System and method for stochastic DT-MRI connectivity mapping on the GPU

1. A graphics processing unit implemented method for fiber tract mapping from diffusion tensor magnetic resonance imaging data, said method comprising the steps of:

providing a digitized brain image volume comprising a plurality of 3×3 diffusion tensors corresponding to a 3-dimensional grid of points;

initializing a set of fiber positions in a 3D set of points, fiber displacements, and a posterior distribution for an updated fiber displacement in terms of the initial displacements and diffusion tensors, wherein said posterior distribution is

exp

⁡

[

-

(

λ

^

1

-

λ

^

2

)

⁢

(

v

·

e

1

)

2

-

λ

^

2

⁢



v

i



2

+

κ

⁡

(

v

i



v

i



)

T

⁢

(

v

i

-

1



v

i

-

1



)

]

,

 wherein c is a normalization constant, v i and v i−1 are, respectively, the updated and initial fiber displacements, e 1 is a dominant eigenvalue of the diffusion tensor at a given point,

λ

^

1

=

1

λ

1

 wherein λ 1 is the eigenvalue associated with e 1 ,

λ

^

2

=

2

λ

2

+

λ

3

 wherein λ 2 and λ 3 are the other eigenvalues of the diffusion tensor, and κ is parameter that characterizes how tightly said posterior distribution is dispersed about a mean direction;

randomly sampling a set of updated fiber displacements from said posterior distribution;

computing a new set of fiber positions from said initial fiber positions and said updated fiber displacements, wherein a fiber path comprises a set of fiber points connected by successive fiber displacements;

accumulating connectivity values in each point of said 3D set of points by additive alpha-blending a scaled value if a fiber path has passed through a point and adding zero if not; and

rendering said connectivity values,

wherein said steps of initializing a set of fiber positions, randomly sampling a set of updated fiber displacements, computing a new set of fiber positions, accumulating connectivity values, and rendering said connectivity values are performed by a graphics processing unit.

2. The method of claim 1 , wherein said initial fiber positions and fiber displacements are stored as textures, said new fiber positions are output into a frame buffer object, and further comprising swapping said new fiber positions frame buffer object with a texture.

3. The method of claim 1 , wherein said steps of randomly sampling a set of updated fiber displacements and computing a new set of fiber positions are performed by a fragment program.

4. The method of claim 1 , further comprising correcting said updated fiber displacements from said new set of fiber positions in a fragment program, outputting said corrected displacements to a frame buffer object, and swapping said frame buffer object with a texture.

5. The method of claim 1 , wherein said step of accumulating connectivity values is performed by a vertex program.

6. The method of claim 5 , further comprising combining said initial fiber positions and said new fiber positions into a vertex buffer object that is an argument to said vertex program, outputting said connectivity values into a frame buffer object, and swapping said frame buffer object with a texture.

7. The method of claim 1 , wherein rendering said connectivity values includes mapping colors to said connectivity values.

8. The method of claim 1 , further comprising terminating fiber tract mapping of a fiber path when said fiber path exits said 3D set of points or if a normalized variance of the eigenvalues of the diffusion tensor of a point on the fiber path falls below a pre-determined threshold, and randomly reinitializing said terminated fiber path within the 3D set of points.

9. The method of claim 1 , further comprising adjusting said randomly sampled displacements according to γv+(1−γ) v , wherein v is said randomly sampled displacement, v is an average value of said displacements, and γ is a pre-determined parameter that corrects for undersampling said distribution.

10. The method of claim 1 , wherein initializing said posterior distribution includes stochastically interpolating values for v i−1 , e 1 , {circumflex over (λ)} 1 and {circumflex over (λ)} 2 .

11. The method of claim 1 , further comprising stochastically interpolating a diffusion tensor D(x) when a new fiber position x falls between set points, wherein

D

⁡

(

x

)

=

{

D

⁡

(

⌊

x

⌋

)

,

if

⁢

⁢

U

⁡

(

0

,

1

)

>

x

-

⌊

x

⌋

,

D

⁡

(

⌈

x

⌉

)

,

otherwise

.

and wherein U(0,1) returns a random value uniformly distributed between 0 and 1.

