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Outline Introduction Model of the Forces Solution Results Conclusion A Deformable Balloon for Tomography Motion Artifact Study Damien Rohmer November 21, 2006 Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces Solution Results Conclusion Introduction 1 Presentation Model Used Model of the Forces 2 Notation Pressure Force Membrane Reaction Dampping Force Solution 3 First Method Evolving Method Results 4 Deformations Artifacts Simulations Conclusion 5 Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces Presentation Solution Model Used Results Conclusion Introduction of the Problem Mechanical system for deformation of a balloon Simulate simply the deformation of a heart Enable tomography measurement of the artifacts $$$ Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces Presentation Solution Model Used Results Conclusion Goal Model the Equation of the deformation. Solve (numerically) to observe the behavior of the balloon. Perform a CT acquisition. Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces Presentation Solution Model Used Results Conclusion Model used Table: Dimensions of the balloons h (cm) R (cm) interior 8 . 5 1 . 75 exterior 8 . 5 2 . 75 Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces Presentation Solution Model Used Results Conclusion Approximations The problem is supposed to be planar (isotropic in the circumferential direction). The problem solved is static (no dynamic fluid mechanic). Gravity effect are neglected. R 1 R 0 h Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Notation Model of the Forces Pressure Force Solution Membrane Reaction Results Dampping Force Conclusion Notation of the curve One membrane is considered The 2D profil is parameterized with s . Every position on the profil is defined by the curve c such that c ( s ) = ( c x ( s ) , c y ( s )) Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Notation Model of the Forces Pressure Force Solution Membrane Reaction Results Dampping Force Conclusion Pressure Action The action is constantly normal to the curve. The magnitude is constant (no gravity effect). The normal is called n ( s ) so F p ( s ) = f n ( s ). The normal can be expressed with the curve: � − c ′ � y ( s ) c ′ x ( s ) F p ( s ) = f � 2 ( s ) + c ′ 2 ( s ) c ′ x y f can be expressed as the total force F divided by the area element f = F α 1 2 π � 2 ( u ) + c ′ 2 ( u ) d u � c c y ( u ) c ′ x y Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Notation Model of the Forces Pressure Force Solution Membrane Reaction Results Dampping Force Conclusion Reaction of the Membrane The elastic membrane tend to limit the deformations. Tend to reach the initial shape at rest. Can be expressed (linear approximation) in the case of a constant stiffness λ by F e ( s ) = λ ( c − c 0 ) ′′ ( s ) Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Notation Model of the Forces Pressure Force Solution Membrane Reaction Results Dampping Force Conclusion Addition of the damping Force A Damping force to decrease the Energy. Simulate by a fluid friction force. Does not change the final state. F d = − µ∂ c ∂ t ( s , t ) Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Static Equation Want to solve directly the final state. The Damping force is not used. There is no curve evolution through time F p ( c ) + F e ( c ) = 0 Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Equation to solve c ′ y ( s )  � ′′ ( s ) = 0 c x − c 0 � − f + λ  x  �  2 ( s ) + c ′ 2 ( s ) c ′   x y c ′ x ( s ) � ′′ ( s ) = 0 c y − c 0 � + λ f   y  � 2 ( s ) + c ′ 2 ( s )  c ′  x y Or, calling z = c x + ic y and µ = f λ , the equation is given in complex form z ′′ + i µ z ′ | z ′ | = z 0 ′′ Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Discretization The equation is spatially discretized to give the non linear system  � � F ( c x i , c y i ) i ∈ [ = 0 [2 , N − 1] ]   ( c x 1 , c y 1 ) = c 0  ( c x 1 , c y 1 ) = c N  − → Fp ( c x n , c y n ) − → F e 2 − → F e 1 ( c x n +1 , c y n +1 ) ( c x n − 1 , c y n − 1 ) Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Numerical Solution Z such that Z 2 n +1 = c x n and Z 2 n = c y n , the non linear system with 2 N unknown is solved by Newton’s method . Z i +1 = Z i − D F − 1 ( Z i ) F ( Z i )     D F ij ( Z k ) = ∂ F i ( Z i )   ∂ Z i  j Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Problems The convergence is slow for large deformations. Oscillations spoil the stability during the iterations. The path between the initial and final step is not controled. Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Second Method: Evolving Method Idea: To stay close from a physical solution during the iterations. Method: The curve is now evolving through time ∂ 2 c ∂ t 2 ( s , t ) = F p ( c , s , t ) + F e ( c , s , t ) + F d ( c , t , s , t ) Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion New Equation A new equation has to be taken in account − F α 1 c y , s  c x − c 0 � � c x , tt = + λ  x , ss  2 π � �  � c 2 x , s + c 2 x 2 x , s + c 2 s c y y , s d s  y , s     − µ c x , t    F α 1 c x , s  c y − c 0  � � c y , tt = + λ   y 2 π � � , ss  � c 2 x , s + c 2 x 2 x , s + c 2  s c y y , s d s  y , s     − µ c y , t Looks not as good ... Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Matrix Notation The temporal order is decreased by the use of matrix � 0 � c � � � � 1 0 U = , M sys = , F = 0 0 � c , t i F i ⇒ U , t = M sys U + F Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Discretization step The system is discretized in the spatial domaine: PDE ⇒ ODE in time (method of lines) U is a vector of 2 N unknown. M sys is a 2 N × 2 N identity block matrix. 5 4 3 2 1 0 −1 −2 −3 −4 −5 0 2 4 6 8 10 12 Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Method of solution For stability reasons, the parabolic equation equation is solve by an implicit method: � � U ( t + ∆ t ) = U ( t ) + ∆ t M sys U ( t + ∆ t ) + F ( t + ∆ t ) Problem: the new step of the Force is unknow and non-linear Need a linearization. Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Elastic Term Already linear � 0 � 0 � � � � 0 0 U − λδ 2 = , λδ 2 c 0 0 F e where δ 2 is the discrete operator of the second derivative. Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Damping term Linear too; � 0 � 0 � � 0 = − µ U F d 0 I Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study

Outline Introduction Model of the Forces First Method Solution Evolving Method Results Conclusion Pressure Action Need to be linearized. Use of multivariable Taylor expansion: ∂ F p ∂ c j c j � F p ( t + ∆ t ) ≃ F p ( t ) + ∆ t , t ( t + ∆ t ) j � 0 � � � � 0 � 0 0 ⇒ = + ∆ t U ∂ F F p ( t + ∆ t ) F p ( t ) 0 ∂ c Damien Rohmer A Deformable Balloon for Tomography Motion Artifact Study