Department of Management and Engineering, Linköping University, SE-581 83 Linköping, Sweden

Center for Medical Image Science and Visualization (CMIV), Linköping University, SE-581 85 Linköping, Sweden

Department of Medical and Health Sciences, Linköping University, SE-581 85 Linköping, Sweden

Department of Cardiothoracic Surgery, School of Medicine, Stanford University, Stanford, CA 94305, USA

Research Institute of the Palo Alto Medical Foundation, Palo Alto, CA 94305, USA

Department of Science and Technology, Linköping University, SE-581 83 Linköping, Sweden

Abstract

Background

The ability to measure and quantify myocardial motion and deformation provides a useful tool to assist in the diagnosis, prognosis and management of heart disease. The recent development of magnetic resonance imaging methods, such as harmonic phase analysis of tagging and displacement encoding with stimulated echoes (DENSE), make detailed non-invasive 3D kinematic analyses of human myocardium possible in the clinic and for research purposes. A robust analysis method is required, however.

Methods

We propose to estimate strain using a polynomial function which produces local models of the displacement field obtained with DENSE. Given a specific polynomial order, the model is obtained as the least squares fit of the acquired displacement field. These local models are subsequently used to produce estimates of the full strain tensor.

Results

The proposed method is evaluated on a numerical phantom as well as

Conclusions

Strain estimation within a 3D myocardial volume based on polynomial functions yields accurate and robust results when validated on an analytical model. The polynomial field is capable of resolving the measured material positions from the

Background

The pumping behavior of the heart consists of complex sequences that constitute cardiac contraction and relaxation. The kinematic behavior of the heart has been analyzed extensively in order to understand the mechanisms that impair the contractile function of the heart during disease. Until recently, the only method with high enough spatial resolution of three-dimensional (3D) myocardial displacements to resolve transmural behaviors was invasive marker technology

A previously presented polynomial method for cardiac strain quantification from surgically implanted markers and beads enables straightforward 3D strain computation within the myocardium

Methods

Strain analysis

Displacement-encoded MRI acquire displacements **u **at time t_{n }of timeframe n relative to a reference configuration of the myocardium, at the spatial coordinates **s **= (s_{x}, s_{y}, s_{z}). Lower-case letters are used to denote the coordinates of the deformed myocardium. The spatial coordinates **S **= (S_{X}, S_{Y}, S_{Z}) of the myocardium in the reference configuration were derived by subtracting the displacements from the spatial coordinates at the current configuration; **S**(t_{1}) = **s **- **u**(**s**, t_{1}). The upper-case letters are used to denote the coordinates of the reference configuration.

**X **= **X**(**S**) is the coordinate of a material point, defined as an infinitesimal volume of myocardial tissue, in the reference configuration and **x **= **x**(**s**) is the coordinate after a deformation of the body. The mapping from reference to deformed configuration was modeled by a polynomial function **g**(**X**) of the coordinate of the material point in the reference configuration **x **and can be of different order in the different reference coordinate directions, radial (X_{R}), circumferential (X_{C}), and longitudinal (X_{L}), depending on the number of material points along each dimension, and can thus be described as

where **f**_{R}, **f**_{C }and **f**_{L }are polynomial functions of X_{R}, X_{C }and X_{L}, respectively. The Lagrangian strain tensor **E **is a measure of deformation and is connected to the deformation gradient tensor **F **via the definition

where **I **is the identity tensor. The deformation gradient tensor is given by differentiation, with respect to reference position, of the mapping from reference to deformed configuration, **F **= ∂**x**/∂**X**.

The principal strains E_{1}, E_{2 }and E_{3 }are obtained by diagonalization of the strain tensor, and their magnitudes are independent of any reference coordinate system. Principal strain E_{1 }represents maximum lengthening and E_{3 }represents maximum shortening.

In the subsequent analytical and _{C }defined as tangential to the surface of the volume and parallel with the short axis planes, X_{L }defined as orthogonal to the short axis planes and oriented apex-to-base, and X_{R }defined parallel with the short axis planes, orthogonal to X_{C }and oriented outwards from the volume.

Four orders of the polynomial function **g**(**X**) were analyzed. 1) A bilinear-quadratic polynomial (BLQ) was bilinear within the X_{C}-X_{L }plane and quadratic in X_{R}, 2) a bilinear-cubic polynomial (BLC) was bilinear within the X_{C}-X_{L }plane and cubic in X_{R}, 3) a linear-quadratic-cubic polynomial (LQC) was cubic in X_{R}, quadratic in X_{C }and linear in X_{L }and 4) a biquadratic-cubic polynomial (BQC) was biquadratic within the X_{C}-X_{L }plane and cubic in X_{R}. For example, for the LQC polynomial field, the estimated coordinates were given by

where _{ki }_{li }

where _{i }_{li}**x**) and estimated (

Analytical evaluation

A previously presented analytical model

Strains were computed throughout the model using the method described above. For all polynomial orders, the estimated strains were compared with the analytical strains (_{IJ }

Three normalized distributions of noise were analyzed, each with a mean value of zero and with the standard deviations 0.05 mm, 0.10 mm and 0.30 mm, respectively. The noise levels evaluated were chosen to represent practical measurement errors in DENSE displacement measurements corresponding to SNR in the order of 35, 17 and 6.

