Driven by wide recognition that target tissue (or cell) dose is the most appropriate metric of dose for risk and safety assessment, there has been continued growth in the development of computational tools for estimating target tissue dose in vivo for a wide range of materials: volatile and non-volatile materials [42–44] organic chemicals [45, 46], pharmaceuticals [47, 48], and particulates and ultrafine particulates [49–53]. It is surprising, then, given the growing importance of in vitro studies to chemical and particle risk assessment  that few efforts [43, 54–56] have been directed at developing computational models of dosimetry for in vitro systems. In part, this oversight may be the result of the incorrect belief that in contrast to in vivo, there are no important kinetic or other processes to consider when addressing the issue of target tissue dose in vitro. We have shown this assumption is not universally true for chemicals  and particulates, including nanomaterials . Thus, as with other systems where kinetics influence dose, there is a need for computational models of in vitro nanoparticle and microparticle target tissue dosimetry to supplement experimental measurements or extrapolate across particle type and experimental systems.
ISDD is a computational model of particokinetics (sedimentation, diffusion) and dosimetry for non-interacting spherical particles and their agglomerates in monolayer cell culture systems. Through simulation, ISDD calculates the delivered dose and rate of transport of particles in vitro using readily available parameters (temperature, media height, particle size in solution, agglomeration state and particle density). To our knowledge this is the first computational model extending principles long used to calculate particle deposition in the respiratory tract to a simpler system: static liquid in vitro studies. Application of ISDD opens the door for post-hoc interpretation of published studies, scaling across particle types and doses and development of a more predictive paradigm for nanoparticle in vitro studies.
Without adjustment of any parameters (calibration), ISDD simulated particle transport of monodisperse silica and polystyrene particles corresponded very well with measured transport, differing in most cases by a factor of approximately two or less. This level of accuracy is common to more sophisticated, extensively calibrated models of in vivo pharmacokinetics used in risk assessment, such as physiologically based pharmacokinetic (PBPK) models [46, 57, 58]. There cannot be an expectation that model accuracy is greater than limits imposed by experimental and biological variability, which can be significant. Thus, PBPK and other biokinetic models are commonly considered acceptable and useful if outputs are within a factor of approximately two (or sometimes more) of the data and the dose and time trends (model behavior) are consistent with experimentally measured trends. This general level of error can be compared to the other sources of error inherent in assuming mass media concentrations are, alone, sufficient measures of exposure to comparative toxicity studies for particles: 1) size and density differences in transport rates, approximately 2-10 fold; 2) media height differences in fraction of material transported to cells, up to approximately 10 fold; 3) particle size and density dependent differences in the number and surface area of particles per unit concentration (e.g. 1 μg/ml), 1,000-1,000,000 fold. Thus, the error in ISDD is low relative to the potential errors associated with common assumptions applied to most in vitro particle toxicity studies.
Beyond the accuracy issue, ISDD allows users to explore expected trends in delivered dose to determine if delivery processes are potentially important. For example, ISDD predicted a six-fold increase in transport of 35 nm amorphous silica as media heights were reduced from 4.5 mm to 1.1 mm, but measured cell associated silica was relatively constant. The experimental finding could be the result of either constant transport rates, or one or more other experimental factors affecting cellular uptake such as saturated uptake of silica. For example, it is plausible that lower levels of nutrients may have affected cell function, uptake of particles, or the number of cells. Since ISDD shows that significant differences in particle transport and delivered dose are expected, we arrive at the hypothesis that cellular uptake might be saturated. Arriving at this insight, and the experimentally testable hypothesis it produces, is not feasible if the experimentalist is unaware of differences in particle transport under the experimental conditions or incorrectly assumes static particle concentration is all one needs to know for dose-response assessment.
With calibration of the parameter DF, ISDD also simulated the transport of a dispersion of more complicated agglomerates of 30 nm iron oxide particles, providing greater confidence that the generalization of particle transport represented by the model can be widely applied to nanoparticle and microparticle solutions including agglomerates. Large errors in model predictions were not observed, with one exception (low volume, silica transport experiment), providing good evidence that one or more of the simplifying assumptions used in the formulation of the model were not violated.
There are additional published experimental data that support the general accuracy of ISDD. Limbach and Stark were the first to carefully explore the relationship between particle size and delivered dose to cells in vitro, firmly and convincingly establishing that particle size and agglomeration state has a significant impact on dose to the cell . Using ceria particles with mean diameters of 20-50 nm or 250-400 nm they exposed lung fibroblasts and measured ceria content by ICP-MS at 6-10 times over the 300 minute experiment. Comparing cellular ceria content and calculated ceria transport rates they showed that cellular ceria uptake of 20-50 nm particles was consistent with diffusion limited transport, and uptake of 250-400 nm particles was primarily controlled by sedimentation. They also showed that for a constant nominal mass media concentration the mass of ceria in cells was related to particle size, with greater mass amounts of ceria but smaller surface area and number concentrations found in cells exposed to larger, more rapidly sedimenting particles. The authors concluded that particle size was the most important factor determining the amount of material on or in cells and that particle concentration and total surface area were of "minor" importance. Limbach and Stark's findings  demonstrate the appropriateness of the Stokes-Einstein equation for predicting transport in vitro, and the importance of addressing these transport issues in vitro nanomaterial toxicity studies.
