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Gonzalo Vegas-S´anchez-Ferrero

Antonio Trist´an-Vega

Lucilio Cordero-Grande

Pablo Casaseca-de-la-Higuera

Santiago Aja-Fern´andez

Marcos Mart´ın-Fern´andez

Carlos Alberola-L´opez.

Laboratory of Image Processing. University of Valladolid. Spain


In this paper we propose an alternative method to es-timate and visualize the Strain Rate Tensor (ST) in Mag-netic Resonance Images (MRI) when Phase Contrast MRI (PCMRI) and Tagged MRI (TMRI) are not available. This alternative is based on image processing techniques. Con-cretely, an elastic image registration algorithm is used to estimate the movement of the myocardium at each point. Our experiments with real data prove that the registration algorithm provides a useful deformation field to estimate the ST fields. A classification between regional normal and dysfunctional contraction patterns, as compared with pro-fessional diagnosis, points out that the parameters extracted from the estimated ST can represent these patterns.

1. Introduction

Mechanical properties of the heart provide a way to de-termine early diagnosis and better patient follow-up. Ac-tually, local motion abnormalities (measurable by means of mechanical anomalies) could precede electrocardiogram disorders [5]. Impaired myocardial perfusion appears as a consequence of reduced blood flow of the heart muscle. This is often analysed with nuclear medicine imaging tech-niques or with MRI.

MRI is a very suitable imaging modality for blood flow and tissue motion measurement. It provides excellent con-trast between soft tissues, and images can be acquired at positions and orientations freely defined by the practitioner. From a temporal sequence of MR images, boundaries and edges of tissues can be tracked by image processing tech-niques [10].

The local deformation of the myocardium can be de-scribed by the ST which gives a measure of the deformation of an object relative to its original length.

This work was supported by grants PI-041483 from FIS, Spain; and TEC2007-67073/TCM from CICYT, Spain.

There are different possibilities to estimate and visualize the ST. Among those modalities involving MRI, TMRI and PCMRI are direct approaches to obtain information about the tensor. The former allows to derive a motion model of the underlying tissue by tracking a temporal sequence of images which have been previously marked by a pattern of dark lines (called tags). This pattern is achieved by modu-lation of the image intensity with a magnetic presaturation pulse [10]. The deformation field of the crossing line points can be calculated just following the temporal trajectories. On the other hand, PCMRI provides a measure of the veloc-ity field by means of phase shifts induced in the transverse magnetization. The phase of the signal is directly related to the velocity of the material within each voxel. When TMRI or PCMRI are not available, the estimation of the strain rate tensor becomes a difficult task, even more if direct 3-D in-formation is not available.

This paper is focused on an alternative to hardware tech-niques which need a TMRI or PCMRI scanner. This alter-native is based on a registration algorithm which is able to estimate a deformation field between two consecutive im-ages deformed in a non rigid way. In case the deformation between adjoining frames is small enough, we can assume that the estimated deformation applies for the physics of the problem, and so we can identify the deformation field with the movement of the myocardial wall. With such a tech-nique, the ST can be estimated from temporal sequences of conventional MRI.

Some non-rigid registration algorithms have been ap-plied in MRI tagging. Concretely, Ledesma et al [8] pro-posed a B-spline registration model that has demonstrated good results with subpixel accuracy. Chandrashekara et al. [3, 4], proposed a 4D B-spline registration model proving that the radial and longitudinal displacement provided by the deformation field shows high correlation coefficients compared to TMRI. Compared to these works, the main contribution of our paper is the estimation of the ST field by means of the deformation field provided by the non para-metric registration algorithm. The posterior analysis of the



ST proves that the registration stage is accurate enough, and moreover the estimated ST field is useful to detect abnor-malities in the behavior of the myocardium.

The remainder of the paper is organized as follows: sec-tion 2 describes the registrasec-tion algorithm we use, together with an explanation of the ST calculation from the deforma-tion field and the method for visualizing the ST. In secdeforma-tion 3 we evaluate the accuracy of the registration algorithm used and the ST field. Finally, the main conclusions extracted from our work are presented in section 4.

