Cardiac deformation mechanics from 4D images

S. Y. Chen

An approach is presented for analyzing cardiac deformation mechanics. Cardiac wall motion is concerned in a physics-based deformable model which is trained from a set of 4D images. The mechanics of myocardium are quantitatively analyzed and the working conditions of the heart are statistically evaluated by observing their deviations from normal values.

*Introduction**:* The
research to diagnose and prevent cardiovascular diseases becomes more important
than ever [1]. Estimating the global functions of the heart accurately, and in
a clinically useful way, is very important yet an open research problem. Over
the past decade, there have been a great deal of model-based
approaches to the analysis of the complicated motion of the heart [2][3]. The
limitation of such traditional techniques is that they do not provide intuitive
motion parameters to describe the non-rigid motion of the heart. Recently, some
attempts are to characterize the cardiac mechanics with finite element methods and
mostly the functional analysis are related to wall motion and wall stress for
identification of myocardial infarction. For example, Wu et al. discuss the
wall motion with infarction in magnetic resonance images (MRIs) [4]. Ledesma-Carbayo
*et al.* contribute strain analysis on the left ventricle (LV) [5]. One
main problem, however still existing, is that people often related dynamical
parameters directly to some cardiac problems without considering locational
effect and individual difference. The heart is in fact a complex system and
conditions may be very differently distributed. For this reason, the functional
evaluation in this research is applied locally and with statistical knowledge.
Furthermore, this research concerns not a single motion parameter, but a set of
combined dynamical factors.

** Method**: Based
on the model-driven techniques for cardiac imaging interpretation developed in
past few years [6], this research proposes a content adaptive object modeling
(CAOM) method to capture the general shape of a heart, as well as its shape
variation, object neighborhood information, and object boundary structure of
the anatomical view from the image volumes. The method is statistically based and
very suitable for medical image interpretation and deformation analysis. In
general, the method works on two stages. The first is the training stage which
includes sample set collection, accurate shape description, discrete
representation, geometrical transformation, statistical processing to learn
shape variation, and creation of the generalized cardiac model. In this
research, the statistical model of the cardiac shape is created from
about 200 cases in three hospitals.

The second stage is instance
interpretation. When a series of images of a new case are read to the system,
it firstly needs to properly locate in the image volume and fit the shape
according to the actual content around the object after active segmentation [7].
The CAOM comprises a generalized object shape and statistical
appearance in the image environment. The former primarily holds information
about the cardiac shape while it allows variation in a statistical model. The
latter is responsible of learning content patterns from the training datasets.
In this research, we concern mostly the left ventricle, represented with about one
thousand 3D points. The cardiac CAOM is expressed as **w** = **v** + **F**b, where
**w** is an instance of cardiac shape which is a vector
containing the 3D surface points, **v** is the normalized
mean shape of the heart which should be constructed during the training stage, **F** is a
matrix containing some principal eigenvectors and each vector represents the
direction of shape variation, and b is a vector of actual variation of an
instance from the trained model.

With the statistically constructed CAOM, a set of regional points of interest in a new case can be tracked accordingly. Some important parameters of regional cardiac dynamics analyzed in this study are: the positional displacement, velocity, acceleration, force, stress and strain. Those parameters are the basis of cardiac tissue motion and can help physicians to analyze the cardiac status. Before computing these parameters, a set of points of interest are assigned for tracking in the 4D image sequence. Without special clinical reason, often tens or hundreds of points are automatically selected from all areas of the cardiac surface so that the whole heart can be evaluated.

A myocardial tissue is tracked by
CAOM fitting and interpretation so that it knows where a point comes from in
the last phase and where it goes in the next phase. In order to measure the
deformation in the myocardium, we may start with an estimation of voxel displacement.
Consider a number of phasic shapes constructed in one cardiac cycle. Each phase
is sampled by a fixed frequency (e.g. 100 ms per frame). Practically, a 4D
cardiac image set is represented as I = *f*(*x, y, z, t*), where a
point in the sequence is M=(*x, y, z*;* t*).

The radial displacements on the
ventricular myocardium are obtained by using the tissue-tracking algorithm in
CAOM. For a point M_{1} at time t_{1}, which is mapped to a
point M_{2} in the deformed myocardium at time t_{2}, the
radial displacement is then d = **u**_{1}¡Á(M_{2}
- M_{1}), where **u**_{1} is the unit radial vector at
M_{1}. Here we have assumed Dt = (t_{2} ¨C
t_{1}) ® 0. The excursion is defined the maximum
radial displacement and is importantly measured in this study.

