Multi-exponential decomposition for MR imaging of HCC and

fibrosis

Giovanni Motta

Jan 7, 2005

Sequences

• UTE• Fat saturation• 4 echoes• 20 sequences 256x256 (4) or 320x320 (16)

– TE = 0.08, 3.25, 6.42 and 9.59ms (2)– TE = 0.08, 4.53, 8.98 and 13.5ms (11) – TE = 0.08, 5.81, 11.6 and 17.4ms (4)– TE = 0.08, 6.90, 13.8 and 19.6ms (3)

• One slice from each sequence

ExampleUTE_0015: TE = 0.08, 5.81, 11.6 and 17.4ms, 320x320 pixels

Model

• Voxel value is proportional to the transverse magnetization of the corresponding volume

• Transverse magnetization decays exponentially with TE

• The time behavior of a voxel can be described by a linear combination of exponentials (plus a residual error)

0 1 1

, , , , , , ,, ,...,

( )k

TE

tx y z x y z t x y z

t t t t

M TE M e E TE

Model• Exponentials are the basis functions used in

the decomposition

Method• Given the four echoes

• We want to solve

• With respect to and

, 1 , 2 , 3 , 4( ), ( ), ( ), ( )x y x y x y x yM TE M TE M TE M TE

1 1

2 2

3 3

4 4

2 2, 1 , , , ,

2 2, 2 , , , ,

2 2, 3 , , , ,

2 2, 4 , , , ,

( )

( )

( )

( )

A B

A B

A B

A B

TE TE

T Tx y x y A x y B

TE TE

T Tx y x y A x y B

TE TE

T Tx y x y A x y B

TE TE

T Tx y x y A x y B

M TE M e M e

M TE M e M e

M TE M e M e

M TE M e M e

, ,x y AM , ,x y BM

Method• An exact solution is not always possible,

so we look for an approximation that minimizes the error

• Where 1 1

2 2

3 3

2 21 1 , 1 , , , ,

2 22 2 , 2 , , , ,

2 23 3 , 3 , , , ,

4 4 , 4 , ,

( ) ( )

( ) ( )

( ) ( )

( ) ( )

A B

A B

A B

TE TE

T Tx y x y A x y B

TE TE

T Tx y x y A x y B

TE TE

T Tx y x y A x y B

TE

x y x y A

E TE M TE M e M e

E TE M TE M e M e

E TE M TE M e M e

E TE M TE M e

4 4

2 2, ,

A B

TE

T Tx y BM e

2

1 4iTE

i

E E

ExampleUTE_0015: voxel of coordinates (208, 63)

Best fit with T2A=13 and T2B=6:

13 6, , , ,210 8 ( )

TE TE

x y z x y zM TE e e E TE

Advantages• Short Term

– Allows generation of synthetic images for arbitrary TE

– Exponentials and reconstruction error can be isolated and imaged individually

– Subtracting the reconstruction error from the image provides a form of denoising

• Long Term– The parameters of this representation can be used in

the classification of the voxels

13 6, , , ,210 8 ( )

TE TE

x y z x y zM TE e e E TE

Main Assumption

The parameters of this decomposition are an

advantageous way of representing all the information necessary for the

classification

Experiments with two exponentials

2BT B2AT, ,x y AM , ,x y BM

Set 1 Var Var Var Var 0 0

Set 2 Var Var 13 6 0 0

Set 3 Var Var 11 5 Var (±2)

Var (±2)

A

• In experiment sets 1 and 3 the system is non linear

• Unknowns can only assume non negative values

Experiments with four exponentials

• T2 =100, 20, 10 and 5ms.

• System of equations is linear

• Non negativity constraints on M

Supervised Classification

• Rudimentary nearest neighborhood classification

• Voxels are represented by the parameters of the decomposition

• Fixed a set of parameters (target), find the voxels that that have parameters closer than a predetermined amount (threshold)

• Distance is measured by the squared error between the two sets of parameters

Example

• Matlab program

• Experiment sets 2 and 3

2BT B2AT, ,x y AM , ,x y BM

Set 1 Var Var Var Var 0 0

Set 2 Var Var 13 6 0 0

Set 3 Var Var 11 5 Var (±2)

Var (±2)

A

Unupervised Classification

• Voxels are represented by the parameters of the decomposition

• Fixed the number of classes, partition the voxels into classes so that voxels belonging to the same class have similar parameters

• Distance is measured by the squared error

Unsupervised Classification

• Two exponentials

• T2=20 and 5ms.

• 8 classes (left) and 16 classes (right)

Unsupervised Classification

• Four exponentials

• T2=100, 20, 10 and 5ms.

• 8 classes (left) and 16 classes (right)

What’s Next?

• Speed up decomposition• Verify assumptions (linearity, for ex.)• More echoes• Sub pixel operations• Registration of different sequences• Classification• Integration with anatomic info• …