2012-01-01 to 2016-01-01

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# Vision

## Towards real time multiscale simulation of cutting in heterogeneous materials with applications to surgical simulation and computer guided surgery

Surgeons are trained as apprentices. Some conditions are rarely encountered and surgeons will only be trained in the specific skills associated with a given situation if they come across it. At the end of their residency, it is hoped that they will have faced sufficiently many cases to be competent. Rising healthcare costs, decreasing working hours (European Working Time Directives) and staff shortage will aggravate the situation.

If we were able to reproduce faithfully, in a virtual environment, the audio, visual and haptic expe- rience of a surgeon as they prod, pull and incise tissue, then, surgeons would not have to train on cadavers, phantoms, or on the patients themselves.

Just as civil pilots train on a flight simulator to rehearse complex maneuvers, learn to react to un- foreseen events, days before a surgery, surgeons could operate on a faithful computer-representation of the same patient they would be seeing in the operating room, greatly reducing risks of complications. They could also use a simulator to test alternate operating techniques in a safe environment (surgical planning).

# Aim

The aim of RealTCut is to devise real-time numerical methods for the simulation of cutting. These methods are aimed at surgical training, which has the potential to help surgeons improve their skills without endangering patients nor using cadavers or phantoms.

# Objectives

## Real-time simulation methods for soft-tissues

Aside from the work of Inria (Courtecuisse), all methods known to the PI to simulate cutting had been relying on explicit time integration schemes. This is because implicit schemes require matrix inversions at each iteration within each load step, thereby severely increasing the computational expense compared to explicit schemes. Yet, explicit schemes cannot guarantee a predefined level of accuracy on equilibrium, which is why we decided to tackle the problem using implicit schemes and tackle the computational burden by developing advanced algorithms on CPU/GPU and obtain results in quasi-real-time (about 40 to 50 frames per second, suitable for visual interactivity). We devised an asynchronous preconditioning technique (and its update during topological changes) running on a CPU/GPU pair. The system inversion (through the preconditioned conjugate gradient algorithm) is running on the GPU whilst the preconditioner is computed (with some lag) on the CPU. This enables the real time simulation of contact and cutting of heterogeneous soft tissues at visual interactive rates for meshes of a few tens of thousands of nodes, which is a scientific first and one of the stated objectives of the project.

## Model reduction methods

### Multi-scale methods for fracture and cutting: physics-based model reduction

We developed adaptive (model and discretization) methods for fracture over two scales. The major difficulty involved is that standard semi-concurrent homogenization methods such as the finite element squared of Feyel and coauthors break down at the onset of softening. We thus developed methods which automatically switch from a semi-concurrent method to a concurrent method at the onset of material instability. This was tested using model and discretization error indicators on one particular micro-heterogeneous material. The next step will require statistic approaches to enable confidence intervals to be derived from the simulations. The main difficulty to bring this forward for biomedical simulation is to characterize the tissues and account for the relevant fields (thermal, electrical, poro-elastic, etc.). It may become apparent that such approaches should be replaced by experimentally-derived data sets and suitable machine learning algorithms. This will be the subject of future research.

### Algebraic model order reduction for fracture and cutting

We developed two series of methods to reduce the computational expense associated with the simulation of fracture and cutting of heterogeneous materials. The first class of methods employs some a priori knowledge about the material’s microstructure to perform averaging at the material point level (homogenization), whilst the second approach relies solely upon algebraic approaches akin to machine learning algorithms. In the first approach, we developed “a priori” error bounds on the homogenized quantities for biphasic microstructures materials. We also constructed virtual charts for computational homogenization which drastically reduces computational expenses associated with the finite element squared method (FE2), and which will have direct applications to the next phases of the work. We will now need to extend this work to large incompressible deformations.

## Smoothed extended finite element methods for cutting

One of the key diffculties associated with modeling and simulating soft tissues is the simulation of incompressible or almost incompressible materials, combined with the need to simulate complex organ geometries. Whilst it is relatively easy and automatic to generate tetrahedral meshes, those elements lock for incompressible materials. We thus developed methods which rely on tetrahedral elements, but do not lock, based on gradient (strain) smoothing, the Finite Element analogue of stabilized conforming nodal integration in mesh-free methods. Those methods are known as the Smoothed (eXtended) Finite Element Methods (Sm(X)FEM), which we developed for continuum in small and large deformations and membrane elements.

