cine mri , combined with ( c)spamm ( ( c)spamm = ( complementary ) spatial modulation of magnetization ) encoding technology [ 14 ] , admits disambiguation of local tissue motion , thus enabling the extraction of myocardial deformation and strain , which are known to correlate with cardiac pathologies . in particular , gtte et al . found that strain is more accurate than geometry in discriminating dysfunctional from functional myocardium .
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 , 733 ] .
. aletras et al . [ 2 ) , and ( harp ) harmonic phase , cf .
tagging - based methods using harp technology form our point of departure , compare with figure 1 for an illustration . 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 .
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 , the velocity 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 . .
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 ) .
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 .
in view of the cited work we will de - emphasize multiscale motion extraction and concentrate on subsequent deformation and strain analysis . in section 2
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 .
some experimental results are given in section 3 to demonstrate the feasibility of our approach . 

section 4 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 .
the velocity gradient tensor , with components l relative to a coordinate frame , relates the rate of change of a momentary infinitesimal line element dx to the line element dx itself .
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 th element of a local covector frame , depending on engineering or mathematical mind set . ) from dx=dv it follows , using the chain rule , that  
 ( 1)dx=l  dx  with  l=vx(,=1, 
the numbers l constitute a matrix l with row and column index  , . 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 map f : 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  ) at times t0 , respectively , t  t0 .
the associated differential map , 
 ( 2)fdf : tmxtnx : vf(v)=vi  fi  f , 
							 relative to a basis { f } 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 { x , n } and { x , m } is given by the jacobian 
 ( 3)fi=xxi . 
							 by virtue of the chain rule ,
the relation between deformation and velocity gradient tensors , ( 1 ) and ( 3 ) is given by the first - order ode  
 ( 4)f=l  f , 
							 subject to an initial condition , namely , f(t
= t0 , t0 ) = i. equivalently , 
 ( 5)f(t , t0)=i+t0 t  l(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 : 
 ( 6)f(t , t0)=i+t0tl()d+t0tl()t0l()d  d+. 
							 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 ,  , tn1 separated by tk = tk  tk1 ( k = 1 ,  , n , with tn = t ) , 
 ( 8)f(t , t0)=f(t , tn1)f(t1,t0),with  t0<t1<<tn1<tn = t . 
							 for an infinitesimally narrow time interval [ tk1 , tk ] we have by approximation 
 ( 9)f(tk , tk1)=i+l(tk ) 
tk+higher  order  terms  in  tk , 
							 in which tk * is any point satisfying tk1  tk *  tk . equations
( 8) and ( 9 ) give rise to a representation in terms of a so - called multiplicative integral : 
 ( 10)f(t , t0)=~t0t(i+l()d)=deflim  tk0(i+l(tn)tn)(i+l(t1)t1 ) . 
							
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)=~t0texp  ( l( )  d)=deflim  tk0exp  ( l(tn)tn)exp  ( l(t1 )  t1 ) . 
							
for instance , for square matrices a , b , one has    det  ab = det  adet  b ,   det  ( i +  a ) = 1 + tr  a + ( ) ,   det  exp  a = exp  tr  a.  
 ( 12)det  f(t , t0)=~t0t(1+tr  l()d)=~t0texp  ( tr  l()d ) . 
							 in particular , a divergence free velocity field ( div  v = tr  l = 0 ) preserves volumes : det  f(t , t0 ) = 1 . furthermore , exp  aexp  b = 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 tk = t corresponding to the ( constant ) frame interval of our tagging mri sequence , and tk * = kt . on the basis of the differential map f
,   ( 2 ) , and its transpose f , one defines an intrinsic mapping on tmx , known as the lagrangian strain tensor , 
 ( 13)e12(ftfidtmx):tmxtmx : ve(v)=vieijej , 
							 relative to a basis { ej } of tmx , with mixed tensor components 
 ( 14)eij=12(gjfhfiij ) . 
							 here
g are the components of the dual euclidean metric tensor in domain coordinates { x , m } , and h are those pertaining to codomain coordinates { x , 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 h 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 = gijxixxjxdxdxhdxdx , 
							 we recognize the deformation tensor , see ( 3 ) , and it follows that 
 ( 16)gij = hfifj. 
							 as anticipated for the coordinate frames employed ,   ( 14 ) indeed reduces to 
 ( 17)eij=12(gjhffi(16)ij)=12(gjgiij)=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 .
. 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 . , 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 h and their duals , g and h , 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 .

 figure 2 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 ) = gjkueiv .
figure 3 illustrates temporal evolutions of these quantities spatially averaged over the respective regions of interest , together with their standard deviations .
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 to t1 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 . , 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 .
an advantage of our method is that only off - the - shelf algorithms are needed .
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 , which can then be compared to those established experimentally for synthetic and phantom data .
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 .
the short - axis mr tagging image data used in this study were acquired with a philips intera 1.5  t scanner from three volunteers ( table 1 ) and a patient in basal , midventricular , and apical slices .
the patient ( figure 4 ) had a history of severe stenoses , and small infarction areas confirmed with late - enhancement mri 14 minutes after gadolinium contrast injection .
a 2d multishot gradient - echo with echo planar imaging ( epi factor 9 ) with breath - holding in end - expiration was used .
scan parameters were : te 4.4  ms , tr 19  ms , flip angle 10 , field - of - view 300  mm , scan matrix 128 , acquisition voxel size 2.34  2.68  8  mm reconstructed into 1.17  1.17  8  mm .
 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 . 
1. Introduction
2. Theory
3. Experiment
4. Conclusion and Future Work
 Data Acquisition Details