The proposed myocardial strain quantification method was initially developed for strain computation on data from a surgically implanted transmural bead array. However, this work shows that the method can be extended to be used with displacement data from displacement-encoded MRI.
This work uses a polynomial function to find a differentiable expression from the discrete displacement field. This polynomial function assumes continuous displacement in the myocardium, which reflects the connective properties of the myocardial tissue. This a priori information helps making the estimation more robust to noise.
The optimal order of the polynomial functions in equation (1) depends on the number of material points along each dimension, which in turn depends on the spatial resolution of the data, wall thickness and the sizes of the finite segments. Generally, the number of unknown constants in the polynomial should be less than the number of measured points along each dimension, which implies that a third order polynomial requires at least five measured material points along the corresponding dimension, a second order polynomial requires at least four points and a first order polynomial requires at least three points. Furthermore, to avoid an undetermined problem, the number of data points must be equal or greater than the number of coefficients in the minimization. This implies that the minimum number of required data points depends on the polynomial orders; BLQ requires 12 data point, BLC 16, LQC 24, and BQC 36. Hence, the polynomial order is limited by the number of included data points. This requirement was fulfilled for the in vivo evaluation; however, the margin for the BQC polynomial was small at some locations of the myocardium.
Four different polynomial orders were evaluated. The smallest absolute errors of the estimated strains in the analytical model in the presence of low noise were obtained with a linear-quadratic-cubic polynomial. In the subsequent in vivo validation, the in vivo results of the linear-quadratic-cubic polynomial were analyzed in detail. Given the incumbent spatial resolution, the restriction on fR(XR) was the wall thickness at the late diastolic time frame and the restrictions on fC(XC) and fL(XL) were the width and height of each segment, respectively. The width (π/6 radians) and height (7.5 mm) of each segment were kept small in order to resolve local variations of deformation.
The RMS differences between the acquired in vivo
coordinates and the coordinates estimated using the polynomial fitting method reflect the accuracy of the polynomial fitting, giving a comprehension of the extent to which the coordinates estimated by the polynomial fit to the acquired in vivo
coordinates. For both systolic and diastolic data, the RMS differences of the LQC polynomial were highest in the posterior wall. For the systolic data the region of highest RMS differences was close to the posterior papillary muscle, which might have caused locally irregular displacements. The RMS differences may furthermore be affected by the spatially varying signal-to-noise ratio in the DENSE measurement [18
]. For the diastolic data, the region of highest RMS differences coincided with the region of thinnest wall. The present implementation of the method used the same polynomial order for each segment within the 3D volume. If a data set with large variations in wall thickness is considered, e.g. a 3D slab comprising an infarct with thin wall near the infarcted myocardium, the highest accuracy could be obtained by local adjustment of the size of the segments to the size of the infarcted region and adapting the polynomial order to wall thickness and segment width.
Systolic radial, circumferential and longitudinal strains, as well as systolic circumferential-longitudinal shear, show agreement with systolic strains previously reported for human myocardium [19
: We observed somewhat higher magnitudes of radial-circumferential shear strain than the results of Moore et al. [19
], and we observed the lowest values in the anterior region of the myocardium while Moore et al. reported the lowest values in septum. ERL
: The observed radial-longitudinal shear values fits within mean ± 2SD of the values reported by Moore et al. [19
Diastolic function of the LV is determined by a complex sequence of many interrelated events and parameters including active relaxation, elastic recoil, passive filling characteristics, heart rate and inotropic state. Diastolic LV filling is a highly dynamic process with early and late transmitral inflows. Thus a detailed analysis of myocardial strain during diastole requires resolving the temporal process of diastolic filling.
The highest values of circumferential strain during the first 213 ms of diastolic filling were observed in the postero-lateral wall. The same quantitative behavior has been reported in previous studies of early diastolic strains in normal human hearts [21
]. Radial strain, interpreted as wall thinning during diastole, was most apparent in the lateral wall, which also has been reported by others [21
This work is limited to study the kinematics of the heart, focusing on strain. Strain should preferably be related to an unloaded, stress free reference configuration. Using in vivo data, there is no unloaded, stress free configuration of the heart. Instead, the reference configurations correspond to defined time points in the cardiac cycle. The strain presented in this, and similar articles within the field, therefore disregards the residual strain needed to study cardiac kinetics as opposed to kinematics.
In vivo validation of strain is challenging, which is why an analytical model was included in the validation. The analytical model can however never fully describe the cardiac kinematics. For in vivo estimation, the quality of the strain measurements is highly dependent on the quality of the underlying displacement data. While the polynomial fit reduces sensitivity-to-noise in the measurements, image artifacts or errors in the displacement measurements can deteriorate the strain estimates. Improving the DENSE acquisitions is an active field in the MRI research community, and strain analysis methods like the one presented in this paper will benefit directly from such improvements.