Residence Time Moments Research Articles

Efficient multi-fidelity computation of blood coagulation under flow.

Clot formation is a crucial process that prevents bleeding, but can lead to severe disorders when imbalanced. This process is regulated by the coagulation cascade, a biochemical network that controls the enzyme thrombin, which converts soluble fibrinogen into the fibrin fibers that constitute clots. Coagulation cascade models are typically complex and involve dozens of partial differential equations (PDEs) representing various chemical species' transport, reaction kinetics, and diffusion. Solving these PDE systems computationally is challenging, due to their large size and multi-scale nature. We propose a multi-fidelity strategy to increase the efficiency of coagulation cascade simulations. Leveraging the slower dynamics of molecular diffusion, we transform the governing PDEs into ordinary differential equations (ODEs) representing the evolution of species concentrations versus blood residence time. We then Taylor-expand the ODE solution around the zero-diffusivity limit to obtain spatiotemporal maps of species concentrations in terms of the statistical moments of residence time, [Formula: see text], and provide the governing PDEs for [Formula: see text]. This strategy replaces a high-fidelity system of N PDEs representing the coagulation cascade of N chemical species by N ODEs and p PDEs governing the residence time statistical moments. The multi-fidelity order (p) allows balancing accuracy and computational cost providing a speedup of over N/p compared to high-fidelity models. Moreover, this cost becomes independent of the number of chemical species in the large computational meshes typical of the arterial and cardiac chamber simulations. Using a coagulation network with N = 9 and an idealized aneurysm geometry with a pulsatile flow as a benchmark, we demonstrate favorable accuracy for low-order models of p = 1 and p = 2. The thrombin concentration in these models departs from the high-fidelity solution by under 20% (p = 1) and 2% (p = 2) after 20 cardiac cycles. These multi-fidelity models could enable new coagulation analyses in complex flow scenarios and extensive reaction networks. Furthermore, it could be generalized to advance our understanding of other reacting systems affected by flow.

First-passage Brownian functionals with stochastic resetting

We study the statistical properties of first-passage time functionals of a one dimensional Brownian motion in the presence of stochastic resetting. A first-passage functional is defined as where t f is the first-passage time of a reset Brownian process x(τ), i.e., the first time the process crosses zero. In here, the particle is reset to x R > 0 at a constant rate r starting from x 0 > 0 and we focus on the following functionals: (i) local time , (ii) residence time , and (iii) functionals of the form with n > −2. For first two functionals, we analytically derive the exact expressions for the moments and distributions. Interestingly, the residence time moments reach minima at some optimal resetting rates. A similar phenomena is also observed for the moments of the functional A n . Finally, we show that the distribution of A n for large A n decays exponentially as for all values of n and the corresponding decay length a n is also estimated. In particular, exact distribution for the first passage time under resetting (which corresponds to the n = 0 case) is derived and shown to be exponential at large time limit in accordance with the generic observation. This behavioural drift from the underlying process can be understood as a ramification due to the resetting mechanism which curtails the undesired long Brownian first passage trajectories and leads to an accelerated completion. We confirm our results to high precision by numerical simulations.

Mixing interfaces, fluxes, residence times and redox conditions of the hyporheic zones induced by dune-like bedforms and ambient groundwater flow

Recent studies highlighted the importance of the interface between streams and their surrounding sediment, known as the hyporheic zone, where stream waters flow through the alluvium. These pore water fluxes stem from the interaction among streambed morphology, stream hydraulics and surrounding groundwater flow. We analytically model the hyporheic hydraulics induced by a spatially uniform ambient groundwater flow made of a horizontal, underflow, and a vertical, basal, component, which mimics gaining and losing stream conditions. The proposed analytical solution allows to investigate the control of simple hydromorphological quantities on the extent, residence time and redox conditions of the hyporheic zone, and the thickness of the mixing interface between hyporheic and groundwater cells. Our analysis shows that the location of the mixing zone shallows or deepens in the sediment as a function of bedform geometry, surface hydraulic and groundwater flow. The point of stagnation, where hyporheic flow velocities vanish and where the separation surface passes through, is shallower than or coincides with the deepest point of the hyporheic zone only due to underflow. An increase of the ambient flow causes a reduction of the hyporheic zone volume similarly in both losing and gaining conditions. The hyporheic residence time is lognormally distributed under neutral, losing and gaining conditions, with the residence time moments depending on the same set of parameters describing dune morphology and stream flow.

Age distribution and the degree of mixing in continuous flow stirred tank reactors

New development of mean age theory is discussed for quantitative analysis of mixing and age distribution in steady continuous flow stirred tank reactors. A new relationship between the moments of age and the moments of residence time are derived. With this new relationship the variance of residence time distribution can be computed much more efficiently and accurately. The relationships of three existing variances of age are described and a new set of variances and the degree of mixing are defined. The theory is used to characterize mixing performance in a CFSTR with different layouts of an inlet and an outlet. Mean age and higher moments of age in the reactors are obtained from CFD solutions of their steady transport equations. The spatial distribution of mean age reveals details of the spatial non-uniformity in mixing. Variances of age and the degree of mixing discussed by Danckwerts and Zwietering are computed for the first time in the literature for non-ideal stirred tank reactors. It is found that although these measures are useful, certain key features in non-uniform mixing are not reflected accurately. Results show that the new set of variances and the degree of mixing more accurately characterize the non-uniform mixing in the reactors.

