homomorphisms representations and characteristic polynomials of digraphs the existence of a homomorphism between two digraphs often implies many structural properties we collect in this paper some characterizations of various digraph homomorphisms using matrix equations and fiber partitions we also survey the relationship among the characteristic polynomials of a digraph and its divisors this includes an introduction of the concept of branched coverings of digraphs their voltage assignment representations and a decomposition formula for the characteristic polynomial of a branched cover with branch index 1 some open problems are included

polarization phase velocity and nmo velocity of qp waves in arbitrary weakly anisotropic media we present approximate formulas for the qp wave phase velocity polarization vector and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first order perturbation theory all these quantities are expressed in terms of weak anisotropy wa parameters which represent a natural generalization of parameters introduced by thomsen the formulas presented and the wa parameters have properties of thomsen s formulas and parameters 1 the approximate equations are considerably simpler than exact equations for qp waves 2 the wa parameters are nondimensional quantities and 3 in isotropic media the wa parameters are zero and the corresponding equations reduce to equations for isotropic media in contrast to thomsen s parameters the wa parameters are related linearly to the density normalized elastic parameters for the transversely isotropic media with vertical axis of symmetry the equations presented and the wa parameters reduce to the equations and linearized parameters of thomsen the accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy although anisotropy is rather strong the approximate formulas presented yield satisfactory results

an accurate validation of a computational model of a human lumbosacral segment clinical studies have recently documented that there is sufficient evidence to suggest that abnormal motion may be an indicator of abnormal mechanics of the spine and therefore may be associated with some types of low back pain however designating a motion as abnormal requires knowledge of normal motions this work hence aims to develop an accurate computational model to simulate the bio mechanical response of the whole lumbosacral spinal unit l1 s1 under physiological loadings and constraint conditions in order to meet this objective computed tomography ct scanning protocols finite element fe analysis and accurate constitutive modelling have been integrated then the ranges of motion rom under flexion extension and lateral bending moment were measured and compared with experimental data finding an excellent agreement in particular the ability of the model to reproduce the relative rotation between each couple of vertebrae was proved finally the shear stresses for the most extreme load cases were reported in order to predict which are the most risky conditions and where the maximum damage would be located the results indicate that the greater values of the stresses were located at l4 s1 levels just in the interfaces between disc and vertebrae across the posterior and posterolateral zone this result can be clinically correlated with the existence of damage exactly where the stresses were maximal in the proposed finite element model

the effect of control strategies for an active back support exoskeleton on spine loading and kinematics during lifting with mechanical loading as the main risk factor for lbp exoskeletons exo are designed to reduce the load on the back by taking over part of the moment normally generated by back muscles the present study investigated the effect of an active exoskeleton controlled using three different control modes inclination emg hybrid on spinal compression forces during lifting with various techniques ten healthy male subjects lifted a 15 kg box with three lifting techniques free squat stoop each of which was performed four times once without exo and once each with the three different control modes using inverse dynamics we calculated l5 s1 joint moments subsequently we estimated spine forces using an emg assisted trunk model peak compression forces substantially decreased by 17 8 when wearing the exo compared to no exo however this reduction was partly by about one third attributable to a reduction of 25 in peak lifting speed when wearing the exo while subtle differences in back load patterns were seen between the three control modes no differences in peak compression forces were found in part this may be related to limitations in the torque generating capacity of the exo therefore with the current limitations of the motors it was impossible to determine which of the control modes was best despite these limitations the exo still reduced both peak and cumulative compression forces by about 18

hypersensitivity of trunk biomechanical model predictions to errors in image based kinematics when using fully displacement control techniques abstract recent advances in medical imaging techniques have allowed pure displacement control trunk models to estimate spinal loads with no need to calculate muscle forces sensitivity of these models to the errors in post imaging evaluation of displacements reported to be 0 4 0 9 and 0 2 0 3 mm in vertebral displacements has not yet been investigated a monte carlo analysis was therefore used to assess the sensitivity of results in both musculoskeletal ms and passive finite element fe spine models to errors in measured displacements six static activities in upright standing flexed and extended postures were initially simulated using a force control hybrid ms fe model computed vertebral displacements were subsequently used to drive two distinct fully displacement control ms and fe models effects of alterations in the reference vertebral displacements at 3 error levels with sd standard deviation 0 1 0 2 and 0 3 mm in input translations together with respectively 0 2 0 4 and 0 6 in input rotations were investigated on the model predictions results indicated that outputs of both models had substantial task dependent sensitivities to errors in the measured vertebral translations for instance l5 s1 intradiscal pressures idps were considerably affected sd values reaching 1 05 mpa and axial compression and shear forces even reversed directions as translation errors increased to 0 3 mm outputs were however generally much less sensitive to errors in measured vertebral rotations accounting for the accuracies in image based kinematics measurements therefore it is concluded that the current measured vertebral translation errors at and beyond 0 1 mm are too large to drive biomechanical models of the spine

