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EARLY DETECTION OF REJECTION IN CARDIAC MRI: A SPECTRAL GRAPH

APPROACH

Hsun-Hsien Chang and Jos´ e M. F. Moura

Electrical and Computer Engineering

Carnegie Mellon University

{hsunhsc, moura}@ece.cmu.edu

Yijen L. Wu and Chien Ho

Pittsburgh NMR Center for Biomedical Research

Carnegie Mellon University

ABSTRACT

This paper develops an algorithm to detect abnormali-

ties of small animals’ transplanted hearts in MRI, at early

stage of rejection when the hearts do not display promi-

nent abnormal features. Existing detection methods require

experts to manually identify these abnormal regions. This

task is time consuming, and the detection criteria are oper-

ator dependent. We present a semi-automatic approach that

needs experts to label only a small portion of the motion

maps. Our algorithm begins with representing the left ven-

tricular motions by a weighted graph that approximates the

manifold where these motions lie. We compute the eigen-

decomposition of the Laplacian of the graph and use these

as basis functions to represent the classifier. The experi-

mental results with synthetic data and real cardiac MRI data

demonstrate the application of our classifier to early detec-

tion of heart rejection.

1. INTRODUCTION

Thecurrentgoldstandardfordiagnosingrejectionafterheart

transplantation is biopsy, which is invasive and prone to

samplingerrors. Cardiacmagneticresonanceimaging(MRI)

is a non-invasive alternative to monitor rejection of an in

vivo heart. Using MRI, we can observe that, in early stages,

heartrejectionstartsfromasmallregion,whichthenspreads,

in late stages, to the entire myocardium [1]. Early diag-

nosis and treatment of rejection increases the survival rate.

However, early stage rejection does not present prominent

abnormal motions. To detect subtle abnormalities, cardiol-

ogists have to carefullyand manuallylabel the images. This

is labor-intensive and the results vary from expert to expert.

To achieve consistent classification, we need a quantitative

algorithm that reduces human involvement.

We present in this paper a semi-supervised classifica-

tion algorithm. The classifier is initially trained with a small

This work was supported by NIH grants, R01EB/AI-00318 and

P41EB001977, to the Pittsburgh NMR Center for Biomedical Research,

Carnegie Mellon University.

number of normal and abnormal motions labeled by an ex-

pert, and then it classifies the remaining unlabeled motions.

The classifier is developed using the framework of spectral

graph theory [2, 3, 4]. We first generate motion maps of

the heart through the cardiac cycle. The motion vectors are

treated as samples of a Riemannian manifold embedded in

the original data space. A graph representation of the data

is an approximationto the manifold. We represent the heart

motionmap as a graphwheresimilar motionsare connected

by an edge. Spectral analysis of the graph Laplacian pro-

vides a basis of functions on the graph. We use this basis to

find the optimal classifier.

This paper is organized as follows. Section 2 develops

in detail our classification algorithm. Section 3 tests the

algorithm with synthetic data and real cardiac MRI data,

demonstrating the good performance of our classifier. Fi-

nally, Section 4 concludes this paper.

2. METHODOLOGY

We process a sequence of cardiac MR images sampling the

cardiac cycle and determine for each phase a dense motion

map; i.e., at each phase n

in the cardiac cycle, every my-

ocardial pixel iis assigned a motion vector ui n . We focus

in this paper on a single MRI heart slice and in a particu-

lar phase of the cardiac cycle, so we consider ui∈ R2and

drop the time dependence n . The classification problem is

to design a classifier h such that the class label ciof pixel i

is

?

Or equivalently,

?

where τcis a threshold.

The2Dmotionvectorsofthemyocardiumaredatapoints

in R2. We can think of these points assembling a Rieman-

nian manifoldM ⊆ R2. In our case, the manifoldM could

cic h ui

1if uiis normal−1if uiis abnormal.

