A new approach for analysing motion and deformation of left ventricle: Post‐processing 3D echocardiography data with finite element method

In the medical community, it is common practice to examine segmental (regional) values. For instance, echocardiography data are analysed by dividing the left ventricle (LV) endocardial surface into 16 or 17 areas (segments), which actually consist of several nodes (vortices) and the nodal values are averaged over the segments. Although, the averaged segment values provide useful information and are widely used, looking at pointwise values might enable a more extensive LV assessment, which has not been attempted so far. For instance, averaging might be ruling out some unusual localized deformations, which can be only detected through a pointwise distribution. Therefore, in this contribution, we introduce a new methodology to analyse 3D echocardiography data from computational modelling perspective which eventually enables a pointwise distribution of deformation related quantities over the LV endocardial surface. To this end, we export the tracked LV endocardial surface from a dedicated software 4D LV‐analysis Tomtec, which enables to export the current coordinates of the nodes on the surface. The coordinates are then integrated into a finite element analysis program and, eventually, one can compute any deformation related quantity and obtain a pointwise distribution over the endocardial surface. To our best knowledge, such an approach has not been introduced before. In order to demonstrate the usability, we compare the pointwise value distribution to traditional averaged segmental values. Additionally, we calculate the stress distribution and the sphericity index in a pointwise manner, which are not provided by commercial LV analysis softwares.

displacements, velocities, rotation, twist, torsion, strains, strain rates are investigated.For this purpose, ultrasound of the heart or, in other words, echocardiography (echo) is one of the most widely used tools in clinical routine.This is because performing echo on a patient is non-invasive, fast and cheap.The data obtained from 3D echo are often analyzed by a dedicated software, which basically tracks speckle patterns in the myocardium.In the medical community, it is common practice to examine the echo data in segmental (regional) values.For example, 3D echo data are analyzed by dividing the LV endocardial surface into 16 or 17 areas (segments), which consist of several nodes (vortices) and the nodal values are averaged over the segments [1].
Apart from dealing with real data in clinics, extensive efforts have been devoted to achieve virtual modelling of the heart with the aim to understand the working mechanisms of the heart in healthy and disease state [2][3][4][5][6].In the field of virtual modelling, the finite element method (FEM) is one of the most utilized numerical tools.
In this study, we aim to analyse real LV motion and deformation data from engineering perspective within a finite element framework.To our best knowledge, such a methodology does not exist in the literature and we believe that this new approach will enable a deeper insight into LV mechanics, particularly, in pathological cases.To this end, we first track the LV endocardial surface from the beginning of systole to the end of diastole with the dedicated software 4D LV-Analysis Tomtec.The software generates 960 triangular elements over 502 vortices (nodes) on the endocardial surface and tracks those nodes through the cycle.Moreover, the current coordinates of the nodes, which are tracked through the cardiac cycle, can be exported as a text file along with the connectivity of the triangular elements.From a computational perspective, it would be correct to say that the displacement field of a deforming body between an initial state (beginning of cycle) and a final state (end of cycle) is computed.The displacements are integrated into a finite element framework and, consequently, we have the ability to compute any deformation related quantity over the endocardium surface in a point-wise manner.
The manuscript is organised as follows.In Section 2, the novel approach for pointwise analysis of echo data is explained.In Section 3, the traditional segmental analysis of 3D echo data is briefly introduced.Section 4 is devoted to demonstrative examples where the capacity of the new approach is presented.We give final remarks in Section 5.

Finite element method
For the sake of completeness, let us shortly introduce the FEM which is one of the most convenient and useful solution tools that is available in literature.The deformation state of a continuum body going through a deformation process is governed by the balance of linear momentum In the equation, ∇ is the nabla operator,  is the Cauchy stress tensor,  is the displacement field,  is mass density and  represents the body forces.After some mathematical operations, the discretized form of the governing Equation ( 1) is solved for the displacement field  in order to obtain the deformation state of the whole body where R and K denote the global residual and global stiffness matrix of the system, respectively.Upon the solution of the system, one can calculate any deformation related variable on the nodes such as stresses, different strain measures, displacements along preferred directions, rotations etc.

