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RECONSTRUCTION OF 3D DENSE CARDIAC MOTION

FROM TAGGED MR SEQUENCES

Hsun-Hsien Chang1 José M.F. Moura1 Yijen L. Wu2 Kazuya Sato2 Chien Ho2

1Department of Electrical and Computer Engineering

2Pittsburgh NMR Center for Biomedical Research

Carnegie Mellon University,

Pittsburgh, Pennsylvania, USA

ABSTRACT

This paper develops an energy minimization algorithm to

reconstruct the 3D motion of transplanted hearts of small

animals (rats) from tagged magnetic resonance (MR) se-

quences. We describe the heart by a layered aggregate of

thin oriented elastic fibers. We use the orientation of myo-

cardial fibers to develop a local dense motion of the heart.

This dense model is fit to the tagged MRI data by mini-

mizing an energy functional with two terms: the first term

is the external energy, derived from matching the image

intensities on the fibers across two consecutive frames; the

second term is the fibers’ internal energy, derived from

biomechanics analysis. This paper illustrates the applica-

tion of the approach to a set of cardiac MR sequences con-

taining four slices of a transplanted rat heart.

1. INTRODUCTION

Studying the left ventricle (LV) is most important in car-

diac function analysis. Cardiologists have widely used the

left ventricle’s global parameters, such as stroke volume

and electrocardiogram, to identify the severity of patients’

cardiac diseases. However, these global parameters are not

good indicators of unhealthy hearts at early stages of dis-

ease. Cardiologists are increasingly interested in identify-

ing the local cardiac malfunction, for example, the local

infarction and ischemia, so that they can start the treatment

of sick hearts as early as possible.

Cardiac magnetic resonance tagging [1] provides an

opportunity for cardiologists to understand the local car-

diac malfunction. The tagged MR images display the myo-

cardium with lines superimposed, as shown in Fig. 1.

When the heart moves, the tag lines deform consistently

with the heart. Sun et al. [2] have developed a method that

propagates the motions of tag lines to all pixels of the

heart and then provides a dense representation for the mo-

tions in each 2D slice of the heart, see Fig. 2.

At common MRI resolutions, heart slices are 10 to 15

pixels apart. To reconstruct dense 3D motions, we propose

here to use a cardiac fiber based model [3, 4]. The heart is

represented by layers of thin elastic fibers that stretch and

compress through the cardiac cycle. The fibers are de-

scribed as a concatenation of many small segments. We

use this model in an energy minimization approach to re-

construct the 3D motion of the left ventricle.

In this paper, Section 2 describes the myocardial fiber

structure. Section 3 introduces a motion model for the fi-

bers. Section 4 develops the energy functional to be mini-

mized. Section 5 presents our experimental results in re-

constructing the 3D motion of the left ventricle of a rat

heart. Finally, Section 6 concludes the paper.

2. MYOCARDIAL STRUCTURE

The cardiac muscle cells of the myocardium are arranged

in layers that are tightly bounded together and completely

encircle the blood-filled chambers, as shown in Fig. 3(a).

Streeter [3, 4] quantitatively analyzed the myocardial fiber

orientations. He excised a block of a canine myocardium,

and cut the parallel cross sections from endocardial to

epicardial surfaces through this block. His studies con-

cluded that the fibers are almost tangential to the endocar-

Fig. 1. A sequence of tagged cardiac MR images in a transversal

plane of the left ventricle over a cardiac cycle.

(a) Two frames at

end-systole

Fig. 2. 2D dense displacements of the left ventricle.

(b) The sparse and dense

displacements of (a)

_______________________________________________________

This work was supported by the National Institute of Health under

Grants R01EB/AI-00318 and P4EB001977.

In International Symposium on BioImaging, Crystal City, VA, April 2004.

Page 2

dial and epicardial surfaces. Furthermore, the fiber orien-

tations vary from –60º at epicardium to +60º at endocar-

dium. This pattern is shown in Fig. 3(b).

Physiologists usually view the myocardial fibers as

small pieces of cylinders and take into account fiber orien-

tations in cardiac structural analysis [5]. This viewpoint is

usually ignored in image analysis. In this paper, we use the

orientations and cylindrical shapes of the myocardial fi-

bers to reconstruct the left ventricular motion from tagged

MR sequences.

We model the left ventricle as a thick, flexible cylin-

der with the height of h and with the outer radius of ρ,

when the heart fully extends at the end-diastole, as shown

in Fig. 4. This cylinder is composed of many fibers. Each

fiber is one pixel thick and extends from the bottom slice

to the top slice at an orientation η. From Streeter’s studies,

η = –60º at epicardium, so that the length of a fiber is ap-

proximately 60 pixels, assuming four slices and 10-pixel

slice thickness.