(1)

ci1if h ui≥ τc−1if h ui< τc(2)

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be a plane, a contour, or a point. Associated with M is

the Laplace-Beltrami operator ∆ that acts on differentiable

functions on M, see [2]. When M is compact, ∆ has a dis-

crete spectrum and its eigenfunctions {er}, also called har-

monic functions,providean orthogonalbasis for the Hilbert

space of square integrable functions on M. Since the clas-

sifier h is square integrable on M, it can be represented in

terms of the harmonic functions; namely,

∀x ∈ M

h x

∞

?

r=1

arer x

.

(3)

Hence,ourtaskbecomestheproblemoffindingafunctionh

on M such that c h x

→ {−11}, where 1 indicates nor-

mal and −1 abnormal.

We have to approximate the manifold M because the

numberof available data points is finite. Belkin and Niyogi,

[3, 4], suggest using a graph model to represent the data;

that is, the graph approximates the manifold M where the

data lies. The Laplace-Beltrami operator is accordingly the

discrete Laplacian, see [2]. In Section 2.1, we describe the

graph model for our data. Similar to (3), the spectral analy-

sis of the graph Laplacian L provides the basis function for

the classifier, see [4]. Once we have a representation for the

classifier, we use the labeled data to find the coefficients ar,

and develop a rule for classifying abnormalities in the unla-

beled data. Section 2.2 details the classifier design.

2.1. Graph Representation

Assume that a slice of the myocardiumis imaged by N pix-

els. From two consecutive phases, we estimate a motion

map U that collects all motion vectors ui:

U

{u1u2···

uq···

uN} .

(4)

We assume that the first q motion vectors in U have been

labeled by human experts. In the framework of spectral

graph theory, we represent the set U of motion vectors by

a graph G . In G , vertices vivjcorresponding to motions

uiujare connected by an edge if the distance κijbetween

uiand uj is smaller than a threshold τκ. For simplicity,

here, we adopt the distance measure κijbetween two vec-

tors uiand ujto be the Euclidean distance, see Fig. 1,

κij?ui− uj? .

(5)

With reference to Fig. 1, κijis the distance between uiand

uj, and τκdefines the neighborhood.

Agraphcanberepresentedbyanadjacencymatrix,whose

entry i jis one if the vertices vivjare connected, and it

is zero otherwise. To take account for the differentvalues of

κij, we use a weighted graph and a weight matrix W. The

entries of the matrix W in this paper are taken to be

?

W

ije−κijif κij≤ τκ

0if κij> τκ .

(6)

6 @ @

?

?

?

?? ?

&%

ui

uj

'$

κij

τκ

Fig. 1. Distance measure κijbetween vectors uiand uj.

TheLaplacianL forthe graphG is definedas, see [2, 4],

L

D − W

(7)

where D is a diagonal matrix with the i th diagonal entry

?

tices of the graph. Then, we solve the eigenfunction prob-

lem:

Dii

jW

ji. The Laplacian is a symmetric, positive

semidefiniteoperatoractingonfunctionsdefinedonthever-

Le

λe

(8)

whereeisaneigenfunctionandλisthecorrespondingeigen-

value. We indextheeigenfunctionsaccordingtotheireigen-

values in ascending order, 0 ≤ λ1≤ λ2≤ ··· ≤ λN. The

eigenfunctions with small eigenvalues are low frequency

harmonics on the graph G .

In order to approximate functions on the graph G , we

pick the first peigenfunctionsand form them into the eigen-

matrix E:

E

e1e2···

ep.

(9)

Hence a function h on G can be written as

h

p

?

r=1

arerEa .

(10)

The meaning of Eq. (10) is clear. The i th entry h iof h is

a function mapping the motion vector uito a real number;

ui→ R. namely, h i2.2. Classifier

Our task of designing a classifier reduces to finding a func-

tion h in (10) such that the class cito which the motion ui

belongs is determined by the following rule:

?

ci1if h i≥ τc−1if h i< τc.

