Supervised learning for bone shape and cortical thickness estimation from CT images for finite element analysis.

Knowledge about the thickness of the cortical bone is of high interest for fracture risk assessment. Most finite element model solutions overlook this information because of the coarse resolution of the CT images. To circumvent this limitation, a three-steps approach is proposed. 1) Two initial surface meshes approximating the outer and inner cortical surfaces are generated via a shape regression based on morphometric features and statistical shape model parameters. 2) The meshes are then corrected locally using a supervised learning model build from image features extracted from pairs of QCT (0.3-1 mm resolution) and HRpQCT images (82 µm resolution). As the resulting meshes better follow the cortical surfaces, the cortical thickness can be estimated at sub-voxel precision. 3) The meshes are finally regularized by a Gaussian process model featuring a two-kernel model, which seamlessly enables smoothness and shape-awareness priors during regularization. The resulting meshes yield high-quality mesh element properties, suitable for construction of tetrahedral meshes and finite element simulations. This pipeline was applied to 36 pairs of proximal femurs (17 males, 19 females, 76 ±â€¯12 years) scanned under QCT and HRpQCT modalities. On a set of leave-one-out experiments, we quantified accuracy (root mean square error = 0.36 ±â€¯0.29 mm) and robustness (Hausdorff distance = 3.90 ±â€¯1.57 mm) of the outer surface meshes. The error in the estimated cortical thickness (0.05 ±â€¯0.40 mm), and the tetrahedral mesh quality (aspect ratio = 1.4 ±â€¯0.02) are also reported. The proposed pipeline produces finite element meshes with patient-specific bone shape and sub-voxel cortical thickness directly from CT scans. It also ensures that the nodes and elements numbering remains consistent and independent of the morphology, which is a distinct advantage in population studies.

Gaussian Process Morphable Models.

Models of shape variations have become a central component for the automated analysis of images. An important class of shape models are point distribution models (PDMs). These models represent a class of shapes as a normal distribution of point variations, whose parameters are estimated from example shapes. Principal component analysis (PCA) is applied to obtain a low-dimensional representation of the shape variation in terms of the leading principal components. In this paper, we propose a generalization of PDMs, which we refer to as Gaussian Process Morphable Models (GPMMs). We model the shape variations with a Gaussian process, which we represent using the leading components of its Karhunen-Loève expansion. To compute the expansion, we make use of an approximation scheme based on the Nyström method. The resulting model can be seen as a continuous analog of a standard PDM. However, while for PDMs the shape variation is restricted to the linear span of the example data, with GPMMs we can define the shape variation using any Gaussian process. For example, we can build shape models that correspond to classical spline models and thus do not require any example data. Furthermore, Gaussian processes make it possible to combine different models. For example, a PDM can be extended with a spline model, to obtain a model that incorporates learned shape characteristics but is flexible enough to explain shapes that cannot be represented by the PDM. We introduce a simple algorithm for fitting a GPMM to a surface or image. This results in a non-rigid registration approach whose regularization properties are defined by a GPMM. We show how we can obtain different registration schemes, including methods for multi-scale or hybrid registration, by constructing an appropriate GPMM. As our approach strictly separates modeling from the fitting process, this is all achieved without changes to the fitting algorithm. To demonstrate the applicability and versatility of GPMMs, we perform a set of experiments in typical usage scenarios in medical image analysis and computer vision: The model-based segmentation of 3D forearm images and the building of a statistical model of the face. To complement the paper, we have made all our methods available as open source.

Spatially varying registration using Gaussian processes.

In this paper we propose a new approach for spatially-varying registration using Gaussian process priors. The method is based on the idea of spectral tempering, i.e. the spectrum of the Gaussian process is modified depending on a user defined tempering function. The result is a non-stationary Gaussian process, which induces different amount of smoothness in different areas. In contrast to most other schemes for spatially-varying registration, our approach does not require any change in the registration algorithm itself, but only affects the prior model. Thus we can obtain spatially-varying versions of any registration method whose deformation prior can be formulated in terms of a Gaussian process. This includes for example most spline-based models, but also statistical shape or deformation models. We present results for the problem of atlas based skull-registration of cone beam CT images. These datasets are difficult to register as they contain a large amount of noise around the teeth. We show that with our method we can become robust against noise, but still obtain accurate correspondence where the data is clean.

Posterior shape models.

