Search

Finite element models (FEMs) of the spine commonly use a limited number of simplified geometries. Nevertheless, the geometric features of the spine are important in determining its FEM outcomes. The link between a spinal segment’s shape and its biomechanical response has been studied, but the co-variances of the shape features have been omitted. We used a principal component (PCA)-based statistical shape modelling (SSM) approach to investigate the contribution of shape features to the intradiscal pressure (IDP) and the facets contact pressure (FCP) in a cohort of synthetic L4/L5 functional spinal units under axial compression. We quantified the uncertainty in the FEM results, and the contribution of individual shape modes to these results. This parameterisation approach is able to capture the variability in the correlated anatomical features in a real population and sample plausible synthetic geometries. The first shape mode ($$\phi _1$$ ϕ 1 ) explained 22.6% of the shape variation in the subject-specific cohort used to train the SSM, and had the largest correlation with, and contribution to IDP (17%) and FCP (11%). The largest geometric variation in ($$\phi _1$$ ϕ 1 ) was in the annulus-nucleus ratio.

How prevalent is spontaneous thrombosis (ST) in intracranial aneurysms (IAs) for an all-size pop- ulation? How can we calibrate computational models of thrombosis based on published data from size-specific aneurysms cohorts? How does ST differ in normo- and hypertensive subjects? We ad- dress the first question by a thorough analysis of published that provide ST rates across different patient demographics and aneurysm characteristics. This analysis provides data for a subgroup of the gen- eral population, viz. large and giant aneurysms (>10 mm). Based on these observed ST rates, our novel computational modelling platform enables the first in-silico observational study of ST prevalence across a broader set of IA phenotypes. We generate 109 virtual patients and use a novel approach to calibrate two trigger thresholds: residence time (RT) and shear rate (SR), thus addressing the second question. We then address the third question by utilising this calibrated thrombosis model to provide new insights on the effects of hypertension on ST. We demonstrate how a mechanistic ST model calibrated on a reduced IA cohort can help esti- mate ST prevalence in a broader IA population. This study was enabled through a comprehensive and fully automatic multi-scale modeling pipeline. We use an emerging property, viz. ST, as an indirect population-level validation of a complex computational modeling framework. Furthermore, our frame- work allows exploration of the influence of hypertension in ST. This lays the foundation for in-silico clinical trials of cerebrovascular devices in high-risk populations, e.g. assessing the performance of flow diverters in cerebral aneurysms for hypertensive patients.

Finite element models (FEMs) of the spine commonly use a limited number of simplified geometries. Nevertheless, the geometric features of the spine are important in determining its FEM outcomes. The link between a spinal segment's shape and its biomechanical response has been studied, but the co-variances of the shape features have been omitted. We used a principal component (PCA)-based statistical shape modelling (SSM) approach to investigate the contribution of shape features to the intradiscal pressure (IDP) and the facets contact pressure (FCP) in a cohort of synthetic L4/L5 functional spinal units under axial compression. We quantified the uncertainty in the FEM results, and the contribution of individual shape modes to these results. This parameterisation approach is able to capture the variability in the correlated anatomical features in a real population and sample plausible synthetic geometries. The first shape mode ([Formula: see text]) explained 22.6% of the shape variation in the subject-specific cohort used to train the SSM, and had the largest correlation with, and contribution to IDP (17%) and FCP (11%). The largest geometric variation in ([Formula: see text]) was in the annulus-nucleus ratio.

Deformable image registration (DIR) can be used to track cardiac motion. Conventional DIR algorithms aim to establish a dense and non-linear correspondence between independent pairs of images. They are, nevertheless, computationally intensive and do not consider temporal dependencies to regulate the estimated motion in a cardiac cycle. In this paper, leveraging deep learning methods, we formulate a novel hierarchical probabilistic model, termed DragNet, for fast and reliable spatio-temporal registration in cine cardiac magnetic resonance (CMR) images and for generating synthetic heart motion sequences. DragNet is a variational inference framework, which takes an image from the sequence in combination with the hidden states of a recurrent neural network (RNN) as inputs to an inference network per time step. As part of this framework, we condition the prior probability of the latent variables on the hidden states of the RNN utilised to capture temporal dependencies. We further condition the posterior of the motion field on a latent variable from hierarchy and features from the moving image. Subsequently, the RNN updates the hidden state variables based on the feature maps of the fixed image and the latent variables. Different from traditional methods, DragNet performs registration on unseen sequences in a forward pass, which significantly expedites the registration process. Besides, DragNet enables generating a large number of realistic synthetic image sequences given only one frame, where the corresponding deformations are also retrieved. The probabilistic framework allows for computing spatio-temporal uncertainties in the estimated motion fields. Our results show that DragNet performance is comparable with state-of-the-art methods in terms of registration accuracy, with the advantage of offering analytical pixel-wise motion uncertainty estimation across a cardiac cycle and being a motion generator. We will make our code publicly available.

