Rapid D-Affine Biventricular Cardiac Function with Polar Prediction

Abstract

Although many solutions have been proposed for left ventricular functional analysis of the heart, right and left (bi-) ventricular function has been problematic due to the complex geometry and large motions. Biventricular function is particularly important in congenital heart disease, the most common type of birth defects. We describe a rapid interactive analysis tool for biventricular function which incorporates 1) a 3D+ time finite element model of biventricular geometry, 2) a fast prediction step which estimates an initial geometry in a polar coordinate system, and 3) a Cartesian update which penalizes deviations from affine transformations (D-Affine) from a prior. Solution times were very rapid, enabling interaction in real time using guide point modeling. The method was applied to 13 patients with congenital heart disease and compared with the clinical gold standard of manual tracing. Results between the methods showed good correlation (R² > 0.9) and good precision (volume<17ml; mass<11g) for both chambers.

1 Introduction

Congenital Heart Disease (CHD) is a cardiac defect which moderately or severely affects 6 in every 1000 infants [6]. These children have improved mortality due to better interventional procedures early in life, however heart failure is a significant problem in adulthood. Both left and right ventricular (LV and RV) function is important in maintaining overall cardiovascular function, particularly in CHD [5]. However, the simultaneous analysis of both ventricles is difficult due to the complex and variable geometry and large deformations undergone in the RV. The current gold standard for evaluating RV function in CHD is MRI with manual tracing of contours in each slice. Figure 1 shows single frames from MR cines from three different CHD cases. MRI does not use harmful ionizing radiation and is non-invasive. It has the ability to image all parts of the heart at high resolution without the influence of other structures [4]. MRI also has the highest reproducibility of all of the main alternatives (cardiac catheterization and X-ray angiography, multi-detector CT and radionuclide ventriculography) and lower risk.

Fig. 1.

Examples of short axis frames at end-diastole from MR cines from three different CHD patients (left: atrial septal defect, partial anomalous pulmonary venous drainage and pulmonary hypertension, middle: teratology of Fallot and, right: dextotransposition of the great arteries). The RV is the chamber on the left hand side (yellow/orange). Note the wide variation of shapes in these examples. Contours were generated by intersection of model with image plane. The arrow refers to in-plane shift due to correction of breath-hold mis-registration.

The manual analysis of biventricular function takes substantial time, causing delay to the assessment of the patient. Manual contouring is problematic in slices which obliquely cut the anatomy, particularly at the base and apex [8]. Rapid analysis is critical for clinical throughput and automated methods are currently not robust enough for biventricular analysis. User interaction is therefore required and must be intuitive and minimal. Guide point modeling has been shown to work well for the LV [9] using a 3D+time finite element model of the geometry, incorporation of all slice orientations, and non-rigid registration for motion tracking. This method provides interactive updates enabling intuitive correction of errors during the customization process, but does not include the RV. Right ventricular models have been proposed using subdivision surfaces [7] or free-form deformations [11]. However, these are not suitable for interactive analysis.

Regularization of the problem is required due to the sparse nature of the MRI slice data. Sobolev regularization has been used in many applications but this penalizes rotations and other large deformations undergone in the right ventricle. Poly-affine models [10] or locally affine transformations [3] constrain the deformation to interpolate affine motion between discrete regions of the heart. These are invariant to rotations but require a priori specification of the number of affine transformations.

In this paper, we present an interactive 3D+time finite element biventricular modeling solution incorporating two novel features: 1) an initial prediction step involving a fast linear fit with a reduced number of parameters in a polar coordinate system, and 2) a Cartesian update which penalizes the deviation in strain from affine transformations of the prior (D-Affine deformation). The first step takes advantage of the regular geometry particularly of the LV surface and improves the initial projection of the data onto the Cartesian model. The second step utilizes a regularization term initially proposed for tagged MRI motion recovery [14], but is applied here in the context of geometric fitting. Like poly-affine models, D-Affine regularization is invariant to large affine transformations (including rotation and uniform scaling) and ensures smoothly varying deformation fields. However, D-Affine regularization does not require prior specification of discrete affine transformations. Transformations are not constrained to be affine, but deviations are penalized to as a smoothly varying deformation. Also, D-Affine regularization is computationally efficient since a solution can be found using linear least squares for each coordinate field independently.

While semi-automated methods for bi-ventricular strain and deformation recovery [1,13] are being developed the MRI images are still manually segmented. The method developed in this paper would replace this inital time consuming step.

2 Methods

2.1 Finite Element Model

The model was constructed of 82 3D elements with bicubic Bezier interpolation in the circumferential and longitudinal directions and linear interpolation in the transmural direction, with C¹ continuity between elements (figure 2). The model explicitly includes the four valves (aortic, mitral, tricuspid and pulmonary) and comprises 597 parameters in each x,y,z Cartesian coordinate field. The time-varying model has Fourier temporal basis functions with five harmonics (11 parameters). Total parameters is therefore 6567.

