A lung retention model based on Michaelis-Menten-like kinetics.

A Michaelis-Menten (MM)-like kinetic model for pulmonary clearance and retention of insoluble dusts was developed and validated by comparing our predictions with experimental data from F344 rats. Published data from inhalation studies involving accumulation and elimination of photocopy test toner, antimony trioxide, carbon black, and diesel exhaust particles were investigated. Numerical integration techniques were used to solve mass balance relationships based upon dust retention in a single lung compartment and clearance via an MM-like kinetic process. The model fit most of the experimental data well. The parameters of MM-like clearance kinetics, which had been derived strictly from the elimination phase, accurately predicted dust retention during the elimination as well as accumulation phases. Furthermore, parameters estimated from one study could accurately predict retention of the same dust in other studies. Particle density and gender of the animals had no effect on the goodness of fit of model predictions. This study suggests that MM-like kinetics offer a reasonable description of particle clearance from the pulmonary region of the rat lung that is more parsimonious than existing particle-clearance models and therefore more suitable for use with small amounts of data. Imagesp496-aFigure 1. AFigure 1. BFigure 2.Figure 3.Figure 4. AFigure 4. BFigure 4. CFigure 4. DFigure 5. AFigure 5. BFigure 6.Figure 7.Figure 8.Figure 9.Figure 10.

Lung clearance and retention of spherical particles was substantially quantified by the comprehensive model published by the Task Group on Lung Dynamics in 1966 (1). The model postulated that alveolar clearance involved multicompartmental linear processes. In the ensuing three decades, these underlying assumptions have been challenged by numerous investigations that point to nonlinear behavior of clearance as a function of lung burden. In particular, recent models of pulmonary retention and clearance of dusts have focused upon three nonlinear processes, namely, sequestration, overloading, and redistribution. Sequestration, first recognized by Soderholm (2), describes a phenomenon whereby the alveolar clearance of very large lung burdens essentially ceases. This concept has been used to explain the clearance and retention behavior of diesel exhaust particles (DEP) (3)(4)(5), carbon black (CB) (6), quartz (7), titanium dioxide (7), photocopy test toner (PTT) (7), and amosite fibers (8). However, because the characteristics of particle sequestration have not been fully elucidated, various types of models have been used to account for the phenomenon. For example, some investigations considered sequestration to be a process in which clearance ceased completely (3,4,(8)(9)(10)(11), while others regarded clearance as being merely very slow (5)(6)(7)12), or not applicable in cases involving low exposure (13).
Dust overloading, hypothesized first by Bolton et al. (14) and subsequently by Morrow (15,16), is a phenomenon in which the lung maintains its normal clearance rate until the burden reaches a threshold, whereupon clearance progressively slows. This phenomenon is based upon empirical observations that, above such a threshold, the alveolar clearance of exposed animals becomes significantly slower than those of control groups. As a result, compartmental models have included threshold values that defined the transition from normal to overloading conditions (5,9,10).
Because clearance and retention models, which explicitly account for sequestration, overloading, and/or redistribution, have become increasingly complex, we propose herein an alternative nonlinear model based upon Michaelis-Menten (MM)-like kinetics. This model is consistent with the treatment by Smith (12), who implicidy recognized the appropriateness of MM-like kinetics for alveolar clearance. We previously presented the rationale underlying the clearance portion of this model and developed a linear relationship between pulmonary clearance half-time and lung burdens to evaluate the fit of the model to published data of PTT, antimony trioxide (Sb203), DEP, and polyvinyl chloride powder (27).
To further validate the full nmodel (including both accumulation and elimination phases), we tested the model predictions of the temporal behavior of lung burdens, which were experimentally determined in a variety of inhalation studies in F344 rats.
Lung retention model. Insoluble particles are deposited in various portions of the rat lung and cleared by several mechanisms. Relatively large particles are likely to be deposited in the conducting airways and are then removed rapidly (over a period of hours to a few days) by mucociliary clearance. Smaller particles tend to be deposited in the alveoli and are then cleared predominately by AMs, which phagocytize the particles and transport them to the ciliated epithelium. A secondary mechanism for clearance in the deep lung involves transport to lung-associated lymph nodes, possibly mediated by AMs or via direct penetration of particles through the interstitium.