(11)

We use the first q labeled motions in U to find the opti-

mal a in (10). Let c be the q-dimensional vector denoting

the classes of these data points; i.e., the i th entry of c is

?

uiis labeled abnormal.

∀i≤ qci1uiis labeled normal

−1(12)

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Since abnormal motions are not easily distinguishable, we

introduce a parameter αito indicate the confidence level of

the experts who classify the motion ui. We let the values

of the confidence level range from 0 to 1. The higher the

value, the more confident the experts. We modify the la-

beled class c to be ¯c with

¯ciciαi.

(13)

Vector ¯c trains the classifier by finding out the coeffi-

cients a in Eq. (10). The optimal a minimizes the quadratic

error

ε?¯c − Elaba?2(14)

whereElabis the eigenmatrixcorrespondingto the qlabeled

data points obtained from the first qrows of E. Minimizing

εgives aopt:

aopt ET

labElab−1ET

lab¯c.

(15)

Theremainingunlabeledmotiondatais classified according

to the rule in Eq. (11) for all i> q.

3. EXPERIMENTS

Inthis section,we presentexperimentalresults. We usesyn-

thetic data to demonstrate the feasibility of the algorithm

and study its performance. We then apply our algorithm to

real MRI sequence of a transplanted rat heart. To run the

classifier, we have to adjust two parameters: the number p

of eigenfunctions in Eq. (9) and the decision threshold τcin

Eq. (11).

SyntheticData: We simulate the heart as a circular hol-

lowdiskandsynthesizea mapof612motionvectors,shown

in Fig. 2(a). Each pixel has a unit motion vector in the cir-

cumferential direction. We generate three regions of abnor-

malities marked by rectangles in Fig. 2(a). In rectangle 1,

we halve the length of the vectors. In rectangle 2, we per-

turb the angles of the vectors by adding to them Gaussian

noise with zero mean and unit variance. In rectangle 3, we

increase the length of the vectors by the factor 1. 2. An ex-

pert labels a few pixels in the normal and the two abnormal

regions, inside the ellipses in Fig. 2(a). The labeled regions

A, B, C with confidence levels α 10. 50. 75, respec-

tively, see Eq. (13), are used to train the classifier. We use

the first 42 eigenfunctions to build the classifier and set the

−0. 4. Fig. 2(b) shows the classi-

fication results, where the classified abnormal motions are

marked by dots. The motion vectors in the three rectan-

gle boxes are correctly classified. Abnormal regions 1 and

2 that were partially identified by the expert are correctly

classified. The detector can also recognize part of the third

abnormal region.

We study the performanceby plotting the receiver oper-

ating characteristic (ROC). The ROC curve is a plot of the

decision threshold τc

probabilityofhitPHversustheprobabilityoffalsealarmPF,

see [5]. In this paper, we define PHand PFas:

PH

number of correctly classified abnormal pixels

total number of abnormal pixels

number of misclassified abnormal pixels

total number of normal pixels

PF

.

(16)

For a given τc, we get a pair of PHand PF. We ob-

tain the ROC curve in Fig. 2(c) by changing sequentially

the threshold from −1 to 1 in increments of 0. 02; i.e., τc∈

1}. This plot shows that the best operat-

{−1−0. 98···

ing point of the curve is at PH0. 75 and PF0. 07.

Cardiac MRI Data: Transplanted rats were studied by

using heterotopicworkinghearts, adoptingDA to BN trans-

plant pairs. The transplanted hearts receive proper pressure

pre-load and exhibit similar cardiac outputs and ventricular

pressure, close to those in native hearts. We use ECG and

respiration gated cine MRI to obtain images with resolution

of 156µ m. We adopt a modified DANTE sequence for MR

tagging. We cover the heart with 8 transversal slices at 10

time phases through the cardiac cycle. All MRI scans were

performed on a Bruker AVANCE DRX 4.7-T system. Each

image has 256 × 256 pixels. We use MATLABR ?to imple-

ment all the algorithms.

