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1.
J Cosmet Dermatol ; 22(12): 3246-3251, 2023 Dec.
Article in English | MEDLINE | ID: mdl-37786282

ABSTRACT

INTRODUCTION: Filler injections in the periocular region are regarded as a challenging and advanced maneuver in a high-risk area. Adverse events as malar edema due to filler treatment may occur. To evaluate the possible reasons, the ultrasound images, and medical data of patients that were prospectively referred with malar edema were evaluated. MATERIALS AND METHODS: A total of 17 patients (26 eyes) with malar edema after hyaluronic acid filler treatment were included. All cases were assessed with an 18 MHz linear ultrasound device. Exact location of the filler material was noted. Relations with clinical data were analyzed using chi-square tests. RESULTS: Onset of malar edema after treatment showed a wide range from immediate (0 days) to 3 years. Most patients had an early onset N = 13 (76%), a minority showed late onset N = 4 (24%). In 23 eyes, the filler material was found to be located inside the SMAS. In 3 cases filler material was located on the periosteum of the orbital rim. After duplex-ultrasound guided filler removal, restored venous flow could be seen in the superficial and/or deep fatty layer often accompanied by flow piercing through the SMAS. Minutes after treatment, clinical improvement of malar edema was observed. CONCLUSION: Malar edema after by filler treatments in the periocular region may be caused by veno-lymphatic compression by filler deposits.


Subject(s)
Cosmetic Techniques , Dermal Fillers , Humans , Face/diagnostic imaging , Cosmetic Techniques/adverse effects , Eye , Zygoma , Hyaluronic Acid/adverse effects , Edema/chemically induced , Edema/diagnostic imaging , Dermal Fillers/adverse effects
2.
J Phys Condens Matter ; 33(41)2021 Aug 05.
Article in English | MEDLINE | ID: mdl-33662946

ABSTRACT

Magnonics is a budding research field in nanomagnetism and nanoscience that addresses the use of spin waves (magnons) to transmit, store, and process information. The rapid advancements of this field during last one decade in terms of upsurge in research papers, review articles, citations, proposals of devices as well as introduction of new sub-topics prompted us to present the first roadmap on magnonics. This is a collection of 22 sections written by leading experts in this field who review and discuss the current status besides presenting their vision of future perspectives. Today, the principal challenges in applied magnonics are the excitation of sub-100 nm wavelength magnons, their manipulation on the nanoscale and the creation of sub-micrometre devices using low-Gilbert damping magnetic materials and its interconnections to standard electronics. To this end, magnonics offers lower energy consumption, easier integrability and compatibility with CMOS structure, reprogrammability, shorter wavelength, smaller device features, anisotropic properties, negative group velocity, non-reciprocity and efficient tunability by various external stimuli to name a few. Hence, despite being a young research field, magnonics has come a long way since its early inception. This roadmap asserts a milestone for future emerging research directions in magnonics, and hopefully, it will inspire a series of exciting new articles on the same topic in the coming years.

3.
Ann Chir Plast Esthet ; 62(5): 365-374, 2017 Oct.
Article in English | MEDLINE | ID: mdl-29033216

ABSTRACT

BACKGROUND: Understanding the anatomy of the lower eyelid and the lid-cheek junction is important for surgical and non-surgical approaches. It is important to understand the correlation between the clinical presentation and the individual anatomy to direct an adequate treatment. METHODS: A review of the literature based on the authors experience combined with anatomical dissections was conducted to reveal the current concepts of the surgical and non-surgical anatomy. The various anatomical structures important for the understanding of the symptoms and the proposed treatment are described in this article. RESULTS: The anatomy of the lower eyelid and the lid-cheek junction has to be understood as a unit. Structures are continuous from the eyelid to the cheek influencing each other during aging. The concept of superficial, i.e. superficial to the orbicularis oculi muscle and deep facial fat compartments, i.e. deep to the orbicularis oculi muscle has to be applied in order to understand the relevant anatomy regarding the ligaments, fat compartments, muscular and tarsal structures and the vascularization. CONCLUSION: The understanding of the layered arrangement of the lower eyelid and eyelid-cheek junction anatomy enables practitioners to perform safe and effective surgical and non-surgical procedures.


