# Mathematical Formula (mathematical + formula)

Distribution by Scientific Domains

## Selected Abstracts

### Prediction of the Distance from the Skin to the Lumbar Epidural Space in the Greek Population, Using Mathematical Models

PAIN PRACTICE, Issue 2 2005
Emmanouil Stamatakis MD
Abstract Background and Objectives:, The skin to lumbar epidural space distance (SLED) is variable, and therefore the ability to clinically predict the SLED may help increase the success of epidural anesthesia/analgesia. The goal of this study was to determine the relationship between the SLED and demographic/anthropometric variables in the Greek population, and develop a mathematical model for its prediction. Methods:, This prospective randomized study enrolled 406 male and female Greek patients who required an epidural block as part of their anesthetic management. With patients placed in the left lateral and knee-chest position, the lumbar epidural space was located by the loss of resistance to normal saline technique. Statistical analysis was used to identify the relationship between SLED, and the following variables were evaluated: age, weight, height, body mass index, body surface area, intervertebral space used, pregnancy, and geographic origin within Greece. Results:, No adverse events or dural punctures occurred. Mean SLED in the general population was 4.98 ± 0.95 cm, with values significantly higher in males (5.37 ± 0.88 cm) compared with females (4.83 ± 0.93 cm). SLED was best associated with weight, body surface area, and body mass index. Mathematical formulae for prediction of SLED in the general population and the female population were derived from linear regression analysis. These formulae were able to predict approximately half of the observed variability in SLED. Conclusions:, While mathematical models of SLED can be a useful tool, they should not be exclusively relied on in the clinical setting, but rather should be used as an adjunct to standardized techniques to improve the safety and efficacy of epidural anesthesia/analgesia. [source]

### Bond length features of linear carbon chains of finite to infinite size: Visual interpretation from Pauling bond orders

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 3 2003
Zexing Cao
Abstract Schemes for Kekulé structure counting of linear carbon chains are suggested. Mathematical formulas, which calculate the Pauling bond order P(k, N) of a chemical bond numbered by k, are given for the carbon chain with N carbon atoms. By use of the least-squares fitting of a linearity, relationships between Pauling bond orders and bond lengths are obtained, and such correlation of the Pauling bond order,bond length can be qualitatively extended to the excited states. The relative magnitudes of Pauling bond orders in unsaturated carbon chains dominate C,C bond lengths a well as the bond length feature with the chain size increasing. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 144,149, 2003 [source]

### Effect of Graft Size, Angle, and Intergraft Distance on Dense Packing in Hair Transplant

DERMATOLOGIC SURGERY, Issue 6 2005
Background. The maximum number of hair grafts that can be safely implanted in 1 cm2 is still debatable. To our knowledge, no previous report has addressed this issue in three dimensions, taking into account the size, the angle of the graft, and the intergraft distance. Objectives. To study the effect of the size and angle of the graft and the intergraft distance on dense packing. Methods. Using a mathematical formula (the maximum number of hair grafts in 1 cm2 = 33 * cosine), the volume of the recipient area and the volume of the hair graft are calculated, assuming that the surface area of the recipient area is 1 cm2, the diameter of the hair graft is 1 mm, and the intergraft distance is 1.5 mm laterally and 1 mm anteriorly and posteriorly. Results. The maximum number of hair grafts that could be implanted in 1 cm2 at a 90 angle in relation to the skin surface is 33 grafts, at a 60 angle is 28 grafts, and at a 30 angle is 16 grafts. Conclusion. The maximum number of hair grafts that can be implanted in any given recipient area depends on the graft size, the angle or direction of these grafts, and the intergraft distance. Where more space is allowed between the grafts, and the more acute the angle, the fewer hair grafts that can be implanted. [source]

### TEM, FISH and molecular studies in infertile men with pericentric inversion of chromosome 9

ANDROLOGIA, Issue 4 2006
G. Collodel
Summary Pericentric inversions involving the secondary constriction (qh) region of chromosome 9 are considered to be normal variants of human karyotype. A number of investigators have suggested that chromosomal anomalies can contribute to human infertility causing spermatogenetic derangement. The present study was aimed at verifying the influence of chromosome 9 inversion on human spermatogenesis. Semen samples of 18 male carriers of chromosome 9 inversion, analysed by light microscopy, revealed that five patients were azoospermic. PCR analysis demonstrated that two of them also had Y microdeletions. The other 13 showed generally normal sperm concentrations and reduced motility. The morphological characteristics of sperm were studied by TEM and the data were elaborated by a mathematical formula. Sperm pathologies resulted more frequently in the studied group compared to controls, particularly apoptosis. Partial sequences of the A-kinase anchoring protein (Akap) 4 and 3 genes were performed in all patients, as a previous study by our group highlighted Dysplasia of Fibrous Sheath (DFS) defect in two men with inv 9 investigations. The possible effect of chromosome 9 inversion on meiotic chromosome segregation was investigated by FISH, which showed an increased incidence of diploidy. We hypothesized that this inversion could have variable effects on spermatogenesis, from azoospermia to severely altered sperm morphology, motility and meiotic segregation. [source]

