D EPARTAMENTO DE C IENCIAS P OLITÉCNICAS UCAM U NIVERSIDAD C ATÓLICA DE M URCIA D. T ECNOLOGÍAS DE LA I NFORMACIÓN Y LAS C OMUNICACIONES U NIVERSIDAD P OLITÉCNICA DE C ARTAGENA 6th. International Work-Conference on the Interplay between Natural and Artificial Computation - IWINAC 2015 Dynamic Modelling of the Whole Heart Based on a Frequency Formulation and Implementation of Parametric Deformable Models Rafael Berenguer-Vidal Rafael Verdú-Monedero Álvar Legaz-Aparicio
Outline Outline Introduction 1 Multidimensional parametric deformable models 2 Practical implementation of the method 3 Results 4 Conclusions 5 R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 2 / 22
Introduction Outline Introduction 1 Deformable models Multidimensional parametric deformable models 2 Practical implementation of the method 3 Results 4 Conclusions 5 R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 3 / 22
Introduction Deformable models Deformable models in CVD analysis Description Analysis of motion and deformation of the heart Considerable interest in the literature Great impact of cardiovascular disease (CVD) Several approaches proposed Deformable models Deformable models based on B- splines Interesting approach for cardiac characterization Use of frequency-based multidimensional parametric deformable models Efficient implementation R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 4 / 22
Introduction Deformable models Deformable models in CVD analysis Description Analysis of motion and deformation of the heart Considerable interest in the literature Great impact of cardiovascular disease (CVD) Several approaches proposed Deformable models Deformable models based on B- splines Interesting approach for cardiac characterization Use of frequency-based multidimensional parametric deformable models Efficient implementation R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 4 / 22
Multidimensional parametric deformable models Outline Introduction 1 Multidimensional parametric deformable models 2 Static model Dynamic model Spatial and temporal discretization Implementation in the frequency domain Practical implementation of the method 3 Results 4 Conclusions 5 R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 5 / 22
Multidimensional parametric deformable models Static model Parametric deformable model Time-varying parametric Hypersurface defined in R d v ≡ v ( s , t ) = [ v 1 ( s , t ) , v 2 ( s , t ) , . . . , v d ( s , t )] ⊤ , Parametric variables s ≡ [ s 1 , . . . , s e ] with s j ∈ [ 0 , L j ] and e � d Coordinate functions v i ( s , t ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 6 / 22
Multidimensional parametric deformable models Static model Parametric deformable model Time-varying parametric Hypersurface defined in R d v ≡ v ( s , t ) = [ v 1 ( s , t ) , v 2 ( s , t ) , . . . , v d ( s , t )] ⊤ , Parametric variables s ≡ [ s 1 , . . . , s e ] with s j ∈ [ 0 , L j ] and e � d Coordinate functions v i ( s , t ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 6 / 22
Multidimensional parametric deformable models Static model Parametric deformable model Time-varying parametric Hypersurface defined in R d v ≡ v ( s , t ) = [ v 1 ( s , t ) , v 2 ( s , t ) , . . . , v d ( s , t )] ⊤ , Parametric variables s ≡ [ s 1 , . . . , s e ] with s j ∈ [ 0 , L j ] and e � d Coordinate functions v i ( s , t ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 6 / 22
Multidimensional parametric deformable models Static model Parametric deformable model Time-varying parametric Hypersurface defined in R d v ≡ v ( s , t ) = [ v 1 ( s , t ) , v 2 ( s , t ) , . . . , v d ( s , t )] ⊤ , Parametric variables s ≡ [ s 1 , . . . , s e ] with s j ∈ [ 0 , L j ] and e � d Coordinate functions v i ( s , t ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 6 / 22
Multidimensional parametric deformable models Static model Parametric deformable model Time-varying parametric Hypersurface defined in R d v ≡ v ( s , t ) = [ v 1 ( s , t ) , v 2 ( s , t ) , . . . , v d ( s , t )] ⊤ , Parametric variables s ≡ [ s 1 , . . . , s e ] with s j ∈ [ 0 , L j ] and e � d Coordinate functions v i ( s , t ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 6 / 22