Subject(s)
Cheek/anatomy & histology , Eyelids/anatomy & histology , Humans
4.
J Eur Acad Dermatol Venereol ; 31(7): 1088-1095, 2017 Jul.
Article in English | MEDLINE | ID: mdl-28449190

ABSTRACT

Facial fillers play an important role in the correction of facial changes associated with ageing. They offer quick treatments in the outpatient setting with minimal subsequent downtime that provide predictable, natural-looking, long-lasting results. Adverse reactions after hyaluronic acid injections tend to be mild or moderate and rather temporary. However, as with all injected or implanted biomaterials, severe adverse events can occur and patients must be fully informed of potential risks prior to undergoing treatment. A panel of experts from Germany (D), Austria (A) and Switzerland (CH) developed recommendations, and this study provides the 'DACH Consensus Recommendations' from this group specifically on the use of hyaluronic acid fillers. The aim is to help clinicians recognize potential risks and to provide guidance on how best to treat adverse events if they arise. Contraindications to hyaluronic acid fillers are also detailed, and ways to prevent adverse events occurring are discussed. Hyaluronic acid-based products are claimed to be very close to an ideal tissue augmentation agent; nevertheless, profound medical, anatomical and product knowledge are of paramount importance to minimize the occurrence of adverse reactions.


Subject(s)
Cosmetic Techniques/adverse effects , Hyaluronic Acid/administration & dosage , Consensus , Humans , Hyaluronic Acid/adverse effects , Injections, Subcutaneous/adverse effects
5.
Osteoarthritis Cartilage ; 22(10): 1634-8, 2014 Oct.
Article in English | MEDLINE | ID: mdl-25278072

ABSTRACT

OBJECTIVE: To compare cross-sectional and longitudinal side-differences in thigh muscle anatomical cross-sectional areas (ACSAs), muscle strength, and specific strength (strength/ACSA), between knees with early radiographic change vs knees without radiographic knee osteoarthritis (RKOA), in the same person. DESIGN: 55 (of 4796) Osteoarthritis Initiative (OAI) participants fulfilled the inclusion criteria of early RKOA in one limb (definite tibiofemoral osteophytes; no radiographic joint space narrowing [JSN]) vs no RKOA (no osteophyte; no JSN) in the contralateral limb. ACSAs of the thigh muscles and quadriceps heads were determined using axial MRIs at 33%/30% femoral length (distal to proximal). Isometric extensor and flexor muscle strength were measured (Good Strength Chair). Baseline quadriceps ACSA and extensor (specific) strength represented the primary analytic focus, and 2-year changes of quadriceps ACSAs the secondary focus. RESULTS: No statistically significant side-differences in quadriceps (or other thigh muscle) ACSAs, muscle strength, or specific strength were observed between early RKOA vs contralateral limbs without RKOA (P ≥ 0.44), neither in men nor in women. The 2-year reduction in quadriceps ACSA in limbs with early RKOA was -0.9 ± 6% (mean ± standard deviation) vs -0.5 ± 6% in limbs without RKOA (statistical difference P = 0.85). CONCLUSION: Our results do not provide evidence that early unilateral radiographic changes, i.e., presence of osteophytes, are associated with cross-sectional or longitudinal differences in quadriceps muscle status compared with contralateral knees without RKOA. At the stage of early unilateral RKOA there thus appears to be no clinical need for countervailing a potential dys-balance in quadriceps ACSAs and strength between both knees.


Subject(s)
Muscle Strength , Osteoarthritis, Knee/pathology , Quadriceps Muscle/pathology , Aged , Case-Control Studies , Disease Progression , Female , Humans , Magnetic Resonance Imaging , Male , Middle Aged , Muscle, Skeletal/pathology , Muscle, Skeletal/physiopathology , Organ Size , Osteoarthritis, Knee/complications , Osteoarthritis, Knee/physiopathology , Osteophyte/diagnostic imaging , Osteophyte/etiology , Quadriceps Muscle/physiopathology , Radiography , Thigh
6.
Osteoarthritis Cartilage ; 22(12): 2059-66, 2014 Dec.
Article in English | MEDLINE | ID: mdl-25262648