2. Methods

2.1. Demons Registration

Since the deformations of the myocardial wall may be highly irregular, an algorithm for elastic image registration is needed. We choose Thirion’s demons algorithm [12] as a well tested, general purpose technique. It computes the de-formation field (myocardial movement) as the optical flow between the images, and further regularize it with a Gaus-sian convolution. This relies on the assumption that the gray level of corresponding pixels does not vary across the im-ages, which is hardly true in a real case. For this reason, we use the approach in [2], based on the local correlation of the images, which is robust even with highly varying contrasts of the MRI slices.

2.2. Strain Rate Tensor Calculation

Since we deal with 2-D images, the deformation field is a two dimensional vector field, so only a two dimensional ST can be calculated. Although a 3-D interpolation of the image could be done in order to obtain a 3-D vector field, the low resolution in the long axis direction does not allow the registration algorithm to achieve coherent results in the deformation field1.

To compute the ST from the deformation field provided by the registration algorithm, the Jacobian matrixJ is cal-culated: J =∇u= ∂u1 ∂x1 ∂u1 ∂x2 ∂u2 ∂x1 ∂u2 ∂x2 ! (1) whereui withi = 1,2 is the deformation of each voxel

from instantttot+ 1andxjwithj= 1,2is the direction.

To define a measure that contains only information about deformation but not rotation we compute the well known

Lagrangian ST[7]. E= 1

2 (∇u) + (∇u)

T + (∇u)T(∇u)


1High resolution cardiac volume requires acquisition times which are

often too high for common clinical practice.

Figure 1. Deformation of an infinitesimal square without change of area. The strain components are approximately the decrease in angle with the axis of the square.

For small deformations the quadratic terms can be ne-glected giving the final equation of thestrain rate tensor:

Eij = 1 2 ∂u i ∂xj +∂uj ∂xi (3) This tensor is by definition symmetric, so it can be de-composed into real eigenvaluesλ1andλ2with real

eigen-vectorsv1andv2. The eigenvectors are the main directions of the strain where there is no shear strain and the eigenval-ues are the elongation or contraction in the directions de-fined by the eigenvectors.

2.3. Visualization

The interpretation of the ST by its components allows an intuitive way to represent them. The diagonal components (Eii) may be interpreted as the unit elongation or

compres-sion of the material voxel in thexi direction. The other

components (Eij withi 6= j) are the shear strains. So,

in an infinitesimal square area strained without change of area, the shear strains components are approximately the decrease angle between the axes. In Fig. 1 an infinitesimal square is deformed without change of area, showing that the sum of shearing strains are approximately the decrease in angle with the axis of the square element [14].

If every voxel of an image is represented in the same way as an infinitesimal rectangle whose diagonals are oriented in the eigenvector directions and the position of each vertex is related to the eigenvalues, a more intuitive visualization of the tensor field may be achieved.

Since the eigenvalues can be both positive or negative, the positive eigenvalues are represented as elongations in the direction of its corresponding eigenvector, whereas neg-ative eigenvalues are represented as contractions. Both eigenvalues are normalized by the largest absolute value of them. The length of each diagonal is:

SDi= √ 2 L 4 + L 4R i 1+ L 2R2 (4)


Figure 2. Rhombus visualization of a ST. The dashed lines are the minimum and maximum allowed squares.

whereLis the side of the square,i = 1,2,R1 is the

nor-malization ratio computed asRi

1 =


2(λi+ 1)andR2 is

the Euclidean norm of the deformation field normalized in each instant. The minimum possible side of the square is L/4, and the maximum is L.

Obviously, when the eigenvalues are equal the eigenvec-tors do not add much information and the orientation of the square is random. It seems that a representation by an ellip-soid could be better than a square, however when the tensor field is visualized all the tensors are seen as a whole and the orientation of groups of tensors are not affected by this ef-fect and it is worth representing them as deformed squares in the direction of each diagonal in the same way as an in-finitesimal square. Fig. 2 shows a possible representation of a ST.