Multidimensional parametric deformable models Static model Static model Energy functional The model shape is governed by an energy functional: E ( v ) = S ( v ) + P ( v ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 7 / 22
Multidimensional parametric deformable models Static model Static model Energy functional The model shape is governed by an energy functional: E ( v ) = S ( v ) + P ( v ) Internal and external energies Internal deformation energy, S ( v ) : d S ( v ) = 1 �� � α ( s ) �∇ v i ( s ) � 2 + β ( s ) | ∆ v i ( s ) | 2 d s � 2 Ω i = 1 Ω := [ 0 , L 1 ] × [ 0 , L 2 ] × . . . × [ 0 , L e ] α ( s ) and β ( s ) represent elasticidad and rigidity Rest of energies, P ( v ) : External energies, gradient of data, ... Non-lineal restrictions, hard-constraints, ... R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 7 / 22
Multidimensional parametric deformable models Static model Static model Energy functional The model shape is governed by an energy functional: E ( v ) = S ( v ) + P ( v ) Euler-Lagrange equation To minimize E ( v ) , model v ( s ) must satisfy E-L equation. For the multidimensional case, we obtain a system of d PDE’s: � � � � � � −∇ · α ( s ) ∇ v ( s ) + ∆ β ( s )∆ v ( s ) = q v ( s ) where ∈ R d � � � � � � q v ( s ) = −∇ v i P v ( s ) + f v ( s ) Represents the balance between internal and external energies R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 7 / 22
Multidimensional parametric deformable models Dynamic model Dynamic model Temporal evolution Lagrangian mechanics theory Dynamic deformable mechanics Temporal parameters Mass density: µ ( s ) Dumping density: γ ( s ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 8 / 22
Multidimensional parametric deformable models Dynamic model Dynamic model Temporal evolution Lagrangian mechanics theory Dynamic deformable mechanics Temporal parameters Mass density: µ ( s ) Dumping density: γ ( s ) R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 8 / 22
Multidimensional parametric deformable models Dynamic model Dynamic model Temporal evolution Lagrangian mechanics theory Dynamic deformable mechanics Temporal parameters Mass density: µ ( s ) Dumping density: γ ( s ) Dynamic model d decoupled PDE’s system: � � � � µ ( s ) ∂ tt v i ( s , t ) + γ ( s ) ∂ t v i ( s , t ) − ∂ s 1 α ( s ) ∂ s 1 v i ( s , t ) − · · · − ∂ s e α ( s ) ∂ s e v i ( s , t ) + � �� � ∂ s 1 s 1 + · · · + ∂ s e s e β ( s ) ∂ s 1 s 1 v i ( s , t ) + · · · + β ( s ) ∂ s e s e v 1 ( s , t ) = 1 � i � d � � q v ( s ) i ∈ N R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 8 / 22
Multidimensional parametric deformable models Spatial and temporal discretization Spatial discretization Finite elements formulation Parametric domain 0 � s j � L j partitioned into N j finite subdomains. Model represented as the union of N = N 1 N 2 · · · N e elements: N 1 − 1 N e − 1 � � v n v i ( s , t ) = i ( s , t ) , n = [ n 1 , . . . , n e ] , · · · n 1 = 0 n e = 0 Each v i is represented using shape functions and variable vectors: v n i ( s , t ) = N n i ( s ) u n i ( s , t ) , R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 9 / 22
Multidimensional parametric deformable models Spatial and temporal discretization Spatial discretization Finite elements formulation Parametric domain 0 � s j � L j partitioned into N j finite subdomains. Model represented as the union of N = N 1 N 2 · · · N e elements: N 1 − 1 N e − 1 � � v n v i ( s , t ) = i ( s , t ) , n = [ n 1 , . . . , n e ] , · · · n 1 = 0 n e = 0 Each v i is represented using shape functions and variable vectors: v n i ( s , t ) = N n i ( s ) u n i ( s , t ) , R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 9 / 22
Multidimensional parametric deformable models Spatial and temporal discretization Spatial discretization Finite elements formulation Parametric domain 0 � s j � L j partitioned into N j finite subdomains. Model represented as the union of N = N 1 N 2 · · · N e elements: N 1 − 1 N e − 1 � � v n v i ( s , t ) = i ( s , t ) , n = [ n 1 , . . . , n e ] , · · · n 1 = 0 n e = 0 Each v i is represented using shape functions and variable vectors: v n i ( s , t ) = N n i ( s ) u n i ( s , t ) , R. Berenguer-Vidal (UCAM) IWINAC 2015, Elche Spain June 3th 2015 9 / 22
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