upwind finite difference calculation of traveltimes seismic traveltimes can be computed efficiently on a regular grid by an upwind finite difference method the method solves a conservation law that describes changes in the gradient components of the traveltime field the traveltime field itself is easily obtained from the solution of the conservation law by numerical integration the conservation law derives from the eikonal equation and its solution depicts the first arrival time field the upwind finite difference scheme can be implemented in fully vectorized form in contrast to a similar scheme proposed recently by vidale the resulting traveltime field is useful both in kirchhoff migration and modeling and in seismic tomography many reliable methods exist for the numerical solution of conservation laws which appear in fluid mechanics as statements of the conservation of mass momentum etc a first order upwind finite difference scheme proves accurate enough for seismic applications upwind schemes are stable because they mimic the behavior of flu

adjustments to zatsiorsky seluyanov s segment inertia parameters zatsiorsky et al in contemporary problems in biomechanics pp 272 291 crc press massachusetts 1990a obtained by means of a gamma ray scanning technique the relative body segment masses center of mass cm positions and radii of gyration for samples of college aged caucasian males and females although these data are the only available and comprehensive set of inertial parameters regarding young adult caucasians they have been rarely utilized for biomechanical analyses of subjects belonging to the same or a similar population the main reason is probably that zatsiorsky et al used bony landmarks as reference points for locating segment cms and defining segment lengths some of these landmarks were markedly distant from the joint centers currently used by most researchers as reference points the purpose of this study was to adjust the mean relative cm positions and radii of gyration reported by zatsiorsky et al in order to reference them to the joint centers or other commonly used landmarks rather than the original landmarks the adjustments were based on a number of carefully selected sources of anthropometric data

sensitivity of lumbar spine loading to anatomical parameters musculoskeletal simulations of lumbar spine loading rely on a geometrical representation of the anatomy however this data has an inherent inaccuracy this study evaluates the influence of defined geometrical parameters on lumbar spine loading utilising five parametrised musculoskeletal lumbar spine models for four different postures the influence of the dimensions of vertebral body disc posterior parts of the vertebrae as well as the curvature of the lumbar spine was studied additionally simulations with combinations of selected parameters were conducted changes in l4 l5 resultant joint force were used as outcome variable variations of the vertebral body height disc height transverse process width and the curvature of the lumbar spine were the most influential these parameters can be easily acquired from x rays and should be used to morph a musculoskeletal lumbar spine model for subject specific approaches with respect to bone geometry furthermore the model was very sensitive to uncommon configurations and therefore it is advised that stiffness properties of discs and ligaments should be individualised

response of trunk muscle coactivation to changes in spinal stability the goal of this effort was to assess the neuromuscular response to changes in spinal stability biomechanical models suggest that antagonistic co contraction may be related to stability constraints during lifting exertions a two dimensional biomechanical model of spinal equilibrium and stability was developed to predict trunk muscle co contraction as a function of lifting height and external load the model predicted antagonistic co contraction must increase with potential energy of the system even when the external moment was maintained at a constant value predicted trends were compared with measured electromyographic emg data recorded during static trunk extension exertions wherein subjects held weighted barbells at specific horizontal and vertical locations relative to the lumbo sacral spine junction the task was designed to assure the applied moment was identical during each height condition thereby changing potential energy without influencing moment measured emg activity in the trunk flexors increased with height of the external load as predicted by the model gender difference in spinal stability were also noted results empirically demonstrate that the neuromuscular system responds to changes in spinal stability and provide insight into the recruitment of trunk muscle activity

investigation of optimal follower load path generated by trunk muscle coordination it has been reported that the center of rotation of each vertebral body is located posterior to the vertebral body center moreover it has been suggested that an optimized follower load fl acts posterior to the vertebral body center however the optimal position of the fl with respect to typical biomechanical characteristics regarding spinal stabilization such as joint compressive force shear force joint moment and muscle stress has not been studied a variation in the center of rotation of each vertebra was formulated in a three dimensional finite element model of the lumbar spine with 117 pairs of trunk muscles then the optimal translation of the fl path connecting the centers of rotations was estimated by solving the optimization problem that was to simultaneously minimize the compressive forces the shear forces and the joint moments or to minimize the cubic muscle stresses an upright neutral standing position and a standing position with 200 n in both hands were considered the fl path moved posterior regardless of the optimization criteria and loading conditions the fl path moved 5 0 and 7 8 mm posterior in upright standing and 4 1 mm and 7 0 mm posterior in standing with 200 n in hands for each optimization scheme in addition it was presented that the optimal fl path may have advantages in comparison to the body center fl path the present techniques may be important in understanding the spine stabilization function of the trunk muscles

equitable switching and spectra of graphs abstract let g be a simple graph and an equitable partition of the vertex set v g it is well known that the characteristic polynomial g of a quotient graph g divides that of g the seidel swiching g of g with respect to a proper subset v of v g is called equitable if v is a union of some cells in we show that if g is equitable g g g g some applications of this theorem to generalized composition graphs isospectral graphs integral graphs and ramanujan graphs are contained

measurement of a spinal motion segment stiffness matrix the six degrees of freedom elastic behavior of spinal motion segments can be approximated by a stiffness matrix a method is described to measure this stiffness matrix directly with the motion segment held under physiological conditions of axial preload and in an isotonic fluid bath by measuring the forces and moments associated with each of the six orthogonal translations and rotations the stiffness matrix was obtained from the load displacement measurements by linear least squares assuming a symmetric matrix results from a pig lumbar spinal motion segment in an isotonic bath with and without a 500 n axial preload showed a large stiffening effect with axial preload