Real-time augmented reality (AR) for minimally invasive surgery without extra tracking devices is a valuable yet challenging task, especially considering dynamic surgery environments. Multiple different motions between target organs are induced by respiration, cardiac motion or operative tools, and often must be characterized by a moving, manually positioned endoscope. Therefore, a 6DoF motion tracking method that takes advantage of the latest 2D target tracking methods and non-linear pose optimization and tracking loss retrieval in SLAM technologies is proposed and can be embedded into such an AR system. Specifically, the SiamMask deep learning-based target tracking method is incorporated to roughly exclude motion distractions and enable frame matching. This algorithm’s light computation cost makes it possible for the proposed method to run in real-time. A global map and a set of keyframes as in ORB-SLAM are maintained for pose optimization and tracking loss retrieval. The stereo matching and frame matching methods are improved and a new strategy to select reference frames is introduced to make the first-time motion estimation of every arriving frame as accurate as possible. Experiments on both a clinical laparoscopic partial nephrectomy dataset and an ex-vivo porcine kidney dataset are conducted. The results show that the proposed method gives a more robust and accurate performance compared with ORB-SLAM2 in the presence of motion distractions or motion blur; however, heavy smoke still remains a big factor that reduces the tracking accuracy.

An improved skin tissue substitute for use in photoelastic testing is required to enable investigation of the mechanics of needle insertion into soft tissue. Current tissue substitutes are mainly used in large scale testing and can neglect the small scale mechanical properties of soft tissue. A series of experiments on konjac glucomannan are performed to characterise its mechanical properties, and the results are compared to published results from similar experiments on skin tissue. The optical properties of the gel, such as its strain optic coefficient, are also assessed using a grey field polariscope (GFP2500). A concentration of 1.5% konjac to water produced a viscoelastic gel whose mechanical response closely matches published data for skin. A strain optic coefficient was recorded and found ideal for the planned testing with a GFP2500. Overall konjac glucomannan was found to be a potential soft tissue surrogate for use in small scale photoelastic testing.

This work presents a locking-free smoothed finite element method (S-FEM) for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity. The proposed method overcomes well-known issues of standard finite element methods (FEM) in the incompressible limit: the over-estimation of stiffness and sensitivity to severely distorted meshes. The concepts of cell-based, edge-based and node-based S-FEMs are extended in this paper to three-dimensions. Additionally, a cubic bubble function is utilized to improve accuracy and stability. For the bubble function, an additional displacement degree of freedom is added at the centroid of the element. Several numerical studies are performed demonstrating the stability and validity of the proposed approach. The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method.

In this paper, we propose to train deep neural networks with biomechanical simulations, to predict the prostate motion encountered during ultrasound-guided interventions. In this application, unstructured points are sampled from segmented pre-operative MR images to represent the anatomical regions of interest. The point sets are then assigned with point-specific material properties and displacement loads, forming the un-ordered input feature vectors. An adapted PointNet can be trained to predict the nodal displacements, using finite element (FE) simulations as ground-truth data. Furthermore, a versatile bootstrap aggregating mechanism is validated to accommodate the variable number of feature vectors due to different patient geometries, comprised of a training-time bootstrap sampling and a model averaging inference. This results in a fast and accurate approximation to the FE solutions without requiring subject-specific solid meshing. Based on 160,000 nonlinear FE simulations on clinical imaging data from 320 patients, we demonstrate that the trained networks generalise to unstructured point sets sampled directly from holdout patient segmentation, yielding a near real-time inference and an expected error of 0.017 mm in predicted nodal displacement.