ABSTRACT

OBJECTIVE: Anti-catabolic disease modifying drugs (DMOADs) aim to reduce cartilage loss in knee osteoarthritis (KOA). Testing such drugs in clinical trials requires sufficient rates of loss in the study participants to occur, preferably at a mild disease stage where cartilage can be preserved. Here we analyze a "progression" model in mild radiographic KOA (RKOA), based on contra-lateral radiographic status. METHODS: We studied 837 participants (62.4 ± 9 yrs; 30 ± 4.9 kg/m²; 61.8% women) from the Osteoarthritis Initiative (OAI) with mild to moderate RKOA (Kellgren Lawrence grade [KLG] 2-3) and with/without Osteoarthritis Research Society International (OARSI) atlas radiographic joint space narrowing (JSN). These had quantitative measurements of subregional femorotibial cartilage thickness from magnetic resonance imaging (MRI) at baseline and 1-year follow-up. They were stratified by contra-lateral knee status: no (KLG 0/1), definite (KLG2) and moderate RKOA (KLG 3/4). RESULTS: KLG2 knees with JSN and moderate contra-lateral RKOA had (P = 0.008) greater maximum subregional cartilage loss -220 µm [95% confidence interval (CI) -255, -184 µm] than those without contra-lateral RKOA -164 µm [-187, -140 µm]. Their rate of subregional cartilage loss was similar and not significantly different (P = 0.61) to that in KLG 3 knees without contra-lateral RKOA (-232 µm; [-266; -198 µm]). The effect of contra-lateral RKOA status was less in KLG2 knees without JSN, and in KLG3 knees. CONCLUSION: KLG2 knees with JSN and moderate contra-lateral RKOA, display relatively high rates of subregional femorotibial cartilage loss, despite being at a relatively mild stage of RKOA. They may therefore provide a unique opportunity for recruitment in clinical trials that explore the efficacy of anti-catabolic DMOADs on structural progression.


Subject(s)
Cartilage, Articular/pathology , Osteoarthritis, Knee/diagnostic imaging , Osteoarthritis, Knee/etiology , Disease Progression , Female , Femur , Humans , Longitudinal Studies , Male , Middle Aged , Osteoarthritis, Knee/pathology , Prospective Studies , Radiography , Severity of Illness Index , Tibia
7.
Osteoarthritis Cartilage ; 22(10): 1542-9, 2014 Oct.
Article in English | MEDLINE | ID: mdl-24792212

ABSTRACT

OBJECTIVE: Knee replacement (KR) represents a clinically important endpoint of knee osteoarthritis (KOA). Here we examine the 4-year trajectory of femoro-tibial cartilage thickness loss prior to KR vs non-replaced controls. METHODS: A nested case-control study was performed in Osteoarthritis Initiative (OAI) participants: Cases with KR between 12 and 60 month (M) follow-up were each matched with one control (without KR through 60M) by age, sex, and baseline radiographic stage. Femoro-tibial cartilage thickness was measured quantitatively using magnetic resonance imaging (MRI) at the annual visit prior to KR occurrence (T0), and at 1-4 years prior to T0 (T-1 to T-4). Cartilage loss between cases and controls was compared using paired t-tests and conditional logistic regression. RESULTS: One hundred and eighty-nine knees of 164 OAI participants [55% women; age 64 ± 8.7; body mass index (BMI) 29 ± 4.5] had KR and longitudinal cartilage data. Comparison of annualized slopes of change across all time points revealed greater loss in the central medial tibia (primary outcome) in KRs than in controls [94 ± 137 vs 55 ± 104 µm; P = 0.0017 (paired t); odds ratio (OR) 1.36 (95% confidence interval (CI): 1.08-1.70)]. The discrimination was stronger for T-2 → T0 [OR 1.61 (1.33-1.95), n = 127] than for T-1 → T0, and was not statistically significant for intervals prior to T-2 [i.e., T-4 → T-2, OR 0.97 (0.67-1.41), n = 60]. Results were similar for total medial femoro-tibial cartilage loss (secondary outcome), and when adjusting for pain and BMI. CONCLUSIONS: In knees with subsequent replacement, cartilage loss accelerates in the 2 years, and particularly in the year prior to surgery, compared with controls. Whether slowing this cartilage loss can delay KR remains to be determined.


