Weinstein Method Research Articles

Appropriate insertion point for percutaneous pedicle screw placement in the lumbar spine using c-arm fluoroscopy: a cadaveric study

BackgroundWe studied the characteristics and regularity of appropriate insertion points for percutaneous pedicle screw placement in the lumbar spine using C-arm X-ray fluoroscopy. The purpose of this study was to improve the accuracy of percutaneous pedicle screw placement and reduce the incidence of superior-level facet joint violation.MethodsSix normal spinal specimens were included. Three different methods for placing percutaneous pedicle screws in the lumbar spine were applied, including the Roy-Camille method, Magerl method and Weinstein method. The relationships among the insertion point, pedicle projection and proximal facet joint on C-arm X-ray films were studied. The projection morphology of the vertebral pedicle in different segments of the lumbar spine was observed. The relationship between the outer edge of the pedicle projection and the outer edge of the cranial articular process was also studied. The distance between the insertion point and the facet joint (M1), the distance between the insertion point and outer edge of the cranial articular process (M2), and the distance between the insertion point and the projection center of the pedicle (M) were measured.ResultsIn this study, we found that the projection shape of the vertebral pedicle differed across segments of the lumbar spine: the shape for L1-L3 was oval, and that for L4-L5 was round. The radiographic study showed that the outer edge of the cranial articular process was located on the lateral side of the outer edge of the pedicle projection and did not overlap with the pedicle projection. M for the Weinstein group was larger than that for the Roy-Camille group (P < 0.05). M1 for the Weinstein group was larger than that for the Roy-Camille and Magerl groups (P < 0.05). M2 for the Roy-Camille group was negative, M2 for the Magerl group was 0, and M2 for the Weinstein group was positive.ConclusionUnder C-arm X-ray fluoroscopy, we were able to accurately identify the characteristics and regularity of the appropriate insertion point for percutaneous pedicle screw placement in the lumbar spine, which was important for improving the accuracy of percutaneous pedicle screw placement and reducing the incidence of superior-level facet joint violation.

What is the difference in morphologic features of the lumbar vertebrae between Caucasian and Taiwanese subjects? A CT-based study: implications of pedicle screw placement via Roy-Camille or Weinstein method

BackgroundSafe placement of pedicle screws without jeopardizing neurovascular structures medially and anteriorly is important during spine surgery. Inferior breach of pedicle is also dangerous due to low margin of error. Lumbar morphology and identical pedicle orientation at L1 to L5 shown on CT scan of young Taiwanese patients (90 patients) were analyzed and compared with findings reported for Caucasian subjects.MethodsPreviously reported techniques were employed to quantitatively elucidate the parameters regarding lumbar morphology and identical pedicle orientation at each vertebra. The parameters for pedicle angle (PA), pedicle diameter (PD), pedicle axis distance (PAD), midline axis distance (MAD), transverse pedicle axis distance (TPAD) and transverse intertangential angle (TITA) were measured.ResultsTaiwanese subjects had different PA, PD, PAD, MAD at L1 to L5 and TITA at L3 to L5 compared with Caucasian subjects. L5 had the most convergent pedicle axis, the widest PD and the shortest antero-posterior axis morphology.ConclusionsThis study provides detailed information for identifying pedicle orientation during pedicle screw placement and elucidate racial differences in lumbar morphology and pedicle orientation between Taiwanese and Caucasian populations.

Technical factors related to the incidence of adjacent superior segment facet joint violation after transpedicular instrumentation in the lumbar spine

