Asymptotic Progress Model

A common pattern of disease progress provides for the patient s return to health or recovery without treatment intervention. For example, the time course of postoperative pain can be expected to start at a baseline state, which would be expected to involve intense levels of pain. However, over a few days the level of pain experienced by the patient would be expected to decrease until eventually pain is no longer perceived. This recovery can be approximated by an exponential model with an asymptote of zero, indicating the absence of pain. As shown in Equation 20.8, the parameters of this model are the baseline pain status So and the half-life of progression, Tprog  

The asymptote model is particularly useful for illustrating one of the primary potential drawbacks of not accounting for disease progress. Because patients are expected to improve over time, a simple minimalist approach to the comparison of different drug effects would be expected to be dependent on the time of comparative assessment. If the comparison were made at a point in time where recovery has largely occurred, the difference between treatments would probably be undetectable. 

As with the linear model of disease progress, the consequences of therapeutic intervention on the asymptotic model of disease progress can be described by including terms to account for the expected action of a drug. Drugs may exert an immediate and transient symptomatic effect, they may act to alter the progress of the disease, such as shortening the time to recovery, or they may do both. 

As shown in Equation 20.9, drug action models based on the zero-asymptote model can be extended to include an offset term [EoFF (Ce,A)l the model of progress describing symptomatic benefit such as the relief of pain from a simple analgesic  

As with the offset model for the linear disease progress model, the effect of drug would be expected to disappear on cessation of therapy in this offset model. Again, a delay to the onset of drug effect can be incorporated with the use of an effect compartment component. 

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The effects of a therapeutic agent (Ejp) on the progress of a disease may include both an immediate palliative effect and a reduction in the overall recovery time. Equation 20.11 describes the combination of these actions on the zero-asymptote disease progress model ... 

A portion of the NONMEM code that wifi describe an asymptotic exponential disease progression model with a symptomatic drug effect model is provided in Table 21.4. A portion of the code for the same disease progression model with the drug effect described as disease modifying is provided in Table 21.5. 

FIGURE 21.6 Example profile of asymptotic exponential model with disease progression modifying drug action. 

TABLE 21.4 Example NONMEM Code 3 Asymptotic Exponential Model of Disease Progression with Symptomatic Drng Effect... 

FIGURE 21.9 Example profile of asymptotic disease progression model with aU forms of drug action. 

An alternative quadratic truncation test (QT) has also been suggested with associated asymptotic superlinear convergence.116 This criterion monitors, instead of the relative residual, the sufficient decrease of the quadratic model, 4 (p). Specifically, it checks whether qk p) has decreased sufficiently from one inner iteration to the next, in relation to the progress realized per inner iteration ... 

Figure 20.6 illustrates the expected changes in the progress of a disease, which can be described using the zero-asymptote model. 

Figure 20.7 illustrates the non-zero-asymptote model with all three patterns of disease progress influenced by drug effect. Patterns similar to this have been described in patients with Parkinson s disease treated with levodopa (11 A). Drug exposure starts at 1.0 time units and is stopped at 8.0 time units. In Equation 20.16 the effects of symptomatic improvement and the two functions describing the action of drug both on the burned out state and on the time to reach this state have been included ... 

FIGURE 20.7 Non-zero-asymptote model with natural history (thick line) offset pattern (thin line), and tu o types of protective pattern drug effects SSS, effect on steady-state burned out state (dashed line), and Tprog, effect on half-life of disease progress (dotted line). 

In some cases, therefore, the linear model can be a reasonable approximation for disease progression when the disease is observed over a relatively short period of time or if the number of individuals experiencing complete recovery is limited. For example, the trajectory of the United Parkinson s Disease Rating Score (UPDRS— a measure of Parkinson s disease) has been described using Gompertz functions (20), which exhibit an asymptotic increase to a maximum score. However, over... 

The asymptotic models are so named because the models contain an inherent maximal or minimal value that the function slowly approaches as time increases (e.g., the asymptote). There have been several such models proposed for various disease states where there is a natural limit in the progression of disease. 

Exponential Function In some cases of disease progression, such as recovery from an injury or some other temporary disease state, the model should be able to describe the improvement over time. In such cases, recovery can be approximated by an exponential function parameterized for the baseline status So and the rate constant of recovery kprog. The exponential function has the property of asymptotically approaching 0 and so is best used in situations where the severity scores have a minimum value of 0 or, in the case of some biomarkers, do not occur in the nondiseased state. 

With the exponential asymptotic model, the effect of drug can be described as symptomatic or as disease modifying or as a combination of both. In the case of the symptomatic benefit, as was seen with the linear model for progression, the drug effect, E(t), is added directly to the function for disease status. 

Emax Functions Another familiar asymptotic function is the max function, which has a natural limit S , . The E ax model for disease progression has been used to describe the progression of several different disease scores that have a natural hmit associated with the score. Anderson et al. (23) used the E ax model to describe pain resolution in pediatric patients and Taylor et al. (24) used this model to describe recovery from ischemic stroke using the National Institutes of Health Stroke Score. This model adequately described the trajectory of both markers of disease progression and was able to describe wide interpatient variability in disease progression and response. 

When the adsorption sites are randomly distributed in the framework, one has a distribution of jump lengths. This was found to be the case for ammonia in silicalite [18]. The broadenings of the spectra did not show a maximum, but converged progressively to an asymptotic value. All the spectra could be fitted simultaneously with the SS model, yielding a mean jump length of 5 A. 

In the above expression, [TOC j is the asymptotic residual organic carbon, which cannot be oxidized (Rcl) further. Agreement between the experimental data for combined thermolysis, catalytic, and non-catalytic WO and the model prediction is shown in Figure 5.5. From this plot one can conclude that the oxidation progress is terminated once the catalyst is deactivated due to the adsorption of carbonaceous intermediates on its surface. However, for practical design purposes one should use the lump kinetic approach based on the triangular reaction scheme such as depicted in Figure 5.1. It is believed that the rate laws can be expressed mostly by a simple power function. 

The transfer of ideas from the theory of critical phenomena to polymer science has led to considerable progress in modelling polymers and understanding their universal properties since P. G. de Gennes seminal work (in 1972) [1] provided a first link. In the theory of critical phenomena one is interested in the peculiarities of the behavior in the vicinity of the critical temperature Tc > 0. The critical temperature essentially depends on microscopic details of the model. However, asymptotically close to Tc the thermodynamic and correlation functions for very different systems may depend on the thermodynamic parameters by the same power-like functions primarily characterized by their critical exponents (scaling laws). In contrast to the critical temperature itself, the vadues of the critical exponents do not depend on any of the microscopic details of the stem and tire determined only by the global properties of the model such cts the (lattice) dimension d, the dimension m and other synunetry properties of the order p l lmeter. Due to the fact that the critical exponents may be identical for systems with very different... 

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