Fitting parametrized polynomials with scattered surface data.

Currently used joint-surface models require the measurements to be structured according to a grid. With the currently available tracking devices a large quantity of unstructured surface points can be measured in a relatively short time. In this paper a method is presented to fit polynomial functions to three-dimensional unstructured data points. To test the method spherical, cylindrical, parabolic, hyperbolic, exponential, logarithmic, and sellar surfaces with different undulations were used. The resulting polynomials were compared with the original shapes. The results show that even complex joint surfaces can be modelled with polynomial functions. In addition, the influence of noise and the number of data points was also analyzed. From a surface (diam: 20 mm) which is measured with a precision of 0.2 mm a model can be constructed with a precision of 0.02 mm.     

Local Polynomial Estimation of Heteroscedasticity in a Multivariate Linear Regression Model and Its Applications in Economics

Multivariate local polynomial fitting is applied to the multivariate linear heteroscedastic regression model. Firstly, the local polynomial fitting is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. One noteworthy feature of our approach is that we avoid the testing for heteroscedasticity by improving the traditional two-stage method. Due to non-parametric technique of local polynomial estimation, it is unnecessary to know the form of heteroscedastic function. Therefore, we can improve the estimation precision, when the heteroscedastic function is unknown. Furthermore, we verify that the regression coefficients is asymptotic normal based on numerical simulations and normal Q-Q plots of residuals. Finally, the simulation results and the local polynomial estimation of real data indicate that our approach is surely effective in finite-sample situations.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Experimental evaluation of a new morphological approximation of the articular surfaces of the ankle joint

The mechanical characteristics of the ankle such as its kinematics and load transfer properties are influenced by the geometry of the articulating surfaces. A recent, image-based study found that these surfaces can be approximated by a saddle-shaped, skewed, truncated cone with its apex oriented laterally. The goal of this study was to establish a reliable experimental technique to study the relationship between the geometry of the articular surfaces of the ankle and its mobility and stability characteristics and to use this technique to determine if morphological approximations of the ankle surfaces based on recent discoveries, produce close to normal behavior. The study was performed on ten cadavers. For each specimen, a process based on medical imaging, modeling and 3D printing was used to produce two subject specific artificial implantable sets of the ankle surfaces. One set was a replica of the natural surfaces. The second approximated the ankle surfaces as an original saddle-shaped truncated cone with apex oriented laterally. Testing under cyclic loading conditions was then performed on each specimen following a previously established technique to determine its mobility and stability characteristics under three different conditions: natural surfaces; artificial surfaces replicating the natural surface morphology; and artificial approximation based on the saddle-shaped truncated cone concept. A repeated measure analysis of variance was then used to compare between the three conditions. The results show that (1): the artificial surfaces replicating natural morphology produce close to natural mobility and stability behavior thus establishing the reliability of the technique; and (2): the approximated surfaces based on saddle-shaped truncated cone concept produce mobility and stability behavior close to the ankle with natural surfaces.     

Response surface optimization for joint contact model evaluation

When optimization is used to evaluate a joint contact model's ability to reproduce experimental measurements, the high computational cost of repeated contact analysis can be a limiting factor. This paper presents a computationally-efficient response surface optimization methodology to address this limitation. Quadratic response surfaces were fit to contact quantities (contact force, maximum pressure, average pressure, and contact area) predicted by a discrete element contact model of the tibiofemoral joint for various combinations of material modulus and relative bone pose (i.e., position and orientation). The response surfaces were then used as surrogates for costly contact analyses in optimizations that minimized differences between measured and predicted contact quantities. The methodology was evaluated theoretically using six sets of synthetic (i.e., computer-generated) contact data, and practically using one set of experimental contact data. For the synthetic cases, the response surface optimizations recovered all contact quantities to within 3.4% error. For the experimental case, they matched all contact quantities to within 6.3% error except for maximum contact pressure, which was in error by up to 50%. Response surface optimization provides rapid evaluation of joint contact models within a limited range of relative bone poses and can help identify potential weaknesses in contact model formulation and/or experimental data quality.     