Accurately determining the spatial relationship between the pelvis and acetabulum is challenging due to their inherently complex three-dimensional (3D) anatomy. A standardized 3D pelvic coordinate system (PCS) and the precise assessment of acetabular orientation would enable the relationship to be determined. We present a surface-based method to establish a reliable PCS and develop software for semi-automatic measurement of acetabular spatial parameters. Vertices on the acetabular rim were manually extracted as an eigenpoint set after 3D models were imported into the software. A reliable PCS consisting of the anterior pelvic plane, midsagittal pelvic plane, and transverse pelvic plane was then computed by iteration on mesh data. A spatial circle was fitted as a succinct description of the acetabular rim. Finally, a series of mutual spatial parameters between the pelvis and acetabulum were determined semi-automatically, including the center of rotation, radius, and acetabular orientation. Pelvic models were reconstructed based on high-resolution computed tomography images. Inter- and intra-rater correlations for measurements of mutual spatial parameters were almost perfect, showing our method affords very reproducible measurements. The approach will thus be useful for analyzing anatomic data and has potential applications for preoperative planning in individuals receiving total hip arthroplasty.

This paper offers a new insight into the computational modelling of crack and delamination of carbon fiber composite. Both transversal cracks (intralaminar) and delamination (interlaminar) are modelled with Extended Finite Element Method (XFEM). Constitutive and fracture laws are integrated to model the initiation of crack or delamination, and their subsequent evolution. The study includes the size effect assessment of composite due to the increment of composite thickness. The results are in close agreement between the experimental and analytical data of each specimen modelled based on the size of the carbon fiber composite volume.

Numerous problems of relevance in physiology and biomechanics, have at their core, the presence of a deformable solid matrix which experiences flow-induced strain. Often, this fluid–structure interaction (FSI) is directed the opposite way, i.e. it is solid deformation that creates flow, with the heart being the most prominent example. In many cases, this interaction of fluid and solid is genuinely bidirectional and strongly coupled, with solid deformation inducing flow and fluid pressure deforming the solid. Although an FSI problem, numerous cases in biomechanics are not tractable via the traditional FSI methodologies: in the internal flows that are of interest to use, the number and range of fluid passages is so vast that the direct approach of a deterministically defined boundary between fluid and solid is impossible to apply. In these cases, homogenisation and statistical treatment of the material-fluid system is possibly the only way forward. Such homogenisation,quite common to flow-only systems through porous media considerations, is also possible for FSI systems, where the loading is effectively internal to the material. A prominent technique of this type is that of poroelasticity. In this paper, we discuss a class of poroelastic theory techniques that allow for the co-existence of a multitude of – always statistically treated – channels and passages of widely different properties: termed multiple-network poroelasticity (or multicompartmental poroelasticity). This paradigm is particularly suitable for the study of living tissue, that is invariably permeated – perfused – by fluids, often different in nature and across a wide range of scales. Multicompartmental poroelasticity is capable of accounting for bidirectional coupling between the fluids and the solid matrix and allows us to track transport of a multitude of substances together with the deformation of the solid material that this transport gives rise to or is caused by, or both. For the purposes of demonstration, we utilise a complex and physiologically very important system, the human brain (specifically, we target the hippocampus), to exemplify the qualities and efficacy of this methodology during the course of Alzheimer’s Disease. The methodology we present has been implemented through the Finite Element Method, in a general manner, allowing for the co-existence of an arbitrary number of compartments. For the applications used in this paper to exemplify the method, a four-compartment implementation is used. A unified pipeline is used on a cohort of 35 subjects to provide statistically meaningful insight into the underlying mechanisms of the neurovascular unit (NVU) in the hippocampus, and to ascertain whether physical activity would have an influence in both swelling and drainage by taking into account both the scaled strain field and the proportion of perfused blood injected into the brain tissue. A key result garnered from his study is the statistically significant differences in right hemisphere hippocampal NVU swelling between males in the control group and females with mild cognitive impairment during high and low activity states.

