Fast Ewald summation for electrostatic potentials with arbitrary periodicity.

A unified treatment for the fast and spectrally accurate evaluation of electrostatic potentials with periodic boundary conditions in any or none of the three spatial dimensions is presented. Ewald decomposition is used to split the problem into real-space and Fourier-space parts, and the Fast Fourier Transform (FFT)-based Spectral Ewald (SE) method is used to accelerate computation of the latter, yielding the total runtime O(N⁡log(N)) for N sources. A key component is a new FFT-based solution technique for the free-space Poisson problem. The computational cost is further reduced by a new adaptive FFT for the doubly and singly periodic cases, allowing for different local upsampling factors. The SE method is most efficient in the triply periodic case where the cost of computing FFTs is the lowest, whereas the rest of the algorithm is essentially independent of periodicity. We show that removing periodic boundary conditions from one or two directions out of three will only moderately increase the total runtime, and in the free-space case, the runtime is around four times that of the triply periodic case. The Gaussian window function previously used in the SE method is compared with a new piecewise polynomial approximation of the Kaiser-Bessel window, which further reduces the runtime. We present error estimates and a parameter selection scheme for all parameters of the method, including a new estimate for the shape parameter of the Kaiser-Bessel window. Finally, we consider methods for force computation and compare the runtime of the SE method with that of the fast multipole method.

Physical activity level in Achilles tendinosis is associated with blood levels of pain-related factors: a pilot study.

Physical activity affects the pain symptoms for Achilles tendinosis patients. Brain-derived neurotrophic factor (BDNF), tumor necrosis factor-alpha (TNF-α) and their receptors have been detected in human Achilles tendon. This pilot study aimed to compare serum BDNF and soluble tumor necrosis factor receptor I (sTNFRI) levels in Achilles tendinosis patients and healthy controls and to examine the influence of physical activity, and BMI and gender, on these levels. Physical activity was measured with a validated questionnaire, total physical activity being the parameter analyzed. Physical activity was strongly correlated with BDNF among tendinosis women [Spearman's rho (ρ)=0.90, P<0.01] but not among control women (ρ=-0.08, P=0.83), or among tendinosis and control men. Physical activity was significantly correlated with sTNFRI in the entire tendinosis group and among tendinosis men (ρ=0.65, P=0.01), but not in the entire control group or among control men (ρ=0.04, P=0.91). Thus, the physical activity pattern is related to the TNF and BDNF systems for tendinosis patients but not controls, the relationship being gender dependent. This is new information concerning the relationship between physical activity and Achilles tendinosis, which may be related to pain for the patients. This aspect should be further evaluated using larger patient materials.