Joint surface modeling with thin-plate splines.

Mathematical joint surface models based on experimentally determined data points can be used to investigate joint characteristics such as curvature, congruency, cartilage thickness, joint contact areas, as well as to provide geometric information well suited for finite element analysis. Commonly, surface modeling methods are based on B-splines, which involve tensor products. These methods have had success; however, they are limited due to the complex organizational aspect of working with surface patches, and modeling unordered, scattered experimental data points. An alternative method for mathematical joint surface modeling is presented based on the thin-plate spline (TPS). It has the advantage that it does not involve surface patches, and can model scattered data points without experimental data preparation. An analytical surface was developed and modeled with the TPS to quantify its interpolating and smoothing characteristics. Some limitations of the TPS include discontinuity of curvature at exactly the experimental surface data points, and numerical problems dealing with data sets in excess of 2000 points. However, suggestions for overcoming these limitations are presented. Testing the TPS with real experimental data, the patellofemoral joint of a cat was measured with multistation digital photogrammetry and modeled using the TPS to determine cartilage thicknesses and surface curvature. The cartilage thickness distribution ranged between 100 to 550 microns on the patella, and 100 to 300 microns on the femur. It was found that the TPS was an effective tool for modeling joint surfaces because no preparation of the experimental data points was necessary, and the resulting unique function representing the entire surface does not involve surface patches. A detailed algorithm is presented for implementation of the TPS.     

Estimation of musculotendon kinematics in large musculoskeletal models using multidimensional B-splines

We present a robust and computationally inexpensive method to estimate the lengths and three-dimensional moment arms for a large number of musculotendon actuators of the human lower limb. Using a musculoskeletal model of the lower extremity, a set of values was established for the length of each musculotendon actuator for different lower limb generalized coordinates (joint angles). A multidimensional spline function was then used to fit these data. Muscle moment arms were obtained by differentiating the musculotendon length spline function with respect to the generalized coordinate of interest. This new method was then compared to a previously used polynomial regression method. Compared to the polynomial regression method, the multidimensional spline method produced lower errors for estimating musculotendon lengths and moment arms throughout the whole generalized coordinate workspace. The fitting accuracy was also less affected by the number of dependent degrees of freedom and by the amount of experimental data available. The spline method only required information on musculotendon lengths to estimate both musculotendon lengths and moment arms, thus relaxing data input requirements, whereas the polynomial regression requires different equations to be used for both musculotendon lengths and moment arms. Finally, we used the spline method in conjunction with an electromyography driven musculoskeletal model to estimate muscle forces under different contractile conditions, which showed that the method is suitable for the integration into large scale neuromusculoskeletal models.     

Identification of protein functional surfaces by the concept of a split pocket

The function of a protein is often fulfilled via molecular interactions on its surfaces, so identifying the functional surface(s) of a protein is helpful for understanding its function. Here, we introduce the concept of a split pocket, which is a pocket that is split by a cognate ligand. We use a geometric approach that is site‐specific. Specifically, we first compute a set of all pockets in the protein with its ligand(s) and a set of all pockets with the ligand(s) removed and then compare the two sets of pockets to identify the split pocket(s) of the protein. To reduce the search space and expedite the process of surface partitioning, we design probe radii according to the physicochemical textures of molecules. Our method achieves a success rate of 96% on a benchmark test set. We conduct a large‐scale computation to identify ～19,000 split pockets from 11,328 structures (1.16 million potential pockets); for each pocket, we obtain residue composition, solvent‐accessible area, and molecular volume. With this database of split pockets, our method can be used to predict the functional surfaces of unbound structures. Indeed, the functional surface of an unbound protein may often be found from its similarity to remotely related bound forms that belong to distinct folds. Finally, we apply our method to identify glucose‐binding proteins, including unbound structures. Our study demonstrates the power of geometric and evolutionary matching for studying protein functional evolution and provides a framework for classifying protein functions by local spatial patterns of functional surfaces. Proteins 2009. © 2009 Wiley‐Liss, Inc.     

