Fast calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements and the reciprocity theorem

Magnetic induction tomography of biological tissue is used to reconstruct the changes in the complex conductivity distribution by measuring the perturbation of an alternating primary magnetic field. To facilitate the sensitivity analysis and the solution of the inverse problem a fast calculation of the sensitivity matrix, i.e. the Jacobian matrix, which maps the changes of the conductivity distribution onto the changes of the voltage induced in a receiver coil, is needed. The use of finite differences to determine the entries of the sensitivity matrix does not represent a feasible solution because of the high computational costs of the basic eddy current problem. Therefore, the reciprocity theorem was exploited. The basic eddy current problem was simulated by the finite element method using symmetric tetrahedral edge elements of second order. To test the method various simulations were carried out and discussed.     

Talairach–Tournoux brain atlas registration using a metalforming principle-based finite element method

In this paper, a novel non-rigid registration method is proposed for registration of the Talairach-Tournoux brain atlas with MRI images and the Schaltenbrand-Wahren brain atlas. A metalforming principle-based finite element method with the large deformation problem is used to find the local deformation, in which finite element equations are governed by constraints in the form of displacements derived from the correspondence relationship between extracted feature points. Some detectable substructures, such as the cortical surface, ventricles and corpus callosum, are first extracted from MRI, forming feature points which are classified into different groups. The softassign method is used to establish the correspondence relationship between feature points within each group and to obtain the global transformation concurrently. The displacement constraints are then derived from the correspondence relationship. A metalforming principle-based finite element method with the large deformation problem is used in which finite element equations are reorganized and simplified by integrating the displacement constraints into the system equations. Our method not only matches the model to the data efficiently, but also decreases the degrees of freedom of the system and consequently reduces the computational cost. The method is illustrated by matching the Talairach-Tournoux brain atlas to MRI normal and pathological data and to the Schaltenbrand-Wahren brain atlas. We compare the results quantitatively between the force assignment-based method and the proposed method. The results show that the proposed method yields more accurate results in a fraction of the time taken by the previous method.     

A unified statistical approach to deformation-based morphometry

We present a unified statistical framework for analyzing temporally varying brain morphology using the 3D displacement vector field from a nonlinear deformation required to register a subject's brain to an atlas brain. The unification comes from a single model for structural change, rather than two separate models, one for displacement and one for volume changes. The displacement velocity field rather than the displacement itself is used to set up a linear model to account for temporal variations. By introducing the rate of the Jacobian change of the deformation, the local volume change at each voxel can be computed and used to measure possible brain tissue growth or loss. We have applied this method to detecting regions of a morphological change in a group of children and adolescents. Using structural magnetic resonance images for 28 children and adolescents taken at different time intervals, we demonstrate how this method works.     

Serial registration of intraoperative MR images of the brain

The increased use of image-guided surgery systems during neurosurgery has brought to prominence the inaccuracies of conventional intraoperative navigation systems caused by shape changes such as those due to brain shift. We propose a method to track the deformation of the brain and update preoperative images using intraoperative MR images acquired at different crucial time points during surgery. We use a deformable surface matching algorithm to capture the deformation of boundaries of key structures (cortical surface, ventricles and tumor) throughout the neurosurgical procedure, and a linear finite element elastic model to infer a volumetric deformation. The boundary data are extracted from intraoperative MR images using a real-time intraoperative segmentation algorithm. The algorithm has been applied to a sequence of intraoperative MR images of the brain exhibiting brain shift and tumor resection. Our results characterize the brain shift after opening of the dura and at the different stages of tumor resection, and brain swelling afterwards. Analysis of the average deformation capture was assessed by comparing landmarks identified manually and the results indicate an accuracy of 0.7+/-0.6 mm (mean+/-S.D.) for boundary surface landmarks, of 0.9+/-0.6 mm for landmarks inside the boundary surfaces, and 1.6+/-0.9 mm for landmarks in the vicinity of the tumor.     

