Average Number Of Levels Research Articles

Surgical Treatment of Ossifications of the Cervical Anterior Longitudinal Ligament: A Retrospective Cohort Study.

Study Design:Retrospective cohort studyObjectives:The study aims to evaluate anterior cervical discectomy and fusion (ACDF) in the treatment of patients with ossification of the anterior longitudinal ligament (OALL).Methods:We retrospectively reviewed cases performed at our institution between January 2015 and December 2018; adult (age ≥18 years) patients who underwent anterior cervical decompression and fusion in the presence of dysphagia and OALL. Ten patients (9 male, 1 female, mean age 64.4 years) with OALL who underwent ACDF were included. Charts were reviewed for demographics and comorbidities. Primary outcomes assessed were intra- and postoperative complications. Secondary outcomes were fusion rates, instrumentation failure, postsurgical instability/deformity, and readmission rates.Results:The average duration of symptoms prior to surgery was 12.3 months. All patients presented with dysphagia (mean Bazaz score 2.0). The average number of levels with OALL was 4.7 (±1.67). All patients underwent ACDF and 3 patients underwent additional posterior cervical fusion for kyphotic deformity correction or when extensive laminectomy was required. We did not encounter any intraoperative complications. Eight patients (72%) had solid fusion demonstrated on the lateral x-rays and no evidence of progressive kyphotic deformity. We did not encounter any instrumentation failure or loosening. Two patients developed recurrence of dysphagia (Bazaz scores 2 and 3 respectively).Conclusion:ACDF for OALL with dysphagia and concomitant myelopathy in our small series of 10 patients demonstrate good fusion and clinical outcomes. Larger studies will be necessary to determine the optimal treatment for patients with dysphagia due to OALL.

Non-Wigner-Dyson level statistics and fractal wave function of disordered Weyl semimetals

Finding fingerprints of disordered Weyl semimetals (WSMs) is an unsolved task. Here we report such findings in the level statistics and the fractal nature of electron wave function around Weyl nodes (WNs) of disordered WSMs. The nearest-neighbor level spacing follows a new universal distribution ${P}_{c}(s)={C}_{1}{s}^{2}exp[\ensuremath{-}{C}_{2}{s}^{2\ensuremath{-}{\ensuremath{\gamma}}_{0}}]$ originally proposed for the level statistics of critical states in the integer quantum Hall systems or normal dirty metals (diffusive metals) at metal-to-insulator transitions, instead of the Wigner-Dyson distribution for diffusive metals. Numerically we find ${\ensuremath{\gamma}}_{0}=0.62\ifmmode\pm\else\textpm\fi{}0.07$. In contrast to the Bloch wave functions of clean WSMs that uniformly distribute over the whole space of $(D=3)$ at large length scale, the wave function of disordered WSMs at a WN occupies a fractal space of dimension $D=2.18\ifmmode\pm\else\textpm\fi{}0.05$. Away from the WN, wave function is a fractal at a length scale below a correlation length diverging at the WN as $\ensuremath{\xi}\ensuremath{\propto}{|E|}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=0.89\ifmmode\pm\else\textpm\fi{}0.05$. Beyond the length scale, the wave function is homogeneous. In the ergodic limit, the level number variance ${\mathrm{\ensuremath{\Sigma}}}_{2}$ around Weyl nodes increases linearly with the average level number $N,{\mathrm{\ensuremath{\Sigma}}}_{2}=\ensuremath{\chi}N$, where $\ensuremath{\chi}=0.2\ifmmode\pm\else\textpm\fi{}0.1$ is independent of system sizes and disorder strengths.

Local trigonal modes and the suppression of the charge density wave in TiSe2−xTex

Finding fingerprints of disordered Weyl semimetals (WSMs) is an unsolved task. Here we report such findings in the level statistics and the fractal nature of electron wavefunction around Weyl nodes of disordered WSMs. The nearest-neighbor level spacing follows a new universal distribution $P_c(s)=C_1 s^2\exp[-C_2 s^{2-\gamma_0}]$ originally proposed for the level statistics of critical states in the integer quantum Hall systems or normal dirty metals (diffusive metals) at metal-to-insulator transitions, instead of the Wigner-Dyson distribution for diffusive metals. Numerically, we find $\gamma_0=0.62\pm0.07$. In contrast to the Bloch wavefuntions of clean WSMs that uniformly distribute over the whole space of ($D=3$) at large length scale, the wavefunction of disordered WSMs at a Weyl node occupies a fractal space of dimension $D=2.18\pm 0.05$. The finite size scaling of the inverse participation ratio suggests that the correlation length of wavefunctions at Weyl nodes ($E=0$) diverges as $\xi\propto |E|^{-\nu}$ with $\nu=0.89\pm0.05$. In the ergodic limit,the level number variance $\Sigma_2$ around Weyl nodes increases linearly with the average level number $N$, $\Sigma_2=\chi N$, where $\chi= 0.2\pm0.1$ is independent of system sizes and disorder strengths.

