The mean surface area of the boxes circumscribed about a convex body

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

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URINARY CATECHOLAMINE EXCRETION BY HEALTHY CHILDREN

PEDIATRICS ◽

10.1542/peds.39.2.252 ◽

1967 ◽

Vol 39 (2) ◽

pp. 252-257

Author(s):

Mary L. Voorhess

Keyword(s):

Urinary Excretion ◽

Surface Area ◽

Body Surface Area ◽

Body Surface ◽

Age Groups ◽

Healthy Children ◽

Urinary Catecholamine ◽

Catecholamine Excretion ◽

Daily Output ◽

The Mean

There is increase in the daily urinary excretion of dopamine (DA), norepinephrine (NE), epinephrine (E), and 3-methoxy-4-hydroxymandelic acid (VMA) with age. The mean output in micrograms per 24 hours for various age groups is as follows: birth to 1 year-DA 60.9 (± 24.3), NE 10.6 (± 3.4), E 1.3 (± 1.2), VMA 569 (± 309); 1 through 5 years—DA 124.1 (± 40.7), NE (18.8 ± 7.0), E 3.2 (± 2.7), VMA 1348 (± 433); 6 through 15 years—DA 169.3 (± 72.6), NE 37.4 (± 16.6), E 4.8 (± 2.4), VMA 2373 (± 698); over 15 years—DA 249.1 (± 74.9), NE 50.7 (± 15.7), E 7.1 (± 3.3), VMA 3192 (± 699). The studies suggest that the daily output of these compounds in the various age groups is similar when related to body surface area after infancy.

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Desmosomes in hamster cheek pouch epithelium: their quantitative characterization during epithelial differentiation

Journal of Cell Science ◽

10.1242/jcs.66.1.411 ◽

1984 ◽

Vol 66 (1) ◽

pp. 411-429

Author(s):

F.H. White ◽

K. Gohari

Keyword(s):

Plasma Membrane ◽

Surface Area ◽

Cheek Pouch ◽

Unit Surface Area ◽

Granular Cells ◽

Hamster Cheek Pouch ◽

Average Cell ◽

Membrane Area ◽

The Mean ◽

Granular Layers

Desmosomes in stratified squamous epithelia appear to exhibit quantitative alterations during differentiation. In this work we use stereological and other morphometric methods to quantify these structures in epithelial cells from defined basal, spinous and granular strata. Hamster cheek pouch mucosa from five animals was processed for electron microscopy using strictly standardized techniques and a stratified random sampling procedure was used to obtain micrographs of cells from basal, spinous and granular layers. Stereological intersection counting techniques were used to determine for each layer the relative surface area of plasma membrane occupied by desmosomes (Ss), the number of desmosomes per unit surface area of plasma membrane (Ns), the mean individual desmosomal diameter (delta) and the mean individual desmosomal surface area (s). In addition, estimates of nuclear volume were obtained by direct measurement of nuclear profiles and volume-to-surface ratios were obtained by a combination of point and intersection counting, which enabled estimates for the volume (Vcell) and plasma membrane surface area (SPM) of the ‘average’ cell within each stratum to be acquired. Using this information, it was then possible to calculate both the total surface area (S) and the number (N) of desmosomes on the plasma membranes of average cells. The parameters Ss and Ns showed progressive increases between basal and granular layers, whereas values for delta and s were lower in granular cells when compared with basal and spinous cells. The parameters Vcell, SPM, S and N all increased progressively and significantly during differentiation. Between basal and granular layers, the mean cell volume and surface area had each increased approximately threefold, whereas the surface area and number of desmosomes on the average cell plasma membrane had increased approximately seven- and eleven-fold, respectively. Granular cells thus possess more numerous desmosomes, which occupy a greater proportion of the plasma membrane area but which are individually smaller, when compared with basal and spinous layers.

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Evaluating the Subtalar Joint in Tibiotalocalcaneal Nail

Foot & Ankle Orthopaedics ◽

10.1177/2473011418s00400 ◽

2018 ◽

Vol 3 (3) ◽

pp. 2473011418S0040

Author(s):

Megan Reilly ◽

Kurosh Darvish ◽

Soroush Assari ◽

John Cole ◽

Tyler Wilps ◽

...