Segmental pedicle screw fixation is rapidly becoming a popular method of spinal instrumentation. Few studies have investigated the rates of adjacent superior segment facet joint violation. The purpose of our study were to investigate the incidence of superior segment facet joint violation after pedicle screw instrumentation in the lumbar spine and to evaluate technical factors related to the incidence. A prospective study including 96 patients who underwent lumbar and lumbosacral fusion was conducted between March 2006 and December 2007. All patients had bilateral or unilateral posterior pedicle screw-rod instrumentation with either CD-Horizon (top-loading screw) or TSRH (side-connecting screw) implants. Pedicle screws were instrumented according to the methods advocated by Roy-Camille (Group 1, 20 cases) or Weinstein (Group 2, 76 cases). All patients had computed tomography scan at 1 week post operation. CT scans were reviewed blind by an experienced spine research fellow and a consultant radiologist to determine violation of the adjacent superior segment facet joint. Superior segment facet joint violation occurred in all of the 20 patients (100%) and all of the top-level screws (100%) in Group 1. The spinal research fellow noted the incidence of facet joint violation to be present in 23.8% of the screws and 32.9% of the patients in Group 2, whereas the consultant radiologist noted this to be the case in 25.2 and 35.5%, respectively. The incidence of facet joint violation in patients with CD-Horizon screws was far lower than patients with TSRH screws (P < 0.001). In conclusion, it seems that meticulous surgical dissection without injuring the top-level facet joints, proper instrumentation of pedicle screws with the appropriate entry site (Weinstein's method), trajectory, and use of top-loading screw heads are some ways that surgeons could minimize the risk of top-level facet joint violation.

Bounds for the Heat Diffusion through Windows of Given Area

We consider the lowest eigenvalue for the Laplacian in a given Lipschitz domain under mixed boundary conditions: Dirichlet in a subset of the boundary (noninsulated window), Neumann otherwise. This eigenvalue can be interpreted as heat leakage rate due to diffusion. We give an explicit calculation for a model problem, a rigorous lower bound that depends only on the area, but not on the geometry of the window. This bound confirms the observations from the model problem. Finally, we show that no nontrivial upper bound is possible; i.e., any small area for the window being prescribed, its geometry can be made bad enough to cause heat leak rates arbitrarily close to the ones for no insulation anywhere. The most important techniques are the Aronszajn–Weinstein method of intermediate variational problems and the Gaussian upper bounds for the heat kernel by E. B. Davies.

Monotone convergence theorems for variational triples with applications to intermediate problems

SynopsisThe variational eigenvalue problem for a real quadratic form b with respect to a positive definite form a on a vector space V may be represented by the triple (b, a, V). Methods of intermediate problems provide approximations from below to the lower eigenvalues of (b, a, V) using monotone increasing sequences of such triples. It is shown that every such approximation method is canonically equivalent to Weinstein's method. General convergence theorems are proved for such methods. These results generalise known convergence results for increasing sequences of quadratic forms. The results are applied to some specific approximation methods and are illustrated using a differential eigenvalue problem.

A rigorous justification and estimation of the rate of convergence for the partial domain method in two-dimensional eigenvalue problems for the Laplace operator

The partial domain method for problems of the type indicated in the title of this paper is treated as a version either of Weinstein's method of intermediate problems or of the Ritz method. This yields a rigorous justification of the method and enables one to estimate its rate of convergence. The justification technique is demonstrated in detail for the problem of the normal modes of an L-shaped membrane clamped at its edges.

The two-dimensional helmholtz equation − Δu = λu for a certain class of mutli-connected regions

The numerical calculation of the eigenvalues and eigenfunctions of − Δμ = γμ is considered for multi-connected regions composed of unions of rectangles. Numerical examples are dealt with for coaxial rectangles. The boundary conditions include three types; the Dirichlet condition ( u = 0), the Neumann condition ( ∂u ∂n = 0 ) and certain mixed conditions. The method used is the Weinstein method which gives lower bounds for exact eigenvalues, as opposed to the Rayleigh-Ritz variational method which gives upper bounds. The method consists of decomposing the given region into rectangles and relaxing the boundary condition (from Dirichlet to Neumann) or the continuity condition of the solution function space, and of choosing trial functions p i 's which play the role of strengthening the boundary condition (from Neumann to Dirichlet) or the discontinuity condition to the continuity condition, and then approximating the original solution function space. The numerical results are considered to be reasonably precise. Furthermore, extrapolated values obtained from remarkably regular convergence results are believed to be extremely precise. The application of the method to an eccentric-axial rectangule or a rectangle with two inner rectangles would be straightforward.

