# Framework to Estimate Total Particulate Mass and Nicotine Delivered to E-cig Users from Natural Environment Monitoring Data

We define the cumulative yield, YAC, of individual Aerosol Constituents (AC) of emissions delivered from an ENDS to the mouth of a user as the integral of the product of the time dependent mass ratio of the aerosol constituent, the Total Particulate Matter (TPM) concentration of the whole aerosol, and the user’s volumetric flow rate:

$${Y}_{{rm{AC}}}equiv {int }_{{t}_{{rm{initial}}}}^{{t}_{{rm{final}}}}{f}_{{rm{AC}}}(t){C}_{{rm{TPM}}}(t)dot{v}(t)dt$$

(1)

where the mass ratio of the constituent fAC(t) = mAC/mTPM [mg/mg] and TPM Concentration CTPM(t) = mTPM/v [mg/mL] (mass per volume) vary with time as a user changes puffing patterns, tobacco product choices, and user-selectable device settings. The constituents in the ENDS aerosol may be present in the un-puffed e-liquid, or generated as decomposition products. We normalize all aerosol constituents (including vapor phase constituents, compounds originating in the e-liquid, and thermal decomposition products) by the mass of TPM emissions to facilitate separation of variables between the fAC and CTPM terms. We posit the TPM concentration and mass ratio of constituents may be expressed as linearly independent functions of Product Characteristics (PC) and User Behavior Characteristics (UBC). Numerous ENDS PC may affect the CTPM including but not limited to the device operating power (reflected in coil wattage, amperage, or temperature), flow path geometry, coil design, and aspiration features. Additionally, the solvent composition of the e-liquid consumable (such as the PG/VG ratio which directly impacts the saturation temperature of the e-liquid) may impact CTPM. Additional consumable PC impacting fAC may include nicotine concentration, flavor additives, viscosity, and pH. Furthermore, a variety of UBC may affect the CTPM and/or fAC including puff duration, d, flow rate, q, volume, v, and interval, i:

$${C}_{{rm{TPM}}}={ {mathcal F} }_{{rm{TPM}}}(PC,UBC)$$

(2)

$${f}_{{rm{AC}}}={ {mathcal F} }_{{rm{AC}}}(PC,UBC)$$

(3)

For the current study we limit variability in PC by selecting a single ENDS with no user-adjustable settings and a single e-liquid. We thus focus on the interaction between UBC and the flow path PC, reflected by the topography parameters q and d, and consider a single AC, nicotine, to illustrate the approach. Prior work4 demonstrated a power law relationship between CTPM [mg/mL] and puff flow rate, q [mL/s]. Therefore, we propose the form of Eq. 4 to account for puff flow rate, q [mL/s], puff duration, d [s], and the product of those terms, which has physical significance as the puff volume v = q d [mL]. A transformation of variables enables a linear systems model describing the model-predicted TPM concentration, ({hat{{rm{C}}}}_{{rm{TPM}}}), of a single puff in terms of a set of empirical coefficients, b:

$$mathrm{ln}({hat{C}}_{{rm{TPM}}})={b}_{1}+{b}_{2},mathrm{ln}(q)+{b}_{3}(d)+{b}_{4}(mathrm{ln},{(q)}^{2})+{b}_{5},mathrm{ln}(d/1000)+{b}_{6}(qcdot d)$$

(4)

The experimental observations of CTPM can be computed as the ratio of the mass emissions captured on a filter pad per measured volume of aerosol passing through the pad. The coefficients in Eq. 4 may be estimated using ordinary least squares (OLS), weighted least squares (WLS), or other regression techniques. OLS regression is employed in the current work. Since the ENDS device chosen for the study does not have a user-selectable power setting, and the nicotine concentration of the e-liquid is held constant across all trials, we hypothesize a first order linear model, Eq. 5, for the model-predicted nicotine mass ratio, ({hat{{rm{f}}}}_{{rm{NIC}}}), as a function of puff flow rate, q.

$$widehat{{f}_{{rm{NIC}}}}={beta }_{1}+{beta }_{2}(q)$$

(5)

The regression coefficients, β, are also determined using OLS.