Outline Diffusion Process MRI Change of Coordinates Numerical Solution A Bloch Torrey Equation for Diffusion in a Deforming Media Damien Rohmer November 21, 2006 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Diffusion Process Introduction to the Diffusion Diffusion Equation Illustrations of the Diffusion Process MRI Introduction Static Case Dynamic Case Change of Coordinates Curvilinear Coordinates Prolate Spheroidal Coordinates Numerical Solution Implicit Method Numerical Solution for the Bloch-Torrey Equation A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction to the Diffusion Diffusion Process ◮ Link Between Microscopical and Macroscopical Behavior. ◮ Expressed with the Diffusion Coefficient ◮ Scalar Case: 6 τ D = [ x ( t + τ ) − x ( t )] 2 ◮ Vectorial Case: � 6 τ D = uu T u = x ( t + τ ) − x ( t ) x ( t + τ ) u x ( t ) A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction to the Diffusion The Diffusion Tensor ◮ D is a Symetric Definite Positive matrix by definition. 3 � λ i e i e T i = R Λ R T D = i =1 λ 3 e 3 λ 1 e 1 D λ 2 e 2 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Diffusion Equation The Diffusion Equation ◮ For a scalar φ ∂φ ∂ t = ∇ · ( D ∇ φ ) � �� � flux density ◮ For a vector φ = φ i e i ∂φ i � D ∇ φ i � ∂ t = ∇ · ◮ General Solution ( D independant of t with boundary conditions sent to infinity.) 1 e − x T D − 1 x φ ( x , t ) = ∗ φ ( x , 0) 4 t � N | D | (4 π t ) 2 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Illustrations of the Diffusion Process Illustration of the Diffusion Process Exemple of the Action of the Orientation of the Diffusion Tensor: 1. Original Distribution 2. Filtered Distribution 3. Main Orientation of the Tensors A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Illustrations of the Diffusion Process Illustration of the Diffusion Process (II) Exemple of the Action of the Inhomogeneous Diffusion Phenomena Applied to the Filtering. 1. Original 2. Noisy 3. Homogeneous Gaussian Filtering 4. Inhomogeneous Diffusion A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction Bloch Equation ◮ 1 H atoms abundant in the water possess a nuclear angular momentum: the Spin . ◮ The orientation of the Spin is given by M . ◮ Under a Magnetic Field B , the momentum rotates around B at the pulsation γ � B � : M , t = M × γ B A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction Bloch Equation (II) ◮ In order to acquire the momentum M , a large Magnetic Field B 0 is applied along the axis z : e 3 , and M is flipped in the ( x , y ) plane by a special field. � − M 3 − M 3 M , t = M × γ B − M 1 e 1 + M 2 e 2 0 e 3 T 2 T 1 M ( x , 0) = M 0 e 3 B 0 M 2 e 2 M M 1 e 1 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction Bloch-Torrey Equation The Diffusive term ∇ · ( D ∇ ) is added: − M 3 − M 3 M , t = M × γ B − M 1 e 1 + M 2 e 2 e 3 + ∇ · ( D ∇ M ) 0 T 2 T 1 Where ∇ · ( D ∇ M ) has to be understood componentwise. A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction Attenuation Expression It is first supposed that ◮ D does not depends on t , then for every position D = const . ◮ The diffusion seen by each molecule is constant along its displacement. A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Introduction Attenuation Expression ◮ Only the ( x , y ) Components are Tacken in Account: M = M 1 + i M 2 ◮ The Magnetization Vector is