Integration of echo data into the FEM
4D LV-Analysis Tomtec discretizes the tracked surface over 501 nodes with 960 triangular elements.A sample of the exported data file is demonstrated in Figure 1A.The software enables to export the current coordinates of the nodes from the beginning of the systole to the ending of the diastole.Namely, the deformation state of the LV endocardial surface, in other words the displacement field is available at each time frame.From a computational perspective, the exported data from the software provides the solution of Equation (1) for .What is further done is to embedding the displacement field into the open source Finite Element Analysis Program (FEAP) [7].To this end, the software is re-programmed so that the text files from 4D LV-Analysis Tomtec are read and the displacement field is calculated by considering the configuration at the end-diastole as the reference configuration.Once the nodal displacements  are provided, one can compute any deformation related quantity (displacements along preferred directions, strains, rotations etc.) on the nodes as post-processing.
A pointwise displacement distribution along -direction is demonstrated in Figure 1B after processing a sample echo data into FEAP.

TRADITIONAL APPROACH: SEGMENTAL ANALYSIS
Conventionally, LV function is assessed by some global and regional metrics by cardiologists.Some examples to global metrics are stroke volume, ejection fraction, global longitudinal strain and twist.On the other hand, the regional metrics provide a more detailed evaluation of LV function.To this end, the LV is divided into three equal regions along the long axis which are named as basal, mid and and apical in the light of anatomical findings and autopsy investigations [8].Furthermore, the basal and mid regions are split into six segments and the apical region is further split into four segments, created in total 16 segments.However, 17 segment analysis is also common where the additional segment is created in the apical region (split into five segments).The configuration of vortices over the whole LV surface are tracked during the cardiac cycle.Then, the interested deformation related quantity, for example longitudinal displacement, in a segment is calculated by averaging the measured values of vortices falling on that segment.The measurements are depicted in a graph over time.A demonstration of echo analysis is shown in Figure 2.

DEMONSTRATIVE EXAMPLES
This section demonstrates the capacity of the new methodology to analyse echo data.In the following, we present examples of point-wise distribution of some deformation dependent quantities, compare the values between the nodal distribution and segmental values and compute stress and curvature evolution during a cardiac cycle.

Pointwise distribution
In this part, the nodal distribution of rotation and radial displacement on a healthy LV is presented, see Figure 3. Positive and negative signs of related deformation measures are illustrated in Figure 4.As the LV contracts, the apical segments start to rotate in positive direction while the rotation in the basal segments occurs dominantly in negative direction.However, the snapshots reveal that the rotation distribution is not completely homogeneous in the apical and basal segments.For instance, there are some nodes rotating in negative direction although the dominant rotation is in positive direction.
On the other hand, the radial displacement distribution demonstrates that the endocardial surface moves towards the LV center as observed in healthy subjects.Moreover, a homogeneous distribution is observed which designates that the LV deforms as a whole along radial direction.

Pointwise versus segmental values
Another sensible investigation would be definitely to compare the averaged segmental value to the nodal values on that segment.Thereby, one can assess whether the deformation state of that region is thoroughly represented by the averaged segmental value.
For this purpose, we use the echo data of a dilated LV.After processing the echo data into the FEM framework and obtaining the pointwise values, we additionally split the LV endocardial surface into 16 segments and calculate the averaged values in each segment, as performed by 4D LV-Analysis Tomtec.A comparison of pointwise values to the averaged segmental values is depicted in Figure 5 for the longitudinal displacement   and the circumferential displacement   .
Looking at the comparison of   , one observes that the averaged segmental values cannot fully represent the actual longitudinal displacement on the endocardial surface.For instance, according to the segmental value in the first diagram of Figure 5A, the segment has a negative longitudinal displacement while the node on that segment has a positive longitudinal displacement.Namely, averaging nodal values over the segment misses out the actual longitudinal displacement value of that node.Moreover, although the general deformation tendency is preserved between the averaged segmental values and the nodal values in the 3rd, 4th and 5th diagrams, a remarkable disagreement still exists.In the 2nd diagram, however, one perceives a satisfactory agreement between the nodal value and averaged segmental value.
Similar observations are made in Figure 5B where the circumferential displacement   is investigated.In the 1st 2nd and 3rd diagrams, the averaged segmental value and nodal value significantly deviate from each other so that pointing out completely different deformation behavior.On the other hand, the nodal values in the 4th and 5th diagrams follow a similar deformation pattern with the averaged segmental value.

