Internal exposures calculation

Computation of internal exposures (internal substance amounts and concentrations) requires a kinetic model to translate external doses, possibly from multiple routes, to internal doses at the target compartment/organ of interest.

Calculation of internal concentrations using absorption factors

In the simplest form, internal concentrations are derived from external exposure concentrations using multiplication factors (or, absorption factors) that can be specified by substance and by route. That is, for a given substance, the internal exposure \(\mathit{exp}_{\mathtt{int}}\) is computed as

\[\mathit{exp}_{\mathtt{int}} = \sum_{r \in \mathit{Routes}} \mathit{f}_{\mathtt{abs},r} \cdot \mathit{exp}_{\mathtt{ext},r}\]

Here, \(\mathit{Routes}\) denotes the set external exposure routes, \(\mathit{exp}_{\mathtt{ext},r}\) denotes the external exposure for route \(r\) and \(\mathit{f}_{\mathtt{abs}, r}\) denotes the absorption factor of route \(r\). Note that this model assumes that both external and internal exposures refer to amounts or concentrations depending on the dietary exposures setting (External exposure: substance amount per individual, or substance amount divided by body weight; internal exposure: substance amount per organ, or substance amount divided by organ weight.) Also, both external and internal exposures are expressed per day.

Calculation of internal concentrations using kinetic models

A more detailed alternative to using absorption factors is to use one of the advanced kinetic models available in MCRA. In this approach, for each substance independently, the external exposures of an individual (chronic) or individual-day (acute) are presented for a number of simulated day to a PBK model of the individual. This yields a time course of the internal substance amount at the specified target compartment/organ from which a long term average substance amount (chronic) or peak substance amount (acute) can be obtained. An example of such a time course is given in Figure 35 for acute exposure assessments, and in Figure 36 for chronic exposure assessments. By dividing this substance amount by the weight of the compartment, an internal concentration is obtained. Notice that this procedure also changes the unit of the exposures from exposure per day to long term exposure.

Figure 35 Time course of the internal substance amount when applying the same single dose on each day. The acute internal concentration is derived as the peak substance amount (the green line in the figure) divided by the compartment weight. The vertical line at 50 indicates the selected end of an assumed non-stationary period, defining a burn-in period that is to be ignored for computing the peak substance amount.

Figure 36 Time course of the internal substance amount when randomly applying one of the individual-day doses for a number days. The chronic internal concentration is derived as the average substance amount (the blue line in the figure), divided by the compartment weight. The vertical line at 50 indicates the selected end of an assumed non-stationary period, defining a burn-in period that is to be ignored for computing the average substance amount.

Mathematically, the calculation of the peak substance amount (\(\mathit{d}_{\mathtt{peak}}\)) for deriving acute internal exposures is as follows:

\[\mathit{d}_{\mathtt{peak}} = \max_{i = 0,\ldots,n_{\mathtt{stop}}} \left \{ d(t_{\mathtt{start}} + i \Delta t ) \right \} .\]

Here, \(d(t)\) denotes the substance amount at time \(t\), \(t_{\mathtt{start}}\) denotes the starting time of the evaluation window (defined by the non-stationary period), \(\Delta t\) denotes the time resolution of the kinetic model (e.g., hours or minutes), and \(n_{\mathtt{stop}}\) denotes the total number of time-points, marking the end of the evaluation window (defined by the specified number of simulation days), which is computed as

\[n_{\mathtt{stop}} = \left\lfloor \frac{t_{\mathtt{stop}} - t_{\mathtt{start}}}{\Delta t} \right\rfloor .\]

Likewise, chronic long term average substance amounts (\(\mathit{d}_{\mathtt{avg}}\)) are computed as:

\[\mathit{d}_{\mathtt{avg}} = \frac{\sum_{i=0}^{n_{\mathtt{stop}}} d(t_{\mathtt{start}} + i \Delta t)}{n_{\mathtt{stop}}} .\]

Dosing patterns

In MCRA, the dietary and non-dietary exposures are computed at the level of exposures per day. However, when applying advanced PBK models, dosing patterns may be specified at a much finer resolution (e.g., hours or minutes). For this, a method is needed to translate external exposures provided per day to dosing patterns of substance amounts during the day. The simplest, yet not very realistic model is to apply, per route, the full exposure amount in one single dose at the beginning of the day. Alternatively, MCRA offers the possibility to specify, per route, the number of exposure events per day. If it is specified to use multiple doses per day, then the total substance amount of each day is divided into equal portions which are applied at regular time-intervals during the day.

Non-stationary period

Especially in the case of chronic exposure assessments, where a long term average exposure is computed based on the simulated time-course, it is important to realise that at time zero, the substance is commonly considered to be completely absent in the simulated system. However, this is not a realistic assumption. It is much more likely that the substance was already present in the system, and that the level is equal to the level obtained from applying the same chronic exposures to the system. For this, a specification of the number of days skipped (or burn-in period) is required in order to come to these initial concentration levels. This period is not used for computing the long term average or peak exposures, but just to determine initial (background) concentration levels.