Expressed as: M ( x , t ) = A x ( t ) e − α ( t ) e i ϕ ( x , t ) ◮ The matrix B is defined: � B ( x , t ) = ( ∇ ϕ ) ( ∇ ϕ ) T � t 0 x · G ( t ′ ) d t ′ !!! ϕ = γ ◮ The Attenuation A x is given by: ��� t � A x ( t ) � � � B ( x , t ′ ) d t ′ ln = − tr D A x (0) 0 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Static Case Special Pulse Sequence � � � A x ( t ) = − ∆ k T D k ln A x (0) k = γ δ G d TE TE 2 2 90 ◦ 180 ◦ ✞✠✟ t ✂✁✂ �✁� ✄✁✄ ☎✁☎ ✂✁✂ �✁� ✄✁✄ ☎✁☎ ✂✁✂ �✁� ☎✁☎ ✄✁✄ ✂✁✂ �✁� ✄✁✄ ☎✁☎ ✂✁✂ �✁� ☎✁☎ ✄✁✄ �✁� ✂✁✂ ☎✁☎ ✄✁✄ �✁� ✄✁✄ ✂✁✂ ☎✁☎ G d �✁� ✄✁✄ t ✂✁✂ ☎✁☎ �✁� ✂✁✂ ✄✁✄ ☎✁☎ �✁� ✂✁✂ ✄✁✄ ☎✁☎ δ d δ d ∆ ✞☛✡✌☞✎✍ ✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝ t ✆✁✆✁✆✁✆ ✝✁✝✁✝✁✝ ✝✁✝✁✝✁✝ ✆✁✆✁✆✁✆ t = 0 t = τ A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Now the material is dynamic ◮ The position x is depending on the time. ◮ Use of an original Underformed Referential given by ( e 1 , e 2 , e 3 ) and X = X i e i . ◮ Addition of a Deformed Referential using the Curvilinear Coordinate system given by ( g 1 , g 2 , g 3 ) and ξ = ξ ( X , t ). − → g 3 − → g 2 − → g 1 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Deformed Referential ◮ The deformation is characterized by the tensorial Deformation Gradient : F = ∂ξ i ∂ X j ◮ And follow the relation: d ξ = F d X A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Expression of the gradient of phase ◮ The spatial phase variation has to be expressed in the fixed referential where the phase is: � t ϕ ( ξ , t ) = γ X ( ξ , t ′ ) · G ( t ′ ) d t ′ 0 ◮ It is assumed a smooth deformation: ∇ T ϕ ( ξ , t ) d ξ = ∇ T ϕ ( X , t ) d X ◮ Using the deformation Gradient F : ∇ ϕ ( ξ , t ) = F − T ( X , t ) ∇ ϕ ( X , t ) d X A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Expression of the Diffusion tensor The component of the tensor depends on the basis: ◮ The tensor expressed in the original referential: D ◮ The tensor expressed in the deformed referential: D ◮ They are linked by the relation: i = ∂ξ i ∂ X l k D j ∂ξ j D l ∂ X k ⇒ D = F D F − 1 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Expression of the Attenuation ◮ The Attenuation is Expressed with the Components of the Initial Referential: � t � A X ( t ) � ( ∇ ϕ ) T D F − 2 ∇ ϕ d t ′ ln = A X (0) 0 ◮ The Right Stretch tensor is introduced such that: F T F = U 2 � t � A X � ( ∇ ϕ ) T D U − 2 ∇ ϕ d t ′ ln = A X 0 A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Dynamic Case Aquisition Sequence � � � A X ( τ ) = − ∆ k T D obs k ln A X (0) � ∆ 0 D U − 2 d t D obs = 1 ∆ ∆ t 0 t 0 t ✏✒✑✠✓ TE TE 2 2 90 ◦ 90 ◦ 90 ◦ t ✞✠✟ ✝✁✝✁✝ ✆✁✆✁✆ ✄✁✄✁✄ ☎✁☎ ✆✁✆✁✆ ✄✁✄✁✄ ✝✁✝✁✝ ☎✁☎ ✆✁✆✁✆ ✝✁✝✁✝ ✄✁✄✁✄ ☎✁☎ ✆✁✆✁✆ ✝✁✝✁✝ ✄✁✄✁✄ ☎✁☎ ✆✁✆✁✆ ✝✁✝✁✝ ✄✁✄✁✄ ☎✁☎ ✝✁✝✁✝ ✆✁✆✁✆ ✄✁✄✁✄ ☎✁☎ t G d ✆✁✆✁✆ ✄✁✄✁✄ ✝✁✝✁✝ ☎✁☎ ✝✁✝✁✝ ✆✁✆✁✆ ✄✁✄✁✄ ☎✁☎ ✝✁✝✁✝ ✆✁✆✁✆ ✄✁✄✁✄ ☎✁☎ δ d δ d ∆ ✞☛✡✌☞✎✍ ✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁� t ✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂ �✁�✁�✁�✁� �✁�✁�✁�✁� ✂✁✂✁✂✁✂✁✂ t = 0 t = τ A Bloch Torrey Equation for Diffusion in a Deforming Media
Outline Diffusion Process MRI Change of Coordinates Numerical Solution Curvilinear Coordinates Use of the Curvilinear Coordinates ◮ A change of coordinates: ( ξ 1 , ξ 2 , ξ 3 ) = φ ( x 1 , x 2 , x 3 ) ξ 1 = const − → V ( ξ 1 , ξ 2 ) ξ 2 = const A Bloch Torrey Equation for Diffusion in a Deforming Media
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