impact of material and morphological parameters on the mechanical response of the lumbar spine a finite element sensitivity study finite element models are frequently used to study lumbar spinal biomechanics deterministic models are used to reflect a certain configuration including the means of geometrical and material properties while probabilistic models account for the inherent variability in the population because model parameters are generally uncertain their predictive power is frequently questioned in the present study we determined the sensitivities of spinal forces and motions to material parameters of intervertebral discs vertebrae and ligaments and to lumbar morphology we performed 1200 model simulations using a generic model of the human lumbar spine loaded under pure moments coefficients of determination and of variation were determined for all parameter and response combinations material properties of the vertebrae displayed the least impact on results whereas those of the discs and morphology impacted most the most affected results were the axial compression forces in the vertebral body and in several ligaments during flexion and the facet joint forces during extension intervertebral rotations were considerably affected only when several parameters were varied simultaneously results can be used to decide which model parameters require careful consideration in deterministic models and which parameters might be omitted in probabilistic studies findings allow quantitative estimation of a model s precision

a non optimized follower load path may cause considerable intervertebral rotations abstract osseoligamentous spinal specimens buckle under even a small vertical compressive force to allow higher axial forces a compressive follower load fl was suggested previously that approximates the curvature of the spine without inducing intervertebral rotation in both the frontal and the sagittal planes in in vitro experiments and finite element analyses the location of the fl path is subjected to estimation by the investigator such non optimized fls may induce bending and so far it is still unknown how this affects the results of the study and their comparability a symmetrical finite element model of the lumbar spine was employed to simulate upright standing while applying a follower load in analogy to in vitro experiments the path of this fl was estimated seven times by different members of our institute s spine group additionally an optimized fl path was determined and additional moments of 7 5 nm were applied to simulate flexion and extension application of the optimized 500 n compressive fl causes only a marginal alteration of the curvature cardan angle l1 s1 in sagittal plane a preceding optimization of the location of the fl path would increase the comparability of different studies

the effect of three dimensional geometrical changes during adolescent growth on the biomechanics of a spinal motion segment abstract during adolescent growth vertebrae and intervertebral discs undergo various geometrical changes although such changes in geometry are well known their effects on spinal stiffness remains poorly understood however this understanding is essential in the treatment of spinal abnormalities during growth such as scoliosis a finite element model of an l3 l4 motion segment was developed validated and applied to study the quantitative effects of changing geometry during adolescent growth on spinal stiffness in flexion extension lateral bending and axial rotation height width and depth of the vertebrae and intervertebral disc were varied as were the width of the transverse processes the length of the spinous process the size of the nucleus facet joint areas and ligament size these variations were based on average growth data for girls as reported in literature overall adolescent growth increases the stiffness with 36 lateral bending and extension to 44 flexion two thirds of this increase occurs between 10 and 14 years of age and the last third between 14 years of age and maturity although the height is the largest geometrical change during adolescent growth its effect on the biomechanics is small the depth increase of the disc and vertebrae significantly affects the stiffness in all directions while the width increase mainly affects the lateral bending stiffness hence when analysing the biomechanics of the growing adolescent spine for instance in scoliosis research the inclusion of depth and width changes in addition to the usually implemented height change is essential

effects of inter individual lumbar spine geometry variation on load sharing geometrically personalized finite element study there is a large at times contradictory body of research relating spinal curvature to low back pain lbp mechanical load is considered as important factor in lbp etiology geometry of the spinal structures and sagittal curvature of the lumbar spine govern its mechanical behavior thus understanding how inter individual geometry particularly sagittal curvature variation affects the spinal load sharing becomes of high importance in lbp assessment this study calculated and compared kinematics and load sharing in three ligamentous lumbosacral spines one hypo lordotic hypo l with low lordosis one normal lordotic norm l with normal lordosis and one hyper lordotic hyper l with high lordosis in flexed and extended postures using 3d nonlinear finite element fe modeling these postures were simulated by applying follower load fl combined with flexion or extension moment the hypo l spine demonstrated stiffer behavior in flexion but more flexible response to extension compared to the norm l spine the excessive lordosis stiffened response of the hyper l spine to extension but did not affect its resistance to flexion compared to the norm l spine despite the different resisting actions of the posterior ligaments to flexion moment the increase of disc compression was similar in all the spines leading to similar load sharing however resistance of the facet joints to extension was more important in the norm and hyper l spines which reduced the disc compression the spinal curvature strongly influenced the magnitude and location of load on the spinal components and also altered the kinematics and load sharing particularly in extension consideration of the subject specific geometry and sagittal curvature should be an integral part of mechanical analysis of the lumbar spine

