFjβ.
As anticipated for the coordinate frames employed,  (14) indeed reduces to(17)Eij=12(gℓjhαβFℓαFiβ︸(16)-δij)=12(gℓjgℓi-δij)=0.
In practice one of course carries out computations relative to fixed (deformation independent) coordinate frames for undeformed and deformed configurations, but there is no reason to assume input and output frames to coincide a priori, nor to restrict oneself to Cartesian coordinate frames. Equation (14) permits us to use whatever bases we may consider convenient. This may be beneficial in certain situations, for example, those that suggest spherical, cylindrical or other coordinate systems depending on the configuration of acquisition data, compare with the phantom study by Young et al. [37], and the desired output representation.Below we consider a single Cartesian coordinate system for both domain and codomain. (In such a system the representations of the various metric tensors,gij and hαβ and their duals, gij and hαβ, all simplify to identity matrices.) Our analysis is confined to a single short-axis plane, whence n=2, that is, we only account for in-plane motion components. The theory trivially generalizes to n=3 and non-Cartesian coordinate grids.


## 3. Experiment

Figure2 illustrates various scalar fields extracted from the strain tensor field, (13) and (14), by contraction with a pair of local unit vectors, namely radial (r = radial) and azimuthal (c = circumferential) basis vectors of the polar coordinate system centered at the midpoint of the region of interest, and those defining the strain tensor's eigensystem. If u,v∈TMX are two such unit vectors, then the local scalar quantity derived from the strain tensor is given by the inner product (u,E(v))=(E(u),v)=gjkukEijvi. Tensor components are evaluated in Cartesian coordinates.Color-coded strain fields in short axis mid-slice cross-sections, time frames1,6,11,16,21 (a full heart cycle subtends 42 frames), regularized through Gaussian convolution with spatial scale σ=1.0, for one healthy volunteer. First row: Circumferential strain Ecc. Second row: Radial strain Err. Third row: Shear strain Ecr. Fourth row: Minimal strain eigenvalue Emin. Fifth row: Maximal strain eigenvalue Emax.
(a)(b)(c)(d)(e)Figure3 illustrates temporal evolutions of these quantities spatially averaged over the respective regions of interest, together with their standard deviations. (Thus one should not confuse standard deviations with uncertainties in this plot.) Table 1 shows statistics for three healthy volunteers. Figure 4 shows that strain fields have the potential to detect local pathology.Table 1
Spatial average and standard deviation of strains from three healthy volunteers (labels I, II, III) over base, mid, and apex short-axis cross-sectional ROIs at that time frame at whichEmax has attained its maximum in mid-slice (tI=16, tII=15, tIII=14). Rows correspond to, from top to bottom, Ecc, Err, Ecr, Emax, and Emin.
BaseMIDAPEXIIIIIIIIIIIIIIIIII−0.16±0.13−0.17 ± 0.11−0.19 ± 0.11−0.19 ± 0.09−0.17 ± 0.08−0.15 ± 0.09−0.16 ± 0.16−0.20 ± 0.09−0.19 ± 0.130.20 ± 0.250.16 ± 0.210.14 ± 0.250.04 ± 0.200.14 ± 0.150.03 ± 0.150.07 ± 0.230.20 ± 0.23−0.02 ± 0.180.02 ± 0.15−0.04 ± 0.12−0.07 ± 0.130.01 ± 0.07−0.02 ± 0.09−0.06 ± 0.110.07 ± 0.14−0.03 ± 0.11−0.05 ± 0.120.25 ± 0.220.19 ± 0.200.18 ± 0.230.07 ± 0.170.16 ± 0.150.09 ± 0.160.15 ± 0.210.22 ± 0.220.05 ± 0.19−0.22 ± 0.11−0.21 ± 0.09−0.24 ± 0.09−0.23 ± 0.09−0.19 ± 0.08−0.21 ± 0.07−0.26 ± 0.12−0.23 ± 0.09−0.27 ± 0.09Temporal evolution of scalar strain quantities over the first half of the heart cycle (i.e., mostly systolic) in mid-slice cross-section. Error bars indicate standard deviations over the spatial ROI in each time frame, thus capturing all sources of variation due to (primarily) actual spatial variability, noise, and numerics. Legends explain the various graphs. Notice the strong correlation between the extrinsic (polar system related) and instrinsic (eigensystem related) strains. (In the eigensystem, shear strain vanishes identically.)
(a)(b)Radial (a) and circumferential (b) colour-coded strains in a basal short-axis slice at approximately end-systole for a patient. Red arrows on the left indicate regions where the respective strain values deviate from normal. These regions were confirmed to be static by a cardiologist. The patient's anamnesis mentions small infarcts on either side of these regions. Their locations, indicated by black arrows, are highlighted in late-enhancement MRI.
(a)(b)


## 4. Conclusion and Future Work

We have proposed a novel, simple and robust linear model for extracting myocardial deformation and strain. Material motion and gradient velocity are determined by a multiscale linear system of algebraic equations for the phase images extracted from the tagging MRI data. These phase images are scalars, not densities, and are not hampered by tag fading due toT1 relaxation.We have analytically solved the linear matrix-ODE governing myocardial deformation. By discretizing the closed-form solution we have subsequently solved for the induced Lagrangian strain tensor field, yielding results that are typical for healthy volunteers, compare with Garot et al. [14], and atypical for a patient with a medical history of small infarcts on either side of a deviatory region and confirmed by late-enhancement MRI. This demonstrates the feasibility of our method. An advantage of our method is that only off-the-shelf algorithms are needed. This serves clarity, facilitates optimization, and enables implementation on dedicated hardware. The method is applicable in any spatial dimension, does not require conversion to a Cartesian sampling grid, is optimally conditioned due to local scale selection, and is readily adapted to other modalities such as velocity encoding MRI.Our quantitative results in this feasibility study demand further evaluation so as to reveal their statistical significance and clinical value. Furthermore, experiments based on carefully controlled synthetic or phantom data will allow us to disclose the physical significance and numerical tolerance of the mathematical framework in terms of ground truth deformation properties. Moreover, by virtue of the transparent mathematical theory, it is also possible to assess tolerances on the basis of theoretical error propagation models [38], which can then be compared to those established experimentally for synthetic and phantom data. Consistency will greatly increase the confidence of the method for in vivo studies. Finally, although the local scale selection criterion exploited in our approach guarantees optimal conditioning, it does not necessarily yield optimal results due to the intrinsically parallel and thus spatiotemporally uncorrelated nature of the selected neighbouring scales. The effect of spatiotemporal regularization on the pattern of locally selected scales needs to be investigated. Future investigation will be needed to pursue all these recommendations.



---

*Source: 341242-2010-03-01.xml*