Tanaka Formula Research Articles

Association of Genetic Variation in the Epithelial Sodium Channel Gene with Urinary Sodium Excretion and Blood Pressure.

This study was performed to investigate whether genetic variation in the epithelial sodium channel (ENaC) is associated with 24-h urinary sodium excretion and blood pressure. A total of 3345 participants of the KoGES_Ansan and Ansung study were eligible for this study. Genomic DNA samples were isolated from peripheral blood and genotyped on the Affymetrix Genome-Wide Human SNP Array 5.0. Thirty-four single nucleotide polymorphisms (SNPs) were extracted for gene regions (SCNN1A, SCNN1B, and SCNN1G) as additive components by using Plink. Twenty-four-hour sodium excretions were estimated from spot urine samples using the Tanaka formula. The general linear model (GLM) was applied to assess the association between SNPs and urinary sodium excretion or blood pressure. In the SCNN1G gene, six SNPs (rs4073291, rs12934362, rs7404408, rs4494543, rs5735, and rs6497657) were significantly different in 24-h urinary sodium excretion according to gene variants. However, no difference was found in blood pressure among participants with gene variants of ENaC. Our finding indicated that 24-h urinary sodium excretions were different according to variants of the SCNN1G gene in large samples. Further studies to replicate these findings are warranted.

Comparison of three formulas to estimate 24-hour urinary sodium and potassium excretion in patients hospitalized in a hypertension unit

Measurements of 24-hour urinary sodium (24hrUNa) and potassium (24hrUK) excretion are useful in hypertensives but 24-hour urine collection may be difficult or unreliable. We compared three formulas (Tanaka, Kawasaki, Pan American Health Organization [PAHO]) proposed to estimate 24hrUNa and 24hrUK based on spot urine measurements. We studied 382 patients admitted to a hypertension unit. Sodium, potassium, and creatinine levels were measured using standard laboratory methods in a morning urine sample, followed by 24-hour urinary collection. Agreement between estimated and measured 24hrUNa and 24hrUK was evaluated using the Pearson correlation and Bland-Altman plots. Measured 24hrUNa was 158 ± 75 mmol/d and 24hrUK was 54 ± 24 mmol/d. The correlation coefficient was r = 0.53 for estimated versus measured 24hrUNa, r = 0.69–0.73 for estimated versus measured 24hrUK (all P < .001). The mean bias for 24hrUNa was significantly smaller for Tanaka (10.5 mmol/d) and PAHO (11.5 mmol/d) compared with Kawasaki formula (−29.9 mmol/d). The mean bias for 24hrUK was significantly smaller for Kawasaki (7.3 mmol/d) and PAHO (8.3 mmol/d) compared with Tanaka formula (16.5 mmol/d). Using a single morning urine sample, we found the PAHO formula to be the best for predicting mean 24hrUK and 24hrUNa in hospitalized hypertensive patients. However, precision and accuracy of all the evaluated formulas was inadequate.

Age-Predicted Maximal Heart Rate in Recreational Marathon Runners: A Cross-Sectional Study on Fox's and Tanaka's Equations.

