Uncertainty in model predictions of exposure response at low exposures is a problem for risk assessment. A particular interest is the internal concentration of an agent in biological systems as a function of external exposure concentrations. Physiologically based pharmacokinetic (PBPK) models permit estimation of internal exposure concentrations in target tissues but most assume that model parameters are either fixed or instantaneously dose-dependent. Taking into account response times for biological regulatory mechanisms introduces new dynamic behaviors that have implications for low-dose exposure response in chronic exposure. A simple one-compartment simulation model is described in which internal concentrations summed over time exhibit significant nonlinearity and nonmonotonicity in relation to external concentrations due to delayed up- or downregulation of a metabolic pathway. These behaviors could be the mechanistic basis for homeostasis and for some apparent hormetic effects.

It is rarely possible to measure attributable health effects at environmental or occupational exposure levels corresponding to target risks in human populations. This is because of insufficient statistical power, measurement error, residual confounding, and inadequate specification of the exposure metric or the model itself. Therefore, low-dose (low-exposure) extrapolation from statistical models of exposure response is an important issue. For some end points, linear extrapolation is often the default either from a point-of-departure (POD) in animal studies (with uncertainty factors) or with linear models of exposure response (XR) in human studies. Deviation from linear response behavior at low exposure concentrations could impact risk assessment in either direction.

In traditional physiologically based pharmacokinetic (PBPK) modeling methodology, fixed, experimentally derived parameters are specified governing flow rates among multiple compartments and a system of differential equations is solved to describe concentrations within compartments over time in response to external concentrations. In the case of manganese PBPK models, a good prediction of Mn levels in the primate brain required (i) a saturable storage element, i.e., Mn binding, and (ii) asymmetrical diffusion rate constants (energy-dependent transport) for the brain compartment (

Much of the research addressing kinetics in transcription regulatory pathways has been motivated by medical interest in predicting, achieving, and sustaining sufficient levels of therapeutic agents in the context of inducible, interfering metabolic pathways (

To illustrate the potential importance of time-dependent metabolic regulation involving protective pathways, a pure simulation example is presented here. This is a nonstatistical, mechanistic simulation as is usually the case in PBPK studies. The possibilities when time-delays are introduced in biological regulation include attainment of (i) homeostatic conditions and (ii) apparent hormetic effects when none are actually present.

Examples of dynamic complexity are presented arising from a deterministic simulation using a simple model. A protective (or toxifying) metabolic pathway is induced as a function of increasing internal dose of an external agent. Model specifications:

one compartment,

one pathway into compartment from environment with fixed influx proportional to external concentration,

one inducible pathway out of compartment (could be transport, degradation, and conjugation) above a fixed first-order pathway out,

the inducible function depends on internal current concentration, but with a delay, and varies between a normal low value (downregulated, constitutive) and a high value (maximum upregulated), and

the accumulated upregulated entity (enzyme, transporter, and so on) concentration follows an negative exponential decline in time (described with a time constant).

Differential equation describing internal concentration, _{0} to

Using a finite difference equation, the internal concentration was calculated iteratively over time,

In this calculation, the following variables were defined:

The time units were 1/25 of a day; 1,000 units ~ 40 days.

If

If

^{Y}

The construction of this simulation model, particularly choices of constants, was by trial-and-error to produce the illustrative dynamics, a process analogous to biological selection of advantageous traits in organisms. There was minimal application of prior mechanistic design concepts. Different patterns of internal concentration,

Instead of an inducible protective effect for a toxic agent, a toxigenic pathway could be induced if there is a toxifying upregulation, e.g., production of toxic metabolic or conjugation products, or errorprone DNA repair. This behavior was investigated using a model in which the level of toxigenic upregulation depends on internal concentrations of the external agent, with a time delay. In this model, the external agent is transformed to a toxic metabolite that accumulates and is rapidly removed by a simple first-order kinetic process.

Most choices of model regulatory response times and time constants for downregulation produced nonlinear but monotonically increasing average internal steady-state concentrations of the external agent with increasing, fixed in time, external concentrations. Here, average internal concentration is equal to the increment in cumulative internal concentration (following attainment of steady state) divided by the specified period of accumulation—320 days; it is the time integral of the time-dependent internal concentration over the period 2,000—10,000 time units. For other choices, a nonmonotonic cumulative (average) internal concentration resulted as the fixed external concentration levels were increased (

The ratio of internal to external exposure at low external exposures would have consequences for risk assessment when determining risks at external exposures in or below the transition range, in these examples below 5,000 (

Another scenario examined was exposures that, like those in an industrial setting, occur typically for eight hours per day followed by 16 hours of no exposure as compared to continuous 24 hours exposure at one-third the concentration. Modeling the internal concentration over time with fixed but periodic external concentrations equivalent to about eight hours per day (9 out of 25 time units were used) produced very similar results but with slightly larger average internal concentrations compared to model with continuous exposure (in 25 contributions per day).