Stress and sphericity index
Since the deformation gradient at each node is known, we have the freedom to compute any deformation related value and even to deduce novel quantities reflecting LV function.For demonstrative purposes, stress and sphericity on the LV endocardial surface are calculated in the following.For the stress response, the hyperleastic material model developed for the passive myocardium behavior [9] is adopted, where the Cauchy stress expression reads Therein,  = FF  is the left Cauchy-Green strain tensor and F = ∕ represents the deformation gradient, where  is the current coordinates and  is the reference coordinates.Moreover,  and  are the deformed vectors along fibre and sheet directions, respectively, and  is the hydrostatic pressure.The deformation dependent stress coefficients are defined as Ψ 4s ∶= 2 s ( 4s − 1) exp[ s ( 4s − 1) 2 ], Ψ 8fs ∶= 2 fs  8fs exp[ fs  2 8fs ], where , ,  f ,  f ,  s ,  s ,  fs ,  fs are non-negative material constants describing the deformation state of the isotropic and orthotropic microstructure of the myocardium along with the invariants  1 ∶= tr ,  4f ∶=  0 ⋅  0 ,  4s ∶=  0 ⋅  0 ,  8fs ∶=  0 ⋅  0 and the right Cauchy-Green tensor  =   , see the recent work [5] regarding the details of the model.
In the calculation of stress, we consider that the myocardium bears compressive load along fibre and sheet directions in contrast to the original model [9].Moreover, the sphericity index (SI) is considered as an essential marker to identify geometrical abnormalities in LV.For instance, a high SI usually designates an enlarged LV cavity volume or in other words a dilated LV.In literature, SI is calculated as a single global value and is defined as the ratio of the longitudinal axis diameter to the minor axis diameter at mid-cavity level [10].However, in this work we calculate the SI in a pointwise manner as   /  , where is the distance to the longitudinal axis in short axis plane and   = √  2 +  2 +  2 is the distance to the tip point of the apex which is located at coordinates  = 0,  = 0,  = 0, see Figure 4 for visualization.This ratio indicates the steepness of LV endocardial surface in its short axis cross-section.
The stress distribution along fibre direction and SI distribution are presented in Figure 6 for a normal LV in (A) and for a dilated LV in (B).The fibre orientation in the normal LV is assumed to have 70 • and 50 • in the dilated LV on the endocardial surface.
The stress distribution in the normal LV, see Figure 6A, is mainly homogeneous without large fluctuations.It is worth to mention that the stress values along the fibre direction are always negative as if the myocardium is exposed to pressure.However, the myocardium in reality experiences tension.This contradiction in the results can be explained as follows.Note that the stress expression in (3) describes the passive myocardium and the deformation gradient that we acquire from the echo data leads to the shortening of myocardial fibres as expected.In reality, however, the myocardial fibres shorten due to the active contraction which produces tensile forces.
Since we do not distinguish the active contraction from the total deformation in this work, the contribution to the total stress coming from the active deformation is neglected.Consequently, we obtain negative stress values over the endocardial surface in the normal LV.However, one does not observe only negative stress values but also positive stress values appear in the dilated LV, see Figure 6B, which indicates the extension of the myocardium in some regions leading to positive tensile forces.This result might be attributed to pathological loading conditions in the dilated LV.Another noteworthy observation is the significantly lower magnitude of stress values in the dilated LV than in the normal LV.The cause is undoubtedly the reduced deformation capacity of the dilated LV.
Moreover, the SI distribution demonstrates that surface sphericity has maximum values at basal regions and gradually decreases towards apex.This observation is valid at every instant of the cardiac cycle.As the normal LV contracts, the SI slightly decreases, nevertheless, the homogeneous distribution is preserved.On the other hand, the SI distribution on the dilated LV reveals considerably larger sphericity values compared to the normal LV particularly at basal regions.It is worth noting that SI distribution in the dilated LV is not conspicuously altered throughout the cardiac cycle due to the diminished contractility of the LV wall.