Fig. 2.

Finite element model showing four valves and element boundaries. Model coordinates (ξ₁, ξ₂, ξ₃) were defined as shown.

The model was interactively fitted to guide points placed on the images by the user. Points were first placed on the valve annuli and the left ventricular apex in the long axis images, followed by additional points placed on the inner (endocardial) and outer (epicardial) surfaces of the heart as necessary. Each edited model contour, as well as all valve and apex points, were automatically tracked through the cardiac cycle using non-rigid registration [5] and the resulting tracked contour points were included in the model fit at each frame. The solution proceeded iteratively, with the following steps per iteration:

1. The polar model of the epicardium and left ventricular endocardial surface was fitted to user placed guide points and automatically tracked contour points.

2. The biventricular Cartesian model was fitted to the predicted points, sampled from the polar model surfaces, with a high smoothing penalty.

3. User defined guide points and contour points were projected onto the closest biventricular model point and the fit updated using a low smoothing weight.

4. After each interaction, tracked contour points were regenerated for all frames and the time-varying model updated.

2.2 Polar Prediction

While the RV is typically crescent shaped, the LV endocardium approximates an ellipse and is typically convex. The biventricular epicardium is also convex and well modeled in a polar coordinate system. A prolate spheroidal coordinate system was constructed with the origin in the middle of the LV, a radial coordinate (λ) and two angular coordinates (μ, θ) and as follows:

$$\begin{array}{c}\hfill x=fcosh\left(\lambda \right)cos\left(\mu \right)\hfill \\ \hfill y=fsinh\left(\lambda \right)sin\left(\mu \right)cos\left(\theta \right)\hfill \\ \hfill z=fsinh\left(\lambda \right)sin\left(\mu \right)sin\left(\theta \right)\hfill \end{array}$$ (1)

The focal length, f, was chosen to make the apex point λ =1. The LV endocardial surface and the biventricular epicardial surface were modeled using 16 bicubic C¹ elements. A base plane calculated from the location of the mitral valve points on each frame determined the extent of the model in the direction. The field was fitted to the LV endocardium and epicardium points by linear least squares (132 parameters).

A uniform sample of 1739 points was obtained from the biventricular Cartesian model surfaces and mapped into the polar surfaces by polar projection in the first iteration of the first frame, after registration of global pose and scale. This mapping was then fixed for all subsequent frames and iterations. The constant mapping gives additional stability in the presence of large motions since the heart contracts approxi-mately uniformly in the prolate coordinate system during the cardiac cycle.

This step resulted in a set of predicted points, sampled from the polar geometry and mapped into the biventricular Cartesian geometry using an invariant map. This polar projection map provides a useful a priori correspondence between model and data, which works well in the LV [9].

2.3 D-Affine Regularization

The biventricular Cartesian model was fitted to the guide points, predicted points, and non-rigid registration tracked contour points by minimizing the following error function:

$$E=S\left(\mathbf{u}\right)+\sum _g{w}_g\Vert \mathbf{x}\left({\xi }_{\mathit{g}}\right)-{{\mathbf{x}}_g}^2\Vert$$ (2)

with respect to the unknown model parameters, were u is the displacement of the model from a prior (template) shape X(u = x − X), x_g are the data points, ξ_g are the model coordinates corresponding to the data, w_g are the data weights and S(u) is the regularization term. The model coordinates were found by the iterated closest point algorithm [2]. At each iteration, the predicted points (with fixed correspondence) were first fitted by the biventricular model using a high smoothing weight for regularization. Then the guide points and tracked points were projected onto the closest model point and another fit performed using a low smoothing weight. This was repeated for each user interaction.

The D-Affine regularization term was defined to be:

$$S\left(\mathbf{u}\right)=\sum _{k\in \{x,y,z\}}{\int }_{\Omega }{\alpha }_k\Vert \frac{\partial \mathbf{J}}{\partial {\xi }_k}{\Vert }_F^{2}d\Omega$$ (3)

Where $\Vert {\Vert }_F^2$ is the Frobenius norm, J is the Jacobian of the motion (i.e. the deformation gradient tensor) and Ω represents the model domain. α_k represents the smoothing weight in the k^th direction.

$$J_{ij}=\frac{\partial x_i}{\partial X_j}=\sum _l\frac{\partial x_i}{\partial {\xi }_l}\frac{\partial {\xi }_l}{\partial X_j}=\sum _l\frac{\partial u_i}{\partial {\xi }_l}\frac{\partial {\xi }_i}{\partial X_j}+{\delta }_{ij}$$ (4)

Is the deformation gradient tensor (i.e. the Jacobian of the motion) and δ_ij is the Kronecker delta. For affine motions, J is a constant with respect to model coordinates ξ. S(u) is minimized by any global affine transformation, and is invariant to superimposed rigid body motions. It is also quadratic in the displacement parameters, leading to a linear least squares minimization (each coordinate field being solved separately using the same system).