Keyword(s):

Surface Area ◽

Subtalar Joint ◽

Articular Surface ◽

Joint Surface ◽

Central Dome ◽

Below The Knee ◽

Joint Surfaces ◽

Circular Surface ◽

The Mean ◽

Pass Through

Category: Hindfoot Introduction/Purpose: In tibiotalocalcaneal nails for arthrodesis, the path of the nail through the subtalar joint has not been well documented. Ideally, the defect caused by reaming and the nail does not pass through the joint surface so that the amount of bony contact between the talus and calcaneus is maintained in order to optimize fusion. Our hypothesis is that the TTC nail does not destroy a significant amount contact area between the talus and calcaneus. However, using larger diameter nails (which are inherently stronger) will have more of an effect on the contact surface. Methods: Five cadaveric below the knee specimens were obtained. The ankle was disarticulated on each specimen. Subsequently, a guidepin was drilled from the central dome of the talus down to the calcaneus. The 11 mm reamer was then passed over the guidepin through the calcaneus to simulate retrograde reaming of a TTC nail. Then, the subtalar joint was dissected open and the articular surface was documented in comparison with the area that was reamed out. Measurements were then made, using software that calculated two dimensional surface area to determine the percentage of actual subtalar joint area that was reamed out. The mean percentage of articular area that was removed with the reamer was then calculated. Results: Among the five specimens, in the calcaneus, the mean total articular area was 599mm2±113 and the mean drilled articular area was 21mm2±16. The percentage of the calcaneal articular surface that was removed with the reamer was 3.4%±1.9. In the talus, the mean total articular area was 782mm2±130 and the mean drilled articular area was 39mm2±18. The percentage of the talar articular surface that was removed with the reamer was 5.0%±2.3. Additionally, an 11 mm reamer makes a circular surface area of 95mm2, and the statistics above indicate that a significant portion of the reamed area is nonarticular, within the calcaneal sulcus or the talar sulcus. Conclusion: In a tibiotalocalcaneal nail the subtalar joint is typically incompletely visualized, however this anatomic study demonstrates that the 11 mm reamer eliminates about 3.4% of the calcaneal articular surface and about 5% of the talar articular surface. Therefore, the majority of the articular surface is left intact, which is ideal in optimizing arthrodesis outcomes. Furthermore, this study could extrapolate the effects of a larger nail on the availability of joint surface. It could also be used to argue for cartilage stripping of the affected joint surfaces in arthrodesis preparation, because the majority of the articular surface is, in fact, left intact.

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The anatomic rationale for transforaminal endoscopic interbody fusion: a cadaveric analysis

Neurosurgical FOCUS ◽

10.3171/2015.10.focus15389 ◽

2016 ◽

Vol 40 (2) ◽

pp. E12 ◽

Cited By ~ 15

Author(s):

Mitchell Hardenbrook ◽

Sergio Lombardo ◽

Miles C. Wilson ◽

Albert E. Telfeian

Keyword(s):

Surface Area ◽

Soft Tissues ◽

Safe Zone ◽

Large Area ◽

Nerve Roots ◽

Disc Space ◽

Endoscopic Techniques ◽

Pars Interarticularis ◽

The Mean ◽

Cadaveric Analysis

OBJECTIVE The authors describe a cadaveric analysis to determine the ideal dimensions and trajectory for considering endoscopic transforaminal interbody implantation. METHODS The soft tissues of 8 human cadavers were removed from L-1 to the sacrum, exposing the posterior bony elements. Facetectomies were performed bilaterally at each lumbar level with resection of the pars interarticularis, revealing the pedicles, nerve roots, and interbody disc space. Each level was digitally photographed with a marker for scale and evaluated with digital analysis software. The traversing and exiting nerve roots and pedicle margins were identified, and the distances between these structures and their relationships to the surrounding structures were documented. RESULTS The dimensions of 2 areas were measured: the working triangle and safe zone. The working triangle is the triangle between the exiting and traversing nerve roots above the superior margin of the inferior pedicle. The safe zone is the trapezoid bounded by the widths of the superior and inferior pedicles between the exiting and traversing nerve roots. The mean surface area for the working triangle was 1.83 cm2, with L5–S1 having the largest area at 2.19 cm2. The mean surface area of the safe zone was 1.19 cm2, with L5–S1 having the largest area at 1.26 cm2. At the medial border of the pedicle extending superiorly, there were no nerve structures within 1.19 cm at any level. On the lateral border of the pedicle, the exiting nerve root was closer superiorly, with the closest being 0.3 cm. CONCLUSIONS The working triangle is a relatively large area. The safe zone, just superior to the pedicle, is free of nerve structures. By utilizing the superior border of the pedicle, the disc space can be accessed within this safe zone without risk of injury to the nerves. A thorough understanding of foraminal anatomy is fundamental for considering how to safely access the disc space, thereby utilizing less invasive endoscopic techniques, and is an important first step in considering what shapes and sizes of interbody implants and retractors are feasible for use in the foramen.

