Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

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1 Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI

2 Next lecture, DTI For this lecture, think in terms of a single voxel

3 We re still looking only at a single voxel experiment This Last Lecture: Multiple Single diffusion encoding directions to to estimate a a diffusion coefficient tensor DD

4 G x B(x) x B(x) x time time x

5 Phases of diffusing spins 90 TEê2 180 TEê2 echo d d»g» e D = + ( )= 0 G(t)x(t) dt

6 Diffusion phase in a Bipolar Pulse 100 x(t 1 ) x(t 1 ) r x(t 2 ) G(t) G x '(x, t) = G {z} q G x [ x(t 2 ) x(t 1 )] {z } r = q r t

7 The Estimation Problem for Gaussian Diffusion measured signal Z S(') S(b) = =S(0) dx P e(x, bd t) + e i'(x,t) non-diffusion weighted signal (b=0) Gaussian b = q 2 noise object of our desire! pulse sequence parameters G(t) G x q = G = 3 G x t

8 The signal from 1D Gaussian Diffusion s(b i )=s 0 e b id + i s 1.0 where s 0 s(b = 0) b

9 Consider only two measurements and write data in vector form s(b1 ) s(b 2 ) exp( b1 D) = s 0 exp( b 2 D) = s 0 exp apple b1 D b 2 D This clearly generalizes to n measurements

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11 Recall: gradients add like vectors k x = k x x + k y y = G x tx + G y ty G y (t) t y G x (t) x t spatial modulation of the phase

12 Directional Diffusion Encoding 90 TEê2 180 TEê2 echo d d»gx» e D z y»gy» z y x x G (x,y)

13 Ideal b-matrix G i G j 2 q b ij = Gq ii q i = G i G j j ( where /3) = /3 q i q j

14 Ideal b-matrix G i G j b ij = q i q j where q i = G i = /3

15 The b-matrix b ij ( )= 0 q i (t)q j (t)dt i=(x,y,z) where q = g(t) dt For constant diffusion gradients b ij ( ) =q i q j

16 The NMR signal for 1D Gaussian diffusion s(q, )=s(0) P ( r, )e iq r d r P ( r, ) = 1 4 D e r 2 /(4D ) s(q, )=s(0)e bd

17 The NMR signal for 3D Gaussian diffusion s(q, ) = Z P ( r, )e iq r d r P ( r, ) = 1 p (4 )3 D e rt D 1 r/4 s(q, ) =s(0)e bd

18 b and D qx 2 q x q y q x q z b = q y q x qy 2 q y q z q z q x q z q y qz 2 known D xx D xy D xz D = D yx D yy D yz D zx D zy D zz desired

19 The NMR signal 3D Gaussian diffusion s(q, )=s(0)e bd s(b) = s(0) exp 3 3 b ij D ij i j bd = q 2 D bd = q t D q

20 A single diffusion-weighting direction G x y z G y x G z

21 1 i j bd b ij D ij = q 2 xd xx + q x q y D xy + q x q z D xz + q y q x D yx + q 2 yd yy + q y q z D yz + q z q x D yx + q z q y D zy + q 2 zd zz

22 Rearranging the directions bd = q t D q q = qû q q q q û qû bd = q 2 u t D u

23 The NMR signal bd = q 2 u t D u D s(q, )=s(0)e bd = e q2 D s(q, )=e b D where b = q 2

24 Measuring the Diffusion Tensor S(b, r ) = S(0)e bd + y r = 3 D= Dx 0 0 Dy cos sin x t 2 2 D = r Dr = Dx cos + Dy sin projection of an ellipsoid! not like projection of a vector

25 Measuring the Diffusion Tensor b=1000 b=0 1.0 y x -1 fiber axis S(b, ) = S(0)e bd( ) + D( ) = Dx cos2 + Dy sin2

26 The Shape of Diffusion fiber signal S b ( ) D app ( ) = 1 b log Sb S 0

27 What is the meaning of D? D u t D u It is the projection of D along û D û D

28 Diffusion Tensor is Symmetric D xx D xy D xz D yx D yy D yz D zx D zy D zz = D xx D xy D xz D xy D yy D yz D xz D yz D zz D = D t matrix form D ij = D ji component form

