Osteoarthritis is one of the most common diseases, and it affects 12% of the population around the world. Although the disease is chronic, it significantly reduces the patient’s quality of life. At present, stem cell therapy is considered to be an efficient approach for treating this condition. Mesenchymal stem cells (MSCs) show the most potential for stem cell therapy of osteoarthritis. In fact, MSCs can differentiate into certain mesodermal tissues such as cartilage and bone. Therefore, in the present study, we applied adipose tissue-derived MSCs to osteoarthritis treatment. This study aimed to evaluate the clinical efficiency of autologous adipose tissue-derived MSC transplantation in patients with confirmed osteoarthritis at grade II and III. Adipose tissue was isolated from the belly, and used for extraction of the stromal vascular fraction (SVF). The SVF was mixed with activated platelet-rich plasma before injection. The clinical efficiencies were evaluated by the pain score (VAS), Lysholm score, and MRI findings. We performed the procedure in 21 cases from 2012 to 2013. All 21 patients showed improved joint function after 8.5 months. The pain score decreased from 7.6±0.5 before injection to 3.5±0.7 at 3 months and 1.5±0.5 at 6 months after injection. The Lysholm score increased from 61±11 before injection to 82±8.1 after injection. Significant improvements were noted in MRI findings, with increased thickness of the cartilage layer. Moreover, there were no side-effects or complications related to microorganism infection, graft rejection, or tumorigenesis. These results provide a new opportunity for osteoarthritis treatment. Level of evidence: IV.

In this paper, our work is aimed to show the fractional maximal gradient estimates and point-wise gradient estimates for quasilinear divergence form elliptic equations with general Dirichlet boundary data:in terms of the Riesz potentials, where Ω is a Reifenberg flat domain of R n (n ≥ 2), the nonlinearity A is a monotone Carathéodory vector valued function, g belongs to some W 1,p (Ω; R) for p > 1 and f ∈ L p p−1 (Ω; R n ). Our proofs of gradient regularity results are established in the weighted Lorentz spaces. Here, we generalize our earlier results concerning the "good-λ" technique and the study of so-called cut-off fractional maximal functions. Moreover, as an application of point-wise gradient estimates, we also prove the existence of solutions for a generalized quasilinear elliptic equation containing the Riesz potential of gradient term.

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis -which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of us and by Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates, and study them numerically. Two types of methods are presented: methods based on the corrector equation, and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures, and calls for new theoretical developments.

This paper continues the development of regularity results for nonlinear measure data problems −div(A(x, ∇u)) = µ in Ω, u = 0 on ∂Ω,in Lorentz and Lorentz-Morrey spaces, where Ω ⊂ R n (n ≥ 2), µ is a finite Radon measure on Ω, and A is a monotone Carathéodory vector valued operator acting between W 1,p 0 (Ω) and its dual W −1,p ′ (Ω). It emphasizes that this paper covers the 'very singular' case of 1 < p ≤ 3n−2 2n−1 and the problem is considered under the weak assumption that the p-capacity uniform thickness condition is imposed on the complement of domain Ω. There are two main results obtained in our study pertaining to the global gradient estimates of solutions (renormalized solutions), involving the use of maximal and fractional maximal operators in Lorentz and Lorentz-Morrey spaces, respectively. The idea for writing this working paper comes directly from the results of others for the same research topic, where estimates for gradient of solutions cannot be obtained for the 'very singular' case, still remains a challenge. Our main goal here is to develop results and improve methods in [64,66] to the case when 1 < p ≤ 3n−2 2n−1 . In order to derive the gradient estimates in Lorentz and Lorentz-Morrey spaces, our approach is based on the good-λ technique proposed early by Q.-H. Nguyen et al. in [55,56] and our previous works in [64,66].

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