Figure 3(a) shows a rejected rat heart imaged on the

third day after transplantation. In this image, there are 2170

pixels in the myocardium. During the preprocessing step,

we apply segmentation, tag line detection, and motion es-

timation using the algorithms developed in [6]. Fig. 3(b)

shows the heart’s dense motion map. In this figure, the

pixels in the ellipses are labeled with the confidence level

αel0. 8 and the pixels in the square with αsq1. 0. Two

ellipses and one square indicate the labeled abnormal and

normal regions, respectively, with about 80 pixels in total.

We choose 112 out of possible 2170 eigenfunctionsto build

the classifier. The detector then automatically classifies the

remaining unlabeled regions. Fig. 4 shows the detected ab-

normal motions marked by dots. We repeat the experiment

by changing the confidence levels αelαsq ∈ 0. 11 . The

classification results are not sensitive to αel ∈ 0. 51

or

αsq∈ 0. 71 .

To evaluate the performance of the algorithm, we use

contrast MRI to providegroundtruth. We inject the rat with

a contrast agent that binds with abnormal myocardial cells

and displays dark intensities under standard cine MR imag-

ing. Injecting a contrast agent is invasive and not desirable

from a clinical point of view because it may be harmful or

induce allergies.

We segment the dark abnormal regions of contrast MR

images. These are the contours shown in Fig. 4. These

contours provide ground truth. The ground truth illustrates

that the heart has begun experiencing rejection in the epi-

cardium. Although there are small regions of false alarms

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1

3

A

C

B

(a) Motion map.

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(b) Dots represent the detected abnormalities.

0 0.2 0.4 0.60.81

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0.3

0.4

0.5

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probability of false alarm

probability of hit

(c) ROC curve of the classifier.

Fig. 2. Experimental results using synthetic data.

(a) Cardiac MR image. (b) Motion map

Fig. 3. A rat heart imaged on post-transplantation day 3.

marked by triangles in Fig. 4, the epicardial regions show

good agreement with our detected abnormal regions. This

verification shows that our approach is promising to detect

early rejection of heart transplants.

4. CONCLUSIONS

In this paper, we develop a semi-supervised algorithm to

classify early rejection of heart transplants. We use a graph

basedmodelto representthe motiondataanduse this model

to design our algorithm. Classification is achieved by com-

puting the eigenfunctions of the graph Laplacian, which we

use to express the classifier. Human-labeled data trains the

classifier. Experimental results with real cardiac MRI im-

ages demonstrate that our approach will be a helpful tool

for early rejection diagnosis.

5. REFERENCES

[1] Y. L. Wu, K. Sato, J. B. Williams, K. Hitchens, H.-

H. Chang, J. M. F. Moura, and C. Ho,

cardiac allograft rejection is heterogeneous and can be

non-invasivelymonitored in vivo with MRI by the two-

pronged method,” in Proceedings of Annual Scientific

Sessions of the Society for Cardiovascular Magnetic

Resonance, San Francisco, CA, January 2005.

“The acute

Fig. 4. Experimental results using real cardiac MRI data.

[2] F. R. K. Chung,

of CBMS Regional Conference Series in Mathematics,

American Mathematical Society, 1997.

Spectral Graph Theory, vol. 92

[3] M. Belkin and P. Niyogi, “Laplacian eigenmaps for di-

mensionality reduction and data representation,” Neu-

ral Computation, vol. 15, no. 6, pp. 1373–1396,2003.

[4] M. Belkin and P. Niyogi, “Semi-supervised learning

on Riemannian manifolds,” Machine Learning, vol. 56,

pp. 209–239, 2004.

[5] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Clas-

sification, John Wiley & Sons, New York, NY, second

edition, 2001.

[6] H.-H. Chang, J. M. F. Moura, Y. L. Wu, K. Sato, and

C. Ho, “Reconstruction of 3-D dense cardiac motion

from tagged MR sequences,” in Proceedings of IEEE

International Symposium on Biomedical Imaging, Ar-

lington, VA, April 2004, pp. 880–883.

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