Mean Action Time Research Articles

Mean time of diffusion- and reaction-limited loading and unloading

Direct determination of mean diffusion times from the Laplace transform of the spatial average of the diffusing species for p=0 offers the advantage of yielding closed-form expressions rather than sum-type ones obtained otherwise from the time-dependent solution. This is made use of in the present work to determine the mean time of diffusion- and reaction-limited loading and unloading a species into or out of bodies of different shape (plate, cylinder, sphere) for the important type of boundary condition of fixed concentration in the surrounding. This approach particularly pays off for more complex cases when the calculation from the inverse of the Laplace transform becomes more and more laborious. As an example of such type, concomitant trapping and untrapping of the diffusing species within the object during unloading is considered. The obtained solutions are quantitatively discussed with examples from literature. The present concept of the mean time of loading or unloading is compared with other time constants, e.g., mean action time or time lag.

Driving reaction time following periacetabular osteotomy.

Periacetabular osteotomy (PAO) is increasingly being used to treat young adults with symptomatic hip dysplasia. Currently there is a lack of evidence to guide return to driving after this procedure. This study aimed to identify the length of time required after a Periacetabular Osteotomy procedure before a patient can safely return to driving. All patients undergoing PAO were assessed for suitability for the study. Inclusion criteria were: currently driving with a valid licence; and being able to attend follow-up assessment. Baseline driving reaction time was assessed using a driving simulator preoperatively. The simulation was repeated 5 times for each patient and reaction times recorded (Thinking time, Action time and Total reaction time for braking at 30 mph). The driving simulation was repeated using the same methods at 6 weeks and 12 weeks postoperatively. Pre- and postoperative times were compared. 26 patients were included (24 females, 2 males) with a mean age of 32 (range 19-50) years. The mean preoperative times were: Thinking time 0.48, Action time 0.21, Total time 0.69 seconds. At 6 weeks postoperatively, mean Action time increased to 0.26 seconds (p = 0.012) and mean Total time increased to 0.78 seconds (p = 0.013). By 12 weeks post procedure, there was no significant difference in reaction times compared to baseline (mean Thinking time 0.47 seconds, Action time 0.23, Total time 0.72; p > 0.05). Most patients may not be safe to drive at 6 weeks following PAO procedures but should be safe to drive at 12 weeks postoperatively. Individual patient factors should also be taken into consideration.

Contaminated site-induced health risk using Monte Carlo simulation: evaluation from the brownfield in Beijing, China.

The occurrence of contaminated sites has caused serious public health problems, and there are significant health risks. This paper applies Monte Carlo simulations to evaluate the impact of pollutants on human health at a contaminated site in Beijing, China. In this study, a total of 429 soil samples were collected. The exposure routes considered were oral ingestion and skin contact. The log-normal distribution or triangular distribution was adopted to convert exposure parameters into statistical parameters, and the final risk probability was estimated through Monte Carlo simulations. The results show that the 95th percentile risk indexes of As, Ni, Zn, and Hg are 1.22E-1, 5.05E-3, 5.10E-4, and 1.69E-1, respectively, which are all within acceptable levels. The maximum values of As and Hg are 2.15E+0 and 1.04E+0, respectively, with a 5% and 4% probability, respectively, of exceeding the acceptable health risk level. In theory, it is believed that they do not pose a potential threat to human health. Nevertheless, As and Hg in soil are still major pollution sources. The results also show that C (pollutant concentration), AT (mean action time), and ED (exposure duration) are the three parameters with the highest sensitivity to the health risk value. The results of this study can help to improve soil risk control measures and remediation decisions for contaminated sites to reduce the environmental risk in contaminated areas.

Characteristic time scales for diffusion processes through layers and across interfaces.

