Torben Pätz

Simulation of Radiofrequency Ablation Including Water Evaporation

We present a new approach for the simulation of radiofrequency (RF) ablation, which takes the vaporization of water into account. This vaporization has an important influence on the outcome of the ablation process due to the significant smaller electric conductivity in the gaseous domain. Therefore the development of a gaseous domain around the RF probe leads to a decreased induction of heat into the surrounding tissue. We discretize the mathematical model by using composite finite elements to solve the resulting partial differential equations. A level set method is used for the tracking of the moving boundary between the tissue containing gaseous water and the tissue containing liquid water.

Segmentation of Stochastic Images With a Stochastic Random Walker Method

We present an extension of the random walker segmentation to images with uncertain gray values. Such gray-value uncertainty may result from noise or other imaging artifacts or more general from measurement errors in the image acquisition process. The purpose is to quantify the influence of the gray-value uncertainty onto the result when using random walker segmentation. In random walker segmentation, a weighted graph is built from the image, where the edge weights depend on the image gradient between the pixels. For given seed regions, the probability is evaluated for a random walk on this graph starting at a pixel to end in one of the seed regions. Here, we extend this method to images with uncertain gray values. To this end, we consider the pixel values to be random variables (RVs), thus introducing the notion of stochastic images. We end up with stochastic weights for the graph in random walker segmentation and a stochastic partial differential equation (PDE) that has to be solved. We discretize the RVs and the stochastic PDE by the method of generalized polynomial chaos, combining the recent developments in numerical methods for the discretization of stochastic PDEs and an interactive segmentation algorithm. The resulting algorithm allows for the detection of regions where the segmentation result is highly influenced by the uncertain pixel values. Thus, it gives a reliability estimate for the resulting segmentation, and it furthermore allows determining the probability density function of the segmented object volume.

Ambrosio-tortorelli segmentation of stochastic images

We present an extension of the classical Ambrosio-Tortorelli approximation of the Mumford-Shah approach for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows to quantify the robustness of edges in noisy images and under gray value uncertainty. We develop an ansatz space for such images by identifying gray values with random variables. The use of these stochastic images in the minimization of energies of Ambrosio-Tortorelli type leads to stochastic partial differential equations for the stochastic smoothed image and a stochastic phase field for the edge set. For their discretization we utilize the generalized polynomial chaos expansion and the generalized spectral decomposition (GSD) method. We demonstrate the performance of the method on artificial data as well as real medical ultrasound data.

Ambrosio-Tortorelli Segmentation of Stochastic Images: Model Extensions, Theoretical Investigations and Numerical Methods

We discuss an extension of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for the segmentation of images with uncertain gray values resulting from measurement errors and noise. Our approach yields a reliable precision estimate for the segmentation result, and it allows us to quantify the robustness of edges in noisy images and under gray value uncertainty. We develop an ansatz space for such images by identifying gray values with random variables. The use of these stochastic images in the minimization of energies of Ambrosio-Tortorelli type leads to stochastic partial differential equations for a stochastic smoothed version of the original image and a stochastic phase field for the edge set. For the discretization of these equations we utilize the generalized polynomial chaos expansion and the generalized spectral decomposition (GSD) method. In contrast to the simple classical sampling technique, this approach allows for an efficient determination of the stochastic properties of the output image and edge set by computations on an optimally small set of random variables. Also, we use an adaptive grid approach for the spatial dimensions to further improve the performance, and we extend an edge linking method for the classical Ambrosio-Tortorelli model for use with our stochastic model. The performance of the method is demonstrated on artificial data and a data set from a digital camera as well as real medical ultrasound data. A comparison of the intrusive GSD discretization with a stochastic collocation and a Monte Carlo sampling is shown.

Radiofrequency ablation planning beyond simulation

It is a challenging task to plan a radiofrequency (RF) ablation therapy to achieve the best outcome of the treatment and avoid recurrences at the same time. A patient specific simulation in advance that takes the cooling effect of blood vessels into account is a helpful tool for radiologists, but this needs a very high accuracy and thus high computational costs. In this work, we present various methods, which improve and extend the planning of an RF ablation procedure. First, we discuss two extensions of the simulation model to obtain a higher accuracy, including the vaporization of the water in the tissue and identifying the model parameters and to analyze their uncertainty. Furthermore, we discuss an extension of the planning procedure namely the optimization of the probe placement, which optimizes the overlap of the tumor area with the estimated coagulation in order to avoid recurrences. Since the optimization is constrained by the model, we have to take into account the uncertainties in the model parameters for the optimization as well. Finally, applications of our methods to a real RF ablation case are presented.

Fast parameter sensitivity analysis of PDE-based image processing methods

We present a fast parameter sensitivity analysis by combining recent developments from uncertainty quantification with image processing operators. The approach is not based on a sampling strategy, instead we combine the polynomial chaos expansion and stochastic finite elements with PDE-based image processing operators. With our approach and a moderate number of parameters in the models the full sensitivity analysis is obtained at the cost of a few Monte Carlo runs. To demonstrate the efficiency and simplicity of the approach we show a parameter sensitivity analysis for Perona-Malik diffusion, random walker and Ambrosio-Tortorelli segmentation, and discontinuity-preserving optical flow computation.

Composite Finite Elements for a Phase Change Model

We present a model and the related discretization for phase change problems. In particular, we are focused on the evaporation of water. The governing equations inside the domains and conditions on the moving interface are derived. Afterward the numerical methods for the discretization are presented. We use a level set method to capture the interface motion and use composite finite elements (CFEs) to solve the required equations in the whole domain. CFEs are a special kind of finite elements, allowing a fast calculation, because they use structured grids and respect the geometry by adapting the basis functions in the neighborhood of an interface or at the domain boundary. For the special construction of CFEs used in this paper, we present a method to take into account Dirichlet boundary conditions on the complicated domain boundary. Also, the method used for the calculation of the interface conditions within the CFE-grid is presented. We tested the Dirichlet boundary condition method for the CFEs by solvin...

Optimization and Fast Estimation of Vessel Cooling for RF Ablation

In this work, we present a three-dimensional (3D) model for the optimization of the probe placement in radio-frequency ablation (RFA). The model is based on a system of partial differential equations (PDEs) that describe the electric potential of the tissue and the steady state of the heat which is induced into the tissue. The PDE system is solved by a finite element approach and the optimization is performed by minimizing a temperature based objective functional under the constraining PDE systems. A well-known difficulty associated with RFA is the cooling influence of large blood vessels on the ablation result. A method is discussed, which efficiently estimates the cooling effect of those vessels, based on a precalculation of all patient-independent data and tabulation of the results.

Some Use Cases for Composite Finite Elements in Image Based Computing

Many bio-medical simulations involve structures of complicated shape. Frequently, the geometry information is given by radiological images. A particular challenge for model discretization in this context is generating appropriate computational meshes.One efficient approach for Finite Element simulations avoiding meshing is the Composite Finite Element approach that has been developed and implemented for image based simulations during the past decade. In the present paper, we provide an overview of previous own work in this field, summarizing the method and showing selected applications: simulation of radio-frequency ablation including vaporization, simulation of elastic deformation of trabecular bone, and numerical homogenization of material properties for the latter.