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An Anatomically Placed Tibial Tunnel does not Completely Surround a Simulated PCL Reconstruction Graft in the Proximal Tibia

The Journal of Knee Surgery ◽

10.1055/s-0041-1741430 ◽

2022 ◽

Author(s):

Christopher Kim ◽

Dustin Baker ◽

Brian Albers ◽

Scott G. Kaar

Keyword(s):

Surface Area ◽

Cruciate Ligament ◽

Tibial Tunnel ◽

Three Dimensional ◽

Pcl Reconstruction ◽

Posterior Cortex ◽

Modeling Software ◽

The Mean ◽

Tibial Cortex ◽

Area And Volume

Abstract Introduction It is hypothesized that anatomic tunnel placement will create tunnels with violation of the posterior cortex and subsequently an oblique aperture that is not circumferentially surrounded by bone. In this article, we aimed to characterize posterior cruciate ligament (PCL) tibial tunnel using a three-dimensional (3D) computed tomography (CT) model. Methods Ten normal knee CTs with the patella, femur, and fibula removed were used. Simulated 11 mm PCL tibial tunnels were created at 55, 50, 45, and 40 degrees. The morphology of the posterior proximal tibial exit was examined with 3D modeling software. The length of tunnel not circumferentially covered (cortex violation) was measured to where the tibial tunnel became circumferential. The surface area and volume of the cylinder both in contact with the tibial bone and that not in contact with the tibia were determined. The percentages of the stick-out length surface area and volume not in contact with bone were calculated. Results The mean stick-out length of uncovered graft at 55, 50, 45, and 40 degrees were 26.3, 20.5, 17.3, and 12.7 mm, respectively. The mean volume of exposed graft at 55, 50, 45, and 40 degrees were 840.8, 596.2, 425.6, and 302.9 mm3, respectively. The mean percent of volume of exposed graft at 55, 50, 45, and 40 degrees were 32, 29, 25, and 24%, respectively. The mean surface of exposed graft at 55, 50, 45, and 40 degrees were 372.2, 280.4, 208.8, and 153.3 mm2, respectively. The mean percent of surface area of exposed graft at 55, 50, 45, and 40 degrees were 40, 39, 34, and 34%, respectively. Conclusion Anatomic tibial tunnel creation using standard transtibial PCL reconstruction techniques consistently risks posterior tibial cortex violation and creation of an oblique aperture posteriorly. This risk is decreased with decreasing the angle of the tibial tunnel, though the posterior cortex is still compromised with angles as low as 40 degrees. With posterior cortex violation, a surgeon should be aware that a graft within the tunnel or socket posteriorly may not be fully in contact with bone. This is especially relevant with inlay and socket techniques.

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Surgical placement of rectus sheath catheters in a cadaveric cystectomy model

Annals of The Royal College of Surgeons of England ◽

10.1308/rcsann.2017.0169 ◽

2018 ◽

Vol 100 (2) ◽

pp. 120-124 ◽

Cited By ~ 2

Author(s):

ECP Chedgy ◽

G Lowe ◽

R Tang ◽

C Krebs ◽

A Sawka ◽

...

Keyword(s):

Abdominal Surgery ◽

Surface Area ◽

Nerve Root ◽

Rectus Sheath ◽

Rectus Muscle ◽

Midline Incision ◽

Nerve Roots ◽

The Mean ◽

Peritoneal Spread ◽

Nerve Root Involvement

Introduction Surgically inserted rectus sheath catheters (RSCs) are used increasingly for analgesia after cystectomy and other abdominal surgery. Currently, there is little information on the optimal positioning of RSCs to allow maximal spread of local anaesthetic. This study sought to assess the spread of dye injected via RSCs and to highlight the extent of its coverage in a fresh unembalmed cadaveric cystectomy model in order to confirm the nerve endings that are likely to be anaesthetised with RSCs. Methods Four cadavers underwent lower midline incision with limited bladder mobilisation. A RSC was inserted into the eight hemiabdomens. The RSCs were positioned either anterior (n=5) or posterior to the rectus muscle (n=3). Dye was injected down the RSCs to evaluate spread. The eight hemiabdomens were dissected anatomically to determine the surface area of dye spread and nerve root involvement. Results The mean surface area of dye spread with anteriorly placed RSCs was 30.6cm2 anterior and 25.9cm2 posterior to the rectus muscle. The mean surface area of dye spread with posteriorly placed RSCs was 11.3cm2 anterior and 37.3cm2 posterior to the rectus muscle. The mean number of nerve roots stained with anteriorly and posteriorly placed RSCs was 3.8 and 2.7 respectively. Subcutaneous spread of dye was seen with one anterior RSC insertion. Peritoneal spread was seen with one anteriorly positioned RSC. Conclusions This study has demonstrated efficient nerve root infiltration with anteriorly and posteriorly positioned RSCs. It appears that dye spreads between the fibres of the rectus muscle rather than out laterally to the nerve roots when spreading from its initial compartment.

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ISOPERIMETRIC INEQUALITIES FOR Lp GEOMINIMAL SURFACE AREA

Glasgow Mathematical Journal ◽

10.1017/s0017089511000292 ◽

2011 ◽

Vol 53 (3) ◽

pp. 717-726 ◽

Cited By ~ 14

Author(s):

BAOCHENG ZHU ◽

NI LI ◽

JIAZU ZHOU

Keyword(s):

Convex Body ◽

Surface Area ◽

Type Inequality ◽

Isoperimetric Inequalities ◽

Surface Areas ◽

Cyclic Inequality ◽

Geominimal Surface Area

AbstractIn this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

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