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Accurate and efficient calculation of response times for groundwater flow

We study measures of the amount of time required for transient flow in heterogeneous porous media to effectively reach steady state, also known as the response time. Here, we develop a new approach that extends the concept of mean action time. Previous applications of the theory of mean action time to estimate the response time use the first two central moments of the probability density function associated with the transition from the initial condition, at t = 0, to the steady state condition that arises in the long time limit, as t→∞. This previous approach leads to a computationally convenient estimation of the response time, but the accuracy can be poor. Here, we outline a powerful extension using the first k raw moments, showing how to produce an extremely accurate estimate by making use of asymptotic properties of the cumulative distribution function. Results are validated using an existing laboratory-scale data set describing flow in a homogeneous porous medium. In addition, we demonstrate how the results also apply to flow in heterogeneous porous media. Overall, the new method is: (i) extremely accurate; and (ii) computationally inexpensive. In fact, the computational cost of the new method is orders of magnitude less than the computational effort required to study the response time by solving the transient flow equation. Furthermore, the approach provides a rigorous mathematical connection with the heuristic argument that the response time for flow in a homogeneous porous medium is proportional to L2/D, where L is a relevant length scale, and D is the aquifer diffusivity. Here, we extend such heuristic arguments by providing a clear mathematical definition of the proportionality constant.

Calculating Groundwater Response Times for Flow in Heterogeneous Porous Media.

Predicting the amount of time required for a transient groundwater response to take place is a practical question that is of interest in many situations. This time scale is often called the response time. In the groundwater hydrology literature, there are two main methods used to calculate the response time: (1) both the transient and steady-state groundwater flow equations are solved, and the response time is taken to be amount of time required for the transient solution to approach the steady solution within some tolerance; and (2) simple scaling arguments are adopted. Certain limitations restrict both of these approaches. In this study, we outline a third method, based on the theory of mean action time. We derive the governing boundary value problem for both the mean and variance of action time for confined flow in two-dimensional heterogeneous porous media. Importantly, we show that these boundary value problems can be solved using widely available software. Applying these methods to a test case reveals the advantages of the theory of mean action time relative to standard methods.

Calculating how long it takes for a diffusion process to effectively reach steady state without computing the transient solution.

Mathematically, it takes an infinite amount of time for the transient solution of a diffusion equation to transition from initial to steady state. Calculating a finite transition time, defined as the time required for the transient solution to transition to within a small prescribed tolerance of the steady-state solution, is much more useful in practice. In this paper, we study estimates of finite transition times that avoid explicit calculation of the transient solution by using the property that the transition to steady state defines a cumulative distribution function when time is treated as a random variable. In total, three approaches are studied: (i) mean action time, (ii) mean plus one standard deviation of action time, and (iii) an approach we derive by approximating the large time asymptotic behavior of the cumulative distribution function. Our approach leads to a simple formula for calculating the finite transition time that depends on the prescribed tolerance δ and the (k-1)th and kth moments (k≥1) of the distribution. Results comparing exact and approximate finite transition times lead to two key findings. First, although the first two approaches are useful at characterizing the time scale of the transition, they do not provide accurate estimates for diffusion processes. Second, the new approach allows one to calculate finite transition times accurate to effectively any number of significant digits using only the moments with the accuracy increasing as the index k is increased.

Acetylcholine chloride 1% usage for intraoperative cataract surgery miosis

Abstract Objective: To test the efficacy of Acetylcholine chloride use in obtaining intraoperative miosis on phacoemulsification cataract surgery. Methods: Patients with cataract diagnosis and elected for surgical phacoemulsification procedure were selected. All patients underwent conventional phacoemulsification procedure performed by a single surgeon and all patients had 0.2 ml of Acetylcholine chloride 1% irrigated in the anterior chamber at the end of the surgery. The pupillary diameter was measured immediately before the beginning of surgery, immediately before and two minutes after the use of acetylcholine chloride 1%. Results: A total of 30 eyes from 30 patients were included in the study. 18 were female, and mean age was of 69.5 years with a 7.2y standard deviation on the population study. The mean pupillary diameter immediately before the beginning of surgery was 7.5 mm with a standard deviation of 0.56 mm; the mean pupillary diameter immediately before the acetylcholine chloride 1% use (after the intraocular lens im-plantation) was 7.1 mm with a standard deviation of 0.57 mm. The mean pupillary diameter two minutes after the use of acetylcholine chloride 1% in the anterior chamber was 3.4 mm with standard deviation of 0.66 mm. The mean maximum action time of ACH chloride 1% was 64 seconds, with a standard deviation of 8 seconds. The mean intraocular pressure on the first postoperative day was 19.1 mmHg with a standard deviation of 2.45 mmHg. Conclusion: We conclude that acetylcholine chloride 1% is an important drug to obtaining intraoperative miosis in cataract surgery.

