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Immune Cells Detection of the In Vivo Rejecting Heart in USPIO-Enhanced MRI

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Immune Cells Detection of the In Vivo Rejecting Heart in USPIO-Enhanced MRI

Hsun-Hsien Chang1, José M. F. Moura1,

Yijen L. Wu2, and Chien Ho2

1Department of Electrical and Computer Engineering

2Pittsburgh NMR Center for Biomedical Research

Carnegie Mellon University, Pittsburgh, PA, USA

Work supported by NIH grants (R01EB/AI-00318 and P4EB001977)

- The extreme treatment of the heart failure is transplantation.

- Gold standard diagnosis method (i.e., biopsy) of heart rejection
- is invasive.
- is prone to sampling errors.

- Alternative diagnosis method: contrast-enhanced cardiac MRI
- is non-invasive.
- monitors the wholein vivo heart.

RV

LV

rejecting tissue

- immune cells (e.g. macrophages).

- contrast agents (USPIO, ultra-small super-paramagnetic iron oxide) label the immune cells

- High relaxivity causes low image intensities under T2* weighted MRI.

Identify immune cells (i.e., dark pixels):

- Large number of myocardial pixels
- Manual classification is labor-intensive and time consuming.

- Dispersion of immune cells
- Immune cells accumulate in multiple regions without known patterns.

- Heart motion blurs images
- It is hard to distinguish the boundaries between the USPIO-labeled and unlabeled pixels

Post Operation Day (POD) 3.

- Need an automatic algorithm to classify pixels as USPIO-labeled or unlabeled.

POD 5.

- Main idea:
- Partition the image into USPIO-labeled and unlabeled parts.

- Graph theory approach:
- Describe the image as a graph.
- Find the optimal edge cut.

- Introduction
- Methodology: Graph Partitioning
- Graph Representation of the USPIO Image
- Optimal Edge Cut and the Cheeger Constant
- Optimal Classifier via Optimization

- Results and Conclusions

Graph Representation of the USPIO Image

Classification through an Edge Cut

Optimal Cut from the Cheeger Constant

Optimal Classifier via Energy Minimization

Red dots are the automatically selected USPIO-labeled pixels.

(a) 0.61

(b) 0.89

(c) 0.76

(d) 0.61

Graph Representation of the USPIO Image

Classification through an Edge Cut

(e) 0.46

(f) 1.00

(g) 0.62

(h) 0.51

Optimal Cut from the Cheeger Constant

(i) 0.23

(j) 0.79

(k) 0.38

(l) 0.43

Optimal Classifier via Energy Minimization

(m) 0.00

(n) 0.17

(o) 0.09

(p) 0.28

- Graph: G(V, E).

- a set V of vertices representing pixels.
- a set E of edges linking the vertices according to a prescribed way.

- Edge assignment strategies:
- Geographical neighborhood
- Feature similarities

- Edge cut:

- Partition:

Graph Representation of the USPIO Image

Classification through an Edge Cut

Optimal Cut from the Cheeger Constant

Optimal Classifier via Energy Minimization

- Classification of the pixels into USPIO-labeled or unlabeled is equivalent to partitioning the graph into two disjoint subgraphs.
- Graph partitioning:
- Divide the vertex set V into disjoint subsets S and S’.
- Remove a set of edges, denoted as Edge(S, S’), to make S and S’ disconnected.

8

(a)

(b)

(2+5+3)

X(S) =

Graph Representation of the USPIO Image

8

(a)

(b)

(2+10)c+(10+5+3)d

5

3

2

= 0.33

8

(a)

(b)

3

(c)

(d)

2

- Consider this example:

10

5

5

Classification through an Edge Cut

(c)

(d)

(8+5+10)

3

2

10

X(S) =

(8+5+2)a+(2+10)c

(c)

(d)

10

= 0.85

Optimal Cut from the Cheeger Constant

8

(a)

(b)

8

(a)

(b)

Optimal Classifier via Energy Minimization

5

5

3

2

2

3

(c)

(d)

(c)

(d)

10

10

(2+5+8+3+10)

(8+3+10+2)

X(S) =

X(S) =

(2+10)c+(8+3)b

(8+3)b+(2+10)c

= 1.21

= 1.00

- Cheeger constant:

- Assuming that Vol(S) < Vol(S’).
- |Edge(S, S’)| = sum of the edges in the cut.
- Vol(S) = sum of edges emanating from all the vertices in S.

+1

0

-1

(c)

(d)

Graph Representation of the USPIO Image

8

(a)

(b)

(a)

(b)

3

2

5

Classification through an Edge Cut

(c)

(d)

10

- Derive an objective functional from the Cheeger constant:

Optimal Cut from the Cheeger Constant

Optimal Classifier via Energy Minimization

- Optimal classifier:

- Classifier

- Classifier

- Introduction
- Methodology: Graph Partitioning
- Graph Representation of the USPIO Image
- Optimal Edge Cut and the Cheeger Constant
- Optimal Classifier via Optimization

- Results and Conclusions

RV

Post Operation Day (POD) 3

POD 4

LV

RV

LV

POD 5

POD 6

RV

LV

RV

LV

(Data were presented in Wu et al, PNAS 2006)

POD3

POD4

POD5

POD6

POD7

Fig1: USPIO-enhanced images

Fig2: manual classification (presented in Wu et al, PNAS 2006)

Fig3: automatic classification

Immune cell accumulation area

Immune cell accumulation percentage

- Develop a graph theoretical approach to classifying immune cells in the USPIO-enhanced images
- Represent an image by a graph.
- Consider the Cheeger constant for the optimal cut.
- Adopt the optimization to find the classifier.

Questions and Answers

1. Assign edges to the neighboring pixels.

(a) 0.61

(b) 0.89

(c) 0.76

(d) 0.61

(e) 0.46

(f) 1.00

(g) 0.62

(h) 0.51

2. Assign edges to similar pixels ( d < 0.1).

(i) 0.23

(j) 0.79

(k) 0.38

(l) 0.43

(m) 0.00

(n) 0.17

(o) 0.09

(p) 0.28

3. Repeat the procedure to all other pixels.