properties of the zeroth first and higher order approximations of attributes of elastic waves in weakly anisotropic media use of the perturbation theory in the study of attributes of elastic waves propagating in weakly anisotropic media leads to approximate but transparent and simple formulas which have many applications in forward and inverse wave modeling we present and study such formulas we show that all studied attributes depend on elements of a matrix linearly dependent on parameters of a medium we study this dependence with the goal to understand which parameters of the medium and in which combinations affect individual wave attributes alternative auxiliar vector bases in which the matrix can be specified are proposed and studied the vector bases offer alternative specifications of polarization vectors of qs waves one of the important observations is that the higher order n 2 perturbation formulas for qs waves are obtained separately for qs1 and qs2 waves we also study effects of the use of the perturbation theory on the accuracy of the determination of the acoustical axes in weakly anisotropic media we show that longitudinal directions in the first order approximation are identical with actual ones in singular directions however the first order formulas provide directions which may deviate from the exact ones or they may even indicate false singular directions again the above mentioned matrix depending linearly on the parameters of the medium plays a central role in this study

are there characteristic motion patterns in the lumbar spine during flexion abstract flexion is the main motion of the lumbar spine while in vitro tests with pure moments suggest larger intra segmental rotations for the more caudal segments in vivo results show diverging motion distributions the present study analysed the motion distribution in vivo of 320 asymptomatic subjects the change of the back curvature between standing and upper body flexion was determined using a non invasive measurement device linear bilinear trilinear quadratic and cubic regression models were fitted to the segmental motion distribution over the lengths of the lordosis to categorise characteristic motion patterns simplicity and approximation quality were used to assign the motion distributions to the regression models seventy seven percent of the motion distributions could be explained by a bilinear model a further 12 and 11 could be represented by a trilinear and linear model respectively less than 1 of the distributions could not satisfactorily be represented by the models all of the bilinear models displayed maximum flexion in approximately the middle of lordosis all linear models showed a decreasing rotation from caudal to cranial most of the trilinear models displayed a distribution similar to the bilinear age sex body height and weight did not significantly affect these distributions this in vivo study identified characteristic motion patterns in the lumbar spine during flexion the quantitative results provide a clear description of the healthy condition and may serve to identify spinal motion abnormalities

a dynamic programming approach to first arrival traveltime computation in media with arbitrarily distributed velocities curved ray tomographic traveltime inversion reverse time migration and various other seismic modeling applications require the calculation of traveltime and raypath information throughout a two or three dimensional medium when arbitrary velocity distributions and curved rays are involved traditional ray shooting or bending procedures can be time consuming and error prone a two dimensional dynamic programming traveltime computation technique based upon fermat s principle uses simple calculus techniques and a systematic mapping scheme to determine first arrival times on a uniform grid given an arbitrary discrete velocity distribution it accurately handles large contrast discontinuous velocity distributions including those that generate caustics first arrival seismic energy can travel either as transmitted waves diffracted waves or headwaves and this technique models all types the traveltime computations begin with starting values computed near the source location then the mapping systemat

characteristic polynomials of graphs having a semifree action j h kwak and j lee linear and multilinear algebra 32 1992 61 73 computed the characteristic polynomial of a finite graph g having an abelian automorphism group which acts freely on g for a finite weighted symmetric pseudograph g having an abelian automorphism group which acts semifreely on g k wang linear algebra appl 51 1983 121 125 showed that the characteristic polynomial of g is factorized into a product of a polynomial associated to the orbit graph and a polynomial associated to the free part of the action but he did not explicitly compute the characteristic polynomial of such a graph g in this paper we introduce a new method to construct a finite pseudograph g having an automorphism group which acts semifreely on g and obtain an explicit formula to compute the characteristic polynomial of such a graph by using the construction method 2000 elsevier science inc all rights reserved

geometry strongly influences the response of numerical models of the lumbar spine a probabilistic finite element analysis typical fe models of the human lumbar spine consider a single fixed geometry such models cannot account for potential effects of the natural variability of the spine s geometry in this study we performed a probabilistic uncertainty and sensitivity analysis of a fully parameterized geometrically simplified model of the l3 l4 segment we examined the impact of the uncertainty in all 40 geometry parameters estimated lower and upper bounds for the required sample size and determined the most important geometry parameters the natural variability of the spine s geometry indeed strongly affects intradiscal pressure range of motion and facet joint contact forces deriving generalized statements from fixed geometry models as well as transferring those results to different cases thus can easily lead to wrong conclusions and should only be performed with extreme caution we recommend a sample size of 100 to obtain reasonable accurate point estimates and a sufficient overview of the remaining uncertainties yet only few parameters especially those determining the disc geometry disc height end plate width and depth and the facets position intra articular space pedicle length facet angles proved to be truly important accurate measurement and modeling of those structures should therefore be prioritized