Spatial analysis of aquifer response times for radial flow processes: Nondimensional analysis and laboratory-scale tests

• Nondimensional model of groundwater pumping, injection and recovery is analyzed. • Time scales for pumping, injection and recovery processes to reach steady state is considered. • Closed form expressions for the time scale, including spatial dependence, are given. • Time scales are equivalent for pumping, injection and recovery. • Theoretical predictions are validated with new laboratory measurements. A fundamental concept in groundwater hydrology is the notion of steady state, or equilibrium conditions. When a system at some initial steady state condition is perturbed by pumping, a transient cone of depression will develop and the system will approach a new steady state condition. Understanding the time scale required for the transient process to occur is of practical interest since it would help practitioners decide whether to use a steady state model or a more complicated transient model. Standard approaches to estimate the response time use simple scaling relationships which neglect spatial variations. Alternatively, others define the response time to be the amount of time taken for the difference between the transient and steady state solutions to fall below some arbitrary tolerance level. Here, we present a novel approach and use the concept of mean action time to predict aquifer response time scales in a two-dimensional radial geometry for pumping, injection and recovery processes. Our approach leads to relatively simple closed form expressions that explicitly show how the time scale depends on the hydraulic parameters and position. Furthermore, our dimensionless framework allows us to predict the response time scales for a range of applications including small scale laboratory problems and large scale field problems. Our analysis shows that the response time scales vary spatially, but are equivalent for pumping, injection and associated recovery processes. Furthermore, the time scale is independent of the pumping or injection flow rate. We test these predictions in a laboratory scale aquifer and find that our physical measurements corroborate the theoretical predictions.

An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions

Summary A major challenge in studying coupled groundwater and surface-water interactions arises from the considerable difference in the response time scales of groundwater and surface-water systems affected by external forcings. Although coupled models representing the interaction of groundwater and surface-water systems have been studied for over a century, most have focused on groundwater quantity or quality issues rather than response time. In this study, we present an analytical framework, based on the concept of mean action time (MAT), to estimate the time scale required for groundwater systems to respond to changes in surface-water conditions. MAT can be used to estimate the transient response time scale by analyzing the governing mathematical model. This framework does not require any form of transient solution (either numerical or analytical) to the governing equation, yet it provides a closed form mathematical relationship for the response time as a function of the aquifer geometry, boundary conditions, and flow parameters. Our analysis indicates that aquifer systems have three fundamental time scales: (i) a time scale that depends on the intrinsic properties of the aquifer, (ii) a time scale that depends on the intrinsic properties of the boundary condition, and (iii) a time scale that depends on the properties of the entire system. We discuss two practical scenarios where MAT estimates provide useful insights and we test the MAT predictions using new laboratory-scale experimental data sets.

Mean action time as a measure for fin performance in one dimensional fins of exponential profiles

The aim of this paper is to numerically solve the one dimensional time dependent heat transfer equation while considering different types of exponential profiles. We consider non-linear thermal conductivity via the power law and employ a numerical relaxation scheme due to its simplicity. Furthermore, we obtain the mean action time and propose it as a novel means of analysing the fin performance. Our results are validated through comparing with benchmark results and our new index for fin performance is shown to provide meaningful insight.

