Maximum Principle for Solution of Biharmonic Equation in The Ball

The intention of this paper is to estimate the biharmonic equation  on a bounded domain  in the space for  with the boundary condition: where we made a function  that equivalent the solution of biharmonic equation:   by using  and applying the principle of the maximum and minimum value, we can estimate the solution of biharmonic equation through the boundary value  : where  is a continuous function ,  Laplace operator ,  biharmonic operator ,  is the boundary of the domain   ,   is the derivative for the outward rhyming on the boundary of ,  , and  is Nabla’s operator.  

Spontaneous regression of hepatocellular carcinoma: what three cases of regression and disease reoccurrence can tell US

Hepatocellular carcinoma (HCC) is a highly morbid disease both in the United States and worldwide. Chronic liver inflammation puts people at risk of developing HCC. As chronic liver disease prevalence increases in the United States there can be an expected rise in HCC. Spontaneous regression of HCC is a rare phenomenon but can provide much needed information on how to better understand disease characteristics and progression. The two proposed theories that may explain spontaneous regression are tumor hypoxia and immunologic reaction. In these cases, we describe 3 patients with heavy disease burden at presentation who showed spontaneous regression of cancer. The patient's characteristics correlate most with systemic immunologic reaction resulting in spontaneous regression. Unfortunately, all of these patients had disease recurrence shortly after regression. By studying patient data in cases of spontaneous regression, we can gain a better understanding of disease progression and which exogenous or endogenous factors determine HCC mortality. With this knowledge we hope to better characterize how spontaneous regression occurs, and how we can use this information to help in developing treatment options in the future

Implementation and Implication of Ignoring Small Polyps at Colonoscopy

Background and Aim: CT colonography has promoted a new paradigm, that up-to 2 polyps ≤5mm can be left in-situ and followed. In contrast, endoscopists identify and remove all colorectal polyps, regardless of size. We evaluated whether and how endoscopists might implement a plan of ignoring small polyps in clinical practice.Methods: We prospectively queried endoscopists as they encountered small polyps with a hypothetical question: “If you accepted and believed in a new paradigm that a polyp ≤5mm does not need to be endoscopically removed, would you remove this polyp?” We assessed how the new paradigm would be implemented by gastroenterologists and the pathologic impact of ignoring polyps’ ≤5mm.Results: Of 141 patients undergoing colonoscopy, 55 (39%) had polyps and 35 (24.8%) had only small polyps ≤5mm. Endoscopists were agreeable to implementing the new paradigm of not removing small polyps in 17/35 (48.6%) patients. Of patients with only small polyps where endoscopists agreed to forego removal, 13/17 (76.5%) had ≥1 adenomatous polyp. Among the 18/35 subjects for whom the ndoscopists would remove the small polyp because of appearance or clinical situation, 12/18 (66.7%) had ≥1 denoma (p=0.521). If polyps in subjects with only diminutive polyps were ignored, 35/55 (64%) of colonoscopies with therapy would be obviated, at an impact of not removing small adenomas in approximately 75% (25/35).Conclusion: Ignoring polyps’ ≤5mm reduces therapeutic colonoscopy at a price of missing a substantial number of small adenomas.</p