MR to ultrasound registration for image-guided prostate interventions

A deformable registration method is described that enables automatic alignment of magnetic resonance (MR) and 3D transrectal ultrasound (TRUS) images of the prostate gland. The method employs a novel "model-to-image" registration approach in which a deformable model of the gland surface, derived from an MR image, is registered automatically to a TRUS volume by maximising the likelihood of a particular model shape given a voxel-intensity-based feature that represents an estimate of surface normal vectors at the boundary of the gland. The deformation of the surface model is constrained by a patient-specific statistical model of gland deformation, which is trained using data provided by biomechanical simulations. Each simulation predicts the motion of a volumetric finite element mesh due to the random placement of a TRUS probe in the rectum. The use of biomechanical modelling in this way also allows a dense displacement field to be calculated within the prostate, which is then used to non-rigidly warp the MR image to match the TRUS image. Using data acquired from eight patients, and anatomical landmarks to quantify the registration accuracy, the median final RMS target registration error after performing 100 MR-TRUS registrations for each patient was 2.40 mm.     

Analytical solution to the three-dimensional electrical forward problem for an elliptical cylinder

In electrical impedance tomography, the shape of the object being imaged (such as the human thorax) is often complex. For this reason, numerical techniques, such as finite element method, are often used for solving the forward problem in 3D rather than analytical solutions which can only model simple geometrical shapes. However, an analytical solution to the 3D forward problem can often be useful. This paper will present an analytical solution to the forward problem for an elliptical cylinder whose eccentricity can be easily modified to approximate the shape of the human thorax.     

有限元分析在腰椎生物力学应用中的研究与进展

背景:有限元法运用数学形式概括脊柱的结构形状、材料性能、载荷边界条件等,通过改变其中任一参数以观察其对整个结构的影响。目的:回顾性分析腰椎有限元模型的建立方法、有限元软件在腰椎生物力学中的应用以及有限元分析在腰椎植骨融合后临近节段活动度的改变及应力改变中的应用。方法:由第一作者检索1990/2010年PubMed数据库及CNKI数据库有关有限元法在腰椎建模及腰椎不同植骨融合式对腰椎融合节段、临近节段应力改变影响等方面的文献。结果与结论:有限元分析被越来越广泛的应用于医学,并成为其重要组成部分。在脊柱方面,有限元分析可以全面了解不同植骨融合术式及各种内固定对其生物力学改变的影响,从而可以改进植骨融合术式及优化内固定器械。但有限元毕竟属于理论性的分析,未来的研究应将其与动物实验及长期临床随访相结合,从而可以很好地预测手术远期疗效,并对并发症的预防具有重要临床意义。     

Biomechanical modeling of the lateral decubitus posture during corrective scoliosis surgery

Background

Patient prone positioning in scoliosis surgeries modifies the spinal curves prior to instrumentation. However, the biomechanical effects of the lateral decubitus posture, used in anterior approaches and minimally invasive techniques, have not yet been investigated. The objectives were to develop and validate a finite element model simulating the spinal changes resulting from this positioning.

Methods

The 3D pre-op reconstructed geometries of six adolescent patients with idiopathic scoliosis were used to develop personalized finite element models of the spine, which integrated a three-step method simulating the lateral posture. Clinical indices were measured on pre- and intra-operative radiographs to validate the finite element model.

Findings

The major Cobb angle and apical vertebral translation were reduced by 44% and 37% respectively between the pre- and intra-op postures. Using appropriately oriented gravity forces and boundary conditions, the finite element model simulations represented adequately these changes, with average differences of 4° for the major Cobb angle and 4 mm for the apical vertebral translation with the radiographic values.

Interpretation

Lateral decubitus positioning significantly reduces the spinal deformities prior to instrumentation, as demonstrated by the finite element model. This study is a first step in the development of a modeling tool for the optimal adjustments of intra-operative positioning, which remains to be further investigated with complementary clinical studies.     

距骨缺血性坏死的有限元分析(英文)

背景:对距骨缺血性坏死的具体临床治疗,目前仍有很大争议。目的:观察不同程度距骨坏死时,距骨滑车关节面上的应力及其分布变化。方法:利用8具踝关节标本建立踝关节三维有限元模型,通过改变各模型距骨坏死的体积分析距骨滑车关节面压应力与其分布的变化规律,计算出距骨缺血性坏死可能诱发踝足创伤性关节炎或出现距骨塌陷的临界坏死体积。结果与结论:实验成功建立了踝关节三维有限元模型。分析发现距骨内、外侧坏死程度比较小时,距骨滑车关节面应力分布变化不大;当内侧距骨坏死体积达到(26.6±1.5)%,外侧距骨坏死体积达到(35.0±2.5)%时,距骨坏死组织与正常骨组织边界区域出现应力集中的现象,其应力区域呈不规则形,提示,此时诱发踝足创伤性关节炎或发生距骨体塌陷的危险性很高,应手术治疗。     

Review on solving the forward problem in EEG source analysis

Background

The aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.