Enquête sur les modalités de la curarisation et la décurarisation au Maroc

ObjectiveTo determine current clinical neuromuscular use in Morocco anesthetic practice. Study designClinical practice survey. MethodsAnesthesiologists were invited to answer anonymously to question about neuromuscular blocking agents (NMBA): pharmacology, monitoring and antagonism drug use. Corrected yates test was used to compare qualitative parameters and variance analysis for quantitative parameters. P<0.05 was considered as significant. ResultsWe had 171 answers, 54% were paramedics (n=93) and 46% (n=78) were physicians (20% global effective in Morocco). Fifty-five percent were seniors. Vecuronium was the most NMBA used in Morocco for 51% of the anesthesiologists. Mean action time of NMBA was underestimate in more than 60% of the staff questioned specially for seniors (24.6%) than juniors (19.2%). Residual neuromuscular blockade was evaluated only by clinical and spirometric parameters in 72%. TOF monitor was used in 18% more in military hospitals (36%) and insurance clinics (40%) than private hospitals (12.5%). Administration of reversal drugs was systematic in 5.8% of anesthesiologists, frequent for 36.3% and rare for 55%. ConclusionsThis study show an underestimation of residual neuromuscular blockade, poor neuromuscular monitor use and lack of knowledge of NMBA specially for senior anesthesiologists and in private hospitals. We should elaborate Moroccan guideline for NMBA use and provide formal training programs for our anesthetic staff.

Simplified approach for calculating moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time underapproximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.

How long does it take for aquifer recharge or aquifer discharge processes to reach steady state?

• Mathematical model of aquifer recharge and discharge is analyzed. • Timescale required to effectively reach steady state is considered. • Mean action time used to show how timescale depends on model parameters. • Theoretical timescales compare well with new laboratory measurements. Groundwater flow models are usually characterized as being either transient flow models or steady state flow models. Given that steady state groundwater flow conditions arise as a long time asymptotic limit of a particular transient response, it is natural for us to seek a finite estimate of the amount of time required for a particular transient flow problem to effectively reach steady state. Here, we introduce the concept of mean action time (MAT) to address a fundamental question: how long does it take for a groundwater recharge process or discharge processes to effectively reach steady state? This concept relies on identifying a cumulative distribution function, F ( t ; x ), which varies from F (0; x ) = 0 to F ( t ; x ) → 1 − as t → ∞, thereby providing us with a measurement of the progress of the system towards steady state. The MAT corresponds to the mean of the associated probability density function f ( t ; x ) = d F /d t , and we demonstrate that this framework provides useful analytical insight by explicitly showing how the MAT depends on the parameters in the model and the geometry of the problem. Additional theoretical results relating to the variance of f ( t ; x ), known as the variance of action time (VAT), are also presented. To test our theoretical predictions we include measurements from a laboratory-scale experiment describing flow through a homogeneous porous medium. The laboratory data confirms that the theoretical MAT predictions are in good agreement with measurements from the physical model.

Critical timescales and time intervals for coupled linear processes

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion– reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes. doi:10.1017/S1446181113000059

CRITICAL TIMESCALES AND TIME INTERVALS FOR COUPLED LINEAR PROCESSES

AbstractIn 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. Sample and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?”Biophys. J. 99(2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for example the analysis of models describing coupled chemical decay and cell differentiation processes.

Moments of action provide insight into critical times for advection-diffusion-reaction processes

Berezhkovskii and co-workers introduced the concept of local accumulation time as a finite measure of the time required for the transient solution of a reaction-diffusion equation to effectively reach steady state [Biophys J. 99, L59 (2010); Phys. Rev. E 83, 051906 (2011)]. Berezhkovskii's approach is a particular application of the concept of mean action time (MAT) that was introduced previously by McNabb [IMA J. Appl. Math. 47, 193 (1991)]. Here, we generalize these previous results by presenting a framework to calculate the MAT, as well as the higher moments, which we call the moments of action. The second moment is the variance of action time, the third moment is related to the skew of action time, and so on. We consider a general transition from some initial condition to an associated steady state for a one-dimensional linear advection-diffusion-reaction partial differential equation (PDE). Our results indicate that it is possible to solve for the moments of action exactly without requiring the transient solution of the PDE. We present specific examples that highlight potential weaknesses of previous studies that have considered the MAT alone without considering higher moments. Finally, we also provide a meaningful interpretation of the moments of action by presenting simulation results from a discrete random-walk model together with some analysis of the particle lifetime distribution. This work shows that the moments of action are identical to the moments of the particle lifetime distribution for certain transitions.

