The Young modulus E measures the way the body resists to deformations induced by external forces. It has the dimension of a pressure force: a quite-rigid body has a high modulus, while a malleable one gets a lower one. Fig. 1 shows, for a simple square FE structure laid on a horizontal plane (i.e. with constraints of no-displacement applied to the lower nodes of the square mesh), the deformations induced by vertical forces for two different values of the Young modulus. The Young modulus of the left structure is ten times larger than for the right one. Note here that the Finite Element codes that were developed for this example assume no displacement in the transverse direction (plane strain hypothesis) as well as a small deformation framework (with relative deformations of the structure that do not exceed 10%). For those reasons, the results presented above should be considered as very qualitative ones, which give an idea of the main trends of the soft body deformation, and not as true representations of the reality.

The Poisson ratio ν describes the way a pressure force, applied in a specific direction, induces deformations in other directions. This parameter is sometimes associated with the way the structure accepts variations of its own volume. For example, a ratio close to zero means that the structure deforms along a dimension, without any change along the other dimensions of the system. On the contrary, in the case of a three-dimensional structure, a Poisson ratio close to 0.5 represents a quasi-uncompressible behaviour (at the second order level), i.e. with volume conservation (see 〚1〛 for details). Fig. 2 plots the deformations induced by the pressure forces for two different Poisson ratios.

Orthognathic surgery deals with skeletal deformities of the lower part of the skull. The correction of these deformities requires the cutting and repositioning of bones by osteotomies of the mandible and/or the maxilla. In the course of the European IGOS project (Image Guided Orthopaedic Surgery), the TIMC/CAMI group developed a 3D simulator to plan bone repositioning in maxillofacial surgery 〚3〛. This simulator integrates 3D cephalometric constraints for bony structures repositioning and orthodontic constraints for teeth occlusion. Following the diagnosis for repositioning of the bone structures provided by this simulator, biomechanics of the patient facial soft tissues has to be taken into account, in order to predict the ‘external aspect’ of the patient face after bones displacements. This complex prediction requires the elaboration of a complete 3D biomechanical model of the patient face, integrating the passive skin tissues and the active muscular structures that deform the face.

Building up 3D biomechanical model of the patient face is a complex task, highly demanding from a modelling point of view. Therefore, to avoid the prohibitive amount of manual work required to generate a model for each patient, the following strategy was adopted:

• • a generic face model, or ‘atlas’, was manually elaborated, including a specific geometrical design of the mesh;
• • starting from patient data (Computer Tomography data in this case), a non-rigid registration algorithm, the ‘Mesh-Matching’ 〚4〛, was used to adapt the generic model to the patient morphology, thus generating a Finite Element model of the patient face.

Facial skin has a layered structure composed of the epidermis (0.1 mm thick), dermis (0.5–3.5 mm thick) and hypodermis (fatty tissues connected to the skull). Facial muscles are inserted between those skin layers and the skull. As many orthognathic interventions concern the mandible and the maxilla, the modelling was mainly dedicated to the lower part of the face, with a special emphasis on soft tissues surrounding human lips.

In that perspective, a 3D generic mesh of the human face was manually designed 〚5〛. It is made of two layers of hexahedral and wedge elements (Fig. 3).

Finite Element Method is then used to discretise the linear elasticity equations at the nodes of this 3D mesh. For this, the commercial ANSYS^TM software was used, mainly because of the possibilities this Finite Element package offers in terms of large deformations and hyper-elasticity modelling framework. Biomechanical parameters (Young modulus and Poisson ratio) are chosen to replicate measurements made on human skin tissues. A value close to 0.5 is therefore chosen for the Poisson ratio, in order to model the quasi-incompressibility of tissues (which are mainly composed of water). Concerning the Young modulus values, distinctions were made between elements that were associated with passive skin tissues, and elements labelled as active ‘muscular’ elements. Based on Fung’s measurements 〚6〛, a value of 15 kPa is used for passive skin tissues. Measurements reported by Duck 〚7〛 were used for the mechanical characteristics of ‘active’ muscular elements, with a transverse isotropic stress/strain relationship. This requires two parameters for stiffness: a Young modulus E_fibres in the main direction of the fibres, and another modulus E_ortho in the orthogonal directions. Following Duck’s measurements, a stiffness of 6.2 kPa is used for both Young modulus at rest, while a 110-kPa value is given to E_fibres, when muscle activation is maximal. E_fibres values therefore linearly increase with muscular activation, while E_ortho remains constant.

The main facial muscles surrounding the lips were taken into account during the manual design of the 3D mesh. A set of elements inside this mesh were therefore labelled as muscular elements and associated with the following muscles: Buccinator, Risorius, Zygomaticus major and minor, Depressor anguli oris and Orbicularis oris (see 〚5〛 for details).