An estimate of the rate of convergence of Weinstein's method in the eigenvalue problem for an L-shaped membrane

General estimates for the rate of convergence of Weinstein's method of intermediate problems, obtained previously by the author, are applied particularly to the case of the eigenvalue problem for the L-shaped membrane attached at its edges.

The eigenvalue problem for −Δu=λu with Dirichlet boundary conditions for two-dimensional regions. II. T-shaped, cross-shaped, and H-shaped regions

This article deals with the calculation of the eigenvalues and eigenfunctions of the two-dimensional Laplacian with Dirichlet boundary condition for T-shaped, cross-shaped, and H-shaped regions. The method used is the Weinstein method (the intermediate method). The suitable base problem can be obtained through dividing the given region into rectangles and replacing Dirichlet boundary conditions by Neumann boundary conditions on a part of the boundaries. The intermediate problems can be determined by appropriate trial functions, and by the eigenvalues and eigenfunctions for each decomposed rectangle. The numerical results are considered to be reasonably precise. Special attention is paid to the dependence of the eigenvalues and eigenfunctions on parameters of the region. Some consideration is also given to the curve-veering phenomena observed frequently in the numerical results.

The eigenvalue problem for −Δu=λu with dirichlet boundary conditions for a certain class of two-dimensional regions

By using the Weinstein method, eigenvalues and eigenfunctions of the equation −Δu=λu with Dirichlet boundary conditions are calculated for a certain class of regions. The regions are composed of unions of rectangles, and include L-shaped, single-notched and crossed rectangles. The method consists of determining the zeros of the Weinstein determinant W n (λ) . This function W n (λ) in turn is determined by the eigenvalues and eigenfunctions of −Δu=λu with mixed boundary conditions (Dirichlet and Neumann) for each component rectangle of the given region, which are easily calculated, and by trial functions p 1 , p 2 , …, p n which are easily chosen. Numerical results for two examples, an asymmetrical L-shaped region and a symmetrical single-notched region, are given, and are shown to be reasonably precise by comparison with results available in the literature. The method is applicable to many problems for a certain class of regions.

A hybrid method of modal synthesis using vibration tests

The assembly of structures along continuous boundaries presents great difficulty in the context of modal synthesis. In order to solve such problems, a method is proposed in which a hybrid model is defined reflecting the dynamic behavior of a structure loaded along a boundary. It is based on Weinstein's method and corresponds to a generalization of the impedance matrix method. Generalized boundary co-ordinates are defined from branch modes obtained by introducing mass loading along the boundary. Thus, the hybrid model can be derived from testing as a result of two independent modal identifications. The method permits high precision prediction of the influence of strong structural modifications. Thus, in the case of rectangular plates, it has been possible to find the modes of a cantilever plate and of a plate with stiffeners from the free modes.

Studies in the accuracy of some lower-bound formulae-a simple one-dimensional problem

A comparative study is made of several lower-bound procedures for calculating the eigenvalues of a simple exactly soluble one-dimensional potential, when this potential V(x) is regarded as the sum of two harmonic-oscillator potentials 1/2(R-x)2 and 1/2(R+x)2 centred at x=+or-R. This potential resembles a molecular potential, the wavefunctions for which are to be represented as combinations of the 'atomic' wavefunctions of the two separate potentials 1/2(R+or-x)2. The resulting LCAO-type wavefunctions are chosen to give lower bounds according to the classical methods of Weinstein and Stevenson (1938) and the truncated hamiltonian method of Bazley and Fox (1963); upper bounds are also obtained according to the Rayleigh-Ritz principle. The calculations are made for different values of R. Acceptable energies can be calculated by all methods. Of the methods considered the Rayleigh-Ritz upper-bound method converges most rapidly; the Stevenson lower bound is best if the optimum value of a certain parameter is known; the Weinstein method gives the poorest convergence of all.