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

a fast sweeping method for eikonal equations in this paper a fast sweeping method for computing the numerical solution of eikonal equations on a rectangular grid is presented the method is an iterative method which uses upwind difference for discretization and uses gauss seidel iterations with alternating sweeping ordering to solve the discretized system the crucial idea is that each sweeping ordering follows a family of characteristics of the corresponding eikonal equation in a certain direction simultaneously the method has an optimal complexity of o n for n grid points and is extremely simple to implement in any number of dimensions monotonicity and stability properties of the fast sweeping algorithm are proven convergence and error estimates of the algorithm for computing the distance function is studied in detail it is shown that 2 n gauss seidel iterations is enough for the distance function in n dimensions an estimation of the number of iterations for general eikonal equations is also studied numerical examples are used to verify the analysis

traveltime and amplitude estimation using wavefront construction we have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two dimensional 2 d model the basic idea of this wavefront wf construction approach is to use ray tracing to estimate a new wf from the old one the wf is defined as a curve in 2 d of constant traveltime from the source the ray direction and amplitude will then be a function of s the distance along the front to maintain a sufficiently small sampling distance along the wf it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit as the wavefronts are constructed the data i e traveltimes amplitude coefficients etc are transferred to the receivers by interpolation within the ray cells advantages of the wf construction method are its flexibility robustness and accuracy first second and later arrivals may be found at any point in the model any shape of the initial wavefront is possible the drawb

sensitivity of predicted muscle forces to parameters of the optimization based human leg model revealed by analytical and numerical analyses there are different opinions in the literature on whether the cost functions the sum of muscle stresses squared and the sum of muscle stresses cubed can reasonably predict muscle forces in humans one potential reason for the discrepancy in the results could be that different authors use different sets of model parameters which could substantially affect forces predicted by optimization based models in this study the sensitivity of the optimal solution obtained by minimizing the above cost functions for a planar three degrees of freedom dof model of the leg with nine muscles was investigated analytically for the quadratic function and numerically for the cubic function analytical results revealed that generally the non zero optimal force of each muscle depends in a very complex non linear way on moments at all three joints and moment arms and physiological cross sectional areas pcsas of all muscles deviations of the model parameters moment arms and pcsas from their nominal values within a physiologically feasible range affected not only the magnitude of the forces predicted by both criteria but also the number of non zero forces in the optimal solution and the combination of muscles with non zero predicted forces muscle force magnitudes calculated by both criteria were similar they could change several times as model parameters changed whereas patterns of muscle forces were typically not as sensitive it is concluded that different opinions in the literature about the behavior of optimization based models can be potentially explained by differences in employed model parameters

approximate ray tracing for qp waves in inhomogeneous layered media with weak structural anisotropy we present approximate equations for qp wave ray tracing and paraxial ray tracing in inhomogeneous layered weakly transversely isotropic ti media inside layers the symmetry axis direction of the ti medium is allowed to vary continuously approximate equations are based on perturbation theory in which deviations of anisotropy from isotropy are considered to be first order quantities for qp waves propagating in a ti medium the approximate ray tracing and paraxial ray tracing equations depend on three parameters and two angles defining the direction of the symmetry axis we also present the boundary conditions at interfaces for first order rays and paraxial rays and compute reflection transmission coefficients to the first order all the quantities required for evaluation of the green s function are calculated to the first order except the traveltime that is calculated to the second order the accuracy of the presented algorithm is verified on simple models with respect to exact ray tracing results f

traveltime computation in transversely isotropic media an approach for calculating first arrival traveltimes in a transversely isotropic medium is developed and has the advantage of avoiding shadow zones while still being computationally fast also it works with an arbitrary velocity grid that may have discontinuities the method is based on fermat s principle the traveltime for each point in the grid is calculated several times using previously calculated traveltimes at surrounding grid points until the minimum time is found different ranges of propagation angle are covered in each traveltime calculation such that at the end of the process all propagation angles are covered this guarantees that the first arrival traveltime for a specific grid point is correctly calculated the resulting algorithm is fully vectorizable the method is robust and can accurately determine first arrival traveltimes in heterogeneous media traveltimes are compared to finite difference modeling of transversely isotropic media and are found to be in excellent agreement an appli

role of disc area and trabecular bone density on lumbar spinal column fracture risk curves under vertical impact abstract while studies have been conducted using human cadaver lumbar spines to understand injury biomechanics in terms of stability energy to fracture and physiological responses under pure moment follower loads data are sparse for inferior to superior impacts injuries occur under this mode from underbody blasts objectives determine role of age disc area and trabecular bone density on tolerances risk curves under vertical loading from a controlled group of specimens t12 s1 columns were obtained pretest x rays and cts taken load cells attached to both ends impacts applied at s1 end using custom vertical accelerator device and posttest x ray ct and dissections done bmd of l2 l4 vertebrae were obtained from qct survival analysis based human injury probability curves hipcs were derived using proximal and distal forces age area and bmd were covariates forces were considered uncensored representing the load carrying capacity the akaike information criterion was used to determine optimal distributions the mean forces 95 confidence intervals and normalized confidence interval size ncis were computed the lognormal distribution was the optimal function for both forces age area and bmd were not significant p 0 05 covariates for distal forces while only bmd was significant for proximal forces the ncis was the lowest for force bmd covariate hipc the hipcs for both genders at 35 and 45 years were based on population bmds these hipcs serve as human tolerance criteria for automotive military and other applications in this controlled group of samples bmd is a better predictor covariate that characterizes lumbar column injury under inferior to superior impacts

