Time-Variant Models

This phenomenon is characterized by a reduction in effect intensity after repeated drug administration. The explanation for the diminution of the effect as a function of time is attributed either to a decrease in receptor affinity or a 

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In this chapter, discrete linear-state space models will be discussed and their similarity to ARX models will be shown. In addition Wiener models are introduced. They are suitable for non-linear process modeling and consist of a linear time variant model and a non-linear static model. Several examples show how to develop both types of models. 

In Sections 41.2 and 41.3 we applied a recursive procedure to estimate the model parameters of time-invariant systems. After each new measurement, the model parameters were updated. The updating procedure for time-variant systems consists of two steps. In the first step the system state j - 1) at time /), is extrapolated to the state x(y) at time by applying the system equation (eq. (41.15)) in Table 41.10). At time tj a new measurement is carried out and the result is used to... 

More complex integrated PK/PD models are necessary to link and account for a possible temporal dissociation between the plasma concentration and the observed effect. Four basic attributes may be used to characterize PK/PD models First, the link between measured concentration and the pharmacological response mechanism that mediates the observed effect (direct versus indirect link) second, the response mechanism that mediates the observed effect (direct versus indirect response) third, the information used to establish the link between measured concentration and observed effect (hard versus soft link) and, fourth, the time dependency of the involved PD parameters (time variant versus time invariant) (Danhof et al., 1993 Steimer et al., 1993 Aarons, 1999 Lees et al., 2004). The expanded and early use of PK/PD modeling in drug discovery and development is highly beneficial for increasing the success rate of drug discovery and development and will most likely improve the current state of applied therapeutics. 

Bauer, J., Balthasar, J., and Fung, H., Application of pharmacodynamic modeling for designing time-variant dosing regimens to overcome nitroglycerin tolerance in experimental heart failure, Pharmaceutical Research, Vol. 14, No. 9, 1997, pp. 1140-1145. 

When the effects of time variations in release rates are included, pulse (or instantaneous) and continuous (plume) emissions are the two most common time-variant inputs in transport models. The classic example of a pulse release is a hazardous waste spill. The steady release of contaminants into groundwater from a subsurface contaminant and the continuous release of volatile solvents from an air-stripping tower are examples of plume emissions. 

The PK-PD relationship for G-CSF following IV and SC administration was well characterized in healthy volunteers (53). The PK model was a two-compartment PK model with bisegmental absorption from the site of SC administration, parallel first-order and saturable elimination pathways, and an indirect effect PD model describing the time course of neutrophils. A sigmoidal Emax model was applied for the stimulation of the neutrophil input rate. In addition, a time-variant scahng factor for absolute neutrophil count (ANC) observations was introduced to account for the early transient depression of ANC. 

Time-variant means that the parameters of the system change over time, such as an autoinduction process that increases a drug s hepatic clearance with repeated administration. Time-invariant or stationary parameters do not change over time. It is typically assumed that a drug s pharmacokinetics are stationary over time so that the principle of superposition1 applies. With a static model, the output depends only on the input and does... 

The consideration of time dependent interfacial area is of importance for many experimental investigations. Various models are described The linearised theories for small area changes, which belong to relaxation theories and are outlined in Chapter 6. The flow in the liquid layer adjacent to a time variant surface area is also incorporated into the theoretical models. Systems such as liquid films flowing down an inclined plane or growing bubbles and drops are described quantitatively, assuming laminar and radial flow patterns (MacLeod Radke 1993). 

Recently the same problem has been reanalyzed by Dicus et al. [86], and indeed they confirmed that the survival probability deviates from exponential at long times. This model and its variants have been applied to study the effect of a distant detector (by adding an absorptive potential) [87], anomalous decay from a flat initial state [44], resonant state expansions [3], initial state reconstruction (ISR) [58], or the relevance of the non-Hermitian Hamiltonian concept (associated with a projector formalism for internal and external regions of space) in potential scattering [88]. In Ref. [88] the model was extended to a chain of delta functions to study overlapping resonances. 

The bormdary conditions for the flow model were a fixed potential boundary at the column inflow and a prescribed flux bormdary condition at the outflow side. For the transport modelling a time-variant fixed concentration boundary at the inflow of the column was assumed. The concentrations values at the inflow boundaries were taken from the measurements at the column inlet. 

However, all time-variant terms of the coupled process-structure model are now formulated to be time invariant. By applying a stabihty criterion and by solving the characteristic equation, the stability lobe diagram can be determined. 