Time Savings and Related Economic Benefits of Suction-Curette Device for Transforaminal Lumbar Interbody Fusion Discectomy.

Performing an adequate transforaminal lumbar interbody fusion (TLIF) discectomy requires numerous instrument passes, increasing surgical time and the risk of complications. The purpose of this study was to evaluate the efficacy and efficiency of discectomy and endplate preparation during TLIF using traditional manual instrumentation versus a novel suction discectomy curette. The direct economic benefit with use of the suction discectomy curette is calculated. Three experienced, spine-fellowship-trained surgeons performed TLIF discectomies on 3 cadaveric specimens from T12 to S1 using either traditional manual discectomy instruments or CONCORD Clear (Xtool) devices supplemented with manual discectomy instruments. For each level in which a discectomy was performed, the following were measured: elapsed time, number of instrument passes and the number of instrument exchanges, and estimated tissue volume. Transforaminal lumbar interbody fusion discectomy times improved on average 11:32 minutes per level, which equates to an estimated procedural time savings of 15:85 minutes, using 1.4 levels per TLIF, the average number of levels in a large series. Usage of the CONCORD Clear significantly reduced instrument passes compared to traditional, with a mean of 62.0 for traditional versus 7.1 for CONCORD Clear, an 8.7-fold improvement. Instrument exchanges showed a 5.9-fold improvement, with means of 26.8 and 4.6, respectively. Wet discectomy tissue volume was measured for each discectomy, with a mean of 5.4 cc for traditional versus 12.9 cc for CONCORD Clear, a 2.4-fold improvement. This study estimates that, in a typical TLIF procedure, over 15 minutes should be saved by using the CONCORD Clear l device (a quarter of the time of a traditional discectomy), and by considering the direct cost-benefit associated with this time savings as well as reduced sterilization costs, it is estimated that a hospital could save approximately $1300 in operating room time and sterilization cost with the use of the CONCORD Clear device in a typical 1-level TLIF procedure.

161 Direct Cost Analysis of 38 Cervical Spinal Deformity Operations Across Two Major Spinal Deformity Centers with Implications for Catastrophic Costs and 90-Day Cost Bundles

Abstract INTRODUCTION CMS has recently implemented 90-day cost bundles for joint replacement surgery, and taken steps to expand bundled payment programs. Our goal is to determine the 90-day costs associated with cervical spinal deformity cases across two major spinal deformity centers in order to guide the development of reasonable bundled payment plans. METHODS Retrospective evaluation of adults who underwent cervical deformity surgery 2013–2016 at two major spinal deformity centers (UCSF, Baylor). Inclusion criteria included adult patients with cervical kyphosis (C2-7 Cobb angle >10°°) or cervical scoliosis (coronal Cobb able >10°). Direct hospital costs for the index surgery and all visits within the first 90 days after surgery were determined from hospital billing data. Direct costs include surgical supplies/implants, room/care, pharmacy; they exclude indirect costs like overhead, administration, and utilities. RESULTS >38 patients (25F, 13M; mean age = 60.4 ± 9.8 years) underwent fusions for cervical deformity. 29 procedures were posterior, 1 anterior, 7 anterior-posterior, and 1 posterior-anterior-posterior. Average number of levels fused was 9.3 ± 3.3. Average direct hospital cost for the 90-day period including surgery was $64,480 ± 23,816, and was similar between institutions ($62,569 versus $70,638). 4 of 38 patients (10.5%) were re-admitted within 90-days: re-admission costs constitute the majority of these patients' post-index surgery costs. 2 of 38 (5.3%) patients had catastrophic 90-day costs, greater than two standard deviations above the mean (>$112,112). CONCLUSION The average 90-day direct hospital costs for patients undergoing cervical deformity surgery are $64,480. Readmissions and a greater number of operated levels contribute to higher costs; reasonable 90-day cost bundle plans must account for both of these issues. A limitation is that our analysis includes costs at our main hospital only; it does not include post-operative outpatient medications, physical therapy, rehabilitation, or admissions to outside hospitals. It is therefore imperative that further studies utilizing national databases are performed to understand the full scope of 90-day costs.