Age-based prediction equations of maximal heart rate (HRmax), such as the popular formulas Fox's 220-age, or Tanaka's 208-0.7 × age, have been widely used in various populations. Surprisingly, so far these equations have not been validated in marathon runners, despite the importance of the role of HRmax for training purposes in endurance running. The aim of the present study was to examine the validity of Fox and Tanaka equations in a large sample of women and men recreational marathon runners. Participants (n = 180, age 43.2 ± 8.5 years, VO2max 46.8 mL/min/kg, finishers in at least one marathon during the last year) performed a graded exercise test on a treadmill, where HRmax was measured. Measured HRmax correlated largely with age in the total sample (r = −0.50, p < 0.001), women (r = −0.60, p < 0.001) and men (r = −0.53, p < 0.001). In women, a large main effect of method on HRmax (p = 0.001, η2 = 0.294) was shown with measured HRmax lower than Fox-HRmax (−4.8 bpm; −8.4, −1.3) and Tanaka-HRmax (−4.9 bpm; −8.1, −1.8). In men, a moderate effect of assessment method on HRmax was found (p = 0.001, η2 = 0.066) with measured HRmax higher than Fox-HRmax (+2.8; 1.0, 4.6), Tanaka-HRmax higher than Fox-HRmax (+1.2; 0.7, 1.7). Based on these findings, it was concluded that Fox and Tanaka' formulas overestimated HRmax by ~5 bpm in women, whereas Fox underestimated HRmax in men by ~3 bpm. Thus, we recommend the further use of Tanaka's formula in men marathon runners. In addition, exercise physiologists and sport scientists should consider the observed differences among various assessment methods when performing exercise testing or prescribing training program relying on HR.

Self-reported Slower Eating Is Associated with a Lower Salt Intake: A Population-based Cross-sectional Study.

Objective Evidence suggests that the eating rate is positively associated with the body weight and blood pressure. Furthermore, people who are overweight or obese tend to have higher salt intakes than those of normal weight. To investigate whether or not the eating rate is also associated with the salt intake, a cross-sectional study was conducted using health examination survey data collected in 2014 from 7,941 residents of Sado City, Niigata, Japan. Methods The eating rates were evaluated using a questionnaire; 11.7% of participants rated themselves as slow eaters, 65.6% as normal eaters, and 22.7% as fast eaters. The salt intake was estimated from sodium and creatinine spot urine measurements using Tanaka's formula. Associations with eating rate were evaluated using multivariate linear regression analyses, with normal eaters as the reference (set at 0). Results Self-reported eating rates were positively associated with the salt intake after adjustment for age and sex [β coefficient (95% confidence interval) for slow -0.51 (-0.67, -0.35); fast 0.18 (0.05, 0.30)]. Further adjustment for the body mass index showed that slower eaters had lower salt intakes than normal eaters, but there was no marked difference in the salt intake between normal and fast eaters. The association between slower eating and a lower salt intake persisted after further adjustment for comorbidities [slow -0.33 (-0.49, -0.18)]. Conclusion Our results suggest that reducing eating rates may be an effective strategy for reducing dietary salt intake as well as preventing obesity.

An association of urinary sodium-potassium ratio with insulin resistance among Korean adults.

BACKGROUND/OBJECTIVESThis study was conducted to investigate the effects of sodium-potassium ratio on insulin resistance and sensitivity in Korean adults.SUBJECTS/METHODSSubjects were 3,722 adults (1,632 men and 2,090 women) aged 40–69 years participating in the Korean genome and epidemiology study_Ansan and Ansung study. Insulin resistance was assessed using homeostasis model assessment of insulin resistance (HoMA-IR) and fasting insulin, and insulin sensitivity was assessed by using the quantitative insulin sensitivity check index (QUICKI). The 24-h urinary sodium and potassium excretion were estimated from spot urinary samples using the Tanaka formula. The generalized linear model was applied to determine the association between urinary sodium-potassium ratio and insulin resistance.RESULTSHoMA-IR (P-value = 0.029, P-trend = 0.008) and fasting insulin (P-value = 0.017, P-trend = 0.005) levels were positively associated with 24-h estimated urinary sodium-potassium ratio in the multivariable model. QUICKI was inversely associated with 24-h estimated urinary sodium-potassium ratio in all models (P-value = 0.0002, P-trend < 0.0001 in the multivariate model).CONCLUSIONThe present study suggests that high sodium-potassium ratio is related to high insulin resistance and low insulin sensitivity. Decreasing sodium intake and increasing potassium intake are important for maintaining insulin sensitivity. Further studies are needed to confirm these findings in longitudinal studies.