For precision medicine to be implemented through the lens of in silico technology, it is imperative that biophysical research workflows offer insight into treatments that are specific to a particular illness and to a particular subject. The boundaries of precision medicine can be extended using multiscale, biophysics-centred workflows that consider the fundamental underpinnings of the constituents of cells and tissues and their dynamic environments. Utilising numerical techniques that can capture the broad spectrum of biological flows within complex, deformable and permeable organs and tissues is of paramount importance when considering the core prerequisites of any state-of-the-art precision medicine pipeline. In this work, a succinct breakdown of two precision medicine pipelines developed within two Virtual Physiological Human (VPH) projects are given. The first workflow is targeted on the trajectory of Alzheimer's Disease, and caters for novel hypothesis testing through a multicompartmental poroelastic model which is integrated with a high throughput imaging workflow and subject-specific blood flow variability model. The second workflow gives rise to the patient specific exploration of Aortic Dissections via a multi-scale and compliant model, harnessing imaging, computational fluid-dynamics (CFD) and dynamic boundary conditions. Results relating to the first workflow include some core outputs of the multiporoelastic modelling framework, and the representation of peri-arterial swelling and peri-venous drainage solution fields. The latter solution fields were statistically analysed for a cohort of thirty-five subjects (stratified with respect to disease status, gender and activity level). The second workflow allowed for a better understanding of complex aortic dissection cases utilising both a rigid-wall model informed by minimal and clinically common datasets as well as a moving-wall model informed by rich datasets. ispartof: Morphologie vol:103 issue:343 pages:148-160 ispartof: location:France status: published

A probabilistic framework for registering generalised point sets comprising multiple voxel-wise data features such as positions, orientations and scalar-valued quantities, is proposed. It is employed for the analysis of magnetic resonance diffusion tensor image (DTI)-derived quantities, such as fractional anisotropy (FA) and fibre orientation, across multiple subjects. A hybrid Student’s t-Watson-Gaussian mixture model-based non-rigid registration framework is formulated for the joint registration and clustering of voxel-wise DTI-derived data, acquired from multiple subjects. The proposed approach jointly estimates the non-rigid transformations necessary to register an unbiased mean template (represented as a 7-dimensional hybrid point set comprising spatial positions, fibre orientations and FA values) to white matter regions of interest (ROIs), and approximates the joint distribution of voxel spatial positions, their associated principal diffusion axes, and FA. Specific white matter ROIs, namely, the corpus callosum and cingulum, are analysed across healthy control (HC) subjects (K = 20 samples) and patients diagnosed with mild cognitive impairment (MCI) (K = 20 samples) or Alzheimer’s disease (AD) (K = 20 samples) using the proposed framework, facilitating inter-group comparisons of FA and fibre orientations. Group-wise analyses of the latter is not afforded by conventional approaches such as tract-based spatial statistics (TBSS) and voxel-based morphometry (VBM).

Residual stress has a great influence on the mechanical behaviour of arterial wall. Numerous research groups used the Uniform Stress Hypothesis to allow the inclusion of the effects of residual stress when computing stress distributions in the arterial wall. Nevertheless, the available methods used for this purpose are very computationally expensive, due to their iterative nature. In this paper we present a new method for including the effects of residual stress on the computed stress distribution in the arterial wall.The new method, by using the Uniform Stress Hypothesis, enables computing the effect of residual stress by averaging stresses across the thickness of the arterial wall.Being a post-processing method for the computed stress distributions, the proposed method is computationally inexpensive, and thus, better suited for clinical applications than the previously used ones.The resulting stress distributions and values obtained using the proposed method based on the Uniform Stress Hypothesis are very close to the ones returned by an existing iterative method.

Skin tribology is complex and in situ behaviour of skin varies considerably between test subjects. The main influencing factor, elasticity, varies due to structural and moisture differences. To find a more reliable test platform, for the first time, synthetic and biological (tissue engineered) substitutes were compared to ex vivo skin, epidermis and dermis. Friction initially increased with rising hydration, before decreasing beyond a threshold for all samples. Friction for Synthetic skin and dermis increased at a similar rate to the other samples, but from a different starting point, and friction dropped at lower hydration. Tissue engineered skin could provide a reliable test platform, but the synthetic skin could only be used if the offset in the data is accounted for.

Medical image analysis has grown into a matured field challenged by progress made across all medical\ud imaging technologies and more recent breakthroughs in biological imaging. The cross-fertilisation\ud between medical image analysis, biomedical imaging physics and technology, and domain knowledge\ud from medicine and biology has spurred a truly interdisciplinary effort that stretched outside the original\ud boundaries of the disciplines that gave birth to this field and created stimulating and enriching synergies.\ud Consideration on how the field has evolved and the experience of the work carried out over the last\ud 15 years in our centre, has led us to envision a future emphasis of medical imaging in Precision Imaging.\ud Precision Imaging is not a new discipline but rather a distinct emphasis in medical imaging borne\ud at the cross-roads between, and unifying the efforts behind mechanistic and phenomenological modelbased\ud imaging. It captures three main directions in the effort to deal with the information deluge in\ud imaging sciences, and thus achieve wisdom from data, information, and knowledge. Precision Imaging is\ud finally characterised by being descriptive, predictive and integrative about the imaged object. This paper\ud provides a brief and personal perspective on how the field has evolved, summarises and formalises our\ud vision of Precision Imaging for Precision Medicine, and highlights some connections with past research\ud and current trends in the field.