At low exposures, some specifications of model parameters, revealing nonmonotonic behavior, display steady-state time-averaged internal concentrations that remain relatively constant over a range of external concentration (e.g., concentrations 3,000–9,000 in

The simple model with a toxigenic effect produces a different picture. In this model,

This simulation describes plausible behavior in biological systems undergoing time-dependent regulatory changes. The focus here was on chronic, low-level exposures as arising for environmental hazards. The regulatory response time scale, days in this exercise, could be hours or weeks depending on the specific pathways. Overall, long-term cumulative effects would depend not on delayed instantaneous levels but rather on some time integration of internal concentrations. In real biological populations, in contrast to model cellular or animal systems, the transition structure with protective pathways described here would be blunted by normal biological variability of regulatory pathway kinetics (and more complex path structure). This variability could lead to low-dose supralinearity for the exposure response transitioning toward a plateau and then converging to an increasing linear response (with smaller slope) at higher exposures. The superlinear effects at low concentrations are consistent with predictions from the simulations (

This proof-of-concept mechanistic model was designed to accommodate delays in regulatory pathway responses. Beyond the examination of ranges of values for the model parameters that were considered, a broader validation of the model could be achieved by different choices in the model structure itself such as alternate monotonic functions in place of the exponential/logistic functions used. However, for those alternatives, it is anticipated that there would be parameter choices that again produce the nonmonotonic exposure responses with chronic exposure predicted in this work. A further validation step would be achieved by identifying biological model systems exhibiting the hypothesized nonlinear behavior.

The conclusion that hormetic phenomena could be observed when there is actually a no underlying hormetic response does not imply that hormesis does not exist. As described by

Many metabolic regulatory systems offer opportunities to examine behavior at low exposure concentrations leading to homeostasis or other detoxification mechanisms, as reviewed by

Metabolic regulation of metals is an area where homeostasis is Important (

The treatment here assumes that the end point of interest (adverse effect) is well predicted by a cumulative internal concentration that is appropriate when (1) an internal concentration contributes irreversibly to future risk and (2) the contribution of concentration intensity to the cumulative metric is linear, that is, there is no dose-rate effect. This assumption would be appropriate for some chronic diseases including some cancers but would be inappropriate for others. If internal concentrations contribute to the predicting metric more than, or less than, proportionately as with a dose-rate effect, the upregulation dynamics could be modified substantially.

Lynne Haber and Dale Hattis provided helpful insights and improvements reviewing an earlier draft of this work. The findings and conclusions in this report are those of the author and do not necessarily represent the official position of the National Institute for Occupational Safety and Health, Centers for Disease Control and Prevention.

Internal versus external concentrations of an exposure agent (detoxification upregulation response time: one day; decay time: five days): (a) cumulative internal concentrations (time-averaged over days 81–400) as a function of fixed external concentrations, (b) real-time internal concentration since start of exposure in time units of 1/25 of a day, for external exposure = 3,000, and (c) internal concentration for external exposure = 5,000.

Internal versus external concentrations (upregulation response time: 2 days; decay time: 0.5 days) (a) cumulative internal concentration (time-averaged over days 81–400) as a function of fixed external concentrations, and (b) real-time internal concentration since start of exposure in time units of 1/25 of a day, for external exposure = 3,000.

Internal versus external concentrations (upregulation response time: five days; decay time: two days): (a) cumulative internal concentration (time-averaged over days 81–400) as a function of fixed external concentrations and (b) real-time internal concentration since start of exposure in time units of 1/25 of a day for external exposure = 3,000.

Risk assessment consequences for an upregulated detoxification effect: choices for recommended exposure limit (example: upresponse time: two days; decay time: 0.5 days).

Risk assessment consequences for an upregulated detoxification effect: choices for recommended exposure limit (example: upresponse time: five days; decay time: two days).

Internal concentration over time with fixed external concentrations (upregulation response time: two days; decay time: 0.5 days) cumulative internal concentration (over 81–400 days): possible hormesis interpretation in the absence of unexposed comparison or with significant background exposure.

Toxigenic example: cumulative concentration for a toxic product derived from a fixed external exposure through upregulation: (a) smooth transition (response time: 0.2 days; decay time: one day) and (b) sharp transition (response time: one day; decay time: five days).