Because these clearance mechanisms in the deep lung are relatively slow (working over a period of weeks to months), the vast majority of the total lung burden resides in the alveolar region following chronic exposure to insoluble dusts. Thus, our model treated the rat lung as a single compartment consisting exclusively of the pulmonary region (the alveoli and the supporting structures) and ignored a separate compartment for the tracheobronchial region, which contributes relatively little to the total lung burden (4). Experimental data reporting burdens of dusts in lung-associated lymph nodes were not used for testing our model.
In keeping with our earlier treatment of MM-like clearance (2/), we defined the pulmonary clearance rate coefficient k = ln(2)/1T/2, where T112 is the clearance halftime for each dust, according to the following relationship: where kmax represents the maximum rate coefficient of pulmonary clearance, m1/2 represents the characteristic burden at which k= 1/2-k~,and m represents the lung burden at which k is evaluated.
During the accumulation phase, when rats are continuously exposed to dust according to a particular exposure regimen [e.g., 30 hr/week for 13 weeks in the study by Newton et al. (29)], particles accumulate in the lung during the period of daily exposure (e.g., 6 hr), and the mass balance governing the lung burden is given by where Vt is the ventilation rate (a function of the animal's age), EA is the particle deposition efficiency, and x is the exposure concentration. Immediately following daily exposure, the lung reaches the initial postexposure burden, designated mo. During the period prior to the next daily exposure (e.g., 18 hr), the burden is cleared according to the following relationship: cConcentrations and regimen were used to build up lung burdens. dAgglomerated particles (primary size = 0.04 pm).
Equations 3 and 1' are also applicable during the weekends and after termination of exposure; the latter will be referred to hereafter as the elimination phase. Note that when particles are not deposited in the lung, k (in Equation 1') is a function of mo and is independent of the postexposure time. Values of m were estimated during the accumulation and elimination phases by numerically integrating Equations 2 and 3, as described in Appendix A.
Estimation of model parameters. The parameters to be estimated under the model, defined by mass balance Equations 2 and 3, are Vt, EA, kmax' and mi112 Because Vt depends on the ages of the animals and, in part, on the particular laboratory where each inhalation study was conducted, we attempted to match the method of estimation to the published data as closely as possible. For the study by Strom et al. (5), we used the following function reported by the same authors (6) for male F344 rats: (4) where t is the animal's age in days. Likewise, for the later study from the same laboratory (6), we used a similar relationship reported by the authors: Vt(ml/min) = VMe-In(2)(50)/t (5) respectively, so that postexposure ventilation rates would correspond exactly to the reported values. In other studies, ventilation rates were not reported, so we used the published relationship of Guyton (32) between V, and body weight (bw) in grams, at age tin days: Vt (ml/min) = (2.1)(bw)0 75. (6) For the study by Newton et al. (29), body weight was estimated for male F344 rats from the following relationship given by Strom  For female rats in the same study (29), the body weight from Equation 7 was adjusted by a factor of 84.5%, representing the estimated ratio of the average body weight of 7-week-old female rats to the correspond- for female rats. The MM-like parameters were estimated from the following relationships given by Yu and Rappaport (27)  Observed lung burden (mg) Figure 2. Overall comparison of model predictions and experimental data of Sb203 observed by Newton et al. (29). kmax= ln(2)/a (10) and M/2 (11) where a (common to all dusts) and , (specific to each dust) are, respectively, the estimated values of the intercept and slope of the linear relationship between T1/2 and lung burden. The rationale for development of this linear relationship was discussed previously (27). From Equations 10 and 11, kmax was estimated to be 0.009/day for F344 rats (27) and ml/2 was estimated to be 0.69 mg for Sb203 [reported by Yu and Rappaport (27) Figure 3. Accumulation and elimination of photocopy test toner in F344 female rats exposed for 90 days in a subchronic study by Muhle et al. (30). Solid lines represent model predictions; symbols represent experimental observations. (The Michaelis-Menten-like clearance parameters for the model were estimated from the elimination phase of the same study.) and Rappaport (27)  Note that values of m1/2 were derived from data obtained exclusively from the elimination phase of each experiment and that for PTT and DEP, values of m1/2 that were estimated from one study were applied to other studies involving the same dust.
Finally, EA was determined empirically for each experimental data set [by experimental group, dust, exposure concentration, duration (if applicable), and gender (if applicable)] by fitting Equations 2 and 3 to the data after including all other terms in the model. Table 2 shows the estimated ranges of EA for each of the data sets for the various experimental groups.