characteristic polynomials of symmetric graphs abstract in this paper we study the characteristic polynomials of graphs which admit semifree actions of an abelian group using the method of group matrices we are able to show that the characteristic polynomial of a such a graph is factorized into a product of a polynomial associated to the orbit of the action and a polynomial associated to the free part of the action

effects of eight different ligament property datasets on biomechanics of a lumbar l4 l5 finite element model ligaments assist trunk muscles in balancing external moments and providing spinal stability in absence of the personalized material properties for ligaments finite element fe models use dispersed data from the literature this study aims to investigate the relative effects of eight different ligament property datasets on fe model responses eight l4 l5 models distinct only in ligament properties were constructed and loaded under moment 15nm alone or combined with a compressive follower load fl range of motions rom of the disc alone model matched well in vitro data ligament properties significantly affected only sagittal roms 3 0 7 1 in flexion and 3 8 5 8 in extension at 10nm sequential removal of ligaments shifted sagittal roms in and out of the corresponding in vitro ranges when moment was combined with fl center of rotation matched in vivo data for all models 3 8 0 9mm and 4 3 1 8mm posterior to the disc center in flexion and extension respectively under 15nm sagittal moments ligament strains were often smaller or within the in vitro range in flexion whereas some posterior ligament forces approached their failure forces in some models ligament forces varied substantially within the models and affected the moment sharing and internal forces on the disc and facet joints intradiscal pressure idp had the greatest variation between models in extension none of the datasets yielded results in agreement with all reported measurements results emphasized the important role of ligaments especially under larger moments and the need for their accurate representation in search for valid spinal models

characteristic polynomials of digraphs having a semi free action abstract let be a semi free action of a group g on a finite digraph the front divisor of has as vertex set the set of the vertex orbits of and there are t arcs going from u to v in if from any vertex u of the orbit u there are t arcs of going towards the vertices in v our main result is that the characteristic polynomial of is a product of the characteristic polynomial of its front divisor and polynomials associated with the free part of under this work extends earlier work of lee and kim characteristic polynomials of graphs having a semifree action linear algebra appl 307 2000 35 46

a database of lumbar spinal mechanical behavior for validation of spinal analytical models data from two experimental studies with eight specimens each of spinal motion segments and or intervertebral discs are presented in a form that can be used for comparison with finite element model predictions the data include the effect of compressive preload 0 250 and 500n with quasistatic cyclic loading 0 0115hz and the effect of loading frequency 1 0 1 0 01 and 0 001hz with a physiological compressive preload mean 642n specimens were tested with displacements in each of six degrees of freedom three translations and three rotations about defined anatomical axes the three forces and three moments in the corresponding axis system were recorded during each test linearized stiffness matrices were calculated that could be used in multi segmental biomechanical models of the spine and these matrices were analyzed to determine whether off diagonal terms and symmetry assumptions should be included these databases of lumbar spinal mechanical behavior under physiological conditions quantify behaviors that should be present in finite element model simulations the addition of more specimens to identify sources of variability associated with physical dimensions degeneration and other variables would be beneficial supplementary data provide the recorded data and matlab codes for reading files linearized stiffness matrices derived from the tests at different preloads revealed few significant unexpected off diagonal terms and little evidence of significant matrix asymmetry

finite difference quasi p traveltimes for anisotropic media the first arrival quasi p wave traveltime field in an anisotropic elastic solid solves a first order nonlinear partial differential equation the q p eikonal equation the difficulty in solving this eikonal equation by a finite difference method is that for anisotropic media the ray group velocity direction is not the same as the direction of the traveltime gradient so that the traveltime gradient can no longer serve as an indicator of the group velocity direction in extrapolating the traveltime field however establishing an explicit relation between the ray velocity vector and the phase velocity vector overcomes this difficulty furthermore the solution of the paraxial q p eikonal equation an evolution equation in depth gives the first arrival traveltime along downward propagating rays a second order upwind finite difference scheme solves this paraxial eikonal equation in o n floating point operations where n is the number of grid points numerical experiments using 2 d and 3 d transversely is

characterization of the l4 l5 s1 motion segment using the stepwise reduction method the two aims of this study were to generate data for a more accurate calibration of finite element models including the l5 s1 segment and to find mechanical differences between the l4 l5 and l5 s1 segments then the range of motion rom and facet forces for the l4 s1 segment were measured using the stepwise reduction method this consists of sequentially testing and reducing each segment in nine stages by cutting the ligaments facet capsules and removing the nucleus five l4 s1 human segments median 65 years range 53 84 years sd 11 0 years were loaded under a maximum pure moment of 8nm the rom was measured using stereo photogrammetry via tracking of three markers and the facet contact forces cf were measured using a tekscan system the rom for the l4 l5 segment and all stages showed good agreement with published data the major differences in rom between the l4 l5 and l5 s1 segments were found for lateral bending and all stages for which the l4 l5 rom was about 1 5 3 times higher than that of the l5 s1 segment consistent with l5 s1 facet cf about 1 3 to 4 times higher than those measured for the l4 l5 segment for the other movements and few stages the l4 l5 rom was significantly lower that of the l5 s1 segment rom and cf provide important baseline data for more accurate calibration of fe models and to understand the role that their structures play in lower lumbar spine mechanics