106. Adjacent Segment Disease Following Posterior Instrumentation and Fusion for Degenerative Lumbar Scoliosis

Purpose: Adjacent segment disease (ASD) is major complication following spinal instrumentation and fusion. The purpose of the current study was to determine the prevalence and risk factors of proximal ASD following posterior instrumentation and fusion for degenerative lumbar scoliosis. Materials and Methods: Seventy-two patients (mean age 64.8 years) who had undergone decompression and fusion with pedicle screw instrumentation were evaluated. The average follow-up was 4.7 years. Twenty-five patients had additional interbody fusion at the lower lumbar spine. The average number of levels fused was 5.1 segments (range 1-9). The upper instrumented vertebrae ranged from T9 to L4. The lower instrumented vertebrae were L5 and S1. Results: Proximal ASD developed in 17 (24%) of 72 patients, including compression fractures (n=6), junctional kyphosis (n=5), spinal stenosis (n=4), and symptomatic disc collapse (n=2). The preoperative scoliotic angle, lumbar lordosis, thoracic kyphosis, and coronal and sagittal C7 plumb were not associated with the development of proximal ASD. There was a close correlation between the level of the upper instrumented vertebrae and the development of ASD (p=0.001). When fusion did not extend beyond the lumbar vertebra, ASD occurred in 15 (38.5%) of 39 patients. In contrast, when fusion extended up to the thoracic vertebrae, ASD occurred in 2 (6.1%) of 33 patients. The improvement in the Oswestry score was superior to the group without ASD (p=0.001). Conclusion: The prevalence of symptomatic ASD at the proximal segment was 24% after posterior instrumentation and fusion for degenerative lumbar scoliosis. The cephalad extent of fusion was the most significant risk factor for the development of proximal ASD.

Pulmonary embolism after major spinal deformity corrective surgery in adults

Purpose of study: To determine the incidence of pulmonary embolism (PE) and contributing factors after major spinal deformity corrective surgery in adults. Although pulmonary emboli are known to occur after spinal surgery, there are limited published data on the incidence of PE and its contributing factors following major spinal corrective surgery. Incidence of PE in this patient population has been reported to be as high as 3% to 17%. We report our experience in 417 patients.Methods used: From 1992 to 1999, 417 patients (347 women, 70 men) aged 20 to 87 years, (average, 51 years) underwent spinal deformity corrective surgery (146 primary, 271 revision). Surgical approaches include 23 anterior, 155 posterior, 167 same-day combined and 72 staged combined. All surgeries were performed by the senior author (JPK). Average intraoperative blood loss was 3,100 cc. Comorbidity risk was assessed by ASA score (1, 28; 2, 233; 3, 143; 4, 3; average 2.3). Fusion levels varied as follows: <5, 108; 5 to 10, 192; >10, 107. Prophylaxis included TEDS/SCDS, coumadin, heparin, and low molecular weight heparin. We retrospectively reviewed the patients who had pulmonary emboli on our service and identified possible contributory factors.of findings: the incidence of PE was 2.6% (11 of 417; 2 men, 9 women; aged 26 to 64 years, average, 50 years). Diagnosis by spiral computed tomography or V/Q scan occurred on average on postoperative day 4.4 (range, 1 to 13). Two had prior histories of PE/DVT. Eighty percent were diagnosed on or before post operative day 4. Average INR (2.8) and PTT (3) were elevated at time of diagnosis. Average intraoperative blood loss was 2,700 cc (1,000 to 5,000). Average number of levels was 8.6 (4 to 14). Of the 11, 5 were anterior approaches, 5 were posterior alone and 4 were combined; 7 were revisions. ASA scores averaged 2.6 (range, 2 to 3). Of the 11, 3 died (27%).Relationship between findings and existing knowledge: This is the largest known database of complications to date.Overall significance of findings: the incidence of PE after major spinal corrective surgery in adults is significant. Sex, age, blood loss and revision status were not factors affecting mortality. The majority of PEs were diagnosed early and occurred despite therapeutic laboratory values. The high incidence of mortality in this patient population makes it imperative to further investigate PE in this patient population.Disclosures: No disclosures.Conflict of interest: No conflicts.