CONCLUDING REMARKS
In this work, we introduced a new method analyzing 3D echo data within the finite element framework.The tracked LV endocardial surface, which is discretized with 960 triangular elements over 501 nodes, was exported from the software 4D LV-Analysis Tomtec.The exported data file includes the actual coordinates of the nodes from which one can simply calculate the displacements as well as the deformation gradient.Eventually, we have the solution of the mechanical problem, that is motion of LV during cardiac cycle.The rest was simply the post-processing of the solution within the finite element framework.To the best of our knowledge, such a study does not exist in the literature.The essential aspect of this work is facilitating a pointwise distribution of real deformation data over the LV surface.We exemplified that averaged segmental values might not be correctly reflecting deformation state of LV endocardial surface.In particular, if big fluctuations in pointwise values of a segment exist, significant deviations will be naturally present between the averaged segmental value and pointwise values.As a result, the segmental value will not reflect the actual deformation state of the LV endocardium.This issue might be predominantly arising in pathological cases where local deformation impairments exist in the LV.Moreover, we demonstrated the capability of the approach to calculate pointwise stress distribution and sphericity index over the LV, which are not provided by conventional LV analysis softwares.Certainly, these calculations are elementally performed and require further improvements.Finally, the pointwise analysis of LV deformation might be superior to averaged segmental analysis and might help to develop more efficient screening protocols that can be performed in clinical routine.However, an extensive analysis must be further carried out, which includes various healthy and pathological cases with large dataset, to establish applicability of the methodology.Moreover, we plan to investigate LV epicardial surface within the same framework and provide a deeper understanding of the motion and deformation characteristics of the epicardium surface.

A C K N O W L E D G M E N T S
We gratefully acknowledge the financial support of the German Research Foundation (DFG) under grant KA 1163/49.Open access funding enabled and organized by Projekt DEAL.

F I G U R E 1
(A) View of exported data file from 4D LV-Analysis Tomtec.From top to bottom, the first data set represents the nodal coordinates (501 nodes) while the second data set represents the element connectivity of triangular elements (960 elements).(B) Exported echocardiography data from 4D LV-Analysis Tomtec is processed into the FEM framework.The deformed shape and nodal displacement distribution along the -axis are depicted during the full cardiac cycle.

F I G U R E 2
Conventional analysis of 3D echocardiography data with 4D LV-Analysis Tomtec.The dedicated software 4D LV-Analysis Tomtec provides global markers such as end-diastolic volume (EDV), end-diastolic volume index (EDVi), end-systolic volume (ESV), end-systolic volume index (ESVi), stroke volume (SV), ejection fraction (EF), mass, systolic dyssynchrony index (SDI), global longitudinal strain (GLS), global circumferential straine (GCS), twist and torsion along with averaged segmental values.Each color on bull's eye represents a specific segment whose averaged value is plotted with the corresponding color.The diagram shows averaged longitudinal displacements.The LV endocardial surface is splitted into 16 segments and for each segment an averaged value is plotted versus time.The averaging over a segment is performed by summing up all nodal values and dividing by the number of nodes on that segment.F I G U R E 3 Exemplification of pointwise distribution of rotation (A) and radial displacement (B) throughout the cardiac cycle.The snapshots at time  = 0 ms depict the beginning of the systole and the snapshots at time  = 347 ms are taken at the end of systole.

F I G U R E 4 F I G U R E 5
Demonstration of LV longitudinal axis with respect to which the rotation, circumferential, radial displacement and the curvature is calculated.Viewed from the apex, anti-clockwise direction is considered as a positive rotation direction while clockwise direction has a negative sign.Inward displacement along the radial direction has negative sign while outward displacement is recorded with positive sign.Comparison of (A) nodal longitudinal displacement   and (B) nodal circumferential displacement   to the averaged value on that segment, where the node is located, throughout a cardiac cycle of a dilated LV.Five different locations are chosen.The blue and red curves represent nodal and segmental averaged values, respectively.

F I G U R E 6
The Cauchy stress distribution along fibre direction and sphericity index distribution throughout the cardiac cycle are demonstrated (A) for normal LV and (B) for dilated LV.