Quantal Models

The PD models fall under two categories graded or quantal of fixed-effect model. Graded refers to a continuous response at different concentrations, whereas the quantal model would evaluate discrete response such as dead or alive, desired or undesired and are almost invariably clinical end points. 

The dose-response relahonship in a quantal model can be analyzed with the help of a logistic model where we calculate probability of an event at a given concentrahon or AUC or dose ... 

Figure 15. Penning electron energy distribution for He(23S)-H. Solid line represents result of quantal (model) calculations. Indicated part of calculated distribution at high electron energies belongs to associative (Pgl) and formation of quasibound HeH+ (see text).

Statistical methods are used for explaining the macroscopic properties of bodies from the point of view of thdr microscopic structure. The macroscopic physical quantities are then found to be mean statistical values of parameters characterizing the microscopic processes occurring in the system as a whole. The procedure of statistical physics consists in assunung some microscopic (classical or quantal) model of the body from which, by methods of statistics, we derive its equation of state and describe all its macroscopic, e.g. in the presort context, electromagnetic, properties. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

In connection with electronic strucmre metlrods (i.e. a quantal description of M), the term SCRF is quite generic, and it does not by itself indicate a specific model. Typically, however, the term is used for models where the cavity is either spherical or ellipsoidal, the charge distribution is represented as a multipole expansion, often terminated at quite low orders (for example only including the charge and dipole terms), and the cavity/ dispersion contributions are neglected. Such a treatment can only be used for a qualitative estimate of the solvent effect, although relative values may be reasonably accurate if the molecules are fairly polar (dominance of the dipole electrostatic term) and sufficiently similar in size and shape (cancellation of the cavity/dispersion terms). 

Due to the second criterion, time-to-tumor models were eliminated from consideration. These models require more detailed experimental data than is generally available. Moreover, it is difficult and unproductive to interpret the distribution of time-to-tumor in the context of human exposures. In most cases, the time-to-tumor variable would be integrated over a human lifetime, thus reducing the model to a purely dose-dependent one. Therefore we restrict our attention to quantal response models that estimate lifetime risks. 

The single most important statistical consideration in the design of bioassays in the past was based on the point of view that what was being observed and evaluated was a simple quantal response (cancer occurred or it didn t), and that a sufficient number of animals needed to be used to have reasonable expectations of detecting such an effect. Though the single fact of whether or not the simple incidence of neoplastic tumors is increased due to an agent of concern is of interest, a much more complex model must now be considered. The time-to-tumor, patterns of tumor incidence, effects on survival rate, and age of first tumor all must now be captured in a bioassay and included in an evaluation of the relevant risk to humans. 

As a general rule, simulations based on classical or quantal equations of motion may serve a useful purpose as benchmarks for model calculations. The days where such simulations may be used for routine calculations of stopping parameters are likely to lie quite a few years ahead, even with the present pace of hardware development in mind. Stopping data are potentially needed for 92X92 elemental ion-target combinations over almost ten decades of beam energy and for a considerable number of charge states, and to this adds an unlimited number of compounds and alloys. It seems wise to keep this in mind in a cost-benefit analysis of one s effort. 

Placket, R.L. and P.S. Hewlett. 1963. A unified theory for quantal responses to mixtures of drugs The fitting of data of certain models for two non-interactive drugs with complete positive correlation of tolerances. 

Continuum models are the most efficient way to include condensed-phase effects into quantum-mechanical calculations, and this is typically accomplished by using the self-consistent reaction field (SCRF) approach for the electrostatic component. Therefore it is very common to replace the quantal problem by a classical one in which the electronic energy plus the coulombic interactions of the nuclei, taken together, are modeled by a classical force field—this approach usually called molecular mechanics (MM) (Cramer and Truhlar, 1996). 

Results. In Table 5.1 we compare a few results of classical, semi-classical and quantum moment calculations. An accurate ab initio dipole surface of He-Ar is employed (from Table 4.3 [278]), along with a refined model of the interaction potential [12]. A temperature of 295 K is assumed. The second line, Table 5.1, gives the lowest three quantum moments, computed from Eqs. 5.37, 5.38, 5.39 the numerical precision is believed to be at the 1% level. For comparison, the third line shows the same three moments, obtained from semi-classical formulae, Eqs. 5.47 along with 5.37 with the semi-classical pair distribution function inserted. We find satisfactory agreement. We note that at much lower temperatures, and also for less massive systems, the semi-classical and quantal results have often been found to differ significantly. The agreement seen in Table 5.1 is good because He-Ar at 295 K is a near-classical system. 

As mentioned in Section II.A, the Pgl process is ideal for the application of the optical model. This is clear in the classical and semiclassical Pgl theory,24,25 for which opacity and cross-section formulas are completely equivalent to those given earlier in this chapter. The quantal optical model is also rigorously related to the elastic component of the quantal Pgl theory. Miller49 has shown that T(r), identified in Pgl as the autoionization width of the excited electronic state, may be accurately obtained by a standard Born-Oppenheimer electronic structure calculation as... 

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