3. Results

3.1. Materials

Cine MRI sequences of 17 patients were evaluated in this study. Every segment of the 16-segment model for wall mo-tion recommended by the American Society of Echocardio-graphy Committee on Standards was considered [11]. This model is divided into three different levels: Apical, Mid-heart and Base. The apical area comprises 4 segments (An-terior, Lateral, Inferior and Septal) whereas Mid and Base have 6 segments for each level (Anterior, Antero-lateral, Infero-lateral, Inferior, Infero-septum and Antero-septum). These segments were classified by expert cardiologists2in the following 3 classes: 1. Normal, 2. Hypokinesia (di-minished movement), 3. Akinesia (negligible movement). Among them, 194 segments were classified as normal, 27 as Hypokinesia and 51 as Akinesia.

Demons registration algorithm was performed with 5 multiresolution levels with 50 iterations per level, a

Gaus-2The authors acknowledge the ICICOR (Instituto de Ciencias del

Coraz´on, Valladolid, Spain) and Jos´e Sierra (Centro PET Recoletas Val-ladolid) for the expertise exchange as well as the data provided to carry out this research.

(a) (b)

Figure 3. An example of the performance of the registration algo-rithm. We present a single slice for patient 10 and time slot 18. In (a) we superimpose the contours of time slot 20 before registra-tion, and in (b) after registration. The algorithm is able to recover the heart motion accurately even for non consecutive time slots.

sian regularization filter withσ= 2.8and gradient regular-ization withσ= 0.57. Images size is 512x512 with voxel size 0.8594x0.8594 mm.

3.2. Testing of the registration results

Without a ground-truth of the deformation field to esti-mate (i.e. without TMRI or PCMRI), the validation of the results is difficult, so we must accomplish an indirect esti-mation of the accuracy. To begin with, we show in Fig. 3 an example of the behaviour of the algorithm when we reg-ister a single slice of patient 10, for time slots 18 and 20; it may be seen that the algorithm is able to recover the de-formations even for non consecutive time slots. To assess the accuracy of the registration, we present a set of results based on three image similarity measures:(1)the Structural Similarity (SSIM) index [13],(2)the Quality Index based on Local Variance (QILV) [1] and(3)the Mean Square Er-ror (MSE). The MSE is a standard measure, so it does not need further explanation. Regarding SSIM and QILV, they have been widely applied in the literature as measures of the structural similarity between two given images; both two indices are bounded: the closer to one, the better the image alignment. In Fig. 4 we present the results for patient 10, for each time slot from 0 to 19 (we register each slottwith t+ 1 andt+ 2), for each slice from 1 to 14, and the av-erage values for all slices. We show the results before and after registration, together with anideal value, computed for each slice and time slot by estimating the noise power σ2

nand then denoising the corresponding imageI; then we

corruptIwith two independent Rician noisy processes with powerσ2

n, and blur the second image to simulate the effect

of the interpolation when we register, so that we have two synthetic imagesI1andI2that we compare. This way we

obtain an estimation of the best result we could obtain if the estimation of the deformation field were exact and the registered images differed only because of noise and inter-polation artifacts.


0 2 4 6 8 10 12 14 16 18 0 100 200 300 400 500 600 700 800 Time slot MSE Ideal value Before registration After registration 0 2 4 6 8 10 12 14 16 18 0.7 0.75 0.8 0.85 0.9 0.95 1 Time slot SSIM Ideal value Before registration After registration 0 2 4 6 8 10 12 14 16 18 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 Time slot QILV Ideal value Before registration After registration (a) (b) (c) 0 2 4 6 8 10 12 14 16 18 0 200 400 600 800 1000 1200 1400 1600 Time slot MSE Ideal value Before registration After registration 0 2 4 6 8 10 12 14 16 18 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Time slot SSIM Ideal value Before registration After registration 0 2 4 6 8 10 12 14 16 18 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Time slot QILV Ideal value Before registration After registration (d) (e) (f)

Figure 4. Similarity measures for each time slot and slice (MSE –a, d–, SSIM –b, e– and QILV –c, f–). Top row: we register slottwith

slott+ 1. Bottom row: slottwitht+ 2. For each case, we superimpose the values for the 14 slices in the same time slot together with the

average value for all slices (solid and dashed lines). The results of the registrations are compared to the similarity before the registration and to the ideal value that could be achieved if the estimation of the deformation field were exact.