Critical time scales for advection-diffusion-reaction processes

The concept of local accumulation time (LAT) was introduced by Berezhkovskii and co-workers to give a finite measure of the time required for the transient solution of a reaction-diffusion equation to approach the steady-state solution [A. M. Berezhkovskii, C. Sample, and S. Y. Shvartsman, Biophys. J. 99, L59 (2010); A. M. Berezhkovskii, C. Sample, and S. Y. Shvartsman, Phys. Rev. E 83, 051906 (2011)]. Such a measure is referred to as a critical time. Here, we show that LAT is, in fact, identical to the concept of mean action time (MAT) that was first introduced by McNabb [A. McNabb and G. C. Wake, IMA J. Appl. Math. 47, 193 (1991)]. Although McNabb's initial argument was motivated by considering the mean particle lifetime (MPLT) for a linear death process, he applied the ideas to study diffusion. We extend the work of these authors by deriving expressions for the MAT for a general one-dimensional linear advection-diffusion-reaction problem. Using a combination of continuum and discrete approaches, we show that MAT and MPLT are equivalent for certain uniform-to-uniform transitions; these results provide a practical interpretation for MAT by directly linking the stochastic microscopic processes to a meaningful macroscopic time scale. We find that for more general transitions, the equivalence between MAT and MPLT does not hold. Unlike other critical time definitions, we show that it is possible to evaluate the MAT without solving the underlying partial differential equation (pde). This makes MAT a simple and attractive quantity for practical situations. Finally, our work explores the accuracy of certain approximations derived using MAT, showing that useful approximations for nonlinear kinetic processes can be obtained, again without treating the governing pde directly.

Mean Heart Rate during International Matches Depend on Endurance Performance in Female Top Level Handball Players

PURPOSE: International female handball has increased dramatically in it's dynamic and intensity in the last years. The exact demands of modern female top level handball, however, have not been investigated systematically. Therefore, scientifically well based training programs do not exist and might not be deduced so far. The aim of this investigation therefore is to analyse some physiological demands of actual top level international handball matches and to develop recommendations concerning performance characteristics and training necessities of players. METHODS: 7 matches of the German female national team during the European Championship in 2004 in Hungary were recorded on video. The real play times of each player, e. g. the time when the ball was in the match and the player was on the field, as well as the duration of highly intensive match phases were analysed. 14 of 16 players of the German team (age: 26.6 ± 3.8 yrs, height: 176.0 ± 7.4 cm, weight: 70.4 ± 6.8 kg) agreed to wear heart-rate-monitoring belts (Polar, team system) during the matches. Heart rates (HR) were recorded every 5 sec. For further analysis mean HR of the real playing phases of 5 of the 7 matches of each player were calculated so far in 12 players. Individual maximal HR had been determined prior to the tournament during a shuttle-run test. Velocity corresponding to 4 mmol/l lactate concentration (v4) had been determined in a preceding incremental field test (start at 2 m/s, increase by 0.4 m/s every 5 min). RESULTS: Mean time of one attack from all matches was 23.0 ± 1.9 sec. The highly intensive phases of attack had a mean duration of 5.8 ± 0.4 sec. Number of attacks was 28.2 ± 2.8 per half time. The players had a mean action time of 41.4 ± 15.2 min, mean HR was 161.7 ± 11.9 1/min, corresponding to 86.9 ± 5.5 % of maximum HR. V4 was 3.3 ± 0.3 m/s. Interestingly, we found a high and highly significant negative correlation between v4 and mean percent HR during the matches (r=0.96, p<0.01). CONCLUSIONS: Cardiopulmonary demands were very high in all players in all matches, over wide periods near the individual maximum heart rate. To our knowledge it could be demonstrated for the first time in a team sport, that individual load during a highly demanding tournament is the higher the worse developed is basic endurance capacity (v4). This clearly demonstrates the necessity of basic endurance training and the development of a high maximum oxygen uptake in international top level handball.

Mean action time for diffusive processes

For a number of diffusive processes involving heat and mass transfer, a convenient and easy way to solve for penetration time or depth is to consider an averaged quantity called mean action time. This approach was originally developed by Alex McNabb, in collaboration with other researchers. It is possible to solve for mean action time without actually solving the full diffusion problem, which may be nonlinear, and may have internal moving boundaries. Mean action time satisfies a linear Poisson equation, and only works for finite problems. We review some nice properties of mean action time, and discuss some recent novel applications.