We now discuss the generation of the fiber. The loca-

tion of a fiber on the epicardial surface can be described as

a helix, see Fig. 4:

cos

x

rq

=⋅

, sin

y

r

=⋅

where θ is the rotation angle in the x-y plane, i.e., the slice

plane, and β is the coefficient controlling how fast the z

component of the helix climbs. From prior knowledge, the

fiber follows the known orientation η; this can be used to

determine the parameter β. With reference to Fig. 4, for a

small rotation on the slice plane Δθ,

tan

h

=

It follows that β = ρ⋅tan η. We assume η known and obtain

ρ from MRI data; so, β can be estimated by (2).

3. CONTINUUM MECHANICS MODEL

We now develop a motion model for our fiber based heart.

Since the left ventricle suffers small deformations through

the cardiac cycle, an infinitesimal deformation model is an

appropriate description for the movement. Based on this

assumption, we divide a long fiber into a series of small

segments each 1 pixel long.

q

, and z

b q

⋅ , (1)

=

b

r

q

q

⋅∆

⋅∆

. (2)

3.1. The Jacobian matrix

The left ventricle in 3D space is denoted by B(t). Initially

it is B(0) and deforms into B(t) at time t, as shown in Fig.

5. A pixel a(0) within B(0) moves to a(t) within B(t). The

location of a(t) is related to its initial condition by

( )( )

tt

=+

aua

where u(t) is the displacement. Let (0)

neighboring point of a(0) that moves to ( )

cording to the equation, [6],

( )

( )

(0)

∂

a

The Jacobian matrix is defined by

∂

∂

∂

∂

∂

∂

which is also known as the deformation gradient matrix.

At t = 0, (0)

F = I . By plugging (3) into (5) for each entry,

the Jacobian has components, [6],

( )( )

(

( )

ij

jj

aa

∂∂

(0), (3)

+

a

t

a

(0)

d t

a

d

a

be a

( ) ac-

+

(0)

t

d t

a

d

∂

=

a

a

. (4)

111

123

222

123

333

123

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

t

( )

t

(0)

a t

a

a

a

a t

a

a t

a

a

a

a t

a

∂

a t

a

a

a

a t

a

∂

ttt

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

==

a

F

a

, (5)

(0) ( )

t

(0)(0)(0)

)

iiii

ij

j

tt

auau

a

F t

d

∂∂+∂

∂

===+

, (6)

y

x

z

ρ

θ

[ρ× cos(θ), ρ× sin(θ), β× θ]

η

β× Δθ

ρ× Δθ

h

Fig. 4. A thick cylinder models the shape of the left ventricle,

and a helix describes the location of a myocardial fiber

on the surface of the left ventricle.

y

x

z

u(t)

a(0)

a(t)

da(0)

da(t)

B(0)

B(t)

Fig. 5. The deformation of a left ventricle.

endocardium

mid-wall

epicardium

(a) (b)

Fig. 3. The structure and orientation of the fibers of the heart.

Page 3

where δij is the Kronecker symbol that is 1 when i

is 0 otherwise. From (6), the Jacobian can be rewritten as

∂

∂

∂

∂

∂

∂

j

=

and

111

123

222

123

333

1

23

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

(0)

( )

t

( )

t

u t

a

u

a

u t

a

u t

a

u

a

u t

a

∂

u t

a

u

a

u t

a

∂

ttt

d

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

= +

I

= +

IFF

. (7)

3.2. The strain energy

When a flexible rod is bent, the work applied is stored as

strain energy. By measuring the displacements of the

points of the rod, we can calculate the strain energy accu-

mulated by the rod. For finite deformations, the strain ten-

sor E is defined as

T

1

2(

E F FI

Using infinitesimal analysis, in continuum mechanics,

the strain tensor is usually approximated by the small

strain tensor

T

1

2()

SFF

Approximation (9) follows by replacing (7) into (8) and

ignoring the second order terms. Since S is symmetric, we

vectorize the six entries located on the upper triangle of S

as the vector s:

[

11 223312

,,,,

SSSSS

=

s

The segment da(0) in Fig. 5 is assumed small. When the

small segment deforms to da(t), the strain energy e it ac-

cumulates is computed by a linear elasticity model [6]

e = s Cs, (11)

where C is a 6×6 matrix describing the anisotropic proper-

ties of the myocardial fibers and takes the form

−

where E‖ is the along-fiber stiffness, E⊥ the cross-fiber

stiffness, and n‖ and n⊥ are the corresponding Poisson’s

ratios. Replacing (7) into (9), it follows that the small

strain tensor S is a function of the displacement u. Hence,

the strain energy e is also a function of u.

In our fiber model, we divide an elastic long fiber into

many small segments, as shown in Fig. 6. The fiber’s

strain energy ε is given by the sum of the strain energy of

each segment:

e = ∑ s Cs . (13)

)

−

. (8)

≈+−

I . (9)

]

13 23

,

T

S

. (10)

T

1

1

⊥

000

1

⊥

000

1

E

000

2(1)

00000

2(1)

00000

2(1)

00000

E

n

−⊥

⊥

−

EE

n

−

E

n

E

n

E

E

⊥

E

⊥

EE

E

E

E

n

n

n

⊥

n

n

−

−⊥

⊥

−

+⊥

+

+

=

C

, (12)

T

all segments

It follows that the strain energy ε is a function of the dis-

placements of all segments. With reference to Fig. 6, we

vectorize the displacements un of N segments of the fiber

into the vector

12

,,

energy ε is now a function of U.