A modified mathematical model of the anatomy of the cardiac left ventricle

A modification of the mathematical model of the shape and fiber direction field of the left cardiac ventricle is presented. The model was developed based on the idea of nested spiral surfaces. The ventricle is composed of surfaces that model myocardial layers. Each layer is filled with curves corresponding to myocardial fibers. The tangents to these curves form the myofiber direction field. A modified spherical coordinate system is linked with the model left ventricle, where the ventricular boundaries are coordinate surfaces. The model is based on echocardiographic, computed-tomography, or magnetic-resonance-imaging data. For this purpose, four-chamber and two-chamber echocardiography views or sections along the long axis of the left ventricle from these tomographic data in several positions are approximated with a model profile. To construct a 3D model, we then interpolate model parameters by periodic cubic splines and the vector field of the tangents to the model fibers is calculated. For verification of the model, we used diffusion-tensor magneticresonance-imaging data of the human heart.     

On Genetic Map Functions

Various genetic map functions have been proposed to infer the unobservable genetic distance between two loci from the observable recombination fraction between them. Some map functions were found to fit data better than others. When there are more than three markers, multilocus recombination probabilities cannot be uniquely determined by the defining property of map functions, and different methods have been proposed to permit the use of map functions to analyze multilocus data. If for a given map function, there is a probability model for recombination that can give rise to it, then joint recombination probabilities can be deduced from this model. This provides another way to use map functions in multilocus analysis. In this paper we show that stationary renewal processes give rise to most of the map functions in the literature. Furthermore, we show that the interevent distributions of these renewal processes can all be approximated quite well by gamma distributions.     

Data Identification for Improving Gene Network Inference using Computational Algebra

Identification of models of gene regulatory networks is sensitive to the amount of data used as input. Considering the substantial costs in conducting experiments, it is of value to have an estimate of the amount of data required to infer the network structure. To minimize wasted resources, it is also beneficial to know which data are necessary to identify the network. Knowledge of the data and knowledge of the terms in polynomial models are often required a priori in model identification. In applications, it is unlikely that the structure of a polynomial model will be known, which may force data sets to be unnecessarily large in order to identify a model. Furthermore, none of the known results provides any strategy for constructing data sets to uniquely identify a model. We provide a specialization of an existing criterion for deciding when a set of data points identifies a minimal polynomial model when its monomial terms have been specified. Then, we relax the requirement of the knowledge of the monomials and present results for model identification given only the data. Finally, we present a method for constructing data sets that identify minimal polynomial models.     

HAPLOFREQ--estimating haplotype frequencies efficiently.

A commonly used tool in disease association studies is the search for discrepancies between the haplotype distribution in the case and control populations. In order to find this discrepancy, the haplotypes frequency in each of the populations is estimated from the genotypes. We present a new method HAPLOFREQ to estimate haplotype frequencies over a short genomic region given the genotypes or haplotypes with missing data or sequencing errors. Our approach incorporates a maximum likelihood model based on a simple random generative model which assumes that the genotypes are independently sampled from the population. We first show that if the phased haplotypes are given, possibly with missing data, we can estimate the frequency of the haplotypes in the population by finding the global optimum of the likelihood function in polynomial time. If the haplotypes are not phased, finding the maximum value of the likelihood function is NP-hard. In this case, we define an alternative likelihood function which can be thought of as a relaxed likelihood function. We show that the maximum relaxed likelihood can be found in polynomial time and that the optimal solution of the relaxed likelihood approaches asymptotically to the haplotype frequencies in the population. In contrast to previous approaches, our algorithms are guaranteed to converge in polynomial time to a global maximum of the different likelihood functions. We compared the performance of our algorithm to the widely used program PHASE, and we found that our estimates are at least 10% more accurate than PHASE and about ten times faster than PHASE. Our techniques involve new algorithms in convex optimization. These algorithms may be of independent interest. Particularly, they may be helpful in other maximum likelihood problems arising from survey sampling.     

