Orbital and maxillofacial computer aided surgery: patient-specific finite element models to predict surgical outcomes

This paper addresses an important issue raised for the clinical relevance of Computer-Assisted Surgical applications, namely the methodology used to automatically build patient-specific finite element (FE) models of anatomical structures. From this perspective, a method is proposed, based on a technique called the mesh-matching method, followed by a process that corrects mesh irregularities. The mesh-matching algorithm generates patient-specific volume meshes from an existing generic model. The mesh regularization process is based on the Jacobian matrix transform related to the FE reference element and the current element.This method for generating patient-specific FE models is first applied to computer-assisted maxillofacial surgery, and more precisely, to the FE elastic modelling of patient facial soft tissues. For each patient, the planned bone osteotomies (mandible, maxilla, chin) are used as boundary conditions to deform the FE face model, in order to predict the aesthetic outcome of the surgery. Seven FE patient-specific models were successfully generated by our method. For one patient, the prediction of the FE model is qualitatively compared with the patient's post-operative appearance, measured from a computer tomography scan. Then, our methodology is applied to computer-assisted orbital surgery. It is, therefore, evaluated for the generation of 11 patient-specific FE poroelastic models of the orbital soft tissues. These models are used to predict the consequences of the surgical decompression of the orbit. More precisely, an average law is extrapolated from the simulations carried out for each patient model. This law links the size of the osteotomy (i.e. the surgical gesture) and the backward displacement of the eyeball (the consequence of the surgical gesture).     

Computer-aided hepatic tumour ablation: requirements and preliminary results

Surgical resection of hepatic tumours is not always possible, since it depends on different factors, among which their location inside the liver functional segments. Alternative techniques consist in local use of chemical or physical agents to destroy the tumour. Radio frequency and cryosurgical ablations are examples of such alternative techniques that may be performed percutaneously. This requires a precise localisation of the tumour placement during ablation. Computer-assisted surgery tools may be used in conjunction with these new ablation techniques to improve the therapeutic efficiency, whilst they benefit from minimal invasiveness. This paper introduces the principles of a system for computer-assisted hepatic tumour ablation and describes preliminary experiments focusing on data registration evaluation. To keep close to conventional protocols, we consider registration of pre-operative CT or MRI data to intra-operative echographic data.     

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided.

The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

Non-linear elastic deformable models for real-time surgery simulation

In this article, we describe the latest developments of the minimally invasive hepatic surgery simulator prototype developed at INRIA. A key problem with such a simulator is the physical modelling of soft tissues. We propose a new deformable model based on non-linear elasticity and the finite element method. This model is valid for large displacements, which means in particular that it is invariant with respect to rotations. This property improves the realism of the deformations and solves the problems related to the shortcomings of linear elasticity, which is only valid for small displacements. We also address the problem of volume variations by adding to our model incompressibility constraints. Finally, we demonstrate the relevance of this approach for the real-time simulation of laparoscopic surgical gestures on the liver.     

Modelling of non-linear elastic tissues for surgical simulation

Realistic modelling of the interaction between surgical instruments and human organs has been recognised as a key requirement in the development of high-fidelity surgical simulators. Primarily due to computational considerations, most of the past real-time surgical simulation research has assumed linear elastic behaviour for modelling tissues, even though human soft tissues generally possess non-linear properties. For a non-linear model, the well-known Poynting effect developed during shearing of the tissue results in normal forces not seen in a linear elastic model. Using constitutive equations of non-linear tissue models together with experiments, we show that the Poynting effect results in differences in force magnitude larger than the absolute human perception threshold for force discrimination in some tissues (e.g. myocardial tissues) but not in others (e.g. brain tissue simulants).     

Mathematical modelling of implant in an operated hernia for estimation of the repair persistence

This paper presents mathematical modelling of an implanted surgical mesh used in the repair process of the abdominal hernia. The synthetic implant is simulated by a membrane structure. The author provides a material modelling of the implant based on the dense net model appropriate for technical fabrics. The accuracy of the proposed solution is evaluated by comparing the simulations of the dynamic behaviour of the system with the experiments carried out on physical models of implanted mesh. The model can be used to estimate the repair persistence for different mesh materials, fixing systems and different numbers of tacks to be provided during the surgery in order to resist the cough pressure and required action to avoid hernia recurrence. The persistence of the repaired hernia is assessed on the basis of the values of the forces in the tissue–implant joints because the usual form of the repair failure is due to as the joint disconnection or tissue failure.     