weak elastic anisotropy by perturbation theory we demonstrate the advantages of adopting a wave vectorbased coordinate system wcs for the application of perturbation theory to derive and display approximate expressions for qp and qs wave polarization vectors phase velocities and group velocities in general weakly anisotropic media the advantages stem from two important properties of the christoffel equation when expressed in the wcs 1 each element of the christoffel matrix is identical to a specific stiffness component in the wcs and 2 the christoffel matrix of an isotropic medium is diagonal in the wcs using these properties one can easily identify the small components of the christoffel matrix in the wcs for a weakly anisotropic medium approximate solutions to the christoffel equation are then obtained by straightforward algebraic manipulations which make our perturbation theory solution considerably simpler than previously published methods we compare and contrast our solutions with those discussed by other workers numerical comparisons between the exact first order and zero order qs wave polarization vectors illustrate the accuracy of our approximate formulas the form of the wcs phasevelocity expressions facilitates the derivation of closed form first order expressions for qp and qs wave group velocity vectors providing explicit formulas for the direction of propagation of seismic energy in general weakly anisotropic media numerical evaluation of our group velocity expressions demonstrates their accuracy we discuss problems with the approximate qs wave group velocities and polarizations in neighboring directions of singularities standard methods are used to transform our solutions from the wcs to the acquisition coordinates as illustrated by application to orthorhombic symmetry

3d traveltime computation for quasi p wave in orthorhombic media using dynamic programming abstracta fractured area such as a fault area usually induces orthorhombic anisotropy ignoring orthorhombic anisotropy may degrade the subsurface image by creating a well mistie and blurring the image traveltime computation is essential for many processing techniques such as depth imaging and tomography solving the ray tracing system and eikonal equation are two popular methods for traveltime computation in isotropic media however because the ray tracing system becomes complex and the eikonal equation becomes highly nonlinear their applications in orthorhombic media become complex we have developed an alternative 3d traveltime computation method in orthorhombic media based on dynamic programming to avoid solving the complex ray tracing system and nonlinear eikonal equation it adopts an explicitly expressed group velocity from the moveout approximation to describe the propagation of the wavepath and computes the traveltime by fermat s principle similar to depth extrapolation it computes the t

comparison of eight published static finite element models of the intact lumbar spine predictive power of models improves when combined together finite element fe model studies have made important contributions to our understanding of functional biomechanics of the lumbar spine however if a model is used to answer clinical and biomechanical questions over a certain population their inherently large inter subject variability has to be considered current fe model studies however generally account only for a single distinct spinal geometry with one set of material properties this raises questions concerning their predictive power their range of results and on their agreement with in vitro and in vivo values eight well established fe models of the lumbar spine l1 5 of different research centres around the globe were subjected to pure and combined loading modes and compared to in vitro and in vivo measurements for intervertebral rotations disc pressures and facet joint forces under pure moment loading the predicted l1 5 rotations of almost all models fell within the reported in vitro ranges and their median values differed on average by only 2 for flexion extension 1 for lateral bending and 5 for axial rotation predicted median facet joint forces and disc pressures were also in good agreement with published median in vitro values however the ranges of predictions were larger and exceeded those reported in vitro especially for the facet joint forces for all combined loading modes except for flexion predicted median segmental intervertebral rotations and disc pressures were in good agreement with measured in vivo values in light of high inter subject variability the generalization of results of a single model to a population remains a concern this study demonstrated that the pooled median of individual model results similar to a probabilistic approach can be used as an improved predictive tool in order to estimate the response of the lumbar spine

3 d traveltime computation using the fast marching method we present a fast algorithm for solving the eikonal equation in three dimensions based on the fast marching method the algorithm is of the order o n log n where n is the total number of grid points in the computational domain the algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain the method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity in addition the method resolves any overturning propagation wavefronts we begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details we then show examples of traveltime propagation through the seg eage salt model using point source and planewave initial conditions and analyze the error in constant velocity media the algorithm allows for any shape of the initial wavefront while a point source is the most commonly used initial condition initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth surface to another the algorithm presented here is designed for computing first arrival traveltimes nonetheless since it exploits the fast marching method for solving the eikonal equation we believe it is the fastest of all possible consistent schemes to compute first arrivals

biomechanical properties of human lumbar spine ligaments biomechanical properties of the six major lumbar spine ligaments were determined from 38 fresh human cadaveric subjects for direct incorporation into mathematical and finite element models anterior and posterior longitudinal ligaments joint capsules ligamentum flavum interspinous and supraspinous ligaments were evaluated using the results from in situ isolation tests individual force deflection responses from 132 samples were transformed with a normalization procedure into mean force deflection properties to describe the nonlinear characteristics ligament responses based on the mechanical characteristics as well as anatomical considerations were grouped into t12 l2 l2 l4 and l4 s1 levels maintaining individuality and nonlinearity a total of 18 data curves are presented geometrical measurements of original length and cross sectional area for these six major ligaments were determined using cryomicrotomy techniques derived parameters including failure stress and strain were computed using the strength and geometry information these properties for the lumbar spinal ligaments which are based on identical definitions used in mechanical testing and geometrical assay will permit more realistic and consistent inputs for analytical models

