Predicting Tumour Location by Modelling the Deformation of the Breast using Nonlinear Elasticity

Transcription

1 Predicting Tumour Location by Modelling the Deformation of the Breast using Nonlinear Elasticity November 8th, 2006

2 Outline Motivation Motivation Motivation for Modelling Breast Deformation Mesh Generation Forward, Backward and Contact Problems Results Problems with the Full Model Modelling Skin as a Membrane The Backward Problem for Skin

3 Outline Motivation Motivation Motivation for Modelling Breast Deformation Mesh Generation Forward, Backward and Contact Problems Results Problems with the Full Model Modelling Skin as a Membrane The Backward Problem for Skin

4 Motivation for a Deformable Model of the Breast Problem is to register images and match tumour location The patient is: prone during Magnetic Resonance Imaging (MRI) standing, with the breast compressed during mammography (various possible directions of compression) supine during surgery CC mammogram MLO mammogram MR image

5 Motivation for a Deformable Model of the Breast Given a point in the CC mammogram, what is the equivalent curve in the MLO mammogram? CC mammogram MLO mammogram

6 Motivation for a Deformable Model of the Breast We want to build a patient-specific, anatomically-accurate finite element model of the breast The model can be used to: Predict tumour location during surgery/biopsy Match MRI with mammograms, or CC with MLO Perform temporal matching Plan reconstructive surgery As a teaching/visualisation tool for radiologists

7 Outline Motivation Motivation Motivation for Modelling Breast Deformation Mesh Generation Forward, Backward and Contact Problems Results Problems with the Full Model Modelling Skin as a Membrane The Backward Problem for Skin

8 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

9 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

10 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

11 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

12 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

13 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

14 Mesh Generation Mesh Generation Process MR images: find boundary and segment interior Segment interior into fat, fibroglandular or tumour Fit a cubic-hermite mesh, convert into trilinear mesh Use segmentation data to assign a tissue to each element

15 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

16 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

17 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

18 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

19 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

20 Details for Breast Deformation Modelling Assumptions Assume tissues are isotropic Assume tissues are incompressible Assume hyperelasticity Boundary Conditions Assume the pectoral muscle stays fixed Simulations Normal nonlinear deformations Backward deformations... Constrained deformations...

21 Nonlinear Elasticity

22 Nonlinear Elasticity Basics Given a deformation x x(x): Define the deformation gradient F i M = xi X M Let the true stress (the Cauchy stress) be given by σ ij Cauchy s law Force balance in the deformed body gives σij + ρg i = 0 x j The corresponding weak form is σ ij (δx i) x j + ρg i δx i dv = 0 δx Ω

23 Nonlinear Elasticity Reformulation in terms of X We define the 2nd Piola-Kirchoff stress, T MN as the force acting on the undeformed body per unit undeformed area Cauchy s law can be restated as ) MN xi (T X M X N + ρg i = 0 in Ω 0 The corresponding weak form is MN xi (δx i ) T X Ω0 M X N + ρgi δx i dv 0 = 0 δx T MN and FM i related through the material law

24 The Backward Problem The Forward and Backward Problems Forward problem: given undeformed (unstressed) state, find the deformed state. Backward problem: given deformed, find undeformed. Not been carried out before with breast deformations. For forward problems start from MN xi (δx i ) T X Ω0 M X N + ρgi δx i dv 0 = 0 δx For backward problems start from σ ij (δx i) x j + ρg i δx i dv = 0 δx Ω

25 The Backward Problem Result of a forward simulation on a cube Result of subsequent backward simulation

26 The Backward Problem Speed Initially it took much longer to compute solutions to the backward problem

27 The Backward Problem and GMRES The trivial distinction between σ ij (u(x), p(x)) (δu j) Ω x i dv ± (det(f(u)) 1)δp dv Ω = 0 δu, δp makes a huge impact on GMRES convergence.