Computational modelling and optimisation of soft tissue outcome in cranio-maxillofacial surgery planning

Cranio-maxillofacial (CMF) surgery operations are associated with rearrangement of facial hard and soft tissues, leading to dramatic changes in facial geometry. Often, correction of the aesthetical patient's appearance is the primary objective of the surgical intervention. Due to the complexity of the facial anatomy and the biomechanical behaviour of soft tissues, the result of the surgical impact cannot always be predicted on the basis of surgeon's intuition and experience alone. Computational modelling of soft tissue outcome using individual tomographic data and consistent numerical simulation of soft tissue mechanics can provide valuable information for surgeons during the planning stage. In this article, we present a general framework for computer-assisted planning of CMF surgery interventions that is based on the reconstruction of patient's anatomy from 3D computer tomography images and finite element analysis of soft tissue deformations. Examples from our clinical case studies that deal with the solution of direct and inverse surgical problems (i.e. soft tissue prediction, inverse implant shape design) demonstrate that the developed approach provides a useful tool for accurate prediction and optimisation of aesthetic surgery outcome.     

Mechanistic DNA damage simulations in Geant4-DNA part 1: A parameter study in a simplified geometry

Mechanistic modelling of DNA damage in Monte Carlo simulations is highly sensitive to the parameters that define DNA damage. In this work, we use a simple testing geometry to investigate how different choices of physics models and damage model parameters can change the estimation of DNA damage in a mechanistic DNA damage simulation built in Geant4-DNA. The choice of physics model can lead to variations by up to a factor of two in the yield of physically induced strand breaks, and the parameters that determine scavenging, and physical and chemical single strand break induction can have even larger consequences. Using low energy electrons as primary particles, a variety of parameters are tested in this geometry in order to arrive at a parameter set consistent with past simulation studies. We find that the modelling of scavenging can play an important role in determining results, and speculate that high-scavenging regimes, where only chemical radicals within 1 nm of DNA are simulated, could provide a good means of testing mechanistic DNA simulations.     

Analysis of the influence of modelling assumptions on the prediction of the elastic properties of cardiac fibres

Abstract

Although several numerical models of the human heart have been proposed in the literature, there are still several discrepancies among the results predicted by each model. These discrepancies can be attributed to the fact that each model has a number of assumptions and simplifications, which can limit the scope and precision of the numerical predictions obtained. Moreover, none of the works reported in the literature have assessed the influence of modelling assumptions on the predicted cardiac fiber elastic properties. In this paper a new passive mechanical model that combines the left ventricular (LV) pressure–volume in-vivo measurements with an indirect approach based on the finite element method (FEM), is proposed and used to analyze the influence of different modelling assumptions on the estimated elastic properties of the cardiac fiber. This analysis is carried out by varying modelling assumptions that are common to existing passive mechanical models. The results have shown that although the different modelling assumptions have a significant effect on the predicted value of the fiber elastic properties, they tend to lead to the same results. This suggests that simplified passive numerical models in combination with adjustment factors, are valid in comparison with more refined and complex LV passive models.     

Calibration of crushable foam plasticity models for synthetic bone material for use in finite element analysis of acetabular cup deformation and primary stability

Polyurethane (PU) foam is a material often used in biomechanical experiments and demands for the definition of crushable foam plasticity (CFP) in numerical simulations of the primary stability and deformation of implants, to describe the crushing behaviour appropriately. Material data of PU foams with five different densities (10–40 pounds per cubic foot were ascertained experimentally in uniaxial compression test and used to calibrate CFP models for finite element modelling. Additionally, experimental and numerical deformation, push-out and lever-out tests of press-fit acetabular cups were carried out to assess the influence of the chosen material definition (linear elastic and CFP) on the numerical results. Comparison of the experimentally and numerically determined force–displacement curves of the uniaxial compression test showed a mean deviation of less than 3%. In primary stability testing, the deviation between the experimental and numerical results was in a range of 0%–27% for CFP modelling and 64%–341% for the linear elastic model. The material definition selected, highly influenced the numerical results in the current study. The use of a CFP model is recommended for further numerical simulations, when a deformation of the foam beyond the yield strength is likely to occur.     

Two predators and one prey in a patchy environment: An application of MICMAC modelling

In this paper we apply a new modelling strategy, called MICMAC modelling for studying the possibility of coexistence of two predators, which live in a patchy environment, which feed exclusively on one and the same prey and which do not interact intra- or interspecifically except through this common resource. The MICMAC modelling strategy enables us to model the interaction within each of the patches, the interaction among the patches, and the interaction between these models. The results of the simulations reveal that each of the following factors can cause coexistence by itself or in collaboration with the others: differences in risk of local extinction, dynamical isolation of patches, differences in relative duration of the growth and starvation phase.     

Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches

Diffuse interface (phase field) models are developed for multi-component vesicle membranes with different lipid compositions and membranes with free boundary. These models are used to simulate the deformation of membranes under the elastic bending energy and the line tension energy with prescribed volume and surface area constraints. By comparing our numerical simulations with recent biological experiments, it is demonstrated that the diffuse interface models can effectively capture the rich phenomena associated with the multi-component vesicle transformation and thus offering great functionality in their simulation and modelling.     