Subject(s)
Arthroplasty, Replacement, Knee , Cartilage Diseases/pathology , Cartilage, Articular/pathology , Knee Joint/pathology , Osteoarthritis, Knee/pathology , Aged , Aged, 80 and over , Cartilage Diseases/etiology , Case-Control Studies , Disease Progression , Female , Humans , Longitudinal Studies , Magnetic Resonance Imaging , Male , Middle Aged , Osteoarthritis, Knee/complications , Osteoarthritis, Knee/surgery
8.
Osteoarthritis Cartilage ; 21(9): 1214-22, 2013 Sep.
Article in English | MEDLINE | ID: mdl-23973133

ABSTRACT

OBJECTIVE: Conflicting associations between imaging biomarkers and pain in knee osteoarthritis (OA) have been reported. A relation between pain and denuded areas of subchondral bone (dABs) has been suggested and this study explores this relationship further by relating the presence, phenotype, location and size of dABs to different measures of knee pain. METHODS: 633 right knees from the Osteoarthritis Initiative (OAI) (250 men, age 61.7 ± 9.6 yrs, BMI 29.4 ± 4.7 kg/m(2)) were included. Manual segmentation of the femorotibial cartilage plates was performed on 3 T coronal fast low angle shot with water excitation (FLASHwe) images. dABs were defined as areas where the subchondral bone was uncovered by cartilage. The following measures of pain were used: weightbearing-, non-weightbearing-, moderate-to-severe-, infrequent- and frequent knee pain. RESULTS: Using pain measures from subjects without dABs as a reference, those with at least one dAB had a 1.64-fold higher prevalence ratio [PR, 95% confidence interval (CI) 1.24-2.18] to have frequent and 1.45-fold higher for moderate-to-severe knee pain (95% CI 1.13-1.85). Subjects with dABs in central subregions had a 1.53-fold increased prevalence of having weightbearing pain (95% CI 1.20-1.97), especially when the central subregion was moderately (>10%) denuded (PR 1.81, 95% CI 1.35-2.42). Individuals with cartilage-loss-type dABs had a slightly higher prevalence (PR 1.13, 95% CI 1.00-1.27) of having frequent knee pain compared to individuals with intra-chondral-osteophyte-type dABs. CONCLUSION: This study supports a positive relation between femorotibial dABs and knee pain, especially when the dABs are located centrally (i.e., in weightbearing regions) or when the respective central subregion is moderately denuded.


Subject(s)
Arthralgia/pathology , Knee Joint/pathology , Magnetic Resonance Imaging , Osteoarthritis, Knee/epidemiology , Osteoarthritis, Knee/pathology , Aged , Cartilage/pathology , Female , Femur/pathology , Humans , Male , Middle Aged , Osteophyte/epidemiology , Osteophyte/pathology , Pain Measurement , Phenotype , Prevalence , Severity of Illness Index , Tibia/pathology
9.
Osteoarthritis Cartilage ; 21(11): 1660-7, 2013 Nov.
Article in English | MEDLINE | ID: mdl-23954704

ABSTRACT

BACKGROUND: Treatment of severe osteoarthritis (OA) in relatively young patients is challenging. Although successful, total knee prosthesis has a limited lifespan, with the risk of revision surgery, especially in active young patients. Knee joint distraction (KJD) provides clinical benefit and tissue structure modification at 1-year follow-up. The present study evaluates whether this benefit is preserved during the second year of follow-up. METHODS: Patients included in this study presented with end-stage knee OA and an indication for total knee replacement (TKR); they were less than 60 years old with a VAS pain ≥60 mm (n = 20). KJD was applied for 2 months (range 54-64 days) and clinical parameters assessed using the WOMAC questionnaire and VAS pain score. Changes in cartilage structure were measured using quantitative MRI, radiography, and biochemical analyses of collagen type II turnover (ELISA). RESULTS: Average follow-up was 24 (range 23-25) months. Clinical improvement compared with baseline (BL) was observed at 2-year follow-up: WOMAC improved by 74% (P < 0.001) and VAS pain decreased by 61% (P < 0.001). Cartilage thickness observed by MRI (2.35 mm (95%CI, 2.06-2.65) at BL) was significantly greater at 2-year follow-up (2.78 mm (2.50-3.09); P = 0.03). Radiographic minimum joint space width (JSW) (1.1 mm (0.5-1.7) at BL) was significantly increased at 2-year follow-up as well (1.7 mm (1.1-2.3); P = 0.03). The denuded area of subchondral bone visualized by MRI (22% (95%CI, 12.5-31.5) at BL) was significantly decreased at 2-year follow-up (8% (3.6-12.2); P = 0.004). The ratio of collagen type II synthesis over breakdown was increased at 2-year follow-up (P = 0.07). CONCLUSION: Clinical improvement by KJD treatment is sustained for at least 2 years. Cartilage repair is still present after 2 years (MRI) and the newly formed tissue continues to be mechanically resilient as shown by an increased JSW under weight-bearing conditions.