Statistical theory of linear adsorption capillary chromatography with porous-layer stationary phase

A set of accurate expressions of elution-curve moments are derived from the moments of residence time and displacement in a step based on probability theory. Then the problems about residence time and displacement in a step of a solute molecule in the porous layer of capillary columns and in the moving mobile phase are described by a set of mass-balance equations respectively. The set of equations are solved in Fourier–Laplace domain, and the characteristic functions of residence time of a step, as well as the moments, are obtained by means of computing software Mathematica. At last, using numerical inverse Laplace transform, the elution curves for various conditions are calculated. In the case of large desorption constant the results entirely coincide with those of mass-balance-equation theory and in the case of small desorption constant they are equivalent to those of stochastic theory.

The residence time equation

The residence time of a diffusing particle in a prescribed spatial region recently finds an increasing number of applications in physics and chemistry. A partial differential equation for the residence time moments is derived, as a generalization of the ordinary differential equation for the first passage time moments. When one seeks to calculate only the mean residence time and not its full distribution, this “residence time equation” constitutes a significant simplification over the conventional Feynman–Kac approach. We demonstrate this for a ball in d-dimensional space with an infinite observation time. The present approach may be useful also for other Brownian functionals.

Statistical theory of multiple-site linear wall-adsorption capillary Chromatography

Based on the mass-balance principle, a particular diffusion equation to describe the movement of solute molecules in the stagnant layer of multiple-site solid surfaces is constructed. From the equation, the moments of residence time in a step on multiple-site surfaces are derived. Similarly, the moments in a step in the mobile phase are also derived from a diffusion-drift equation. According to the probability theory, there exists a general relationship between the moments of an elution curve and the moments in a step. Through this relationship, the expressions of the elution-curve moments are derived from the step moments. In this paper, the details related to multiple-site linear wall-adsorption capillary chromatography are described and added in the equations to determine the step moments. The resultant expressions of the elution-curve moments involve various factors, such as adsorption–desorption rate constants, equilibrium constants, axial and radial dispersions in the mobile phase. Afterwards, the moment expressions are used to analyze the peak tailing. The results show that a small quantity of sites with a slow desorption rate will lead to a large peak asymmetry.

Generalized stochastic compartmental models with Erlang transit times

This paper considers the use of semi-Markov process models with Erlang transit times for the description of compartmental systems. The semi-Markov models seem particularly useful for systems with nonhomogeneous "poorly-stirred" compartments. The paper reviews the Markov process models with exponential transit times, and illustrates the application of such models, describing the clearance of calcium in man. The semi-Markov model with Erlang transit times is then developed, and the solutions for its concentration-time curves and residence time moments are given. The use of semi-Markov models is illustrated with the same calcium data, and the results from the two models are compared. The example demonstrates that these semi-Markov models are physiologically more realistic than standard models and may be fitted to pharmacokinetic data using readily available software.

Some clarifications regarding moments of residence times with pharmacokinetic models.

The stochastic formulation of linear kinetic models is elaborated in order to introduce some new concepts and help clarify the meaning and role of residence time moments. Certain conditional moments are introduced. Multicompartment and steady-state dosing within the stochastic context are considered. A general model-independent formula for steady state volume of distribution and a new concept of steady-state moments are presented. A technique for constructing a model of a given topology from its moments is also given.

Use of residence time moments in compartmental analysis.

Residence time moments, particularly mean residence times, can be very useful in the modeling and kinetic analysis of physiological systems. They are defined and illustrated, and their practical utility is then explored by comparing the model formulation based on these moments with three alternative formulations: component half-lives, compartmental model rate coefficients, and multiexponential model coefficients and exponents. The residence time moment formulation is shown to have several advantages: a smaller number of parameters are usually needed to describe a biological system; they have a clear physical interpretation; their statistical "power" to detect certain treatment differences is greater; and they are computationally simpler in many situations.

Elution properties of inhomogeneous chromatographic systems

AbstractThe first two moments of residence time in an inhomogeneous column are calculated by means of first passage time theory. These can be used to find the resolving power of the column for nearly equal amounts of protein. A recently developed perturbation theory allows the calculation of approximate elution profiles.

Contributions to the Stochastic Theory of Chromatographic Kinetics

Abstract The stochastic theory of chromatographic kinetics is extended in two directions: (a) To include diffusion effects, and (b) to treat the n-site adsorption problem. In the first case we show that the solution to a first passage problem yields the moments of residence time in the mobile phase in a rather simple form. For the second problem we show that the central limit theorem allows us to deduce rather general results about the distribution of residence time.