### A mathematical model of the impact of infused targeted cytotoxic agents on brain tumours: implications for detection, design and delivery

CELL PROLIFERATION, Issue 6 2002
Lawrence M. Wein
Motivated by the recent development of highly specific agents for brain tumours, we develop a mathematical model of the spatio-temporal dynamics of a brain tumour that receives an infusion of a highly specific cytotoxic agent (e.g. IL-4-PE, a cytotoxin comprised of IL-4 and a mutated form of Pseudomonas exotoxin). We derive an approximate but accurate mathematical formula for the tumour cure probability in terms of the tumour characteristics (size at time of detection, proliferation rate, diffusion coefficient), drug design (killing rate, loss rate and convection constants for tumour and tissue), and drug delivery (infusion rate, infusion duration). Our results suggest that high specificity is necessary but not sufficient to cure malignant gliomas; a nondispersed spatial profile of pretreatment tumour cells and/or good drug penetration are also required. The most important levers to improve tumour cure appear to be earlier detection, higher infusion rate, lower drug clearance rate and better convection into tumour, but not tissue. In contrast, the tumour cure probability is less sensitive to a longer infusion duration and enhancements in drug potency and drug specificity. [source]

### Large cava septi pellucidi in schizophrenic patients, alcoholics, head-traumatized, and normal individuals: morpholgical features and forensic implications.

ACTA NEUROPSYCHIATRICA, Issue 1 2006
A postmortem study
Background:, The enlarged cava septi pellucidi (CSP = 6 mm in length) have been reported as a reliable marker of an underlying neuropsychiatric disease or disorder. Differences in the dimensions of cava longer than 6 mm associated with a neuropsychiatric impairment could be of possible clinical and forensic significance. Methods:, We obtained 479 brains from autopsied persons (310 males and 169 females, aged 22,89 years) and observed that 110 brains (75 males and 35 females) had CSP, of which the length of CSP was equal to or longer than 6 mm on 69 (49 males and 20 females) of them. These cava were classified into four groups depending on the past medical histories of the autopsied person: five without neuropsychiatric history (asymptomatic CSP), 25 schizophrenic patients, 22 alcoholics, and 17 with a past head trauma (symptomatic CSP). Results:, The linear parameters of CSP (i.e. length, width) of the symptomatic and asymptomatic groups were measured and were statistically analyzed. Analysis revealed that the cava in the group of schizophrenic patients were significantly longer and wider. Conclusions:, Discriminant function analysis was used to derive a mathematical formula to classify CSP into one of the groups obtained based on width measurements of the cavum. [source]

### A comprehensive 3-D analysis of polymer melt flow in slit extrusion dies

ADVANCES IN POLYMER TECHNOLOGY, Issue 2 2004
Yihan Huang
Abstract An understanding of flow behaviour of polymer melts through a slit die is extremely important for optimizing die design and, consequently, for die performance in processing polymer sheets and films. In view of the complex nature and the physical properties of polymer melts as well as of die geometries, such as coat-hanger dies, no simple mathematical formulae can be used to compute the flow regimes within dies. This paper illustrates the development of a three-dimensional (3-D) computer model of an example of a coat-hanger die design using the computational fluid dynamics package, FIDAP, based on the finite element method. A difference of only 3.7% was found when comparing the velocity distribution at the die exit obtained from the 3-D simulation with that calculated using a two-dimensional analytical design procedure, indicating that full 3-D analysis seems to be unnecessary. However it has been shown that unwanted flow phenomena and production problems can be ameliorated by means of visualization and the detailed information obtained from computer simulations. Comparative simulation results with polymers of different rheological properties in the same die are also described. The comprehensive analyses provide a means of interpretation for flow behavior, which allows modification of the die geometry for optimal design. © 2004 Wiley Periodicals, Inc. Adv Polym Techn 23: 111,124, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/adv.20002 [source]