Atomic solvation parameters in the analysis of protein-protein docking results.

Several sets of amino acid surface areas and transfer free energies were used to derive a total of nine sets of atomic solvation parameters (ASPs). We tested the accuracy of each of these sets of parameters in predicting the experimentally determined transfer free energies of the amino acid derivatives from which the parameters were derived. In all cases, the calculated and experimental values correlated well. We then chose three parameter sets and examined the effect of adding an energetic correction for desolvation based on these three parameter sets to the simple potential function used in our multiple start Monte Carlo docking method. A variety of protein-protein interactions and docking results were examined. In the docking simulations studied, the desolvation correction was only applied during the final energy calculation of each simulation. For most of the docking results we analyzed, the use of an octanol-water-based ASP set marginally improved the energetic ranking of the low-energy dockings, whereas the other ASP sets we tested disturbed the ranking of the low-energy dockings in many of the same systems. We also examined the correlation between the experimental free energies of association and our calculated interaction energies for a series of proteinase-inhibitor complexes. Again, the octanol-water-based ASP set was compatible with our standard potential function, whereas ASP sets derived from other solvent systems were not.     

Defining and evaluating wrapping surfaces for MRI-derived spinal muscle paths

Muscle paths can be approximated in biomechanical models by wrapping the path around geometric objects; however, the process for selecting and evaluating wrapping surface parameters is not well defined, especially for spinal muscles. In this study, we defined objective methods to select the shape, orientation, size and location of wrapping surfaces and evaluated the wrapping surfaces using an error metric based on the distance between the modeled muscle path and the centroid path from magnetic resonance imaging (MRI). We applied these methods and the error metric to a model of the neck musculature, where our specific goals were (1) to optimize the vertebral level at which to place a single wrapping surface per muscle; and (2) to define wrapping surface parameters in the neutral posture and evaluate them in other postures. Detailed results are provided for the sternocleidomastoid and the semispinalis capitis muscles. For the sternocleidomastoid, the level where the wrapping surface was placed did not significantly affect the error between the modeled path and the centroid path; use of wrapping surfaces defined from the neutral posture improved the representation of the muscle path compared to a straight line in all postures except contralateral rotation. For the semispinalis capitis, wrapping surfaces placed at C3 or C4 resulted in lower error compared to other levels; and the use of wrapping surfaces significantly improved the muscle path representation in all postures. These methods will be used to improve the estimates of muscle length, moment arm and moment-generating capacity in biomechanical models.     

Structural signatures of enzyme binding pockets from order-independent surface alignment: a study of metalloendopeptidase and NAD binding proteins

Detecting similarities between local binding surfaces can facilitate identification of enzyme binding sites and prediction of enzyme functions, and aid in our understanding of enzyme mechanisms. Constructing a template of local surface characteristics for a specific enzyme function or binding activity is a challenging task, as the size and shape of the binding surfaces of a biochemical function often vary. Here we introduce the concept of signature binding pockets, which captures information on preserved and varied atomic positions at multiresolution levels. For proteins with complex enzyme binding and activity, multiple signatures arise naturally in our model, forming a signature basis set that characterizes this class of proteins. Both signatures and signature basis sets can be automatically constructed by a method called SOLAR (Signature Of Local Active Regions). This method is based on a sequence-order-independent alignment of computed binding surface pockets. SOLAR also provides a structure-based multiple sequence fragment alignment to facilitate the interpretation of computed signatures. By studying a family of evolutionarily related proteins, we show that for metzincin metalloendopeptidase, which has a broad spectrum of substrate binding, signature and basis set pockets can be used to discriminate metzincins from other enzymes, to predict the subclass of metzincins functions, and to identify specific binding surfaces. Studying unrelated proteins that have evolved to bind to the same NAD cofactor, we constructed signatures of NAD binding pockets and used them to predict NAD binding proteins and to locate NAD binding pockets. By measuring preservation ratio and location variation, our method can identify residues and atoms that are important for binding affinity and specificity. In both cases, we show that signatures and signature basis set reveal significant biological insight.     