The radial velocity of a voxel is **v**_{1} = d**u**_{1}/(t_{2}
- t_{1}) and its external force is

*F*_{1} = *r*_{m}D_{V}(**v**_{2}-**
v**_{1})/(t_{2} - t_{1}), where *r*_{m} is the
density of cardiac muscle and D_{V} is the
volume size of the voxel at M_{1}. To analyze the wall
stress, Laplace's law can be applicable to differential areas
of the cardiac 3D shell under a pressure load. The general representation of wall
stress is *V* = *PR*/(2*h*), where *P* is
the pressure applied on the voxel, *R* is the curvature radius at M_{1}
which can be approximated as the distance from the origin of the bisectors to
their intersection on the short-axis view of the ventricle, and *h* is the
thickness of the shell which shall also be measured along the radial direction.
The wall stress is thus inversely proportional to the wall thickness *h*
in a differential segment. The final determinant of wall stress with the use of
a Laplace's law formulation is cavity pressure. The radial component of the wall
stress is *V*_{r} = *V** *sin(D_{q}/2)
where D_{q} is the sphere
angle of the voxel.

The myocardial strain can be
calculated from the dense displacement field using a Green-Lagrange Strain
Tensor [5]. For the 4D image, we first compute the deformation gradient tensor:
**D** = [¶*g*_{i}/¶x, ¶*g*_{i}/¶y, ¶*g*_{i}/¶z] + **I**,
where **I** is an identity matrix and *g*_{i} is the dense
displacement field computed from the image *f* along the three directions.
Finally the strain tensor is **S** = (**D**^{T}**D** - **I**)/2. So far
we can compute the displacement, velocity, force, stress and strain in a 4D
cardiac image.

Previously people attempted to use one or two these parameters directly to judge some cardiac motion problems. It is, however, very difficult to tell the quantities truly related to abnormality or not since all these parameters vary very much from different areas of heart. It is important to use statistical strategies for evaluation of the cardiac functions. This research marks near one thousand featured points on the cardiac shape among which a large part are set on the left ventricle (Fig. 1a). Each of them is statistically modeled. Information learned from the training set during CAOM construction includes its relative position on the heart, its mean values and derivations of excursion, velocity, force, stress, strain, etc. Normally these parameters should be in Gaussian distribution.

The evaluation is carried out to compare some specific parameters with those in the learned database. For example, with the 4D cardiac image of a new case, a few important points on the cardiac shape are selected to compute its properties of dynamics. If the value is relatively far outside the mean, it is known of low active of the myocardium around its area. Especially, the value below two standard deviations is recognized ¡°very inactive¡± and the value below three standard deviations is recognized ¡°abnormally inactive¡±.

** Results**: This
research investigated a series of 4D images for cardiac evaluation. Hundreds of
points are assigned on the cardiac surface to observe the myocardial deformation
mechanics. It is informative to identify the cardiac deformation quantitatively
in vision. Experiments demonstrated the applicability of above technique by illustrating
the dynamical parameters described above.
Figure 1b illustrates some marked points on the cardiac surface for dynamic
tracking. Figure 2 quantitatively visualizes some important dynamic parameters
for a specific point or the distribution on a ventricle. The diagnosis of ventricular
defects can be made on the standard deviations from the statistical
distribution of the training set. The defect area and location can be
illustrated visually for further clinical decision and medical treatment.

**Table I** Example results of
automatic highlighting for abnormal mechanics

Tracked point |
velocity |
force |
strain |
displacement |

Inf1 |
- |
O |
+ |
- |

Lateral1 |
O |
- |
O |
O |

Ant1 |
-- |
- |
- |
- |

Ant2 |
- |
- |
O |
- |

Septum1 |
O |
O |
O |
O |

O tissue mechanics in normal range (within 67% of the statistical population)

- inactive and below one-sigma, -- very inactive and below two-sigma, --- abnormally inactive and below 3-sigma, etc.

+ active and beyond one-sigma, ++ very active and beyond two-sigma, +++ abnormally active and beyond 3-sigma, etc.

In our practical study, a table of all tracked points is constructed to analyze their dynamics and warn on some abnormal points with corresponding spatial positions. Table I displays the evaluation result of a few points around the left ventricle, where we can know the regional maximum velocity, force, strain, and excursion of myocardium through the cardiac cycle. The ¡°not so active¡± areas are labelled with ¡°-¡±, ¡°--¡±, or ¡°---¡± according to the extent of their deviations from normal values.

*Conclusions**: *This letter concerns to analyze
important dynamic parameters based on a statistical model. In the
special physics-based deformable model used in the study,
regional structure and statistical dynamics are available for cardiac shape
fitting and ventricular defect evaluation from the 4D image. Visualized
regional mechanics and labeled defective
status will help us to estimate the working conditions
of the heart and find possible cardiovascular
diseases even in the early stage.

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**Authors¡¯ affiliations:**

S. Y. Chen (Dept of informatics, University of Hamburg; College of Information Eng., Zhejiang University of Technology)

E-mail: sy@ieee.org

**Figure captions:**

Fig. 1 (a) A number of points distributed on the cardiac shape for statistical analysis of dynamical functions. Each point is model with its relative position and statistical values of all dynamical parameters. (b) A few regional tracked points for evaluation of cardiac deformation mechanics. The displacement, velocity, acceleration, force, and stress are derived from 4D image sequence with the tissue tracking method.

Fig. 2 Analysis of correlated radial velocity and myocardial strain in cardiac periods.

Figure 1

(a) (b)

Figure 2