Methods

While other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.

Results

It starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.

Conclusion

Solving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.     

Real-time simulation of contact and cutting of heterogeneous soft-tissues

This paper presents a numerical method for interactive (real-time) simulations, which considerably improves the accuracy of the response of heterogeneous soft-tissue models undergoing contact, cutting and other topological changes. We provide an integrated methodology able to deal both with the ill-conditioning issues associated with material heterogeneities, contact boundary conditions which are one of the main sources of inaccuracies, and cutting which is one of the most challenging issues in interactive simulations. Our approach is based on an implicit time integration of a non-linear finite element model. To enable real-time computations, we propose a new preconditioning technique, based on an asynchronous update at low frequency. The preconditioner is not only used to improve the computation of the deformation of the tissues, but also to simulate the contact response of homogeneous and heterogeneous bodies with the same accuracy. We also address the problem of cutting the heterogeneous structures and propose a method to update the preconditioner according to the topological modifications. Finally, we apply our approach to three challenging demonstrators: (i) a simulation of cataract surgery (ii) a simulation of laparoscopic hepatectomy (iii) a brain tumor surgery.     

Measuring size and shape of the hippocampus in MR images using a deformable shape model.

A method for segmentation and quantification of the shape and size of the hippocampus is proposed, based on an automated image analysis algorithm. The algorithm uses a deformable shape model to locate the hippocampus in magnetic resonance images and to determine a geometric representation of its boundary. The deformable model combines three types of information. First, it employs information about the geometric properties of the hippocampal boundary, from a local and relatively finer scale to a more global and relatively coarser scale. Second, the model includes a statistical characterization of normal shape variation across individuals, serving as prior knowledge to the algorithm. Third, the algorithm utilizes a number of manually defined boundary points, which can help guide the model deformation to the appropriate boundaries, wherever these boundaries are weak or not clearly defined in MR images. Excellent agreement is demonstrated between the algorithm and manual segmentations by well-trained raters, with a correlation coefficient equal to 0.97 and algorithm/rater differences statistically equivalent to interrater differences for manual definitions.     

A new nonlinear parameter in the developed strain-to-applied strain of the soft tissues and its application in ultrasound elasticity imaging

Strain developed under quasi-static deformation has been mostly used in ultrasound elasticity imaging (UEI) to determine the stiffness change of tissues. However, the strain measure in UEI is often less sensitive to a subtle change of stiffness. This is particularly true for Crohn's disease where we have applied strain imaging to the differentiation of acutely inflamed bowel from chronically fibrotic bowel. In this study, a new nonlinear elastic parameter of the soft tissues is proposed to overcome this limit. The purpose of this study is to evaluate the newly proposed method and demonstrate its feasibility in the UEI. A nonlinear characteristic of soft tissues over a relatively large dynamic range of strain was investigated. A simplified tissue model based on a finite element (FE) analysis was integrated with a laboratory developed ultrasound radio-frequency (RF) signal synthesis program. Two-dimensional speckle tracking was applied to this model to simulate the nonlinear behavior of the strain developed in a target inclusion over the applied average strain to the surrounding tissues. A nonlinear empirical equation was formulated and optimized to best match the developed strain-to-applied strain relation obtained from the FE simulation. The proposed nonlinear equation was applied to in vivo measurements and nonlinear parameters were further empirically optimized. For an animal model, acute and chronic inflammatory bowel disease was induced in Lewis rats with trinitrobenzene sulfonic acid (TNBS)-ethanol treatments. After UEI, histopathology and direct mechanical measurements were performed on the excised tissues. The extracted nonlinear parameter from the developed strain-to-applied strain relation differentiated the three different tissue types with 1.96 ± 0.12 for normal, 1.50 ± 0.09 for the acutely inflamed and 1.03 ± 0.08 for the chronically fibrotic tissue. T-tests determined that the nonlinear parameters between normal, acutely inflamed and fibrotic tissue types were statistically significantly different (normal/ fibrotic [p = 0.0000185], normal/acutely inflamed [p = 0.0013] and fibrotic/acutely inflamed [p = 0.0029]). This technique may provide a sensitive and robust tool to assess subtle stiffness changes in tissues such as in acutely inflamed bowel wall.     