28 The Backward Problem and GMRES Consider the Stokes Problem 2 u + p = f ± u = 0 [ ] A B matrix = B T e/values real, some -ve 0 [ ] A B + matrix = B T e/values have +ve real 0 part Latter is much better for GMRES

29 The Backward Problem and GMRES Even though equations are nonlinear, the matrices have structure dominated by the linear part The choice of weak form (for both forward and backward problems) is very important For example, we must choose + in: MN xi (δx i ) T X Ω0 M X N ± (det F 1)δp dv 0 = 0 δx, δp This doesn t seem to be well-known

30 The Contact Problem Simulation of mammography - solve a deformation problem with inequality constraints Formulate equations as an energy minimisation and add a penalty function Used the Augmented Lagrangian Method: 1 ( ) 2 min : total energy + [ λ(x) + Pd(x)]+ ds0 2P skin Here P is a large penalty parameter d measures violation of the constraint λ are parameters which tend to the true Lagrange multipliers

31 Results Motivation CC compression MLO compression

32 Motivation Results simulated CC mammogram, one simulated MLO mammogram, point highlighted (red) corresponding curve in red

33 Results Motivation CC compressed state reference state MLO compressed state

34 Outline Motivation Motivation Motivation for Modelling Breast Deformation Mesh Generation Forward, Backward and Contact Problems Results Problems with the Full Model Modelling Skin as a Membrane The Backward Problem for Skin

35 Problems with the Full Model Badly-shaped skin elements Thin skin elements are badly-shaped - drastically slows Newton convergence Numerical results show little sensitivity to skin stiffness - probably due to lack of residual stress

36 Problems with the Full Model Replace thin 3D skin elements with 2D surface elements - model skin as a membrane Need to compute and include surface integral contribution thin skin membrane skin

37 Modelling Skin as a Membrane Another Reformation... Define new coordinate systems (θ α ) Define tangent vectors (A α ) and metric tensors. Let a αβ and A αβ be the metric tensors on deformed and undeformed surfaces Then strain is E αβ = 1 2 (a αβ A αβ ) Partial deriv.s covariant deriv.s Curvature becomes important

38 Modelling Skin as a Membrane Asymptotic derivation of the membrane equations Perform an asymptotic analysis in the thickness h 0 Analysis allowed us to compute the membrane strain energy, W mem (a αβ ), that is the exact equivalent to the 3D strain W(C MN ) Then considered a fluid-filled membrane model, for which the equation of state was F αβ ;β = 0 - essentially tension = constant F αβ b αβ = P - essentially tension*curvature = pressure

39 Modelling Skin as a Membrane Membrane strain energy worked... Removed the need for thin elements and fixed slow convergence Skin model Time (s) No. Newton iterations thin skin no skin 71 6 membrane thin skin no skin membrane 314 9

40 Modelling Skin as a Membrane However.. Residual stress in the skin was still an issue The fluid-filled membrane model needed residual stress to be solvable This leads to the backward problem for skin: given an unloaded but stressed state, compute the stresses..

41 Backward Problems for Skin (a) membrane with elastic interior, under gravity, in contact, (b) membrane with elastic interior, under gravity, (c) membrane with elastic interior, (d) membrane with fluid interior

42 The Backward Fluid-Filled Membrane Problem Start from: F αβ ;β = 0 F αβ b αβ = P Here: P, the pressure, is known b αβ, deformed curvature, is known F αβ F αβ (a αβ, A αβ ), where a αβ, A αβ are the known and unknown metrics Integrate for F αβ, then solve for A α,β, then integrate for X Even this simplified problem is difficult..

43 The Backward Fluid-Filled Membrane Problem Analysis into this revealed that: An undeformed shape exists (but may not be unique) Don t actually need the undeformed shape, X, just the undeformed metric, A αβ Would have to solve a mixed elliptic-hyperbolic problem One boundary condition is enough (sensible to specify it on the curve of zero Gaussian curvature) Much more research required into: Backward problem on a fluid-filled membrane Backward problem on a membrane with elastic interior

44 Outline Motivation Motivation Motivation for Modelling Breast Deformation Mesh Generation Forward, Backward and Contact Problems Results Problems with the Full Model Modelling Skin as a Membrane The Backward Problem for Skin

45 Conclusions Have shown modelling breast deformation with nonlinear elasticity is computationally tractable Discussed the three types of simulation necessary Used a patient-specific model to simulate surgical and mammographic breast shape Motivated and introduced a membrane skin model Described issues involved with membrane skin

46 Further Work Need proper validation of results, especially with patient studies Further analysis of required complexity of the model Modelling the effect of Cooper s Ligaments Analysis of the backward problem for skin