Anatomical images and displacement data were acquired in a 31-year old healthy volunteer on a 1.5 T MR scanner (Philips Achieva, Philips Medical Systems, Best, The Netherlands). Displacement data were acquired, using an in-house DENSE implementation, in three cardiac phases as illustrated in Figure

Timing of acquisition

**Timing of acquisition**. Definitions of the time frames of end diastole (ED), end systole (ES), mitral valve opening (MVO) and the two time frames during diastolic filling (75 ms and 213 ms after MVO, respectively). ED is defined at the ECG R-wave peak and ES at aortic valve closure (AVC).

Segmentation of the myocardium was performed using the freely available software Segment

Systolic Lagrangian strains were analyzed at ES, with reference configuration at ED. Diastolic Lagrangian strains were analyzed during LV filling with reference configuration at MVO.

The research has been performed with informed consent, approved by the Regional Ethical Review Board in Linköping and carried out in compliance with the Helsinki Declaration.

Results

Analytical evaluation

The size of the errors of estimated strains is dependent on the spatial resolution of the sampled displacements _{IJ }_{IJ }_{IJ}_{IJ}_{RR }= 0.0081 for the radial strain, ε_{CC }= 0.0054, ε_{LL }= 0.0089, ε_{RC }= 0.0072, ε_{RL }= 0.0048 and ε_{CL }= 0.0067. The LQC RMS differences, i.e. the residual of the polynomial mapping, averaged within the analytical model, were RMSx_{R }= 0.030 mm in the radial direction, RMSx_{C }= 0.017 mm in the circumferential direction, and RMSx_{L }= 0.0004 mm in the longitudinal direction. The sensitivities to noise for each polynomial are plotted in Figure

Sensitivities to noise

**Sensitivities to noise**. The mean ± SD absolute error in estimated strain of each polynomial for different extents of noise on the analytical model. BLQ: bilinear-quadratic polynomial, BLC: bilinear-cubic polynomial, LQC: linear-quadratic-cubic polynomial, BQC: biquadratic-cubic polynomial.

The LQC and the BQC polynomials yielded the smallest maximum RMS differences in both systole and diastole. The results from the LQC polynomial are analyzed in further detail below. A mid-section (comprised of the short axis slices 5-7) is shown in Figure

Systolic strains

**Systolic strains**. All components of the end-systolic strain tensor at the mid-section of the 3D volume in a healthy volunteer estimated using the polynomial method.

Systolic strains

**Systolic strains**. All components of the end-systolic strain tensor at the mid-section of the 3D volume in a healthy volunteer estimated using a finite element method.

Diastolic strains

**Diastolic strains**. All components of the diastolic Lagrangian strain tensor at the mid-section of the 3D volume in a healthy volunteer. a) At 75 ms after MVO; b) At 213 ms after MVO.

Diastolic principal strains

**Diastolic principal strains**. Diastolic principal strains at the mid-section of the 3D volume in a healthy volunteer, at 213 ms after MVO. a) E1; b) E2; c) E3.

Systole

The systolic strain components estimated using the polynomial method are shown in Figure

Maximum lengthening (up to 0.74) was found in the subendocardium in the septum and lateral free wall. Maximum shortening (down to -0.35) was found in the subendocardium and was essentially evenly distributed throughout the plane.

For all three directions, the lowest RMS values were found in septum and the highest values in the posterior wall. RMSx_{R }was within the range 0.13 - 0.40 mm, RMSx_{C }0.09 - 0.21 mm, and RMSx_{L }0.09 - 0.24 mm.

Diastole

All diastolic strain components at 75 ms (22% of the filling interval) and 213 ms (62%) after MVO at the mid-section of the 3D volume are shown in Figure

The diastolic principal strains at 213 ms after MVO at the mid-section of the 3D volume are shown in Figure

The highest RMS values were found in the posterior wall at 75 ms after MVO and in the anterior and posterior walls at 213 ms after MVO. The RMS values ranged from 0.07 mm to 0.40 mm for all directions at both diastolic times, except RMSx_{R }at 75 ms after MVO which approached 0.50 mm in a small region in the posterior wall.

Discussion

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

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 **f**_{R}(X_{R}) was the wall thickness at the late diastolic time frame and the restrictions on **f**_{C}(X_{C}) and **f**_{L}(X_{L}) 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

Systolic radial, circumferential and longitudinal strains, as well as systolic circumferential-longitudinal shear, show agreement with systolic strains previously reported for human myocardium _{RC}: We observed somewhat higher magnitudes of radial-circumferential shear strain than the results of Moore et al. _{RL}: The observed radial-longitudinal shear values fits within mean ± 2SD of the values reported by Moore et al.

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

Limitations

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

Conclusions

In conclusion, the proposed method for strain estimation within a 3D myocardial volume yields accurate results when validated on an analytical model. The polynomial field is capable of resolving the measured material positions from the

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

KK developed the analysis method, participated in the study design, carried out the analysis of the data, participated and coordinated the writing of the manuscript. HH participated in the study design, participated in the design of the data acquisition, participated in writing the manuscript. AS participated in the design of the data acquisition, participated in writing the manuscript. JE participated in the design of the data acquisition. NBI participated in the development of the analysis method, participated in the writing of the manuscript. TE participated in the study design, participated in the design of the data acquisition. MK participated in the design of the study. All authors read and approved the final manuscript.

Acknowledgements

The study was supported by grants from Swedish Heart-Lung Foundation and Swedish Research Council.

Pre-publication history

The pre-publication history for this paper can be accessed here: