No abstract

In this paper, we consider an inverse problem to determine a source term in the parabolic equation, once the measured data are obtained at a later time. In general, this problem is ill-posed, therefore the Tikhonov regularization method with a priori and a posteriori parameter choice rule strategies is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solution is obtained. For estimating the errors between the regularized solution and its exact solution, numerical experiments have been carried out. From the numerical results it shows that the a posteriori parameter choice rule method has a better convergence speed in comparison with the a priori parameter choice rule method. MSC: 35K05; 35K99; 47J06; 47H10

Background Spinal cord injury (SCI) leads to various degrees of lifelong functional deficits. Most individuals with incomplete SCI experience a certain degree of functional recovery, especially within the first-year postinjury. However, this is difficult to predict, and surrogate biomarkers are urgently needed. Objective We aimed to (1) determine if routine blood chemistry parameters are related to neurological recovery after SCI, (2) evaluate if such parameters could predict functional recovery, and (3) establish cutoff values that could inform clinical decision-making. Methods We performed a post hoc analysis of routine blood chemistry parameters in patients with traumatic SCI (n = 676). Blood samples were collected between 24 and 72 hours as well as at 1, 2, 4, 8, and 52 weeks postinjury. Linear mixed models, regression analysis, and unbiased recursive partitioning (URP) of blood chemistry data were used to relate to and predict walking recovery 1 year postinjury. Results The temporal profile of platelet counts and serum levels of albumin, alkaline phosphatase, and creatinine differentiated patients who recovered walking from those who remained wheelchair bound. The 4 blood chemistry parameters from the sample collection 8 weeks postinjury predicted functional recovery observed 1 year after incomplete SCI. Finally, URP defined a cutoff for serum albumin at 3.7 g/dL, which in combination with baseline injury severity differentiates individuals who regain ambulation from those not able to walk. Specifically, about 80% of those with albumin >3.7 g/dL recovered walking. Conclusions Routine blood chemistry data from the postacute phase, together with baseline injury severity, predict functional outcome after incomplete SCI.