traveltime calculation and prestack depth migration in tilted transversely isotropic media the principal objective of our work is to develop a technique for prestack depth migration in tilted transversely isotropic tti media in which the axis of symmetry is not vertical and may be spatially varying such models are required to image seismic data in geologically complex regions such as the canadian foothills we have developed a 2d kirchhoff integral based migration algorithm in which the traveltime computation comprises the major task among the existing traveltime computation algorithms such as ray tracing with interpolation ray bending and eikonal solvers a direct or a brute force approach of traveltime computation is generally highly robust we have modified a direct method of first arrival p wave traveltime computation in 2d media that accounts for tti the algorithm requires that the group velocity be computed at each gridpoint using either an analytic solution or by an approximate fourier series expansion the p wave traveltime contours computed for complex geologic models show the pronounced effects from tti media our results using laboratory p wave data collected over a physical model of an anisotropic thrust sheet reveal that a 2d kirchhoff migration based on our traveltime algorithm tti model images the structure beneath the thrust sheet very well the vertical transversely isotropic assuming a vertical axis of symmetry or isotropic imaging introduces false anticlinal structures we compare our results with those obtained by a recursive extrapolation method and find that our approach images the underside of one of the thrust sheets better

computation of trunk equilibrium and stability in free flexion extension movements at different velocities velocity of movement has been suggested as a risk factor for low back disorders the effect of changes in velocity during unconstrained flexion extension movements on muscle activations spinal loads base reaction forces and system stability was computed in vivo measurements of kinematics and ground reaction forces were initially carried out on young asymptomatic subjects the collected kinematics of three subjects representing maximum mean and minimum lumbar rotations were subsequently used in the kinematics driven model to compute results during the entire movements at three different velocities estimated spinal loads and muscle forces were significantly larger in fastest pace as compared to slower ones indicating the effect of inertial forces spinal stability was improved in larger trunk flexion angles and fastest movement partial or full flexion relaxation of global extensor muscles occurred only in slower movements some local lumbar muscles especially in subjects with larger lumbar flexion and at slower paces also demonstrated flexion relaxation results confirmed the crucial role of movement velocity on spinal biomechanics predictions also demonstrated the important role on response of the magnitude of peak lumbar rotation and its temporal variation

moment rotation responses of the human lumbosacral spinal column abstract the objective of this study was to test the hypothesis that the human lumbosacral joint behaves differently from l1 l5 joints and provides primary moment rotation responses under pure moment flexion and extension and left and right lateral bending on a level by level basis in addition range of motion rom and stiffness data were extracted from the moment rotation responses ten t12 s1 column specimens with ages ranging from 27 to 68 years mean 50 6 13 2 were tested at a load level of 4 0 n m nonlinear flexion and extension and left and right lateral bending moment rotation responses at each spinal level are reported in the form of a logarithmic function the mean rom was the greatest at the l5 s1 level under flexion 7 37 3 69 and extension 4 62 2 56 and at the l3 l4 level under lateral bending 4 04 1 11 the mean rom was the least at the l1 l2 level under flexion 2 42 0 90 l2 l3 level under extension 1 58 0 63 and l1 l2 level under lateral bending 2 50 0 75 the present study proved the hypothesis that l5 s1 motions are significantly greater than l1 l5 motions under flexion and extension loadings but the hypothesis was found to be untrue under the lateral bending mode these experimental data are useful in the improved validation of fe models which will increase the confidence of stress analysis and other modeling applications

invariant subspace determinant and characteristic polynomials abstract making use of an elementary fact on invariant subspace and determinant of a linear map and the method of algebraic identities we obtain a factorization formula for a general characteristic polynomial of a matrix this answers a question posed in a deng i sato y wu characteristic polynomials of ramified uniform covering digraphs european j combin 28 2007 1099 1114 the approach of this work can be used to supply alternative proofs of several other earlier results including some results of y teranishi equitable switching and spectra of graphs linear algebra appl 359 2003 121 131

computation of trunk stability in forward perturbations effects of preload perturbation load initial flexion and abdominal preactivation spine stability demand influences active passive coordination of the trunk response especially during sudden perturbations the objective of this study was to look at the role of passive stationary active and reflexive subsystems on spinal stability spine stability was evaluated here during pre and post perturbation phases by computing the minimum i e critical muscle stiffness coefficient required to maintain stability the effects of pre perturbation conditions preloading initial posture and abdominal antagonistic coactivation as well as perturbation magnitude were studied results revealed that higher preload initially flexed trunk posture and abdominal pre activation enhanced pre perturbation stiffness and stability in contrast to the preload however larger sudden load initial flexion and abdominal preactivation significantly increased post perturbation stability margin as a result much lower critical muscle stiffness coefficient was required post perturbation compared to the pre perturbation phase the trunk stiffness and stability substantially increased post perturbation demanding thus a much lower critical muscle stiffness coefficient overall these findings highlight the crucial role of the ligamentous spine and muscles in both passive and active states in augmenting the trunk stiffness and hence stability during pre and post perturbation phases a role much evident in the presence of initial trunk flexion