Fig. 4 plots the face deformations induced by the activation of the Zygomaticus major muscle (with a 1 N force value) in the model.

Data were extracted from a complete CT exam of the patient head. Two datasets of 3D points were automatically extracted: points located on the external surface of the skull (marching cubes algorithm, 〚8〛) and points located on the patient’s skin surface (automatic segmentation procedure). The Mesh-Matching algorithm 〚4〛 was then used to match the nodes of the generic mesh to these two datasets. The external nodes of the generic mesh were transformed to match the points segmented from the patient’s skin surface. In the same way, a non-rigid transformation was calculated to project the internal nodes of the generic mesh onto the points located on the external surface of the patient’s skull. By connecting together the nodes transformed by the Mesh-Matching algorithm, a new 3D mesh was provided for Finite Element Analysis adapted to the patient morphology. Fig. 5 shows the automatically generated model of a first patient.

Fig. 6 shows qualitative results for the simulation of an osteotomy of the patient mandible. A forward 5 mm displacement of the mandible was simulated. Note that relative deformations of the face, in the chin and lips regions can be observed.

To illustrate our approach, Fig. 7 presents in a simple way a constriction inside the upper airways.

An exact analytical solution for the flow through such a constriction is not available. Further, full numerical simulations of the unsteady three-dimensional flow through a deformable structure are still, at present time, impossible even using the recent numerical codes and using powerful computers 〚9〛. For these reasons, and also because the aim of this paper is to provide a qualitative description of a sleep apnoea, we will use in the following a very simplified flow theory based on the following assumptions:

• • as the airflow velocity in the upper airways is, in general, much smaller than the speed of sound (low Mach number flow), it can be assumed that the flow is locally incompressible;
• • it can be reasonably assumed that the time needed for the constriction to collapse (of order of a second) is large compared with typical flow convection times (the time needed for the flow to pass the constriction is of order of a few milliseconds); therefore, it will be assumed that the flow is quasi-steady (low Strouhal number flow).

The principle of mass conservation thus yields the following relationship:

Equations (1) and (2) must be corrected in order to take into account a spectacular viscous effect: flow separation. Indeed, it is expected that the strongest pressure losses be due to the phenomenon of flow separation at the outlet of the constriction. This phenomenon is due to the presence of a strong adverse pressure gradient that causes the flow to decelerate so rapidly that it separates from the walls to form a free jet. Very strong pressure losses, due to the appearance of turbulence downstream of the constriction, are associated with flow separation. As a matter of fact, the pressure recovery past the flow separation point is so small that it can in general be neglected. In the following, we consider that the flow separates from the walls of the constriction at the point where the area reaches 1.2 times the minimum area A₀ (see Fig. 7). This approached value was empirically proposed and constitutes an acceptable approximation of the phenomena 〚10〛.

To summarise, for a given pressure drop (p₁ – p₂), and for a given geometry of the constriction, the volume flow velocity Φ is:

Thus, the force exerted by the airflow onto the walls of the constriction can be computed by integrating the pressure along the x-axis up to the flow separation point. This force induces a deformation of the upper airways soft tissues, thus modifying the airways geometry, and therefore changing the pressure distribution along the airways. Next part describes the model that was developed to simulate this phenomenon.

The soft tissues that constitute the tongue, the soft palate (including the velum) and the pharyngeal walls are partly responsible for the SAS as their deformations can even lead to a total closure of the upper airways. In a first step, we only focused on the velo-pharyngeal region, at the intersection between the nasal and the oral cavity. In this perspective, a continuous model of the velum was elaborated and defined through the Finite Element Method. The codes that were developed for this model assume no displacement in the transverse direction (‘plane strain hypothesis’) as well as a small deformation framework. Again, for reasons of quasi-incompressibility, a value close to 0.5 was chosen for the Poisson ratio. A 10-kPa value was taken for the Young modulus, which seems coherent with values reported for tongue 〚11〛 and vocal folds 〚12〛. The geometry of the model was extracted from a single midsagittal radiography of a patient (Fig. 8).

The upper part of the model represents the velar tissues. Geometrical values were coherent with values reported in the literature 〚13, 14〛: thickness of the velum varies from one extremity to the other (with a mean of 5 mm), while the total length is of order of 30 mm.

For the current simulations, only the velum deformations were taken into account, as the pharyngeal walls (the lower part of the model in Fig. 8) were supposed to be rigid during FE computations. The two points located onto the left part of the velum were also considered as fixed in order to model the velar attachment to the hard palate. Last, simulations are limited to the midsagittal plane, but for the computation of pressure forces, a 30 mm value was taken for the velar thickness in the frontal plane.