Fig. 4 demonstrates that the registration clearly improves the similarity between the images in all cases: mean values for all slices and individual values for each slice are much closer to the ideal values than they are to the values without registration. Is is clear that the registration of non consec-utive slots is more difficult than it is for consecconsec-utive slots, and therefore the registration results are poorer (note that the similarity measures, both before and after registration, are worse for the pairst, t+ 2than for the pairst, t+ 1). Be-sides, not all time slots show the same similarity, but some of them are more difficult (and yield poorer results) than others. Moreover, the shape of the averaged values resem-bles the shape of the cardiac cycle: the peaks of end sys-tole and end diassys-tole correspond to the slots where the my-ocardium moves the slowest, so the images to register are closest to each other (better similarity measures).

Once it is shown that the algorithm is able to recover the deformations even for non consecutive time slots, we adapt theconsistency checkingmethodology in [9] to esti-mate the accuracy of the results: for each time slott, we estimate the deformation fieldDt,t+1between this slot and

slott + 1, and respectively Dt+1,t+2 and Dt,t+2. Then

we compute for each image voxel the magnitude M =

kDt,t+1+Dt+1,t+2− Dt,t+2k. The composition ofDt,t+1

andDt+1,t+2should ideally equalDt,t+2, and thereforeM

should be 0. If we consider thatDare estimates of the true displacement field contaminated with some random noise, the residual M would be the modulus of the addition of three random variables, which in turn should be indepen-dent, since they correspond to three independent experi-ments. All in all, M may be considered as an estimate of the euclidean error of the deformation field D when-ever the estimate ofDis unbiased. The results shown in Figs. 3 and 4 suggest that in fact we have true estimates of the deformation fields, both forDt,t+1 andDt,t+2, so

the proposed methodology makes sense. The correspond-ing results are shown in Fig. 5, where we show as well the mean displacementskDtkrecovered by the registration al-gorithm. Note that the maximum mean displacements cor-respond to early systole and late diastole, where the strain is maximum, which confirms once again that the estima-tion of the displacement fields is adequate. Comparing the mean displacements and the mean consistency errors, the latter are well below (mean errors are below 0.59 mm.), so we may conclude that the registration algorithm is able to recover the deformation field with an error in the order of the mean discrepancies (consistency errors). Note that to compute these discrepancies we accumulate the


(indepen-0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 2 2.5 3 Time slot Quadratic error/deforamtion (mm.)

Mean Squared Error Mean extent of deformations

Figure 5. Consistency checking of the deformation fields. We

present mean values ofM = kDt,t+1+Dt+1,t+2− Dt,t+2k

for all slices and time slots for patient 10, together with the mean

values of the displacements,kDtk. The computation has been

re-stricted to a ROI containing the myocardial wall.

dent) errors of three different fields,Dt,t+1,Dt+1,t+2and Dt,t+2; moreover, at the sight of Fig. 4, the error in the

esti-mation ofDt,t+2is greater , so in fact the mean discrepancy

overestimates thetrueerror. This holds when the estimation ofDtis unbiased; unfortunately, we would need a ground-truth (which is not available) to guarantee this condition, so we must trust the indirect validation given by Fig 4.

