Probability coefficient, stochastic

Estimates of Probability Coefficients for Carcinogens. The nominal probabilities of a stochastic response (primarily cancers) per unit dose used in risk assessments, which are referred to in this Report as probability coefficients, normally differ for radionuclides and chemical carcinogens in regard to the degree of conservatism incorporated in the assumed values and the number of organs or tissues at risk that are taken into account. 

In general, the relationship between dose and response can be represented by a variety of functional forms. At low doses of substances that cause stochastic effects, the dose-response relationship usually is assumed to be linear and, thus, can be expressed as a single probability coefficient. This coefficient is frequently referred to as a risk (or potency factor or unit risk factor or slope factor) in the literature. However, it is really the response (consequence) resulting from a dose of a hazardous substance, and it should not be confused with risk as defined and used in this Report. 

In estimating the total detriment due to stochastic responses in any organ as described above, the probability coefficient for fatal cancers (F) or severe hereditary responses is based on data in humans and animals described in Section 3.2.2.2, and the lethality fraction (k) and relative length of life lost per fatal response (.II) are based on data on responses from all causes in various national populations. The values of F, k, and t/l for different organs, as well as the probability coefficient for severe hereditary responses, assumed by ICRP (1991) and the resulting estimates of total detriment, F((/T)(2 - k ), are summarized in Table 3.2. The two entries for Total in the last row represent the probability coefficient for... 

Table 3.3—Nominal probability coefficients for stochastic responses due to radiation exposure of the general public.a...

Stochastic Responses. A basic principle of health protection for both radionuclides and hazardous chemicals is that the probability of a stochastic response, primarily cancers, should be limited to acceptable levels. For any substance that causes stochastic responses, a linear dose-response relationship, without threshold, generally is assumed for purposes of health protection. However, the probability coefficients for radionuclides and chemicals that induce stochastic responses that are generally assumed for purposes of health protection differ in two potentially important ways. 

First, the dose-response relationships for radiation used for purposes of health protection and the probability coefficients derived from those relationships are intended to be MLEs. In contrast, the dose-response relationships and probability coefficients for chemicals that induce stochastic responses are intended to be upper-bound estimates (UCLs), although MLEs also are available. In animal data from which the probability coefficients for most chemicals that cause stochastic responses are obtained, UCL can be greater than MLE by a factor that ranges from 5 to 100 or more. 

The recommended dose limits for the public define limits on the probability of stochastic responses that are regarded as necessary for protection of public health. Doses above the limits are regarded as intolerable and normally must be reduced regardless of cost or other circumstances, except in the case of accidents or emergencies (see Section 3.3.1). For continuous exposure over a 70 y lifetime, and assuming a nominal probability coefficient for fatal cancers (i.e., the probability of a fatal cancer per unit effective dose) of 5 X 10 2 Sv 1 (ICRP, 1991 NCRP, 1993a), the dose limit for continuous exposure corresponds to an estimated lifetime fatal cancer risk of about 4 X 10 3. However, meeting the dose limits is not sufficient to ensure that routine exposures of the public to man-made sources would be acceptable. 

However, this option presents some difficulties for radionuclides, because studies of radiation effects in human populations have focused on cancer fatalities as the measure of response and probability coefficients for radiation-induced cancer incidence have not yet been developed by ICRP or NCRP for use in radiation protection. Probabilities of cancer incidence in the Japanese atomic-bomb survivors have been obtained in recent studies (see Section 3.2.3.2), but probability coefficients for cancer incidence appropriate for use in radiation protection would need to take into account available data on cancer incidence rates from all causes in human populations of concern, which may not be as reliable as data on cancer fatalities. Thus, in effect, if incidence were used as the measure of stochastic response for radionuclides, the most technically defensible database on radiation effects in human populations available at the present time (the data on fatalities in the Japanese atomic-bomb survivors) would be given less weight in classifying waste. 

Stochastic Responses. Consideration of the dose-response relationships and the nominal probability coefficients for induction of stochastic responses at low doses is important for both radionuclides and hazardous chemicals. 