An iterative process governed the coupling between the airflow pressure forces computation and the deformations of the FE model of the velum. An adaptive Runge–Kutta algorithm was used to solve the dynamical equations that govern the deformations of the model and its coupling with the airflow.

In order to simulate a respiratory cycle, the pressure difference (p₁ – p₂) was approximated by a sinusoid: p₁ – p₂ = p_max sin(4 π t). Fig. 9 shows this pressure command for half a period and with an 800 Pa maximal value.

For computation-time reasons, the time scale was divided by a factor of 10. Fig. 10 shows results of the simulations for the command depicted in Fig. 9. A clear reduction of the constriction can be observed, thus simulating a hypo-apnoea 〚15〛.

These results are consistent with the volume flow velocity limitation observed on patient suffering from SAS. Fig. 11 plots the simulated volume flow velocity for three values of p_max. A flow limitation is clearly identifiable at p_max = 800 Pa but tends to disappear when p_max decreases. This result is qualitatively consistent with the use for the nasal Continuous Positive Airway Pressure 〚16〛, which in case of flow-limitation syndrome is reducing the pressure drop.

Fig. 12 presents the simulated volume flow velocity for three different values of the Young modulus, that is to say for three different stiffness of the velum. As can be clearly noticed here, an increase of the stiffness value of the velum tends to reduce the hypo-apnoea phenomena. This result is consistent with some surgery techniques that try to modify mechanical properties of the velum by burning tissues, thus increasing their stiffness.

Le programme GMCAO (Gestes médico-chirurgicaux assistés par ordinateur) du laboratoire TIMC de l’Imag vise à assister les chirurgiens dans la réalisation de gestes diagnostiques ou thérapeutiques les plus précis et les moins invasifs possibles. Dans ce cadre, les chercheurs du laboratoire sont amenés à développer des modèles d’organes humains (on parle de modèles « biomécaniques » ou de modèles « physiques ») et à utiliser ces modèles pour simuler numériquement les conséquences de certains actes chirurgicaux.

Les structures osseuses ont été les premières à être ainsi modélisées, essentiellement du fait de leur rigidité et donc de la relative facilité à modéliser numériquement de telles structures. C’est plus récemment que les chercheurs ont abordé la modélisation des structures anatomiques molles dans un cadre de chirurgie assistée par ordinateur. Ce papier focalise sur l’intérêt de la mise place de tels outils de modélisation, en prenant pour illustration le domaine de la chirurgie maxillo-faciale assistée par ordinateur. Après une brève introduction sur les outils numériques de modélisation utilisés, deux applications seront plus particulièrement abordées : la chirurgie orthognathique (avec le développement d’un modèle biomécanique de la face) et le syndrome d’apnées du sommeil (avec la mise au point de modèles physiques d’interaction flux d’air/parois déformables).

L’outil mathématique qui semble le plus propice à une modélisation des propriétés d’élasticité continue des structures anatomiques molles est sans doute la méthode des éléments finis (MEF). Cette formulation a en effet le double avantage de rester fidèle aux propriétés élastiques et continues (puisqu’elle consiste, en fait, en une discrétisation des équations d’élasticité), tout en permettant, grâce à la notion d’élément, de définir différentes zones, avec des propriétés mécaniques distinctes. Les points clés de cette modélisation par éléments finis peuvent se résumer par :

• • un partage du domaine en éléments finis standards délimités par un certain nombre de nœuds ;
• • la discrétisation des équations d’élasticité continue en chacun des nœuds du domaine – les déplacements, vitesses et accélérations de ces nœuds, sont alors les inconnues du système, tandis que les variables cinématiques de chacun des points d’un élément sont obtenues après interpolation entre les nœuds – ;
• • la prise en compte des propriétés physiques du matériau à modéliser par l’intermédiaire de deux constantes d’élasticité, à savoir le module d’Young E (caractérisant la raideur du matériau) et le taux de Poisson ν (caractérisant la plus ou moins grande compressibilité du matériau) ;
• • la possibilité offerte alors par la méthode d’associer à chaque élément du domaine des propriétés physiques distinctes (avec, par exemple, des éléments se comportant comme des tissus passifs, tandis que d’autres peuvent simuler des propriétés de fibres musculaires).

3.1 Contexte

3.2 Modélisation biomécanique

4.1 Contexte

4.2 Modélisation physique du SAS

Même si leurs résultats doivent toujours être observés avec précaution, les modèles physiques (biomécanique des tissus mous, aérodynamique de l’écoulement) peuvent apporter, de notre point de vue, des compléments intéressants à la planification des actes chirurgicaux. Nous avons tenté dans ce papier d’en présenter les premières lignes et d’ébaucher des protocoles de chirurgie assistée par ordinateur dans les domaines du syndrome d’apnées du sommeil et de la chirurgie orthognathique.