12. A program storage device readable by a computer, tangibly embodying a program of instructions executable by the computer to perform the method steps for fiber tract mapping from diffusion tensor magnetic resonance imaging data, said method comprising the steps of:

providing a digitized brain image volume comprising a plurality of 3×3 diffusion tensors corresponding to a 3-dimensional grid of points;

initializing a set of fiber positions in a 3D set of points, fiber displacements, and a posterior distribution for an updated fiber displacement in terms of the initial displacements and diffusion tensors, wherein said posterior distribution is

exp

⁡

[

-

(

λ

^

1

-

λ

^

2

)

⁢

(

v

·

e

1

)

2

-

λ

^

2

⁢



v

i



2

+

κ

⁡

(

v

i



v

i



)

T

⁢

(

v

i

-

1



v

i

-

1



)

]

,

 wherein c is a normalization constant, v i and v i−1 are, respectively, the updated and initial fiber displacements, e 1 is a dominant eigenvalue of the diffusion tensor at a given point,

λ

^

1

=

1

λ

1

 wherein λ 1 is the eigenvalue associated with e 1 ,

λ

^

2

=

2

λ

2

+

λ

3

 wherein λ 2 and λ 3 are the other eigenvalues of the diffusion tensor, and κ is parameter that characterizes how tightly said posterior distribution is dispersed about a mean direction;

randomly sampling a set of updated fiber displacements from said posterior distribution;

computing a new set of fiber positions from said initial fiber positions and said updated fiber displacements, wherein a fiber path comprises a set of fiber points connected by successive fiber displacements;

accumulating connectivity values in each point of said 3D set of points by additive alpha-blending a scaled value if a fiber path has passed through a point and adding zero if not; and

rendering said connectivity values,

wherein said steps of sampling updated fiber displacements, computing a new set of fiber positions, accumulating connectivity values, and rendering said connectivity values are performed by a graphics processing unit.

13. The computer readable program storage device of claim 12 , wherein said initial fiber positions and fiber displacements are stored as textures, said new fiber positions are output into a frame buffer object, and further comprising swapping said new fiber positions frame buffer object with a texture.

14. The computer readable program storage device of claim 12 , wherein said steps of randomly sampling a set of updated fiber displacements and computing a new set of fiber positions are performed by a fragment program.

15. The computer readable program storage device of claim 12 , the method further comprising correcting said updated fiber displacements from said new set of fiber positions in a fragment program, outputting said corrected displacements to a frame buffer object, and swapping said frame buffer object with a texture.

16. The computer readable program storage device of claim 12 , wherein said step of accumulating connectivity values is performed by a vertex program.

17. The computer readable program storage device of claim 16 , the method further comprising combining said initial fiber positions and said new fiber positions into a vertex buffer object that is an argument to said vertex program, outputting said connectivity values into a frame buffer object, and swapping said frame buffer object with a texture.

18. The computer readable program storage device of claim 12 , wherein rendering said connectivity values includes mapping colors to said connectivity values.

19. The computer readable program storage device of claim 12 , the method further comprising terminating fiber tract mapping of a fiber path when said fiber path exits said 3D set of points or if a normalized variance of the eigenvalues of the diffusion tensor of a point on the fiber path falls below a pre-determined threshold, and randomly reinitializing said terminated fiber path within the 3D set of points.

20. The computer readable program storage device of claim 12 , the method further comprising adjusting said randomly sampled displacements according to γv+(1−γ) v , wherein v is said randomly sampled displacement, v is an average value of said displacements, and γ is a pre-determined parameter that corrects for undersampling said distribution.

21. The computer readable program storage device of claim 12 , wherein initializing said posterior distribution includes stochastically interpolating values for v i−1 , e 1 , {circumflex over (λ)} 1 and {circumflex over (λ)} 2 .

22. The computer readable program storage device of claim 12 , the method further comprising stochastically interpolating a diffusion tensor D(x) when a new fiber position x falls between set points, wherein

D

⁡

(

x

)

=

{

D

⁡

(

⌊

x

⌋

)

,

if

⁢

⁢

U

⁡

(

0

,

1

)

>

x

-

⌊

x

⌋

,

D

⁡

(

⌈

x

⌉

)

,

otherwise

.

and wherein U(0,1) returns a random value uniformly distributed between 0 and 1.