For substances that cause stochastic effects (radionuclides and hazardous chemicals), specify negligible and acceptable (barely tolerable) risks to be used in classifying waste. Then, establish the corresponding negligible and acceptable dose of each substance of concern based on an assumed probability coefficient (risk per unit dose). 

The use of MLEs of probability coefficients for radionuclides but UCLs for chemicals that induce stochastic responses is the most important issue that would need to be resolved to achieve a consistent approach to estimating risks for the purpose of waste classification. For some chemicals, the difference between MLE and UCL can be a factor of 100 or more. The difference between using fatalities or incidence as the measure of response is unlikely to be important. Use of the linearized, multistage model to extrapolate the dose-response relationship for chemicals that induce stochastic effects, as recommended by NCRP, should be reasonably consistent with estimates of the dose-response relationship for radionuclides, and this model has been used widely in estimating probability coefficients in chemical risk assessments. The difference in the number of organs or tissues that are taken into account, although it cannot be reconciled at the present time, should be unimportant. 

In many respects, the foundations and framework of the proposed risk-based hazardous waste classification system and the recommended approaches to implementation are intended to be neutral in regard to the degree of conservatism in protecting public health. With respect to calculations of risk or dose in the numerator of the risk index, important examples include (1) the recommendation that best estimates (MLEs) of probability coefficients for stochastic responses should be used for all substances that cause stochastic responses in classifying waste, rather than upper bounds (UCLs) as normally used in risk assessments for chemicals that induce stochastic effects, and (2) the recommended approach to estimating threshold doses of substances that induce deterministic effects in humans based on lower confidence limits of benchmark doses obtained from studies in humans or animals. Similarly, NCRP believes that the allowable (negligible or acceptable) risks or doses in the denominator of the risk index should be consistent with values used in health protection of the public in other routine exposure situations. NCRP does not believe that the allowable risks or doses assumed for purposes of waste classification should include margins of safety that are not applied in other situations. 

The stochastic risk corresponding to the estimated dose is obtained using the nominal probability coefficient for fatal cancers of 0.05 Sv-1 (ICRP, 1991 NCRP, 1993a). Since this coefficient is intended to represent a best estimate, rather than a conservative upper bound, the value is not increased by a factor of 10, as in the adjustment of the slope factors for chemicals that induce stochastic effects (Section 7.1.7.5). Therefore, the calculated stochastic risk due to 137Cs in the waste is (5 X 10 4 Sv)(5 X 10 2 Sv-1) = 2.5 X 10 5. 

Table 22.10. Probability coefficients assessed for stochastic effects in detriment (ICRP 1990)...

TABLE II Nominal Probability Coefficients for Stochastic Effects ... 

Next we seek the stochastic equation for the probability distribution of fluctuations in the macroscopic mass velocity for which the excess work, (8.50), gives the stationary distribution. It is the Fokker-Planck equation with constant probability coefficient c = 2akTjf ... 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

As a consistency test of the stochastic model, one can check whether the percentage Nk t) of trajectories propagating on the adiabatic PES Wk is equal to the corresponding adiabatic population probability Pf t). In a SH calculation, the latter quantity may be evaluated by an ensemble average over the squared modulus of the adiabatic electronic coefficients [cf. Eq. (22)], that is. 

Zwanzig s diffusion equation [444], eqn. (211), can be reduced to the stochastic equation used by Clifford et al. [442, 443] [eqn. (183)] to describe the probability that N identical reactant particles exist at time t (see also McQuarrie [502]), Let us consider the case where U — 0, with a static solvent, for a constant homogeneous diffusion coefficient. This is a major simplification of eqn. (211). Now, rather than represent the reaction between two reactants k and j by a boundary condition which requires the... 

As for the problem of a formula for the diffusion coefficient, D, this molecular model has to resort to the theory of stochastic processes (48). In a homogeneous medium in which a penetrant may jump with equal probability in all directions D is given by ... 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

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