29 1 i j bd b ij D ij = q 2 xd xx + q x q y D xy + q x q z D xz + q y q x D yx + q 2 yd yy + q y q z D yz + q z q x D yx + q z q y D zy + q 2 zd zz 1 i j b ij D ij = q 2 xd xx +2q x q y D xy +2q x q z D xz + q 2 yd yy +2q y q z D yz + q 2 zd zz

30 A computational simplification s(b) =s(0)e bd a trick: write log s(0) s(0) = e s(b) =s(0)e bd = e log s(0) e bd = e bd+log s(0)

31 Estimating the Diffusion Tensor d = D xx D yy D zz D xy D xz D yz log s(0) There are 7 unknowns

32 Estimating the Diffusion Tensor B =(q 2 x,q 2 y,q 2 z, 2q x q y, 2q x q z, 2q y q z, 1)

33 Estimating the Diffusion Tensor log s(b) y = (q 2 x,q 2 y,q 2 z, 2q x q y, 2q x q z, 2q y q z, 1) B t D xx D yy D zz D xy D xz D yz log s(0) D But there are 7 unknowns, so we need 7 equations to solve for them

34 Estimating the Diffusion Tensor y = log s(b 1 ) log s(b 2 ). log s(b n ) We make 7 measurements, each with a different direction

35 The B-matrix B t = tensor dimensions ˆq 1,x 2 ˆq 1,y 2 ˆq 2 1,z ˆq 1,xˆq 1,y ˆq 1,xˆq 1,z ˆq 1,y ˆq 1,z 1 ˆq 2,x 2 ˆq 2,y 2 ˆq 2,z 2 ˆq 2,xˆq 1,y ˆq 2,xˆq 2,z ˆq 2,y ˆq 2,z ˆq n,x 2 ˆq n,y 2 ˆq n,z 2 ˆq n,xˆq 1,y ˆq n,xˆq n,z ˆq n,y ˆq n,z 1 gradient directions q j,k = g k = /3 j th direction

36 Angular measurements

37 Estimating the Diffusion Tensor log s(b 1 ) ˆq 1,x 2 ˆq 1,y 2 ˆq 2 1,z ˆq 1,xˆq 1,y ˆq 1,xˆq 1,z ˆq 1,y ˆq 1,z 1 log s(b 2 ).. = ˆq 2,x 2 ˆq 2,y 2 ˆq 2,z 2 ˆq 2,xˆq 1,y ˆq 2,xˆq 2,z ˆq 2,y ˆq 2,z log s(b n ) ˆq n,x 2 ˆq n,y 2 ˆq n,z 2 ˆq n,xˆq 1,y ˆq n,xˆq n,z ˆq n,y ˆq n,z 1 D xx D yy D zz D xy D xz D yz log s(0) y B t d

38 Least Squares The matrix equation y = Bd has dimensions [n 1] = [n m][m 1]

39 Estimating the Diffusion Tensor Solving for the diffusion tensor is reduced to finding the solution to the matrix equation y = Bd diffusion tensor elements data b-matrix

40 Estimating the Diffusion Tensor Matrix equation y = B t d data b-matrix diffusion tensor elements Matrix solution d = B + y pseudo-inverse

41 Least Squares The least-squares solution to the matrix equation y = Ax is? ˆx = A 1 y? NO!

42 Least Squares The least-squares solution to the matrix equation y = Ax is ˆx = A + y (note that ˆx = A 1 y) where A + (A t A) 1 A t This is called the pseudo-inverse of A

43 Estimating the Diffusion Tensor In practice D calculated with 3dDWItoDT (AFNI) eigensystem calculated by: [evals,evecs] = eig(d)