Stochastic Calculus with Respect to Gaussian Processes

Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stochastic integral, It\^o formulas (both for tempered distributions and for functions with sub-exponential growth), a Tanaka Formula as well as a definition, and a short study, of (both weighted and non weighted) local times of elements of G . In that view, a white noise derivative of any Gaussian process G of G is defined and used to integrate, with respect to G, a large class of stochastic processes, using Wick products. A comparison of our integral wrt elements of G to the ones provided by Malliavin calculus in [AMN01] and by It\^o stochastic calculus is also made. Moreover, one shows that the stochastic calculus with respect to Gaussian processes provided in this work generalizes the stochastic calculus originally proposed for fBm in [EVdH03, BS{\O}W04, Ben03a] and for mBm in [LLV14, Leb13, LLVH14]. Likewise, it generalizes results given in [NT06] and some results given in [AMN01]. In addition, it offers alternative conditions to the ones required in [AMN01] when one deals with stochastic calculus with respect to Gaussian processes.

Nash equilibrium in nonzero-sum games of optimal stopping for Brownian motion

Abstract We present solutions to nonzero-sum games of optimal stopping for Brownian motion in [0, 1] absorbed at either 0 or 1. The approach used is based on the double partial superharmonic characterisation of the value functions derived in Attard (2015). In this setting the characterisation of the value functions has a transparent geometrical interpretation of 'pulling two ropes' above 'two obstacles' which must, however, be constrained to pass through certain regions. This is an extension of the analogous result derived by Peskir (2009), (2012) (semiharmonic characterisation) for the value function in zero-sum games of optimal stopping. To derive the value functions we transform the game into a free-boundary problem. The latter is then solved by making use of the double smooth fit principle which was also observed in Attard (2015). Martingale arguments based on the Itô–Tanaka formula will then be used to verify that the solution to the free-boundary problem coincides with the value functions of the game and this will establish the Nash equilibrium.

MPS 06-03 THE EXPLORATORY AND APPLICATION STUDY OF ESTIMATION METHOD OF 24-HOUR URINARY SODIUM EXCRETION BY USING SPOT URINE FOR CHINESE POPULATION

Objective: To develop and validation of the new estimation methods for 24-h urinary sodium excretion by using spot urine for Chinese urban and rural residents. Design and Method: Three populations were involved in this study. 100 healthy participants were enrolled in this study (Xishan population) and they all collected each spot urine sample and mixed finally as the 24-hour urine samples according to the study design and protocol. Data from previous researches were also used in this study, including Prospective Urban Rural Epidemiology study – China urine validation trial population (PURE-China urine trial population) and Application of prevention and treatment technique in community for noncommunicable diseases (Dexing population), for which these two researches were both collected the morning urine and 24-hour urine samples. The validation analysis conducted through correlation, bias distribution, residual analysis and Bland-Altman plots to assess the accuracy and the applicability of New methods in Chinese residents. Compared with the estimation by Kawasaki, INTERSALT, and Tanaka formula. Results: Second morning urine New method: Male: PrCRE24h = 9.39 × weight (kg)-3.567 × height (cm)-4.683 × age +1498.014 Female: PrCRE24h = 8.09 × weight (kg)+4.417 × height (cm)-1.906 × age -159.528 e-24-h USE = 2.41 × PrUCr24h × (NaSMU / CrSMU)0.461 New method underestimated the mean value of 24-h USE about 273 mg/day, which was better for population level estimation. Furthermore, the accuracy of estimation at individual level also was not improved through bias distribution analysis. With the similar effect of K method, New method showed a dispersed randomly Bland-Altman plot and indicated a relative accuracy. Although the gap between estimation values and real measured values was the least by using New method (Estimated – Measured), the bias of individual estimated values were all big no matter which methods used. Conclusions: The estimation for 24-h USE by using spot urine through New methods did not showed a good accuracy at individual level and could be an alternative for 24-h USE estimation at population level to evaluate the salt intake in given population.

