Saddlepoint approximations to sensitivities of tail probabilities of random sums and comparisons with Monte Carlo estimators

This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus, the suggested saddlepoint approximation can be conveniently used in various scientific problems.

Bounding Standard Gaussian Tail Probabilities

We review various inequalities for Mills' ratio (1 - Φ)= Ø, where Ø and Φ denote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.

Preparation of intact bovine tail intervertebral discs for organ culture

The intervertebral disc (IVD) is the joint of the spine connecting vertebra to vertebra. It functions to transmit loading of the spine and give flexibility to the spine. It composes of three compartments: the innermost nucleus pulposus (NP) encompassing by the annulus fibrosus (AF), and two cartilaginous endplates connecting the NP and AF to the vertebral body on both sides. Discogenic pain possibly caused by degenerative intervertebral disc disease (DDD) and disc herniations has been identified as a major problem in our modern society. To study possible mechanisms of IVD degeneration, in vitro organ culture systems with live disc cells are highly appealing. The in vitro culture of intact bovine coccygeal IVDs has advanced to a relevant model system, which allows the study of mechano-biological aspects in a well-controlled physiological and mechanical environment. Bovine tail IVDs can be obtained relatively easy in higher numbers and are very similar to the human lumbar IVDs with respect to cell density, cell population and dimensions. However, previous bovine caudal IVD harvesting techniques retaining cartilaginous endplates and bony endplates failed after 1-2 days of culture since the nutrition pathways were obviously blocked by clotted blood. IVDs are the biggest avascular organs, thus, the nutrients to the cells in the NP are solely dependent on diffusion via the capillary buds from the adjacent vertebral body. Presence of bone debris and clotted blood on the endplate surfaces can hinder nutrient diffusion into the center of the disc and compromise cell viability. Our group established a relatively quick protocol to "crack"-out the IVDs from the tail with a low risk for contamination. We are able to permeabilize the freshly-cut bony endplate surfaces by using a surgical jet lavage system, which removes the blood clots and cutting debris and very efficiently reopens the nutrition diffusion pathway to the center of the IVD. The presence of growth plates on both sides of the vertebral bone has to be avoided and to be removed prior to culture. In this video, we outline the crucial steps during preparation and demonstrate the key to a successful organ culture maintaining high cell viability for 14 days under free swelling culture. The culture time could be extended when appropriate mechanical environment can be maintained by using mechanical loading bioreactor. The technique demonstrated here can be extended to other animal species such as porcine, ovine and leporine caudal and lumbar IVD isolation.