### An evaluation of mathematical models for predicting skin permeability

JOURNAL OF PHARMACEUTICAL SCIENCES, Issue 1 2008
Guoping Lian
Abstract A number of mathematical models have been proposed for predicting skin permeability, mostly empirical and very few are deterministic. Early empirical models use simple lipophilicity parameters. The recent trend is to use more complicated molecular structure descriptors. There has been much debate on which models best predict skin permeability. This article evaluates various mathematical models using a comprehensive experimental dataset of skin permeability for 124 chemical compounds compiled from various sources. Of the seven models compared, the deterministic model of Mitragotri gives the best prediction. The simple quantitative structure permeability relationships (QSPR) model of Potts and Guy gives the second best prediction. The two models have many features in common. Both assume the lipid matrix as the pathway of transdermal permeation. Both use octanol,water partition coefficient and molecular size. Even the mathematical formulae are similar. All other empirical QSPR models that use more complicated molecular structure descriptors fail to provide satisfactory prediction. The molecular structure descriptors in the more complicated QSPR models are empirically related to skin permeation. The mechanism on how these descriptors affect transdermal permeation is not clear. Mathematically it is an ill-defined approach to use many colinearly related parameters rather than fewer independent parameters in multi-linear regression. © 2007 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 97:584,598, 2008 [source]

### Review article: renal function assessment in cirrhosis , difficulties and alternative measurements

ALIMENTARY PHARMACOLOGY & THERAPEUTICS, Issue 7 2007
E. CHOLONGITAS
Summary Background, Renal function in patients with cirrhosis is important prognostically, both before and following liver transplantation. Its prognostic impact is reflected by the inclusion of serum creatinine in the model for end-stage liver disease score, which is now used for recipient prioritization on liver transplantation waiting lists in the USA. Aim, To review the accuracy of the surrogate markers for the assessment of renal function, i.e. glomerular filtration rate, particularly in patients with cirrhosis. Method, We reviewed the available literature in PubMed regarding the markers for GFR evaluation and the factors which affect their accuracy in cirrhosis. Results, Although creatinine is widely available, it is an unreliable marker of glomerular filtration rate, particularly in patients with cirrhosis. Clearance of exogenous markers is considered the ,gold standard', but this methodology has many drawbacks, particularly poor applicability. Several mathematical formulae for estimated glomerular filtration rate are used to overcome some of these limitations: Cockcroft-Gault and Modification of Diet in Renal Disease formulae are the most frequently applied, but they are based on serum creatinine. Conclusions, Due to the inaccuracy of serum creatinine and its derived formulae in estimating glomerular filtration rate, alternative serum markers, such as cystatin C, and new formulae are desirable. These need formal evaluation in patients with cirrhosis so as to have a reliable surrogate of glomerular filtration rate, and to obviate many problems that are associated with using creatinine and estimated glomerular filtration rate. [source]

### Applications: The Mathematics of Electoral District Allocation in New Zealand

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2002
Ian Westbrooke
This paper outlines the implicit conceptual basis of New Zealand's electoral district allocation, as embodied in the Electoral Act 1993. It puts the words of the Act into simple mathematical formulae and describes how these concepts are implemented in the Government Statistician's electoral calculations, with examples from the 1997 calculations. The paper shows how the M,ori Option affects the number of M,ori electoral districts, and describes an unexpected effect of the Option on the number of General electoral districts. [source]

### Complete basis set prediction of methanol isotropic nuclear magnetic shieldings and indirect nuclear spin,spin coupling constants (SSCC) using polarization-consistent and XZP basis sets and B3LYP and BHandH density functionals

MAGNETIC RESONANCE IN CHEMISTRY, Issue 8 2009
Teobald Kupka
Abstract Efficient B3LYP and BHandH density functionals were used to estimate methanol's nuclear magnetic isotropic shieldings and spin,spin coupling constants in the basis set limit. Polarization-consistent pcS- n and pcJ- n (n = 0, 1, 2, 3 and 4), and segmented contracted XZP, where X = D, T, Q and 5, basis sets were used and the results fitted with simple mathematical formulas. The performance of the methods was assessed from comparison with experiment and higher level calculations. 1J(CH) and 3J(HH) values were determined from very diluted solutions in deuterochloroform and compared with theoretical predictions. The agreement between complete basis set (CBS) density functional theory (DFT) predicted isotropic shieldings and spin,spin values and experiment was good. The BHandH/pcS- n methanol shieldings obtained using structures optimized at the same level of theory are approaching the accuracy of the advanced coupled-cluster-singles-doubles-approximate triples (CCSD(T)) calculations. Copyright © 2009 John Wiley & Sons, Ltd. [source]