Subject(s)
Osteoarthritis, Knee/surgery , Osteogenesis, Distraction/methods , Adult , Cartilage, Articular/metabolism , Cartilage, Articular/pathology , Collagen Type II/metabolism , External Fixators , Female , Follow-Up Studies , Humans , Knee Joint/diagnostic imaging , Knee Joint/pathology , Magnetic Resonance Imaging , Male , Middle Aged , Osteoarthritis, Knee/diagnostic imaging , Osteoarthritis, Knee/pathology , Osteogenesis, Distraction/adverse effects , Radiography , Severity of Illness Index , Treatment Outcome
10.
Eur J Appl Physiol ; 110(1): 91-7, 2010 Sep.
Article in English | MEDLINE | ID: mdl-20401666

ABSTRACT

Quantitative data on muscle volume (MV) are important for estimating maximal muscle power. The objective of this study was to determine the correlation between anatomical cross-sectional areas (ACSAs) and the MV in thigh muscles (extensors, flexors, adductors, and sartorius) in perimenopausal women, and to identify at which proximal-to-distal level a single-slice ACSA measurement with magnetic resonance imaging (MRI) displays the highest correlation in each of these. Axial MRIs of the thigh were acquired in 41 perimenopausal women aged 50.8 +/- 3.2 years. Segmentation of the extensors, flexors, adductors, and the sartorius was performed between the femoral neck (0%) and the distal end of the intermediate vastus (100%). MVs were determined by numerical integration, and the ACSA was calculated from three-dimensional reconstructions at 10% intervals from proximal to distal. The extensors contributed 50%, the flexors 19%, the adductors 28%, and the sartorius 3% of the total thigh MV. Maximal correlations between ACSA and MV were observed at the 20-40% proximal-to-distal level in the extensors (R (2) = 0.73), at 30% in the adductors (R (2) = 0.82), and at 70% in the flexors (R (2) = 0.72) and sartorius (R (2) = 0.85), respectively. ACSA at 50% displayed the highest overall correlations (R (2) >or= 0.69) with MV for all muscle groups. Single-slice ACSAs from MRI displayed high correlations with MVs. Although the (proximal to distal) measurement locations with the highest correlation varied between the muscle groups, a single slice at the 50% location achieved the best compromise in terms of correlation between ACSA and MV across extensors, flexors, adductors and the sartorius.


Subject(s)
Magnetic Resonance Imaging , Muscle, Skeletal/anatomy & histology , Perimenopause , Female , Humans , Image Interpretation, Computer-Assisted , Imaging, Three-Dimensional , Middle Aged , Muscle Strength , Numerical Analysis, Computer-Assisted , Organ Size , Quadriceps Muscle/anatomy & histology , Thigh
11.
Osteoarthritis Cartilage ; 18(5): 668-76, 2010 May.
Article in English | MEDLINE | ID: mdl-20175972

ABSTRACT

OBJECTIVE: To assess the presence, location, type and size of denuded areas of subchondral bone (dAB) in the femorotibial joint, measured quantitatively with 3T MRI, in a large subset of OAI participants. METHODS: One knee of 633 subjects (250 men, 383 women, aged 61.7+/-9.6 y) were studied, spanning all radiographic osteoarthritis (OA) stages. dABs were determined quantitatively using segmentations of coronal FLASHwe images, representing areas where the subchondral bone was not covered by cartilage. Post hoc visual examination of segmented images determined whether dABs represented full thickness cartilage loss or internal osteophyte. RESULTS: 7% Of the knees were Kellgren & Lawrence (KL) grade 0, 6% grade 1, 41% grade 2, 41% grade 3, and 5% grade 4. 39% Of the participants (48% of the men and 33% of the women) displayed dABs; 61% of the dABs represented internal osteophytes. 1/47 Participants with KL grade 0 displayed 'any' dAB whereas 29/32 of the KL grade 4 knees were affected. Even as early as KL grade 1, 29% of the participants showed dABs. There were significant relationships of dAB with increasing KL grades (P<0.001) and with ipsi-compartimental JSN (P< or =0.001). Internal osteophytes were more frequent laterally (mainly posterior tibia and internal femur) whereas full thickness cartilage loss was more frequent medially (mainly external tibia and femur). CONCLUSIONS: dABs occur already at earliest stages of radiographic OA (KL grades 1 and 2) and become more common (and larger) with increasing disease severity. Almost all KL grade 4 knees exhibited dABs, with cartilage loss being more frequent than internal osteophytes.


