Eigenvalue Problem in Two Dimensions for an Irregular Boundary II: Neumann Condition

We formulate a systematic elegant perturbative scheme for determining the eigenvalues of the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions when the normal derivative of {\psi} vanishes on an irregular closed curve. Unique feature of this method, unlike other perturbation schemes, is that it does not require a separate formalism to treat degeneracies. Degenerate states are handled equally elegantly as the non-degenerate ones. A real parameter, extracted from the parameters defining the irregular boundary, serves as a perturbation parameter in this scheme as opposed to earlier schemes where the perturbation parameter is an artificial one. The efficacy of the proposed scheme is gauged by calculating the eigenvalues for elliptical and supercircular boundaries and comparing with the results obtained numerically. We also present a simple and interesting semi-empirical formula, determining the eigenspectrum of the 2D Helmholtz equation with the Dirichlet or the Neumann condition for a supercircular boundary. A comparison of the eigenspectrum for several low-lying modes obtained by employing the formula with the corresponding numerical estimates shows good agreement for a wide range of the supercircular exponent.Comment: 26 pages, 12 figure

EFFECT OF BETA-BLOCKERS ON BONE MINERAL DENSITY AND OSTEOPOROTIC FRACTURES

This review presents the current data on the pleiotropic effects of beta-blockers in particular on the effect on bone mass and fracture development by means of increased bone fragility. In this article we discuss the mechanisms of action of beta-blockers on bone metabolism and their possible protective effect on bone tissue in the treatment of cardiovascular disease

EFFECT OF CARDIOVASCULAR DRUGS ON BONE HEALTH AND THE POSSIBILITY OF THEIR USE FOR THE PREVENTION OF OSTEOPOROSIS

Widespread use of antihypertensive and lipid-lowering agents in clinical practice determines the necessity of knowledge of their pleiotropic effects. Results of studies of the effect of cardiac drugs and, first of all beta-blockers, ACE inhibitors, diuretics, and statins on bone are presented. Mode of action on the bone mineral density , markers of bone turnover and ultimately impact on the incidence of fractures associated with osteoporosis are discussed. At the present time there are no sufficient evidences of positive effect of these medications on bone coming from randomized controlled trials. It is not possible to recommend discussed cardiovascular drugs for prevention of osteoporosis and fractures, as well as registration new indications for them. However , knowledge of additional effects on the bone metabolism in cardiovascular drugs, will allow doctors to choose optimal treatment of hypertension and lipid disorders, taking into account the state of bone tissue. At the same time it will also allow to prevent osteoporosis in patients having osteoporosis risk factors or initial signs of bone loss.</p

IZUChENIE PLEYOTROPNYKh EFFEKTOV V-ADRENOBLOKATOROV I INGIBITOROV APF NA KOSTNUYu TKAN'

Objective: To investigate the effect of treatment with betablockers (β-AB), inhibitors of angiotensin converting enzyme (ACEI) on bone mineral density (BMD) depending on the risk factors (RF) of osteoporosis. Material and methods. In a retrospective study included 1129 outpatients (1093 women) aged over 40 years, who had the first DXA examination prior to start of the treatment for osteoporosis. Baseline characteristics of pts including data on osteoporosis risk factors (RF) and medication were obtained at the initial visit which had taken place between 2001 and 2011. BMD at the lumbar spine (LS), femoral neck (FN) and total hip (TH) were measured by DXA (Hologic Delphi W). 384 pts have been taking β-AB, ACEI and their combination not less than 6 months before the DXA examination ("users group”), 745 pts. have not been receiving any therapy ("non-users group”). Results. In the "users group” risk of reduction of BMD was lower than in the non-users [RR=1,6 (95 % CI 1.25-2,022) p<0.001], osteoporosis was diagnosed 1,3 times less frequently, and the BMD in LS, FN and TH were significantly higher than these parameters in "non-users group”. The highest BMD were noted in pts on combined therapy. The risk of BMD reduction not depends in both groups on RF such as age, postmenopause duration, presense of early or surgical menopause, low body weight, physical inactivity, previous fractures, fractures in relatives, rheumatoid arthritis, glucocorticoid use or alcohol abuse. In multivariate regression analysis after adjustment with these RF, BMD at all measured locations in users group maintained significantly higher than in non-users. There was no correlation between BMD and duration of β-AB and ACEI therapy. Conclusion Prolonged use of β-AB, ACEI in combination as well as monotherapy could has a protective effect on bone mass regardless of osteoporosis risk factors

ASSOCIATION BETWEEN ANTIHYPERTENSIVE AND LIPID-LOWERING THERAPY, BONE MASS, AND OSTEOPOROSIS RISK FACTORS

Aim. To evaluate the association between β-adrenoblocker (β-AB), angiotensin-converting enzyme inhibitor (ACEI), and statin therapy, bone mineral density (BMD), and osteoporosis risk factors (RFs).Material and methods. This retrospective study (n=1163) included 42 men and 1121 women aged &gt;40 years, who underwent densitometry before the start of the osteoporosis therapy. The main group (MG) included 418 people receiving β-AB, ACEI, statins, or their combination for at least 6 months before densitometry. The control group (CG) included 745 untreated patients. The data on RFs and pharmacological therapy came from ambulatory case histories and telephone surveys. BMD was measured by X-ray densitometry of lumbar spine and femur.Results. In patients who received antihypertensive (AHT) and lipid-lowering therapy (LLT), the risk of bone mass (BM) reduction was lower than that in CG (odds ratio (OR) 1,6; 95% confidence interval 1,25–2,022; p&lt;0,001); the osteoporosis incidence was 1,3 times lower; and all BMD measurements were significantly higher than in untreated patients. The highest BMD was observed in patients on combined therapy which included statins. The odds of BM reduction by such factors as age, postmenopause duration, early and surgical menopause, low body mass, low physical activity (LPA), previous fractures, fractures in relatives, rheumatoid arthritis, glucocorticoid therapy, and alcohol abuse, were similar in MG and CG. In logistic regression analyses, these factors were not associated with the protective effects of the medications on BMD. The number of peripheral fractures was higher in MG than in CG (36% vs. 26%). This was due to the fact that in the MG, patients were older (mean age 62,5±8,52 years, vs. 57,8±8,2 years in CG), more likely to report LPA (&gt;50%), and potentially, more prone to adverse effects of AHT which could result in more frequent falls.Conclusion. Treatment with β-AB, ACEI, and statins is associated with higher BMD of both lumbar spine and proximal femur. With the exception of age and postmenopause duration, osteoporosis RFs did not influence the effects of cardiovascular treatment