These issues, positive and negative, are reflected in the available correlations. These correlations are both highly useful and also limited. Some are useful because the inputs are easily measured and adjusted as needed however, correlations are mostly empirical or semi-empirical, which means that they are not widely applicable but, rather, are bioreactor design dependent at best. Hence, geometric similarity is very important. Furthermore, most studies are performed in air-water systems while most industrial processes use much more complicated and time-variant liquids. In other words, the airhft bioreactor correlations have similar problems as those for stirred-tank bioreactors and bubble columns and are due to the fact that they share the problem source bubble-bubble interactions. Bubble-bubble interactions are highly variable and lead to hydrodynamics which, in turn, are difficult to quantify and predict. Hence, the result has been that the airlift bioreactor correlations and models are either system dependent or not adequately constrained. 

In joint time-frequency analysis, impedance is presented as function of time. This however, does not mean that time and frequency domains are mixed together. Any impedance at a given time should be considered as time invariant that is true for any time interval approaching zero. The consequence is that the impedance should change negligibly within the smallest given time interval. Sophisticated, nonlinear models for time variant impedances are not considered here. 

The simplified models for the time-variant resistance and traffic load are indicated in Figure 1. Note that extremes related to the reciprocal of the mean renewal rate are assinned to be present imtil a subsequent load renewal. Hence no intermittencies are taken into accoimt. 

Only one aspect of the methodology is dealt with in this paper, namely the way in which time variant mechanical reliability for a mechanical component can be integrated into the functional and dysfunctional modeling of a complex (notably mechatronic) system achieved using Petri Nets. The first part describes how the Petri Nets are applied. The second part reexamines the principle of time variant mechanical reliability, and more specifically the PHI2 outcrossing rate that enables analysis of time variant failure... 

Very often the laws of distribution are associated with the dysfunctional transitions, where knowledge of the various components makes this possible - the exponential law for electronic devices, the Musa s model for software, etc. For mechanical components, the following approach is proposed if one considers the C2 component to be a mechanical component, two steps are required in order to deter-mine the times associated with the T3 transition. The first step consists in determining, by means of a physical model, the stresses apphed to the C2 component. The second step is to use the PHI2 mechanical reliability method to estimate the Pf t) time variant failure probability of the C2 mechanical component. For this, we propose the detailed methods given by Lemaire (2005), Andrieu (2002, 2004) Sudret (2002, 2005, 2007). 

In this paper, we have presented an approach that enables us to assess time variant failure probability using the PH 12 method, which we propose integrating into the Petri Nets rehahihly analysis of a mechatronic system. The physical model of the system under consideration (a mechatronic system) allows us to obtain the functional times of the various system components. These times are used in the functional transitions of... 

ABSTRACT This study addresses the time-variaut reliability assessment in relation to systems exhibiting a non-stationary random process during their operation, such as thermal-hydrauhc passive systems for advanced reactors, relying on natural circulation. The reliability assessment efforts conducted so far don t deal with this specific aspect the dependence upon time is usually ignored, or at most the system unavaUabUity is intended to he assessed per mission time, during which the parameter values, as t-h parameters for instance, are assumed as constant quantities. The paper presents an effort for a consistent approach to model and evaluate the natural circulation passive systems, in terms of time-variant performance parameters, as for instance mass flow-rate and thermal power, to cite any. 

This concept is depicted in Figure 1, with reference to the well known R — S (Resistance - Stress) (Melchers 1999) or load-capacity interference model, within a rehahility physics framework, where the system prohahihty o f failure Pf is evaluated by comparing the distributions of the two quantities, (Burgazzi 2003, Apostolakis et al. 2005). The curves 1 and 2 in Fig. lb represent respectively the system failure prohahihty Pf in case of time variant and time invariant stochastic process. 

As a remark, the above model can be adjusted to reflect different simations (i) where only the most fouled exchangers are selected to clean (ii) where time variant margin is estimated instead of average margin and (iii) other significant costs are introduced. 

Moreno-Rodriguez, G. (2008), "An IIR-Filter Approach to Time Variant PLCChannel Modelling", Proc.ll th Int. Symp. on Pawerline Communications and Its Applications, pp. 87-92, Jeju Island, Korea, April. 

Coloured noise, narrow band noise and periodic impulsive noise are usually modeled as background noise because they remain stationary from seconds to even hours. Periodic impulsive noise synchronous to the mains and asynchronous impulsive noise may cause bit or burst errors over the transmission, although they are time variant. A complete theoretical analysis on shown noises in Figure 2 can be found on [5, 6,11,12,13]. 

FIGURE 20.41 Model of the atmospheric radio channel as a linear time-variant system with additive noise and additive interference. 

Radio waves from a transmitting antenna often reach the receiving antenna through many different paths. Examples include different reflecting layers in the ionosphere, numerous scattering points in the troposphere, reflections from the Earth s surface, and the direct line-of-sight path. Each of these paths will contribute a different attenuation and time delay. Shown in Fig. 20.45 is a model of the time-variant multipath radio channel. For the sake of simplicity, the effect of noise has been ignored. 

In general, the nonlinear time-variant process model and the nonlinear observation... 

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