These findings were further verified by Sun et al. , in an elegant quantitative study of in vitro transport and uptake of fluorescently labeled 100 nm mesoporous silica nanoparticles in human lung cancer cells. The movement and uptake of particles was observed using differential interference contrast microscopy. Transport to the cell through the cell culture media was driven by diffusion. The diffusion rate calculated directly from measured rates of particle movement was 2.9 × 10-8 cm2/s, in very close agreement with the theoretical value calculated from the Stokes-Einstein equation as applied in ISSD: 4.4 × 10-8 cm2/s . Not surprisingly, the diffusion rate slowed an order of magnitude as the nanoparticles neared the cell surface. It should also be noted that the most common approach to measuring particle size, DLS, directly measures the diffusivity of particles in solution and infers the size of the particle using the Stokes-Einstein equation. Thus, it would seem, that if one believes DLS instruments are accurate, belief that that the ISDD calculated diffusional transport rate is accurate should follow. Finally, we point out that the description of particle motion in liquids derived by Navier, Stokes and Einstein and applied in ISSD has been widely and successfully applied across multiple scientific and engineering disciplines for many decades.
There are, however, a number of limitations to be considered when using ISDD. Particle settling must not generate turbulence (low Reynolds numbers) and dynamic agglomeration or other particle interactions are not accounted for in the model. The model may not be appropriate to apply where advection occurs in the cell culture system or where there has been significant advective or mechanical mixing over the course of the experiment. Formulated for spheres or particles that can be adequately described as spheres, ISDD should not be used for fibers without additional modification and testing. Changes in the agglomeration or aggregation state of modeled particles would be expected to lead to larger discrepancies between modeled and observed target cell doses. Uncertainty in many of the parameters for the model is low; particle size, density, agglomeration state and media temperature and viscosity are easy to measure with sufficient accuracy. As noted, the PF and DF are more challenging to obtain experimentally, and represent an area of higher uncertainty. Nonetheless, ISDD provides an excellent approximation of the expected cellular dose as a function of particle size and density, and allows reasonable estimates of the range of errors introduced using metrics of exposure.
ISDD, now tested and verified, provides further quantitative evidence that use of nominal media concentration as a metric of "dose"--its actually exposure--confounds particle comparisons by introducing large errors from the assumption that dose to the target cell or site is proportional to media concentration across particle size, density, and agglomeration state. This erroneous assumption is particularly important where nominal mass media concentrations (μg/mL) are used for dose response analysis, but the biologically relevant dose-metric is target cell dose on a surface area or particle number basis. In this case, particle size and density dependent differences in transport rates are compounded by particle size and density dependent differences in particle number and surface area. Of course, this problem is somewhat mitigated by using nominal surface area concentrations in dose-response experiments. These conclusions, along with those regarding the influence of media height and agglomeration status on particle transport reaffirm the need for a far greater curiosity about target cell dosimetry in vitro and a correspondingly increased role for research on nanoparticle dosimetry for in vitro systems.
The value of organizing and conceptualizing the processes controlling particle transport and dosimetry in cell culture systems and presenting them in the form of a model to the community of biologists and other scientists using in vitro systems should not be overlooked. In the past, similar efforts such as the early publications on PBPK modeling and more recent biologically based dose-response (BBDR) models have led to a deeper and wider understanding of the systems being studied, enabled new biological or toxicological insights, and promoted more accurate study design and interpretation. Experimentalists can use ISDD to explore the potential impact of particle and media characteristics on target cell dose in their systems, and to guide experimental design. Hazard and risk assessors can utilize the model for post-hoc calculation of target cell doses from published studies. As more complete models of biological response to particles are developed, linkages to ISDD will allow inclusion of target cell dose-time vectors, improving the basis for biologically-based dose response analysis and predictive toxicology. Perhaps most importantly, the concepts represented by the ISDD model can be used to define a new paradigm for nanomaterial and particle dosimetry for in vitro systems that parallels the widely accepted paradigm for particle dosimetry in vitro. Absent now, such a paradigm would improve the accuracy and scalability of in vitro systems for hazard screening and exploratory mechanistic work.
The gold standard for particle dosimetry for in vitro nanotoxicology studies should be direct experimental measurement of the cellular content of the studied particle. However, where such measurements are impractical, unfeasible, and before such measurements become common, particle dosimetry models such as ISDD provide a valuable, immediately useful alternative, and eventually an adjunct to such measurements. The model also allows researchers to estimate trends in particle transport to determine if transport processes may be an important factor in the study. Ultimately, ISDD is a computational framework for describing particle transport that can raise awareness of particokinetic issues in vitro, and be revised to improve its accuracy for specific particles and linked to models describing cellular processes affecting uptake of particles.