4. ENERGY FUNCTIONAL

We now define the energy functional E(U) that will be

minimized to derive the dense displacement field U. We

first introduce the notation used to describe the deforma-

tion of a fiber, see Fig. 6. A fiber has N segments. The

endpoints of a segment are referred to as nodes. The node,

an(0) for n≤N, its displacement, un(t) = an(t) an(0), and

its small segment, an+1(0) an(0), play the same roles as

a(0), u(t) and da(0) in Section 3, respectively. We define

the energy functional

( )( )

gg

=+

UU

EE

as the sum of two terms, where γi are the weightings. The

term E1(U) is the external energy that accounts for the

mismatch of the fiber image intensities across two con-

secutive frames. The term E2(U) is the internal energy that

corresponds to the strain energy described in Section 3.2.

External Energy: The intensity of a pixel (i,j,k) in a

fiber at time t is labeled by Iijk(t). We vectorize all pixel

intensities of the fiber at time t by the vector I(t). The ex-

ternal energy is then given by

−+

UII

E

Internal Energy: Because a long fiber is represented

by a series of tiny segments, we can apply the linear elas-

ticity model, (11), to each segment. From (13), the fiber’s

internal energy E2(U) is

∑∑

T

TTT

N

,

=

uuu

U

; hence, the strain

1122

( )

U

E

, (14)

2

1( ) = ( )

t

( 1)

t

. (15)

2

11

( )

U

NN

nnn

nn

e

==

==

T

s Cs

E

. (16)

The two terms of the energy functional attempt to

strike a balance between matching the intensities across

frames and the mechanics of the fiber. If E1(U) were not

a1(0)

a2(0)

aN(0)

a1(t)

a2(t)

.

.

.

.

u1(t)

aN(t)

aN+1(t)

uN(t)

.

.

.

.

.

.

.

.

.

.

aN+1(0)

u2(t)

Fig. 6. The deformation of a myocardial fiber.

Page 4

present, minimization of E2(U) alone would penalize any

displacement and enforce the fiber to move back to its

initial location.

Displacement Constraints: Our 3D image domain

consists of four transversal MR slices. We fill the space

between a pair of MR slices by virtual slices obtained by

interpolation. On the other hand, on the four MR slices,

the movements of the tag lines can be accurately measured

[2]. We treat the motions on the tag lines as a constraint in

the energy functional. These constraints are considered via

Lagrange multiplier λα. The energy functional becomes the

Hamiltonian

( ,)( )( )

a

lgg

=+

UU

EEE

[

0,,0,1,1,1,0,

Θ

entries of the location of the node α on the tag line.

5. EXPERIMENTAL RESULTS

Transplanted rats were studied by using heterotopic work-

ing hearts. 4 transversal slices are taken to cover the heart

at 10 time frames per cardiac cycle. All MRI scans were

performed on a Bruker AVANCE DRX 4.7-T system.

Each image has 256×256 pixels. The segmentation, tag

line detection, and 2D displacement estimation are carried

out by using the algorithms developed in [2].

A left ventricle structure with 3 layers, as shown in

Fig. 7(a), is generated according to the method described

in Section 2. To initialize the minimization of (17), we

assume that the myocardial fibers are oriented according

to the prediction in [3, 4]. Since we only use 3 layers, the

orientations of the fibers are taken to be from –45º at

epicardium to +45º at endocardium. These are shown in

Figs. 7(b) … 7(d).

We apply the method in Section 4 to reconstruct the

3D left ventricle. Figs. 8(a) and 8(b) show the systolic and

diastolic 3D motions of the epicardial left ventricle, re-

spectively. The motions depicted in the figures are quite

realistic. We see that the basal left ventricle (on the top)

has more pronounced motions at the end-systole than at

the end-diastole as it should be. The figures also illustrate

well how the myocardial fibers deform through the cardiac

cycle. The results show not only the fiber displacements,

but also the myocardial rotation, stretch, and compression

that are very useful in clinic studies.

6. CONCLUSIONS

In this paper, we develop a myocardial fiber based model

to describe the left ventricle and use this model to recon-

struct accurately the 3D motion of the left ventricle. The

1122

T

()

aaa

l

+−

UU Θ

u

, (17)

where

]

T

,0

a=

has 1’s only at the

reconstruction is achieved by minimizing an energy func-

tional that combines the fibers’ external energy, obtained

from the images, with an internal energy derived from

biomechanics. Experimental results with real tagged MR

sequences demonstrate that the fiber model based recon-

struction is appropriate to study the myocardial rotation,

stretch, and compression.

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(a) systole (from left to right)

(b) diastole (from left to right)

Fig. 8. The epicardial deformation of the left ventricle.

(a) The cylindri-

cal layers of LV

(b) The epi-

cardial layer

(c) The mid-

wall layer

(d) The endo-

cardial layer

Fig. 7. The initialization of the left ventricle.