2.4 Implementation

The method was implemented in C++ on a Dell OptiPlex 990 running Windows 7 using an Intel^® Core i5 3.30GHz with 4GB of RAM, containing 4 cores. A preconditioned conjugate gradient solution was implemented using the Math Kernel Library (Version 10.1.3.028) using the inbuilt routine with multithreading. A solution was possible in 0.15s per frame (including x, y and z), enabling real time updates of model edits while dragging a guide point.

2.5 Experiments

The method was applied to 13 patients with CHD and the results compared with the clinical gold standard of manual contouring on the short axis images using, Argus (Syngo MR 2004V, Siemens Medical Solutions, Erlangen, Germany). The MRI images were obtained using a 1.5T scanner (Siemens Avanto; Siemens Medical Solutions, Erlangen, Germany) and the study was approved by the institutional ethics committee, with written informed consent from all patients. Images were acquired using either prospectively or retrospectively-gated Steady-state Free Precession (SSFP) cine MRI. Short axis slices were acquired parallel to the tricuspid valve annulus and spanned both ventricles from apex to base. Long axis slices were acquired through both ventricles, including through the pulmonary and tricuspid valve. Typical imaging parameters were 30ms repetition time, 1.6ms echo time, 60deg flip angle 360 ×360mm field of view, 6mm slice thickness, 256 ×208 image matrix, and 30 reconstructed frames.

Trabeculations and papillary muscles were excluded from the mass and included in the blood volume. Smoothing weights, represented by α_k in equation 3, were 10000 in each direction, and α_k increased 100 times for fitting with high smoothing.

3 Results

Analysis with the new method took approximately 30 to 40 minutes per case for all frames compared to approximately 5 hours for the manual gold standard analysis. The polar prediction step worked well for all hearts, including those with a systemic RV, in which the RV geometry is more spherical and the LV more crescent shaped. Figure 1 shows examples of the three cases contoured with the new method.

The regression plots in Figure 3 show good correlation between the two methods, while the Bland–Altman plots in Figure 4 shows that the new method gives higher values for both LV and RV end-diastolic volume (EDV) than the gold standard, but lower values for end-systolic volume (ESV) and mass of both ventricles. This discrepancy is likely due to the different methods of volume calculation (model integration of a curved 3D surface versus summation of stacked slice contours). However, the scatter in all measures was small (standard deviation of the differences <17ml for volume and <11g for mass), indicating that the method can be used clinically.

Fig. 3.

Comparison of LVEDV, RVEDV, LVESV, RVEDV, and mass with the gold standard Argus

Fig. 4.

Bland–Altman plots for LVEDV, RVEDV, LVESV, RVEDV, and mass. The red lines represent 2 standard deviations from the mean. The interobserver error for Argus is shown in green (mean ± 2 standard devations).

The modified Hausdoff distance was 2.5±1.4 and 2.4±1.1 pixels for observer 1 in the LV chamber and 2.5±1.3 and 2.3±0.9 pixels for observer 2 at ED and ES respectively. In the RV chamber the distances were 3.4±2.1 and 3.6±1.8 pixels for observer 1 and 3.7±2.0 and 4.0±1.9 pixels for observer 2 at ED and ES respectively.

4 Discussion and Conclusions

Argus mass and volumes were calculated by slice summation from the short axis slices whereas this method used model integration. A major advantage of the model is the inclusion of all four valves, unlike previous methods which ignore the valves[12]. The high correlation and low scatter of all clinical measures suggests a small correction factor could be used to convert results between manual and modeling methods if this was required.This software has been installed in Auckland City Hospital. Cardiologists report that each analysis takes 12–15 minutes for clinical evaluation of mass and volume (on a DELL Precision T5610 with an Intel^® Xeon^® Processor E5-2650 v2 and 16 GB RAM) .

While the polar step worked well in all cases tested, it is possible that the LV endocardial surface may not always be convex, especially in pathology such as hypertrophic cardiomyopathy. In these cases, the predicted points from the endocardium can be given a small or zero weighting.

The main advantage of the current method over poly-affine methods is that the model transformation is arbitrary and can vary from point to point. This does not require a prior specification of the number of locations of affine transformations. Also, the method enables a linear least squares solution which can be efficiently solved for each field as a separate right hand side vectors using preconditioned conjugate gradients.

Future work includes the implementation of a diffeomorphic version of the transformation, and the calculation of the trabeculae and papillary muscle mass.