Guido David

This paper studies the three-player gambler’s ruin with variable bet size, that is, more than one chip may be transferred from one player to another. Weighted directed multigraphs were constructed to model the transitions between chip states. Linear systems are constructed based on the connections between nodes in these graphs. Solutions for the placing probabilities of each player are obtained from these linear systems. Expected time until ruin is solved by modeling the game as a Markov process. A numerical algorithm is developed to solve the No Limit three-player gambler’s ruin problem for any positive integer chip total.

Asian options are path dependent options whose payoff depends on the average price of an asset. The price of these options does not have closed form solutions. The multilevel Monte Carlo is a method for discretization with the application of the standard Monte Carlo method. This method is easy to implement and has less computational cost than the standard Monte Carlo method. The application of this method in pricing Asian options whose underlying asset follows the geometric Brownian motion has already been studied. However, this has not been applied in pricing Asian options where the underlying asset follows the square root process. We use the Riemann scheme to discretize the temporal integral of the square root process and a simulation algorithm based on the transition density of the square root process to determine the values of the square root process at specified times. We determined that the rate of mean square convergence of the Riemann sum in approximating the temporal integral of the square root is at least of order 1. The results are compared with those obtained using standard Monte Carlo in terms of computational cost. It is shown that the use of this method with the Riemann scheme as discretization scheme is effective in terms of reducing computational cost as compared to the standard Monte Carlo method.

The risk of rupture of intracranial saccular aneurysms is one of the leading dilemmas for patients and neurologists. Although the probability of rupture is small, the consequences of rupture are usually fatal or crippling, and a concern for the patient is whether or not to treat an existing aneurysm. In this paper, an idealized model of saccular aneurysms with assumed Fung material behavior was investigated for rupture potential when the stresses exceeded the maximum wall strength of the aneurysm wall. Numerical simulations used various levels of blood pressure, from normal to hypertensive, in order to determine correlations of aneurysm size and risk of rupture. Results showed that hypertensive individuals harboring cerebral aneurysms with a size of at least 6 mm are at risk.

Continuum biomechanics continues to aid in the design of clinical procedures and biomedical devices. Such design requires detailed information on the biomechanical properties of the tissues of interest as well as appropriate methods of analysis. In this paper, we use a fully nonlinear, virtual work based finite element method to study the normal stress field in the anterior lens capsule of the porcine eye. The analysis shows that recently measured regional variations in material symmetry may combine with regional variations in membrane thickness to yield a nearly uniform and equibiaxial stress field in normalcy. These findings are discussed in terms of potential implications to the underlying mechanobiology and in particular, in terms of perturbations in stress that result from cataract surgery, the most commonly performed surgical procedure in North America.

Hyperglycemia can alter the mechanical properties of tissues through the formation of advanced glycation endproducts in matrix proteins that have long half-lives. We used a custom experimental system and subdomain finite element method to quantify alterations in the regional multiaxial mechanical properties of porcine lens capsules that were cultured for 8 or 14 weeks in high glucose versus control media. Findings revealed that high glucose significantly stiffened the capsules in both the circumferential and the meridional directions, but it did not affect the known regional variations in anisotropy. Such information could be important in the design of both improved clinical procedures and intraocular implants for diabetic patients.

The lens capsule of the eye plays fundamental biomechanical roles in both normal physiological processes and clinical interventions. There has been modest attention given to the mechanical properties of this important membrane, however, and prior studies have focused on 1-D analyses of the data. We present results that suggest that the porcine anterior lens capsule has a complex, regionally dependent, nonlinear, anisotropic behavior. Specifically, using a subdomain inverse finite element method to analyze data collected via a new biplane video-based test system, we found that the lens capsule is nearly isotropic (in-plane) near the pole but progressively stiffer in the circumferential compared to the meridional direction as one approaches the equator. Because the porcine capsule is a good model of the young human capsule, there is strong motivation to determine if similar regional variations exist in the human lens capsule for knowledge of such complexities may allow us to improve the design of surgical procedures and implants.

Clinical interventions can change the mechanical environment of the tissues targeted for therapy. In order to design better procedures, it is important to understand cellular responses to altered mechanical stress. Rigid fixation is one example of a constraint imposed on living tissues as a result of implanted devices. This results in disturbed stress and strain fields, with potentially strong gradients. Herein, we numerically solve the governing nonlinear ordinary differential equation for the stress distribution in a finitely deformed anisotropic circular membrane with a concentric fixation by applying a zero-displacement condition at the inner circumference. Results show that rigid fixations yield distributions of stress and strain that are markedly different from tissue defects with traction-free boundaries. Moreover, the material anisotropy plays a significant role in the manner the stress redistributes regardless of the size of fixation. The present study will contribute to the design of experiments to determine cellular reactions involved in the failure of interventional treatments.