Purpose MR elastography (MRE) of the brain is being explored as a biomarker of neurodegenerative disease such as dementia. However, MRE measures for healthy brain have varied widely. Differing wave delivery methodologies may have influenced this, hence finite element-based simulations were performed to explore this possibility. Methods The natural frequencies of a series of cranial models were calculated, and MRE-associated vibration was simulated for different wave delivery methods at varying frequency, using simple isotropic viscoelastic material models for the brain. Displacement fields and the corresponding brain constitutive properties estimated by standard inversion techniques were compared across delivery methods and frequencies. Results The delivery methods produced widely different MRE displacement fields and inversions. Furthermore, resonances at natural frequencies influenced the displacement patterns. Consequently, some delivery methods led to lower inversion errors than others, and the error on the storage modulus varied by up to 11% between methods. Conclusion Wave delivery has a considerable impact on brain MRE reliability. Assuming small variations in brain biomechanics, as recently reported to accompany neurodegenerative disease (e.g., 7% for Alzheimer's disease), the effect of wave delivery is important. Hence, a consensus should be established on a consistent methodology to ensure diagnostic and prognostic consistency.

Low trauma fractures are amongst the most frequently encountered problems in the clinical assessment and treatment of bones, with dramatic health consequences for individuals and high financial costs for health systems. Consequently, significant research efforts have been dedicated to the development of accurate computational models of bone biomechanics and strength. However, the estimation of the fabric tensors, which describe the microarchitecture of the bone, has proven to be challenging using in vivo imaging. On the other hand, existing research has shown that isotropic models do not produce accurate predictions of stress states within the bone, as the material properties of the trabecular bone are anisotropic. In this paper, we present the first biomechanical study that uses statistically-derived fabric tensors for the estimation of bone strength in order to obtain patient-specific results. We integrate a statistical predictive model of trabecular bone microarchitecture previously constructed from a sample of ex vivo micro-CT datasets within a biomechanical simulation workflow. We assess the accuracy and flexibility of the statistical approach by estimating fracture load for two different databases and bone sites, i.e., for the femur and the T12 vertebra. The results obtained demonstrate good agreement between the statistically-driven and micro-CT-based estimates, with concordance coefficients of 98.6 and 95.5% for the femur and vertebra datasets, respectively.

There is emerging evidence suggesting that Alzheimer's disease is a vascular disorder, caused by impaired cerebral perfusion, which may be promoted by cardiovascular risk factors that are strongly influenced by lifestyle. In order to develop an understanding of the exact nature of such a hypothesis, a biomechanical understanding of the influence of lifestyle factors is pursued. An extended poroelastic model of perfused parenchymal tissue coupled with separate workflows concerning subject-specific meshes, permeability tensor maps and cerebral blood flow variability is used. The subject-specific datasets used in the modelling of this paper were collected as part of prospective data collection. Two cases were simulated involving male, non-smokers (control and mild cognitive impairment (MCI) case) during two states of activity (high and low). Results showed a marginally reduced clearance of cerebrospinal fluid (CSF)/interstitial fluid (ISF), elevated parenchymal tissue displacement and CSF/ISF accumulation and drainage in the MCI case. The peak perfusion remained at 8 mm s −1 between the two cases.