Classification trees: An alternative non‐parametric approach for predicting species distributions

Abstract. The use of Generalized Linear Models (GLM) in vegetation analysis has been advocated to accommodate complex species response curves. This paper investigates the potential advantages of using classification and regression trees (CART), a recursive partitioning method that is free of distributional assumptions. We used multiple logistic regression (a form of GLM) and CART to predict the distribution of three major oak species in California. We compared two types of model: polynomial logistic regression models optimized to account for non‐linearity and factor interactions, and simple CART‐models. Each type of model was developed using learning data sets of 2085 and 410 sample cases, and assessed on test sets containing 2016 and 3691 cases respectively. The responses of the three species to environmental gradients were varied and often non‐homogeneous or context dependent. We tested the methods for predictive accuracy: CART‐models performed significantly better than our polynomial logistic regression models in four of the six cases considered, and as well in the two remaining cases. CART also showed a superior ability to detect factor interactions. Insight gained from CART‐models then helped develop improved parametric models. Although the probabilistic form of logistic regression results is more adapted to test theories about species responses to environmental gradients, we found that CART‐models are intuitive, easy to develop and interpret, and constitute a valuable tool for modeling species distributions.     

Surface Thermodynamics of Mucoadhesive Dry Powder Formulation of Zolmitriptan

Microparticle powders for nasal delivery were formulated to contain the model drug, zolmitriptan, and varying proportions of different polymers. The objective of the study was to investigate the effects of these formulative parameters on the surface chemistry of the spray-dried microparticles and their potential for adhesion to the tested substrates, porcine mucin, and nasal tissue. The polymers used were chitosans of varying ionization states and molecular weights and hydroxypropyl methyl cellulose. The surface energies of the surfaces of the microparticles were determined using contact angle measurements and the van Oss model. The theory of surface thermodynamics was applied to determine the theoretical potential for the different materials to adhere to the substrates. It was found that the drug or polymers alone, as well as the various formulations, were more likely to adhere to mucin than to nasal tissue. Further, there was a trend for higher molecular weight chitosans to adhere better to the substrates than lower molecular weight chitosans. Similarly, adhesion was improved for formulations with a higher content of polymers. These theoretical predictions may be compared with further experimental results and be of use in making informed decisions on the choice of formulations for future expensive bio-studies.     

Variational Reconstruction of Left Cardiac Structure from CMR Images

Cardiovascular Disease (CVD), accounting for 17% of overall deaths in the USA, is the leading cause of death over the world. Advances in medical imaging techniques make the quantitative assessment of both the anatomy and function of heart possible. The cardiac modeling is an invariable prerequisite for quantitative analysis. In this study, a novel method is proposed to reconstruct the left cardiac structure from multi-planed cardiac magnetic resonance (CMR) images and contours. Routine CMR examination was performed to acquire both long axis and short axis images. Trained technologists delineated the endocardial contours. Multiple sets of two dimensional contours were projected into the three dimensional patient-based coordinate system and registered to each other. The union of the registered point sets was applied a variational surface reconstruction algorithm based on Delaunay triangulation and graph-cuts. The resulting triangulated surfaces were further post-processed. Quantitative evaluation on our method was performed via computing the overlapping ratio between the reconstructed model and the manually delineated long axis contours, which validates our method. We envisage that this method could be used by radiographers and cardiologists to diagnose and assess cardiac function in patients with diverse heart diseases.