On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail

Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements.Comment: 17 pages, 2 figure

Identification of Chemotaxis Models with Volume Filling

Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these non-linear parameter functions in a chemotaxis model with volume filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.Comment: Added bibfile missing in v2, no changes on conten

Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

In this work we consider the identifiability of two coefficients $a(u)$ and $c(x)$ in a quasilinear elliptic partial differential equation from observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov [On uniqueness in inverse problems for semilinear parabolic equations. Archive for Rational Mechanics and Analysis, 1993] and special singular solutions to first determine $a(0)$ and $c(x)$ for $x \in \Omega$. Based on this partial result, we are then able to determine $a(u)$ for $u \in \mathbb{R}$ by an adjoint approach.Comment: 10 pages; Proof of Theorem 4.1 correcte

Identification of nonlinear heat conduction laws

We consider the identification of nonlinear heat conduction laws in stationary and instationary heat transfer problems. Only a single additional measurement of the temperature on a curve on the boundary is required to determine the unknown parameter function on the range of observed temperatures. We first present a new proof of Cannon's uniqueness result for the stationary case, then derive a corresponding stability estimate, and finally extend our argument to instationary problems

Untersuchungen zur funktionellen und morphologischen Situation nach Dioden-Laser-Koagulation bei Frühgeborenenretinopathie

Untersucht wurde die funktionelle und morphologische Situation der Augen von Frühgeborenen die mit Dioden-Laser-Koagulation behandelt wurden. Die Dioden-Laser-Koagulation erbrachte sehr gute Ergebnisse und zeigte sich der Kryo-Koagulation in vergleichbaren Studien überlegen

Sports Activity After Short-Stem Hip Arthroplasty

Background: No data are available about the sports activity of patients with bone-conserving short-stem hip implants. Hypothesis: Patients can return to a good level of sports activity after implantation of a short-stem hip implant. Study Design: Case series; Level of evidence, 4. Methods: The sports activity level of 68 patients (76 hips) after short-stem hip arthroplasty was assessed for a minimum of 2 years after implantation. In addition to the clinical examination, a detailed evaluation of the patients’ sports pattern was obtained. Furthermore, the results were analyzed with regard to gender (female and male) and age (55 years). Results: After a mean of 2.7 years, patients showed a Harris Hip Score (HHS) of 93.6, a Western Ontario and McMaster Universities Arthritis Index (WOMAC) score of 9.5, and a University of California, Los Angeles (UCLA) activity score of 7.6, with each individual participating on average in 3.5 different disciplines after surgery compared with 3.9 before surgery. High-impact activities decreased significantly postoperatively, whereas low-impact activities increased significantly. The duration of the sports activities remained stable, while the frequency actually increased. In contrast, men participated preoperatively in more sports than women (4.3 men vs 3.3 women). However, because of a pronounced decrease in high-impact activities by men, both genders participated in an equal number of sports postoperatively (3.5 men vs 3.5 women). Finally, 45% (n = 31) reported at least one activity that they missed. Most of them were disciplines with an intermediate- or high-impact level. Conclusion: Patients with a short-stem hip implant can return to a good level of activity postoperatively. Participation in sports almost reached similar levels as preoperatively but with a shift from high- to low-impact activities. This seems desirable from a surgeon’s point of view but should also be communicated to the patient before hip replacemen

The Incidence and Clinical Relevance of Graft Hypertrophy After Matrix-Based Autologous Chondrocyte Implantation

Background: Graft hypertrophy is the most common complication of periosteal autologous chondrocyte implantation (p-ACI). Purpose: The aim of this prospective study was to analyze the development, the incidence rate, and the persistence of graft hypertrophy after matrix-based autologous chondrocyte implantation (mb-ACI) in the knee joint within a 2-year postoperative course. Study Design: Case series; Level of evidence, 4. Methods: Between 2004 and 2007, a total of 41 patients with 44 isolated cartilage defects of the knee were treated with the mb-ACI technique. The mean age of the patients was 35.8 years (standard deviation [SD], 11.3 years), and the mean body mass index was 25.9 (SD, 4.2; range, 19-35.3). The cartilage defects were arthroscopically classified as Outerbridge grades III and IV. The mean area of the cartilage defect measured 6.14 cm2 (SD, 2.3 cm2). Postoperative clinical and magnetic resonance imaging (MRI) examinations were conducted at 3, 6, 12, and 24 months to analyze the incidence and course of the graft. Results: Graft hypertrophy developed in 25% of the patients treated with mb-ACI within a postoperative course of 1 year; 16% of the patients developed hypertrophy grade 2, and 9% developed hypertrophy grade 1. Graft hypertrophy occurred primarily in the first 12 months and regressed in most cases within 2 years. The International Knee Documentation Committee (IKDC) and visual analog scale (VAS) scores improved during the postoperative follow-up time of 2 years. There was no difference between the clinical results regarding the IKDC and VAS pain scores and the presence of graft hypertrophy. Conclusion: The mb-ACI technique does not lead to graft hypertrophy requiring treatment as opposed to classic p-ACI. The frequency of occurrence of graft hypertrophy after p-ACI and mb-ACI is comparable. Graft hypertrophy can be considered as a temporary excessive growth of regenerative cartilage tissue rather than a true graft hypertrophy. It is therefore usually not a persistent or systematic complication in the treatment of circumscribed cartilage defects with mb-ACI

Effect of the defect localization and size on the success of third-generation autologous chondrocyte implantation in the knee joint

Introduction. Femoral and patellar cartilage defects with a defect size > 2.5 cm2 are a potential indication for an autologous chondrocyte implantation (ACI). However, the influence of the localization and the absolute and relative defect size on the clinical outcome has not yet been determined. The purpose of this study is to analyze the influence of the localization and the absolute and relative defect size on the clinical outcome after third-generation autologous chondrocyte implantation. Methods. A total of 50 patients with cartilage defects of the knee were treated with third-generation autologous chondrocyte implantation (Novocart® 3D). A match paired analysis was performed of 25 treated femoral and 25 treated patella defects with a follow-up of three years. MRI data was used to do the manual segmentation of the cartilage layer throughout the knee joint. The defect size was determined by taking the defect size measured in the MRI in relation to the whole cartilage area. The clinical outcome was measured by the IKDC score and VAS pre-operatively and after six, 12, 24, and 36 months post-operatively. Results. IKDC and VAS scores showed a significant improvement from the baseline in both groups. Femoral cartilage defects showed significantly superior clinical results in the analyzed scores compared to patellar defects. The femoral group improved IKDC from 33.9 (SD 18.1) pre-operatively to 71.5 (SD 17.4) after three years and the VAS from 6.9 (SD 2.9) pre-operatively to 2.4 (SD 2.5) after three years. In the patellar group, IKDC improved from 36.1 (SD 12.6) pre-operatively to 54.7 (SD 20.3) after three years and the VAS improved from 6.7 (SD 2.8) pre-operatively to 3.4 (SD 2.) after three years. Regarding the defect size, results showed that the same absolute defect size at med FC (4.8, range 2–15) and patella (4.6, range 2–12) has a significantly different share of the total cartilaginous size of the joint compartment (med FC: 6.7, range 1.2–13.9; pat: 18.9, range 4.0–47.0). However, there was no significant influence of the relative defect size on the clinical outcome in either patellar or femoral localization. Conclusion. Third-generation autologous chondrocyte implantation in ACI-treated femoral cartilage defects leads to a superior clinical outcome in a follow-up of three years compared with patellar defects. No significant influence of the defect size was found in either femoral or patellar cartilage defects