We present a method to compute the conditional distribution of a statistical shape model given partial data. The result is a "posterior shape model", which is again a statistical shape model of the same form as the original model. This allows its direct use in the variety of algorithms that include prior knowledge about the variability of a class of shapes with a statistical shape model. Posterior shape models then provide a statistically sound yet easy method to integrate partial data into these algorithms. Usually, shape models represent a complete organ, for instance in our experiments the femur bone, modeled by a multivariate normal distribution. But because in many application certain parts of the shape are known a priori, it is of great interest to model the posterior distribution of the whole shape given the known parts. These could be isolated landmark points or larger portions of the shape, like the healthy part of a pathological or damaged organ. However, because for most shape models the dimensionality of the data is much higher than the number of examples, the normal distribution is singular, and the conditional distribution not readily available. In this paper, we present two main contributions: First, we show how the posterior model can be efficiently computed as a statistical shape model in standard form and used in any shape model algorithm. We complement this paper with a freely available implementation of our algorithms. Second, we show that most common approaches put forth in the literature to overcome this are equivalent to probabilistic principal component analysis (PPCA), and Gaussian Process regression. To illustrate the use of posterior shape models, we apply them on two problems from medical image analysis: model-based image segmentation incorporating prior knowledge from landmarks, and the prediction of anatomically correct knee shapes for trochlear dysplasia patients, which constitutes a novel medical application. Our experiments confirm that the use of conditional shape models for image segmentation improves the overall segmentation accuracy and robustness.

Interrelationships between p-coumaric acid, evapotranspiration, soil water content, and leaf expansion.

Increasing concentrations of p-coumaric acid applied to (cucumber seedling)-[Cecil A( p ) soil-sand mixture (or soil)] systems inhibited evapotranspiration (primarily transpiration) and leaf area expansion of cucumber seedlings and increased soil moisture. Higher soil moisture resulting from the inhibition of evapotranspiration lowered soil solution concentrations of p-coumaric acid by 14-40% but did not significantly influence the inhibitory effects of p-coumaric acid on seedlings. Inhibition of evapotranspiration and total leaf area and increases in lowest daily soil water were observed 1-3 d after the first p-coumaric acid treatment, whereas inhibition of absolute and relative rates of leaf expansion was observed within a 24-hr period. Development of the maximum effects of p-coumaric acid required several additional days. Recovery from effects, i.e., return to control levels, after p-coumaric acid depletion from soil solution was a gradual process requiring days for evapotranspiration, lowest daily soil water, and total leaf area, but was slightly faster for leaf area expansion. It appears, at least for short-term studies, that the initial input or treatment concentrations of p-coumaric acid represented a reasonable estimate of dose despite the dynamic nature of soil solution concentrations, and that the lowering of available p-coumaric acid concentrations, associated with the elevation of soil moisture, did not result in a concurrent detectable seedling response. However, increased soil moisture associated with p-coumaric acid treatments of sensitive species suggests a means by which the magnitude of some allelopathic interactions may be modified and resource competition and allelopathy could interact.

Relationships between phenolic acid concentrations, transpiration, water utilization, leaf area expansion, and uptake of phenolic acids: nutrient culture studies.

Phenolic acid treatments of cucumber seedlings (Cucumis sativus cv "Early Green Cluster") inhibited transpiration, water utilization, leaf area, and absolute and relative rates of leaf expansion. The cinnamic acids, ferulic and p-coumaric acids, were two to five times more inhibitory than the benzoic acids, p-hydroxybenzoic acid and vanillic acid. When phenolic acid concentrations were maintained at inhibitory concentrations through multiple successive treatments, percent inhibition of water utilization remained relatively constant for a given concentration and phenolic acid, percent inhibition of leaf area initially increased and then leveled off to a constant percent, and percent inhibition of transpiration and rates of leaf area expansion declined over time. Subsequently, p-coumaric acid was chosen as the model compound for further study. When p-coumaric acid was inhibitory, percent inhibition of transpiration, water utilization, and rates of leaf area expansion of actively growing leaves rapidly declined (i.e., was lost) as p-coumaric acid concentrations surrounding roots decreased. Absolute and relative rates of leaf expansion, for example, declined approximately 12 and 14%, respectively, for every 0.1 mM decline in p-coumaric acid concentration. Uptake of p-coumaric acid by cucumber seedling roots was continuous over the 24- or 36-hr periods monitored, but was not consistently related to the initial p-coumaric acid treatment concentrations. However, declining p-coumaric acid concentrations monitored at 6- or 12-hr intervals over the 24- or 36-hr periods continued to be highly correlated to the initial p-coumaric acid treatment concentrations. A 25% depletion by 1 3-d-old cucumber seedlings took 8.5, 12, 19.5, 25, and 29.5 hr for 0.125-, 0.25-, 0.5-, 0.75-, and 1-mM treatments, respectively. Uptake during periods when phenolic acid concentrations and root uptake (depletion from solution) were related appeared to represent periods dominated by apoplastic movement into the intercellular spaces of roots. Uptake during periods without this relationship likely represented periods dominated by symplastic movement. The ability of cucumber seedlings to modify active phenolic acid concentrations surrounding their roots suggests that cucumber seedling can directly influence the magnitude of primary and secondary effects of phenolic acids through feedback regulation.