Mortality after major spinal deformity corrective surgery in adults

Purpose of study: To determine the incidence of mortality after major spinal deformity corrective surgery in adults and factors that may contribute to postoperative mortality. Incidence of mortality after major spinal deformity corrective surgery has reportedly ranged form 1.4% to 20%. Few studies assess mortality in a large series. We report our experience in 417 patients.Methods used: From 1992 to 1999, 417 patients (347 women, 70 men) aged 20 to 87 years (average, 51 years) underwent spinal deformity corrective surgery (146 primary, 271 revision). Surgical approaches include 23 anterior, 155 posterior, 167 same-day combined and 72 staged combined. Average intraoperative blood loss was 3,100 cc. Comorbidity risk was assessed by ASA score (1, 28; 2, 233; 3, 143; 4, 3; average 2.3). Fusion levels varied as follows: <5, 108; 5–10 192; >10, 107. We retrospectively reviewed the patients who died on our service and identified factors that may have contributed to their demise.of findings: The incidence of mortality was 2.4% (10 of 417; 2 men, 8 women; aged 35 to 70 years, average, 52 years). One death was intraoperative. Death occurred on average on postoperative day 9. Average intraoperative blood loss was 3,699 cc (1,300 to 9,000). Average number of levels was 10.1 (5 to 15). Of the 10, 2 were anterior approaches, 5 were posterior alone and 3 were combined; 5 were revisions. Pedicle subtraction osteotomies were performed in six of eight posterior approaches. ASA scores averaged 3 (range, 2 to 4; 4, 2; 3, 5; 2, 2; 1, 0). Causes of death included pulmonary embolus (3), myocardial infarction, shock, abdominal hemorrhage, seizures, multisystem organ failure, cerebral edema/brain death and ARDS.Relationship between findings and existing knowledge: This is the largest known database of complications to date.Overall significance of findings: The incidence of mortality after major spinal corrective surgery in adults is significant. Sex, age, blood loss and revision status were not factors affecting mortality. Patients who died tended to have higher ASA scores/comorbidities (3 vs. 2.3) and were more likely to have had osteotomies. The high incidence of pulmonary embolus as cause of death makes it imperative to further investigate PE in this patient population.Disclosures: No disclosures.Conflict of interest: No conflicts.

Finite-size scaling of the level compressibility at the Anderson transition

We compute the number level variance $\Sigma_{2}$ and the level compressibility $\chi$ from high precision data for the Anderson model of localization and show that they can be used in order to estimate the critical properties at the metal-insulator transition by means of finite-size scaling. With $N$, $W$, and $L$ denoting, respectively, system size, disorder strength, and the average number of levels in units of the mean level spacing, we find that both $\chi(N,W)$ and the integrated $\Sigma_{2}$ obey finite-size scaling. The high precision data was obtained for an anisotropic three-dimensional Anderson model with disorder given by a box distribution of width $W/2$. We compute the critical exponent as $\nu \approx 1.45 \pm 0.12$ and the critical disorder as $W_{\rm c} \approx 8.59 \pm 0.05$ in agreement with previous transfer-matrix studies in the anisotropic model. Furthermore, we find $\chi\approx 0.28 \pm 0.06$ at the metal-insulator transition in very close agreement with previous results.

On the average hyperoscillations of planted plane trees

Assume that the leaves of a planted plane tree are enumerated from left to right by 1, 2, .... Thej-ths-turn of the tree is defined to be the root of the (unique) subtree of minimal height with leavesj, j+1, ...,j+s−1. If all trees withn nodes are regarded equally likely, the average level number of thej-ths-turn tends to a finite limitα s (j), which is of orderj 1/2. Thej-th ”s-hyperoscillation”α 1(j)−α s+1(j) is given by 1/2α 1(s)+O(j −1/2) and therefore tends (forj → ∞) to a constant behaving like √8/π·s 1/2 fors → ∞. These results are obtained by setting up appropriate generating functions, which are expanded about their (algebraic) singularities nearest to the origin, so that the asymptotic formulas are consequences of the so-called Darboux-Polyamethod.

Random insertion into a priority queue structure

The average number of levels that a new element moves up when inserted into a heap is investigated. Two probabilistic models under which such an average might be computed are proposed. A `Lemma of Conservation of Ignorance' is formulated and used in the derivation of an exact formula for the average in one of these models. It is shown that this average is bounded by a constant and its asymptotic behaviour is discussed. Numerical data for the second model are also provided and analyzed.