3.3. Strain rate tensor estimation

In order to see whether ST estimation provides a good desription of the dynamic behavior of the heart, a classifi-cation was performed using the dynamic range of contrac-tion and the maximum of the strain invariantIt=λ21+λ22

as input data. A Multilayer Perceptron with 3 layers was used with 2 neurons in the first layer, 2 in the second, and 1 in the output layer [6]. The network was trained with a Bayesian Regularization of the weights using the Levenberg-Marquardt algorithm. Testing was performed with a leave-one-out strategy.

A correct rate of 76.67% is reached by classifying into Normal and Abnormal (Hypokinesia or Akinesia) seg-ments. This result makes clear that the strain rate tensor estimation can be considered as a feature for classifying.

In order to test the coherence of the strain estimation, a representation of the invariantIt = λ21+λ22in each of

the zones was performed for a patient who has an Akine-sia diagnosed in the inferior area of the mid zone compared to a normal patient (see Fig. 6). In this representation, the strain rate shows its higher values at the beginning of the cy-cle (early systole) and in the mid heart cycy-cle, when diastole begins. Curves obtained for the inferior and inferolateral, show a low strain through the cardiac cycle even on systole and diastole compared to the normal patient, so the zone with Akinesia presents low strain as it is expected.

(a) Normal Patient

(b) Patient with Akinesia

Figure 6. (a) Normalized strain invariantItof a normal patient in

a cardiac cycle. (b) Normalized strain invariant of a patient with a diagnosed Akinesia in the inferior area. The higher strain at the beginning and in the mid heart cycle shows the systole and diastole moments. In general, normal patient shows higher strain through all the heart cycle. In the Inferior and Inferolateral segments show an even lower strain in the whole cycle, especially in diastole, due to Akinesia.

Fig. 7 shows a sequence of the heart cycle represented with tensors drawn in the myocardial wall. The sequence is zoomed in to analyse the behaviour of tensors in the An-terior and AnAn-terior Septum area of the wall. The intensity of the deformation field is represented by a red level added to the gray level of the image. In the first frame, tensors are oriented to the endocardium contour showing elongations in that direction due to the contraction of the wall in the radial direction. An increased strain in the anterior-septum area is represented in the second frame with overlapped tensors in the direction of the deformation. The third frame shows the myocardium is still contracting as an effect of the inertial movements. This behavior of the strain is coherent with the strain expected all over the cardiac cycle, where the strain peaks are in early diastole (when contraction begins) and in systole (when dilatation begins).

4. Conclusion

The analysis of the registration results shows that the deformation field for motion estimation of endocardium is accurate enough even when non consecutive time slots are considered. Since the computation of the ST is based just on the registration of consecutive slots, even more accurate


Figure 7. Sequence of the heart cycle with tensors represented in the myocardial wall. Right sequence corresponds to the zoomed Anterior and Anterior Septum area. The intensity of the defor-mation field is superimposed to the gray level of the image. The sequence shows how the strain increases in early systole until it reaches its maximum represented in the second frame with over-lapped tensors. After reached its maximum, the strain diminishes.

estimates could be expected than those shown in Fig. 5. Hence, it makes sense to estimate the ST fields from the deformation field provided by the registration algorithm.

The classification between Normal and Abnormal pa-tients using the strain invariant and the dynamic range of contraction as input features provided a correct classifica-tion rate of76.76%. This result confirms that the ST fields estimated from the deformations provided by the registra-tion algorithm can be used to detect abnormalities accord-ing to the professional diagnosis.

Furthermore, the study of strain in a patient with di-agnosed Akinesia shows a coherent behavior of the strain curves that allows to detect systole, diastole and the areas with less strain. Tensor visualization is also consistent with the deformation of the left ventricle wall through the cardiac cycle showing an elongation in the radial direction when endocardium is compressing, and a contraction when endo-cardium is expanding.

Future work will focus on developing a 3Dad hoc reg-istration algorithm. An improvement of the results is ex-pected if the registration algorithm is designed taking into account the physics of the problem at hand. Additionally, we will work on a comparison with other MRI modalities such as TMRI and PCMRI which would confirm the consis-tency analysis in order to quantify the degree of correlation between different motion estimations.


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