## Advanced discretization for plates and shells

In order to deal with thin structures appearing within biological problems, we developed new methods based on strain smoothing for heterogeneous thin structures which do not suffer from shear locking. Some basic results are now available to model cracks within such structures, and will need to be extended to large deformations.

## Model error for multi-scale fracture

We started investigating the evaluation of model error for fracture. The goal of this part of the work is to become able

to automatically switch to a finer model (in terms of spatial scale), when a given error indicator is achieved. For example, a continuum model of the solid is replaced by a finer scale representation when the loss of material stability criterion is reached at one of the integration (material) points in the domain. The work done here was only as a proof-of-concept and needs to be extended to realistic microstructures. One of the diffculties is associated with the computational expense, both of the continuum (FE2) model and that of the micro-scale simulations, which we would like to tackle using machine learning algorithms.

## Estimation of the discretization error (on quantities of interest) for cutting and fracture

We were among the first to develop a posteriori error estimation techniques for extended (enriched) nite element methods. These methods were extended recently to handle quantities of interest including the energy release rate during fracture propagation, in two and three dimensions, but are yet limited to small deformations.

## Isogeometric analysis and geometry-independent field approximation (GIFT)

We developed discretization methods aiming at the generalization of isogeometric analysis, where the boundary description is also used to solve for the unknown fields within the domain. First, we extended the method to boundary elements, which only require knowledge of the boundary. Second, we generalized isogeometric analysis, to allow to choose the geometry and the field approximations as independently as possible. This enables to choose the most adapted approximation field for the unknown fields, whilst keeping geometrical exactness. The method will have to be further tested and studied from a fundamental basis in order to understand its mathematical properties, in particular concerning the patch test.

# Conclusions of the work

The work is ongoing. Current conclusions are that

- It is possible to simulate cutting in real-time using implicit time integration
- Selective domain-decomposition based algebraic model reduction can be use to reduce the computational expense for non-linear fracture problems (several orders of magnitude)
- Geometry and field approximations can be decoupled for elliptic PDEs in an isogeometric (IGA) context
- Adaptive methods for fracture can be derived that minimize both the discretization and model error
- Goal-oriented oct-tree adaptivity methods are efficient to optimize the computational expense in fracture simulations

# Future work

RealTCut raised the following unanswered questions

- How can the methods developed in RealTCut be extended to Computer-Assisted Surgery where accuracy is critical?
- How can patient-specificity be taken into account?
- How can the error in quantities of interest be controlled in real time?
- How can material models be selected and identified in real-time? Should this be done from image data provided during the operation?

# Academic collaborators

- Pierre Kerfriden, Cardiff University: model-order reduction, multi-scale methods and stochastic problems
- Stéphane Cotin, Inria: Real-time simulations
- Christian Duriez, Inria:Real-time contact simulations
- Bruno Lévy, Inria: computational geometry
- Karol Miller, University of Western Australia: brain biomechanics

# Clinical collaborators: a list with links

- Frank Hertel, Luxembourg Neurosurgery
- IHU, Strasbourg

# In the news

- http://www.surgicalproductsmag.com/product-releases/2014/06/realtcut-offers-advanced-materials-surgical-training
- http://www.news-medical.net/news/20120323/Cardiff-team-receives-European-grants-to-develop-RealTCut-surgical-simulation.aspx
- http://wwwen.uni.lu/press/press_releases/2014/25_06_2014_erc_fellow_stephane_bordas_with_exciting_research_project_joins_university_of_luxembourg
- http://www.innovation.public.lu/en/actualites/2014/06/bordas/index.html
- http://www.theengineer.co.uk/more-sectors/medical/news/doctors-training-via-virtual-surgery-wins-research-grant/1018839.article
- http://www.trainhealthcare.com/research-grants-awarded-to-virtual-surgery-training-innovators/
- http://www.science.lu/de/content/forscher-planen-virtuelle-experimente-f%C3%BCr-chirurgen
- http://www.surgical-tribune.com/articles/news/europe/8946_virtual_training_could_improve_surgeons_operating_skills_.html
- http://www.science.lu/fr/content/des-exp%C3%A9riences-virtuelles-pour-les-chirurgiens

# Publications

# Movies

# Meshless brain models

# Brain polyhedral and HEX-dominant meshes

Thanks to Bruno Lévy and Vorpaline

# Tetrahedral brain meshes