Purpose Magnetic resonance elastography (MRE) of the brain has demonstrated potential as a biomarker of neurodegenerative disease such as dementia but requires further evaluation. Cranial anatomical features such as the falx cerebri and tentorium cerebelli membranes may influence MRE measurements through wave reflection and interference and tissue heterogeneity at their boundaries. We sought to determine the influence of these effects via simulation. Methods MRE-associated mechanical stimulation of the brain was simulated using steady state harmonic finite element analysis. Simulations of geometrical models and anthropomorphic brain models derived from anatomical MRI data of healthy individuals were compared. Constitutive parameters were taken from MRE measurements for healthy brain. Viscoelastic moduli were reconstructed from the simulated displacement fields and compared with ground truth. Results Interference patterns from reflections and heterogeneity resulted in artifacts in the reconstructions of viscoelastic moduli. Artifacts typically occurred in the vicinity of boundaries between different tissues within the cranium, with a magnitude of 10%–20%. Conclusion Given that MRE studies for neurodegenerative disease have reported only marginal variations in brain elasticity between controls and patients (e.g., 7% for Alzheimer's disease), the predicted errors are a potential confound to the development of MRE as a biomarker of dementia and other neurodegenerative diseases. Magn Reson Med, 2015. © 2015 Wiley Periodicals, Inc.

This paper describes a constitutive model for ballistic gelatin at the low strain rates experienced, for example, by soft tissues during surgery. While this material is most commonly associated with high speed projectile penetration and impact investigations, it has also been used extensively as a soft tissue simulant in validation studies for surgical technologies (e.g. surgical simulation and guidance systems), for which loading speeds and the corresponding mechanical response of the material are quite different. We conducted mechanical compression experiments on gelatin specimens at strain rates spanning two orders of magnitude ( ~ 0.001 – 0.1 s − 1 ) and observed a nonlinear load–displacement history and strong strain rate-dependence. A compact and efficient visco-hyperelastic constitutive model was then formulated and found to fit the experimental data well. An Ogden type strain energy density function was employed for the elastic component. A single Prony exponential term was found to be adequate to capture the observed rate-dependence of the response over multiple strain rates. The model lends itself to immediate use within many commercial finite element packages.

The personalization of trabecular micro-architecture has been recently shown to be important in patient-specific biomechanical models of the femur. However, high-resolution in vivo imaging of bone micro-architecture using existing modalities is still infeasible in practice due to the associated acquisition times, costs, and X-ray radiation exposure. In this study, we describe a statistical approach for the prediction of the femur micro-architecture based on the more easily extracted subject-specific bone shape and mineral density information. To this end, a training sample of ex vivo micro-CT images is used to learn the existing statistical relationships within the low and high resolution image data. More specifically, optimal bone shape and mineral density features are selected based on their predictive power and used within a partial least square regression model to estimate the unknown trabecular micro-architecture within the anatomical models of new subjects. The experimental results demonstrate the accuracy of the proposed approach, with average errors of 0.07 for both the degree of anisotropy and tensor norms.

Realistic modelling of soft tissue biomechanics and mechanical interactions between tissues is an important part of biomechanically-informed surgical image-guidance and surgical simulation. This submission details a contact-modelling pipeline suitable for implementation in explicit matrix-free FEM solvers. While these FEM algorithms have been shown to be very suitable for simulation of soft tissue biomechanics and successfully used in a number of image-guidance systems, contact modelling specifically for these solvers is rarely addressed, partly because the typically large number of time steps required with this class of FEM solvers has led to a perception of them being a poor choice for simulations requiring complex contact modelling. The presented algorithm is capable of handling most scenarios typically encountered in image-guidance. The contact forces are computed with an evolution of the Lagrange-multiplier method first used by Taylor and Flanagan in PRONTO 3D extended with spatio-temporal smoothing heuristics for improved stability and edge–edge collision handling, and a new friction model. For contact search, a bounding-volume hierarchy (BVH) is employed, which is capable of identifying self-collisions by means of the surface-normal bounding cone of Volino and Magnenat-Thalmann, in turn computed with a novel formula. The BVH is further optimised for the small time steps by reducing the number of bounding-volume refittings between iterations through identification of regions with mostly rigid motion and negligible deformation. Further optimisation is achieved by integrating the self-collision criterion in the BVH creation and updating algorithms. The effectiveness of the algorithm is demonstrated on a number of artificial test cases and meshes derived from medical image data. It is shown that the proposed algorithm reduces the cost of BVH refitting to the point where it becomes a negligible part of the overall computation time of the simulation. It is also shown that the proposed surface-normal cone computation formula leads to about 40 % fewer BVH subtrees that must be checked for self-collisions compared with the widely used method of Provot. The proposed contact-force formulation and friction model are evaluated on artificial test cases that allow for a comparison with a ground truth. The quality of the proposed contact forces is assessed in terms of trajectories and energy conservation; a $$