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Objective: Reducing sodium intake and increasing potassium intake are established essential measures for the prevention of cardiovascular diseases and hypertension. Sodium intake in East Asian populations is still higher in the world. However, there were few reports on the characteristics of individuals with high sodium intake or low potassium intake. The aim of this study is to investigate factors related to urinary sodium and potassium excretion in a representative Japanese population from the National Health and Nutrition Survey. Design and method: This is a cross-sectional study of 2,761 men and women (1,179 men and 1,582 women, aged 20 and older) (the NIPPON DATA2010), who participated in the National Health and Nutrition Survey of Japan conducted in 2010 by the Ministry of Health, Labour and Welfare. They participated from 300 randomly selected districts from all-over Japan. Estimated 24-hour urinary sodium (Na) and potassium (K) excretion was calculated using Tanaka's formula (Tanaka T. et al. J Hum Hypertens. 2002; 16: 97–103) from casual urinary mineral concentration. In order to investigate the association of factors with urinary Na, K and casual urine Na/K ratio, multiple linear regression analysis was performed. Results: Means of estimated 24-hour urine mineral excretions were; Na 176 mmol/day and 172 mmol/day, K 42 mmol/day and 41 mmol/day, and Na/K ratio 3.62 and 3.66 (geometric mean) for men and women, respectively. In men, current smoking was positively associated with urinary Na excretion (ß = 7.121, p = 0.018) and log-transformed Na/K ratio (ß = 0.189, p < 0.001) while it was inversely associated with urinary K excretion (ß = -1.412, p = 0.021) after adjusting for confounding factors. In both men and women, high education status was positively associated with urinary K excretion (men; ß = 2.242, p = 0.001, women; ß = 1.651, p = 0.009) and inversely associated with log-transformed Na/K ratio, (men; ß = -0.136, p = 0.005, women; ß = -0.175, p < 0.001) after adjusting for confounding factors. Conclusions: Non smoking and high education status was independently associated with lower sodium and higher potassium intake estimated using casual urine in general Japanese men and women.

Correlation between Food Frequency Questionnaire Sodium Scoring and Predictive24-Hour Sodium Urine

Background: 24 – hour Sodium urine is the gold standard for estimating daily sodium intake. Although it is the gold standard, it is expensive and difficult to obtain a good sample. Alternatively, we can use food frequency questionnaire, which is relatively cheap and easy to obtain. The purpose of this study is to know the correlation between sodium scoring of a food frequency questionnaire and predictive 24-hour sodium urine. Methods: Thiscross – sectional correlative study was taken from29 people, aged 18–59 years old in Jatinangor. The food list of the questionnaire used in this study was modified based on a qualitative study in Jatinangor. The sodium content of the food was calculated by computer software based on the general recipe of 20people in Jatinangor, which was obtained by the same qualitative study. Afternoon spot urine sample was used as the replacement of 24 – hour urine sample, then the sodium level of the spot urine was converted to 24 – hour sodium urine by Tanaka formula. The correlation was analyzed by using computer software. Results: The correlation between food frequency questionnaire scoring and predictive 24-hour sodium that is analyzed with pearson method was insignificant (r = −0.169, p = 0.381) Conclusions: The correlation is insignificant, this result may arise due to some confounders and limitation of the study.

Pathwise Integrals and Itô–Tanaka Formula for Gaussian Processes

We prove an Itô–Tanaka formula and existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue–Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of the Itô–Tanaka formula for pricing and hedging of financial derivatives.

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L2 and L3 moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.