An adaptive snake model for ultrasound image segmentation: modified trimmed mean filter, ramp integration and adaptive weighting parameters

The snake model is a widely-used approach to finding the boundary of the object of interest in an ultrasound image. However, due to the speckles, the weak edges and the tissue-related textures in an ultrasound image, conventional snake models usually cannot obtain the desired boundary satisfactorily. In this paper, we propose a new adaptive snake model for ultrasound image segmentation. The proposed snake model is composed of three major techniques, namely, the modified trimmed mean (MTM) filtering, ramp integration and adaptive weighting parameters. With the advantages of the mean and median filters, the MTM filter is employed to alleviate the speckle interference in the segmentation process. The weak edge enhancement by ramp integration attempts to capture the slowly varying edges, which are hard to capture by conventional snake models. The adaptive weighting parameter allows weighting of each energy term to change adaptively during the deformation process. The proposed snake model has been verified on the phantom and clinical ultrasound images. The experimental results showed that the proposed snake model achieves a reasonable performance with an initial contour placed 10 to 20 pixels away from the desired boundary. The mean minimal distances from the derived boundary to the desired boundary have been shown to be less than 3.5 (for CNR > or = 0.5) and 2.5 pixels, respectively, for the phantom and ultrasound images.     

C_4～C_6颈椎有限元模型的建立和多点力学加载方法

背景：人体颈椎运动是多节段之间相互力学及位移关系的变化,建立多节段有限元模型及多点力学加载方法可以为颈椎生物力学研究提供高精度的模型和科学的计算分析方法。目的：建立人体C4~C6颈椎三维有限元模型,并在此基础上提出多点力学加载方法。方法：以正常人C4~C6颈椎CT图像作为数据源,利用Mimics10.0、Ansys11.0有限元分析软件建立三维有限元模型,对其进行多点力学加载测试,模拟生理状态时颈椎的轴向、屈曲、后伸、侧弯、扭转运动,分析各运动状态下C4~C6颈椎关节突和椎间盘的应力和位移改变。结果与结论：建立的C4~C6颈椎有限元模型几何形态逼真,重现了C4~C6颈椎节段解剖结构外形,整体显示直观,表面无过多简化,建成后的三维有限元模型与实体组织具有良好的几何相似性。应力Se、Sz在不同加载工况时,前屈/后伸〉侧弯〉轴向加载。轴向加载载荷明显小,导致应力位移水平低。提示应用正常人体原始资料构建的C4~C6颈椎有限元模型以及多点力学加载分析的方法科学有效,为颈椎的生物力学研究提供了高精度模型和科学的计算分析方法。     

C4～C6颈椎有限元模型的建立和多点力学加载方法

背景:人体颈椎运动是多节段之间相互力学及位移关系的变化,建立多节段有限元模型及多点力学加载方法可以为颈椎生物力学研究提供高精度的模型和科学的计算分析方法。目的:建立人体C4~C6颈椎三维有限元模型,并在此基础上提出多点力学加载方法。方法:以正常人C4~C6颈椎CT图像作为数据源,利用Mimics10.0、Ansys11.0有限元分析软件建立三维有限元模型,对其进行多点力学加载测试,模拟生理状态时颈椎的轴向、屈曲、后伸、侧弯、扭转运动,分析各运动状态下C4~C6颈椎关节突和椎间盘的应力和位移改变。结果与结论:建立的C4~C6颈椎有限元模型几何形态逼真,重现了C4~C6颈椎节段解剖结构外形,整体显示直观,表面无过多简化,建成后的三维有限元模型与实体组织具有良好的几何相似性。应力Se、Sz在不同加载工况时,前屈/后伸>侧弯>轴向加载。轴向加载载荷明显小,导致应力位移水平低。提示应用正常人体原始资料构建的C4~C6颈椎有限元模型以及多点力学加载分析的方法科学有效,为颈椎的生物力学研究提供了高精度模型和科学的计算分析方法。     

