Tumor Growth Simulation

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TABLE 5.1

DTI Scalar Indices

Scalar Index Name

Equation

Fractional Anisotropy (FA)

FA =

√

3((λ1−λ)

2+(λ2−λ)

2+(λ3−λ)

2)

√

2(λ1²+λ2²+λ3²)

Rational Anisotropy (RA)

RA =

√

(^λ1^−λ)

2+(λ2−λ)

2+(λ3−λ)

2

√

3λ

Mean Diffusivity (MD)

MD = (λ1 + λ2 + λ3) /3

Linear Anisotropy(cl)

cl =

λ1−λ2

λ1+λ2+λ3

spherical Anisotropy(cs)

cs =

3λ3

λ1+λ2+λ3

Volume ratio (VR)

V R =

λ1λ2λ3

[λ]

3

Angular Anisotropy (AA)

AA = arccos



3

i=1

λi

√∑3

j=1 ^{λj 2}

λ

√∑3

j=1 ^λ

2



Diffusion Volume (DV)

DV = ⁴

3^πλ1λ2λ3

Aitchison

Anisotropy(Aita)

AitA =

3

i=1 ^{(ln (λi) −}

1

3

3

j=1 ^{ln (λj))}

Matusita Anisotropy (MA)

MA =

√

∑3

i=1

(√λi−

√

λ

)

√∑3

i=1 ^λi

Kullback-Leibler

Anisotropy (KLA)

KLA =



2

3



ln(



1

3

3

i=1

λi

λ

 

1

3

3

i=1 ^{ln( λi}

λ ⁾



)

VR might be used to distinguish between WM and non-WM regions, while

eigenvalues or MD could be used to categorize CSF and non-CSF regions.

K-Means [33], is used for segmenting brain to WM/non-WM from FA images

(Figure 5.2) and to CSF/non-CSF from MD images (Figure 5.3).

The John Hopkins Medical Institute’s Laboratory of Brain Anatomical

MRI website provided the datasets utilized in this study. These files provide

large amounts of DTI brain data. There are 50 slices of 256×256 voxels in each

brain DTI volume. To calculate tensor data, each dataset has a file with 35

gradient orientations. The voxel width and height in each slice are 0.9375 mm