Modeling In Vivo Interactions of Engineered Nanoparticles in the Pulmonary Alveolar Lining Fluid

Increasing use of engineered nanomaterials (ENMs) in consumer products may result in widespread human inhalation exposures. Due to their high surface area per unit mass, inhaled ENMs interact with multiple components of the pulmonary system, and these interactions affect their ultimate fate in the body. Modeling of ENM transport and clearance in vivo has traditionally treated tissues as well-mixed compartments, without consideration of nanoscale interaction and transformation mechanisms. ENM agglomeration, dissolution and transport, along with adsorption of biomolecules, such as surfactant lipids and proteins, cause irreversible changes to ENM morphology and surface properties. The model presented in this article quantifies ENM transformation and transport in the alveolar air to liquid interface and estimates eventual alveolar cell dosimetry. This formulation brings together established concepts from colloidal and surface science, physics, and biochemistry to provide a stochastic framework capable of capturing essential in vivo processes in the pulmonary alveolar lining layer. The model has been implemented for in vitro solutions with parameters estimated from relevant published in vitro measurements and has been extended here to in vivo systems simulating human inhalation exposures. Applications are presented for four different ENMs, and relevant kinetic rates are estimated, demonstrating an approach for improving human in vivo pulmonary dosimetry.

The size distribution of lipid vesicles is constructed based on available information regarding vesicular size in mammalian alveolar fluid. There are generally three broad types of vesicles based on size: small unilamellar vesicles (SUVs), large unilamellar vesicles (LUVs) and giant unilamellar vesicles (GUVs). The available information has been summarized in Table S1.
The upper and lower limits of the size ranges are taken as the 95% and 5% values of a log-normal distribution for vesicular size. Then, the subsequent mean and standard deviation for each type of vesicle are estimated. The measures for the respective log-normal distributions are combined with the respective percent compositions to construct an overall population of vesicles having SUVs, LUVs and GUVs with a size distribution representative of mammalian alveolar fluid. The figure shows the overall size distribution of lipid vesicles.

Estimation of Steric Stabilization
Steric effects have been considered for coating molecules using the method shown by Damodaran [3], who used the following equation to estimate the repulsive energy due to steric effects: where k is Boltzmann's constant, T is the absolute temperature, nm is the number of coating molecules per unit surface area of the ENM, L is the chain length of the coating molecule, s is the mean distance Nanomaterials 2015, 5

S2
between coating molecules and d is the mean distance between interacting ENMs. Here, the mean distance between coating molecules, s, is given by 1/ m s n = . The distance between interacting ENMs, d, varies as two ENMs approach each other, and hence, the total interaction energy is estimated by integrating over the entire distance, as was done by Mukherjee et al. [4] for attractive and repulsive electrical potentials. Parameters L and nm in Equation (S1) for citrate and PVP molecules are summarized in Table S2. The number of coating molecules per unit ENM surface area (nm) is generally known to vary with ENM size. The values of nm for various sizes of ENMs have been estimated by Mukherjee et al. [4]. The molecular lengths of citrate and PVP molecules are taken from the literature (PVP from Zeng et al. [5] and citrate from Appelblat and Manzurola [6]). * Based on an apparent molar volume of 96.24 cm 3 /mol for trisodium citrate from Appelblat and Manzurola [6]; ** Based on 10 kDa and 40 kDa PVP chain length from Zeng et al. [5].
The steric repulsive potential estimated from Equation (S1) is normalized by the steric potential created by an uncoated Ag ENM. The molar volume of Ag is 10.335 cm 3 per mole. Considering an Avogadro number of molecules making up that volume and assuming an Ag atom as roughly spherical, the mean radius of the Ag atom, rAg, can be estimated as 1.6 A°. Empirical and calculated values of the atomic radius of Ag are also reported to be around 160 pm or 1.6 A° (Source: Webelements [7]). Therefore, using L = 3.2 A° and nm = 1.24 × 10 19 atoms/m 2 , we can get from Equation (S1) the value of EAg. The steric effect is quantified as The mutual interaction of ENMs consists of an attractive van der Waals' interaction potential and a repulsive interaction potential. Detailed expressions for these interaction potentials are provided in the original ADSRM article (Mukherjee et al. [4]). The repulsive interaction potential between the agglomerates can be expressed via the electric double layer (EDL) interaction potential equation that was developed using the linear superposition principle by Gregory [8] as: Here, ε is the permittivity of the medium, Z is the valence of ions in the medium, e is the elementary charge, kB is the Boltzmann's constant, κ is the Debye-Hückel parameter and γ is the reduced surface potential, which is a function of the surface zeta potential ζ of the particles. The effective repulsive potential, φe, is obtained by adjusting the electric repulsive potential φR by the steric effect represented by λST as: Nanomaterials 2015, 5

Estimation of Zeta Potentials
The zeta potential of lipid-adsorbed ENMs has been estimated using a weighed function as: where ζ0 is the zeta potential of the ENM without lipid adsorption, ζL is the reduced zeta potential due to lipid adsorption and θ is the fractional surface coverage of the ENM by lipids. Allam et al. [9] measured zeta potentials of nanoparticles, with and without the presence of lipids. Table S3. Surface zeta potential (in mV) for NPs. Measured values of zeta potential for coated and uncoated NPs from Allam et al. [9]; From these measurements, the average value of ζL/ζ0 was estimated as 0.34; Zeta potentials of various types of uncoated ENMs are summarized in the main article.

Protein Adsorption and Desorption
Zhdanov and Kasemo [10] modeled protein adsorption and desorption on lipid bilayers and how the adsorption and desorption rates (ka and kd) varied with lipid surface coverage. The measurements for adsorption and desorption were used to fit a function for ka and kd as follows:

Calculation of Protein Diffusion Coefficients
The estimation of diffusion coefficients of proteins involves multiple parameters, such as the molecular weight M, partial specific volume � and the sedimentation coefficient, s, which are related by the Svedberg equation [11] as shown below: (1 ) The sedimentation parameter, s, is given by: s = v/a, where v/a is the ratio of the volume-to-area of the protein molecule concerned. Young et al. [11] used the Svedberg equation to derive an empirical correlation for proteins, which can be written as: 8 1/3 (7.51 10 ) where � is the partial specific volume of the protein under consideration, η is the viscosity of the medium and T is the absolute temperature.