The interplay between chromosome stability and cell cycle control explored through gene–gene interaction and computational simulation

Chromosome stability models are usually qualitative models derived from molecular-genetic mechanisms for DNA repair, DNA synthesis, and cell division. While qualitative models are informative, they are also challenging to reformulate as precise quantitative models. In this report we explore how (A) laboratory experiments, (B) quantitative simulation, and (C) seriation algorithms can inform models of chromosome stability. Laboratory experiments were used to identify 19 genes that when over-expressed cause chromosome instability in the yeast Saccharomyces cerevisiae. To better understand the molecular mechanisms by which these genes act, we explored their genetic interactions with 18 deletion mutations known to cause chromosome instability. Quantitative simulations based on a mathematical model of the cell cycle were used to predict the consequences of several genetic interactions. These simulations lead us to suspect that the chromosome instability genes cause cell-cycle perturbations. Cell-cycle involvement was confirmed using a seriation algorithm, which was used to analyze the genetic interaction matrix to reveal an underlying cyclical pattern. The seriation algorithm searched over 10¹⁴ possible arrangements of rows and columns to find one optimal arrangement, which correctly reflects events during cell cycle phases. To conclude, we illustrate how the molecular mechanisms behind these cell cycle events are consistent with established molecular interaction maps.     

Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interactions: a positive (resp. negative) arrow from a gene to another represents an activation (resp. inhibition) of the expression of the latter gene by some product of the former. Once such an interaction graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. There mathematical tools, among others, can be of some help. In this paper we discuss a rule proposed by Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential [C. Soulé, Graphic requirements for multistationarity, ComPlexUs 1 (2003) 123-133] or Boolean [E. Rémy, P. Ruet, D. Thieffry, Graphic requirements for multistability and attractive cycles in a boolean dynamical framework, 2005, Preprint] type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models. We then discuss open problems.     

Comparing parametric solid modelling/reconfiguration,global shape modelling and free-form deformation for the generation of 3D digital models of femurs from X-ray images

At present, computer assisted surgery systems help orthopaedic surgeons both plan and perform surgical procedures. To enable these systems to function, it is crucial to have at one's disposal 3D models of anatomical structures, surgical tools and prostheses (if required). This paper analyses and compares three methods for generating 3D digital models of anatomical structures starting from X-ray images: parametric solid modelling/reconfiguration, global shape modelling and free-form deformation. Seven experiences involving the generation of a femur model were conducted by software developers and different skilled users. These experiences are described in detail and compared at different stages and from different points of view.     

Vacuum Effects over the Closing of Enterocutaneous Fistulae: A Mathematical Modeling Approach

Enterocutaneous fistulae are pathological communications between the intestinal lumen and the abdominal skin. Under surgery the mortality of this pathology is very high, therefore a vacuum applying system has been carried previously on attempting to close these fistulae. The objective of this article is the understanding of how these treatments might work through deterministic mathematical modelling. Four models are here proposed based on several assumptions involving: the conservation of the flow in the fistula, a low enough Reynolds number justifying a laminar flow, the use of Poiseuille law to model the movement of the fistulous liquid, as well as phenomenological equations including the fistula tissue and intermediate chamber compressibility. Interestingly, the four models show fistulae closing behaviour during experimental time (t<60 sec). To compare the models, both, simulations and pressure measurements, carried out on the vacuum connected to the patients, are performed. Time course of pressure are then simulated (from each model) and fitted to the experimental data. The model which best describes actual measurements shows exponential pumping flux kinetics. Applying this model, numerical relationship between the fistula compressibility and closure time is presented. The models here developed would contribute to clarify the treatment mechanism and, eventually, improve the fistulae treatment.     

Soft tissue modelling for applications in virtual surgery and surgical robotics

Soft tissue modelling has gained a great deal of importance, for a large part due to its application in surgical training simulators for minimally invasive surgery (MIS). This article provides a structured overview of different continuum-mechanical models that have been developed over the years. It aims at facilitating model choice for specific soft tissue modelling applications. According to the complexity of the model, different features of soft biological tissue will be incorporated, i.e. nonlinearity, viscoelasticity, anisotropy, heterogeneity and finally, tissue damage during deformation. A brief summary of experimental methods for material characterisation and an introduction to methods for geometric modelling are also provided. The overview is non-exhaustive, focusing on the most important general models and models with specific biological applications. A trade-off in complexity must be made for enabling real-time simulation, but still maintaining realistic representation of the organ deformation. Depending on the organ and tissue types, different models with emphasis on certain features will prove to be more appropriate, meaning the optimal model choice is organ and tissue-dependent.     

Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

Plankton-toxin interaction with a variable input nutrient

A simple model of phytoplankton–zooplankton interaction with a periodic input nutrient is presented. The model is then used to study a nutrient–plankton interaction with a toxic substance that inhibits the growth rate of plankton populations. The effects of the toxin upon the existence, magnitude, and stability of the periodic solutions are discussed. Numerical simulations are also provided to illustrate analytical results and to compare more complicated dynamical behaviour.     

Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.