Geodesic shape regression with multiple geometries and sparse parameters

Many problems in medicine are inherently dynamic processes which include the aspect of change over time, such as childhood development, aging, and disease progression. From medical images, numerous geometric structures can be extracted with various representations, such as landmarks, point clouds, curves, and surfaces. Different sources of geometry may characterize different aspects of the anatomy, such as fiber tracts from DTI and subcortical shapes from structural MRI, and therefore require a modeling scheme which can include various shape representations in any combination. In this paper, we present a geodesic regression model in the large deformation (LDDMM) framework applicable to multi-object complexes in a variety of shape representations. Our model decouples the deformation parameters from the specific shape representations, allowing the complexity of the model to reflect the nature of the shape changes, rather than the sampling of the data. As a consequence, the sparse representation of diffeomorphic flow allows for the straightforward embedding of a variety of geometry in different combinations, which all contribute towards the estimation of a single deformation of the ambient space. Additionally, the sparse representation along with the geodesic constraint results in a compact statistical model of shape change by a small number of parameters defined by the user. Experimental validation on multi-object complexes demonstrate robust model estimation across a variety of parameter settings. We further demonstrate the utility of our method to support the analysis of derived shape features, such as volume, and explore shape model extrapolation. Our method is freely available in the software package deformetrica which can be downloaded at www.deformetrica.org.     

膝下残肢界面应力准动态有限元模型研究

目的:为计算小腿截肢患者行走过程中残肢和接受腔之间的界面压力.方法:根据残肢、骨骼和假肢接受腔的真实几何形状建立了准动态有限元模型,采用了自动面面接触模型模拟皮肤和接受腔衬套之间的边界摩擦和滑动,并在考虑行走过程中外力和惯性载荷的同时,考虑了膝关节刚体位移和残肢弹性变形之间的耦合.结果:对一个步态周期内的残肢界面压力进行了仿真计算,结果表明界面压力主要分布在髌韧带区、腘窝区、胫骨内外侧.在站立相,除髌韧带区外,其他区域的压力具有与地面反力相似的变化规律.结论:本文所建立的准动态模型考虑了患者行走过程中膝关节位移,更接近于人体真实运动,能够用于假肢接受腔界面特性的预测和现代假肢的数字化设计.     

Quantifying anatomical shape variations in neurological disorders

We develop a multivariate analysis of brain anatomy to identify the relevant shape deformation patterns and quantify the shape changes that explain corresponding variations in clinical neuropsychological measures. We use kernel Partial Least Squares (PLS) and formulate a regression model in the tangent space of the manifold of diffeomorphisms characterized by deformation momenta. The scalar deformation momenta completely encode the diffeomorphic changes in anatomical shape. In this model, the clinical measures are the response variables, while the anatomical variability is treated as the independent variable. To better understand the “shape—clinical response” relationship, we also control for demographic confounders, such as age, gender, and years of education in our regression model. We evaluate the proposed methodology on the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database using baseline structural MR imaging data and neuropsychological evaluation test scores. We demonstrate the ability of our model to quantify the anatomical deformations in units of clinical response. Our results also demonstrate that the proposed method is generic and generates reliable shape deformations both in terms of the extracted patterns and the amount of shape changes. We found that while the hippocampus and amygdala emerge as mainly responsible for changes in test scores for global measures of dementia and memory function, they are not a determinant factor for executive function. Another critical finding was the appearance of thalamus and putamen as most important regions that relate to executive function. These resulting anatomical regions were consistent with very high confidence irrespective of the size of the population used in the study. This data-driven global analysis of brain anatomy was able to reach similar conclusions as other studies in Alzheimer’s disease based on predefined ROIs, together with the identification of other new patterns of deformation. The proposed methodology thus holds promise for discovering new patterns of shape changes in the human brain that could add to our understanding of disease progression in neurological disorders.     

An intensity consistent filtering approach to the analysis of deformation tensor derived maps of brain shape

Deformation tensor morphometry makes use of the derivatives of spatial transformations between anatomies, to provide highly localized volumetric maps of relative anatomical size. The analysis of such maps, however, has the challenge of describing the data in a way that allows the spatial scale and extent of the local shape properties to match those induced by the disease process being studied. This study examines an approach to the spatial filtering of transformation Jacobian maps created in multisubject studies of brain anatomy, which constrains the filter neighborhood within common structural boundaries present in the spatially normalized image data. The filtering incorporates information derived from the spatial normalization process, using a statistical framework to introduce a measure of uncertainty in local regional intensity correspondence following spatial normalisation. The proposed filtering approach is compared to the use of spatially invariant Gaussian filtering in the analysis of Jacobian determinant maps of brain shape and shape change in Alzheimer's disease and normal aging. Results show significantly improved delineation of fine scale patterns of shape difference (in cross-sectional studies) and shape change (from multiple serial magnetic resonance imaging studies).