Validation and comparison of three formulae to estimate sodium and potassium excretion from a single morning fasting urine compared to 24-h measures in 11 countries

Although 24-h urinary measure to estimate sodium and potassium excretion is the gold standard, it is not practical for large studies. We compared estimates of 24-h sodium and potassium excretion from a single morning fasting urine (MFU) using three different formulae in healthy individuals. We studied 1083 individuals aged 35-70 years from the general population in 11 countries. A 24-h urine and MFU specimen were obtained from each individual. A subset of 448 individuals repeated the measures after 30-90 days. The Kawasaki, Tanaka, and INTERSALT formulae were used to estimate urinary excretion from a MFU specimen. The intraclass correlation coefficient (ICC) between estimated and measured sodium excretion was higher with Kawasaki (0.71; 95% confidence interval, CI: 0.65-0.76) compared with INTERSALT (0.49; 95% CI: 0.29-0.62) and Tanaka (0.54; 95% CI: 0.42-0.62) formulae (P <0.001). For potassium, the ICC was higher with the Kawasaki (0.55; 95% CI: 0.31-0.69) than the Tanaka (0.36; 95% CI: -0.07 to 0.60; P <0.05) formula (no INTERSALT formula exists for potassium). The degree of bias (vs. the 24-h urine) for sodium was smaller with Kawasaki (+313 mg/day; 95% CI: +182 to +444) compared with INTERSALT (-872 mg/day; 95% CI: -728 to -1016) and Tanaka (-548 mg/day; 95% CI: -408 to -688) formulae (P <0.001 and P = 0.02, respectively). Similarly for potassium, the Kawasaki formula provided the best agreement and least bias. Blood pressure correlated most closely and similarly with the 24-h and Kawasaki estimates for sodium compared with the other two formulae. In a diverse population, the Kawasaki formula is the most valid and least biased method of estimating 24-h sodium excretion from a single MFU and is suitable for population studies.

A Note on Itô's Formula for Discrete-Time Quantum Walk

We present an Ito's formula for the one-dimensional discrete-time quantum walk and give some examples including a Tanaka's formula by using the formula. Moreover we discuss integrals for the quantum walk.

White noise-based stochastic calculus with respect to multifractional Brownian motion

Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is an extension of fBm enabling to control the local regularity of the process. It is obtained by replacing the constant Hurst parameter H of fBm by a function h(t), thus allowing for a finer modelling of various phenomena.In this work we extend to mBm the construction of the Wick–Itô stochastic integral with respect to fBm, as originally proposed in Bender (Stoch. Process. Appl. 104 (2003), pp. 81–106), Bender (Bernouilli 9(6) (2003), pp. 955–983), Biagini et al. (Proceedings of Royal Society, special issue on stochastic analysis and applications, 2004, pp. 347–372) and Elliott and Van der Hoek (Math. Finance 13(2) (2003), pp. 301–330). In that view, a multifractional white noise is defined and used to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are obtained, as well as a Tanaka Formula.

Local time and Tanaka formula for the [formula omitted]-Brownian motion

In this paper, we study the notion of local time and obtain the Tanaka formula for the G-Brownian motion. Moreover, the joint continuity of the local time of the G-Brownian motion is obtained and its quadratic variation is proven. As an application, we generalize the Itô formula with respect to the G-Brownian motion to convex functions.

QUANTUM TANAKA FORMULA IN TERMS OF QUANTUM BROWNIAN MOTION

AbstractA quantum local time, which is a generalized operator-valued process, is defined for quantum Brownian motion, and a quantum analogue of the classical Tanaka formula is then established.

Brownian Motion, Reflection Groups and Tanaka Formula

In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanaka's formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times of the distances from the initial process to the facets.

Kunita–Watanabe inequalities and Tanaka formula for multi-dimensional G-Brownian motion

In this paper, some new properties and interesting estimations of mutual variation process for G-Brownian motion are presented, Kunita–Watanabe inequalities and Tanaka formula for multi-dimensional G-Brownian motion are also obtained.

Local time and Tanaka formula for a Volterra-type multifractional Gaussian process

The stochastic calculus for Gaussian processes is applied to obtain a Tanaka formula for a Volterra-type multifractional Gaussian process. The existence and regularity properties of the local time of this process are obtained by means of Berman's Fourier analytic approach.