4.5.1 Synthetic 2D sequence
In order to quantitatively assess the accuracy of our method, we created a test sequence using a Cardiac MRI sequence simulator. This simulator is based on a cardiac motion model [124] applied to true MR images to generate a realistic cardiac MR sequence. The sequence is generated as follows : A 2D MR short axis slice of the heart at end diastole is taken as input data. This image is warped using the motion predicted by the model at each of the cycle instants. The result is a series of MR images that closely resemble a real MR sequence, but for which the motion is completely known. Figure 4.2 shows two sample images generated using this simulator.
We applied the dynamic elastic model to this sequence, using a ring as the initial shape of the model. In fact, the shape of a short axis slice of the myocardium in healthy patients is very close to a perfect ring. This ring was meshed with a very simple method : first divide the ring into quadrangles using sectors and concentric rings, then divide each quadrangle into two triangles. Better meshing methods could be used, but the triangles generated by our simple method proved good enough for this application. Figure 4.2 shows a sample mesh generated by our method.
The force field was generated with the following algorithm : – Preprocess image with a median filter.
– Extract the contours with a simple Sobel edge detector.
– Smooth the contours with a Gaussian filter.
– Compute the gradient of the resulting image.
– Apply a diffusion process to the resulting vector field
– Apply a Geman-McClure function to the vector norms. The vector norm is set as σ2d+d2 2, where d is a distance computed as a distance map of the previously extracted contour map. This step ensures that forces are null on the contours. In our experiments,σwas set to 2.5 pixels.
The resulting force map is a vector image with vector attributes at each node
FIGURE 4.2 – On the left, sample ring mesh used for 2D short-axis cardiac MR analysis. On the right, two sample images generated using the Cardiac MRI se- quence simulator, respectively in end-diastole and end-systole.
FIGURE 4.3 – Radial error maps at (a) mid-systole and (b) end-systole. Note that the error is larger at end-systole because the total motion (computed relatively to end-diastole) is largest at end-systole.
of a regular grid (image). To compute the forces at the arbitrary locations of mesh nodes from this force map, linear interpolation between the 4 nearest pixels was used. Using linear interpolation instead of nearest neighbor ensures a smooth tran- sition at contours and helps avoid oscillation problems.
After running the model on the synthetic sequence, we compared the estimated radial motion to the real radial motion. The radial motion is the projection of the real motion at one point on the line that joins the points to the center of the heart.
We only consider radial motion since in the absence of additional data (such as Tagged MRI), our model does not naturally capture the torsion motion. Figure 4.3 shows radial error maps for two sample instants of the cardiac cycle. On average over the sequence, the radial error was of 0.561 mm, with a standard deviation of 0.452. The mean radial error at end-systole (the instant of peak contraction), was of 1.02 mm, with a standard deviation of 0.694.
4.5.2 Results on real Cardiac MR images
In order to test the model on real data, we used a publicly available 4D heart database1. This database is comprised of cardiac MR images of 18 patients, toge- ther with two expert segmentations at end-diastole and end-systole.
Experiments were conducted on 2D image sequences corresponding to a basal slice of each of the 18 patients. The ring mesh described in the previous section was used as the initial shape of the model.
The force field was generated with the same algorithm used for the synthetic sequence, except that we used a method based on discrete watersheds to extract contours, directly on the 4D image. More details about this segmentation method can be found in [54]. This method yields contour maps that are close to the actual myocardial contours.
The resulting force map is a vector image with vector attributes at each node of a regular grid (Image). To compute the forces at the arbitrary locations of mesh nodes from this force map, linear interpolation between the 4 nearest pixels was used. Using linear interpolation instead of nearest neighbor ensures a smooth tran- sition at contours and helps avoid oscillation problems. We found that using two distinct force fields, one for the endocardium and one for the epicardium leads to better results.
Figure 4.4 shows the model after convergence, superimposed with the image data, for two sample patients (also see attached movies for a dynamic view of the deformation process and the results). The results show good adequation between the image and model contours. One remaining problem is the accurate segmenta- tion of papillary muscles. In this experiment, the papillary muscles were removed using alternate morphological openings and closings. This method performs well when the papillaries don’t touch the myocardial wall. Better methods for remo- ving the papillary muscles have already been presented [125] and could be used to improve this point.
It is interesting to evaluate the effect of the different parameters of the model : the mechanical parameters, namely the Young modulus and the Poisson coeffi- cient, the damping factor and the number of Fourier harmonics used to describe the motion of nodal points throughout the cardiac cycle.
The Young modulus represents the balance between prior knowledge and image data. If set too high, the model will not deform at all. If set too low, the model will deform to match perfectly all image features, real structures and arti- facts alike. In practice, for the analysis of normal MR sequences of the heart, we used values of the Young modulus between 0.2 and 0.6, with forces of a maximal norm of 1.
The Poisson coefficient characterizes the ability of the material to be com- pressed, ie. to change volume when under stress. It is generally admitted that the myocardium is nearly incompressible. However, since we do not know the volume of the patients myocardium prior to analyzing the images, it is better to allow the volume to change during the adaptation of the model to the patient’s images. An idea would be to do a first run with a compressible material to estimate the vo- lume, then rerun with the Poisson coefficient set to 0.5. This could be interesting,
1. 4D Heart Database : http ://www.laurentnajman.org/heart/index.html
FIGURE 4.4 – Model and images superimposed, on two sequences of a median slice of the heart of two patients over the cardiac cycle. Sequences should be read like a book, with end-diastole on the top-left corner(Only one in two frames are shown.) The pink mesh is the model. It is set to be translucent so that correspon- dence between model and image contours can be appreciated.
FIGURE 4.5 – Effect of Fourier filtering. Each row shows several frames of a dynamic sequence, uniformly sampled over the cardiac cycle, with first frame corresponding to end-diastole and the second frame corresponding to end-systole.
For each row, the forces are filtered with a different number of harmonics. Top row, 2 harmonics. Second row, 3 harmonics. Third row, 4 harmonics, Fourth row, 5 harmonics, and last row, no filtering at all.
especially to deal with the problem of papillary muscles touching the myocardial wall during systole. However, due to the problem of the motion of the myocardium across the image slices, it is not guaranteed that the imaged area of myocardium stays the same. In practice, an intermediary value such as 0.2 was found to work well.
Figure 4.5 shows the effect of filtering on tracking results. The top row shows that using two harmonics is clearly insufficient for accurately capturing the cardiac motion. However, as few as 4 to 5 harmonics are enough to capture most of the cardiac motion, while filtering a significant part of the noise introduced by the low-level segmentation method. The last row shows the results in the absence of filtering : while motion is captured accurately, the borders are not smooth and temporal consistency is reduced.
Damping acts in a complementary way with Fourier filtering : while Fourier
FIGURE 4.6 – Effect of the constrained scheme : on the left, normal scheme ; on the right, constrained scheme. Notice that the constrained scheme follows borders more accurately, at the expense of contour regularity.
filtering ensures global periodicity and smoothness, it does not filter local and smooth oscillations. Damping allows to enforce temporal consistency at the local level. In our experiments on the synthetic sequence, best results were obtained with damping set to 0.6.
Figure 4.6 shows the effect of the constrained scheme on an instant of the real MR sequence. The constrained scheme allows to perfectly match borders at the expense of contour regularity. It should thus be used only when the pre- segmentation is very good and the model is used mostly for motion estimation.
Our implementation of the model was written in Python with some performance-critical sections of the code rewritten in C, without paying too much attention to optimization. The program takes about 1 minute on a standard PC to compute the solution of the 2D dynamic problem on a sequence of 20 images.