Subject(s)
Cartilage, Articular/pathology , Knee Joint/pathology , Osteoarthritis, Knee/pathology , Aged , Cohort Studies , Cross-Sectional Studies , Female , Humans , Knee Joint/diagnostic imaging , Magnetic Resonance Imaging , Male , Middle Aged , Osteoarthritis, Knee/complications , Osteoarthritis, Knee/diagnostic imaging , Radiography
12.
Cells Tissues Organs ; 192(1): 64-72, 2010.
Article in English | MEDLINE | ID: mdl-20185897

ABSTRACT

INTRODUCTION: Epidemiological studies show a positive relationship between physical activity and cartilage volume, suggesting that exercise may protect against osteoarthritis. Cross-sectional experimental studies, however, have failed to show significant differences in knee cartilage morphology between athletes and nonathletic controls. The aim of the study was to test the hypothesis that knee cartilage morphology, specifically regional cartilage thickness and global subchondral bone area, is modified in sedentary, untrained adult women who increased their physical fitness during a 3-month supervised exercise intervention. MATERIALS AND METHODS: Thirty-eight untrained women, aged 45-55 years, were randomly assigned to: endurance training (n = 18), strength training (n = 15) and autogenic training (control group, n = 5). Patellar and femorotibial knee cartilage morphology was determined before and after the 3-month supervised training intervention, using magnetic resonance imaging and state-of-the-art image analysis. RESULTS: Whereas the endurance training group showed a significant increase in cardiorespiratory fitness and the strength training group a significant increase in the maximal voluntary isometric contraction force of the leg, this study did not show significant differences in knee cartilage thickness and volume, subchondral bone area or regional cartilage thickness between baseline and follow-up acquisitions within any intervention group. DISCUSSION: This randomized longitudinal study provides no evidence that a 3-month exercise intervention in untrained middle-aged women can significantly alter knee joint morphology. Longitudinal evidence supporting that a training program, successful in increasing physical fitness, can also induce functional adaptation of articular tissues and may protect against knee osteoarthritis, remains to be presented.


Subject(s)
Exercise , Knee Joint/pathology , Knee/physiology , Physical Endurance , Resistance Training , Cross-Sectional Studies , Female , Humans , Knee Joint/anatomy & histology , Longitudinal Studies , Magnetic Resonance Imaging/methods , Middle Aged , Osteoarthritis/prevention & control
13.
IEEE Trans Neural Netw ; 9(6): 1118-28, 1998.
Article in English | MEDLINE | ID: mdl-18255796

ABSTRACT

This paper investigates threshold based neural networks for periodic symmetric Boolean functions and some related operations. It is shown that any n-input variable periodic symmetric Boolean function can be implemented with a feedforward linear threshold-based neural network with size of O(log n) and depth also of O(log n), both measured in terms of neurons. The maximum weight and fan-in values are in the order of O(n). Under the same assumptions on weight and fan-in values, an asymptotic bound of O(log n) for both size and depth of the network is also derived for symmetric Boolean functions that can be decomposed into a constant number of periodic symmetric Boolean subfunctions. Based on this results neural networks for serial binary addition and multiplication of n-bit operands are also proposed. It is shown that the serial addition can be computed with polynomially bounded weights and a maximum fan-in in the order of O(log n) in O(n= log n) serial cycles, where a serial cycle comprises a neural gate and a latch. The implementation cost is in the order of O(log n), in terms of neural gates, and in the order of O(log2 n), in terms of latches. Finally, it is shown that the serial multiplication can be computed in O(n) serial cycles with O(log n) size neural gate network, and with O(n log n) latches. The maximum weight value in the network is in the order of O(n2) and the maximum fan-in is in the order of O(n log n).

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