Results
Antimony trioxide. Newton et al. (29) investigated both accumulation (for 13 weeks) and elimination (for an additional 28 weeks) of Sb2O3 in rats exposed to four exposure concentrations in a subchronic inhalation study. Figure 1 shows accumulation and elimination of the dust over time in male (Fig. 1A) and female (Fig. 1B) rats using values of km. and mI/2 that were estimated from the elimination phase of the same study. The model predicted the behavior of all experimental data quite well. Furthermore, the residuals were found to be unremarkable with no apparent gender effect on model predictions (Fig. 2).
Photocopy test toner. Muhle et al. (30) investigated PTT accumulation (up to 90 days) and elimination (for an additional 75 days), whereas Bellmann et al. (8) examined only accumulation of PTT (for 24 months). Figure 3 depicts the predicted and observed burdens in female rats exposed to different dust concentrations, Figure 4 shows the burdens in males exposed to various concentrations of PTT in the study by Muhle et al. (30), and Figure 5 shows the results from the study by Bellmann et al. (8). In each case, kmax and m1/2 were estimated from the elimination phase of the study by Muhle et  from the lungs of F344 rats in a study by Griffis et al. (31). Solid lines represent model predictions. Exposure time (months) Figure 5. Accumulation of photocopy test toner in F344 female (A) and male (B) rats in a chronic study by Bellmann et al. (9). Solid lines represent model predictions; symbols represent experimental data. [The Michaelis-Menten-like clearance parameters for the model were estimated from the elimination phase of the study by Muhle et al. (30).] the lowest concentration (Fig. 4D). Again, no effect on model fit by gender was observed (data not shown). The model, given the kmax and min/2 estimated from the elimination phase, accurately predicted the burdens during the accumulation and elimination phases in the same study by Muhle et al. (30). Furthermore, the parameters estimated from the study by Muhle et al. Time (days) Figure 6. Experimental observations (symbols) on the elimination of carbon black from the lungs of F344 male rats in a study by Strom et al. (6). Solid lines represent model predictions. (The Michaelis-Menten-like clearance parameters for the model were estimated from the elimination phase of the same study.) accumulation of a range of burdens in male F344 rats exposed at about 7 mg/m3 for 7-41 days. Results are shown in Figure 6, using values of kmaX and mI/2 that were estimated from the same study. The overall fit of the model was good, despite some underestimation of lung burdens at early stages (<60 days) of the experiment in which the animals were exposed for 7 days.
Diesel exhaust particle. Three studies involved inhalation of DEP: Griffis et al. Exposure time (months) Figure 8. Accumulation of diesel exhaust particulate in F344 rats in a chronic study by Wolff et al. (4). Solid lines represent model predictions; symbols represent experimental observations. [The Michaelis-Menten-like clearance parameters for the model were estimated from the elimination phase of the study by Griffis et al. (31).] compared favorably with the data. The parameters kmax and ml1/2, which had been estimated from the elimination phase of the study by Griffis et al. (31), were valid in predicting burdens in both the accumulation and elimination phases of all studies. However, the model tended to underestimate burdens at later stages of the experiments (1 year after termination of exposure).

Discussion
As noted in the introduction, investigators have long sought to develop an accurate model for the retention of insoluble particles in the lung. Indeed, many retention models have been published for the rat lung which, as shown in Table 3, contain varying numbers of compartments (1-7) and clearance-related parameters (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13). As computational power and speed have improved, complex physiologically based models have become increasingly popular because they Environmental Health Perspectives * Volume 105, Number  ance-related parameters, kma and m1/2* We previously showed that kmax and mi1/2 are easily estimated and evaluated by applying simple linear regression techniques to modest amounts of clearance data (27). We regard this as a particular strength given the sparseness of data that can currently be used to assess the fit of any dust-retention model. Despite its simplicity, our model reasonably described the mass-dependent behavior of lung burdens in rats exposed to four dusts (Sb203, PTT, CB, and DEP) over a wide   (22). range of particle sizes and experimental protocols. Thus, we conclude that MM-like kinetics captured the essential features of saturable AM-mediated clearance, which largely governed both the accumulation and elimination of lung burdens in inhalation studies involving insoluble dusts. During the accumulation phase of inhalation experiments, our model relies upon an empirically derived constant (EA) to account for the efficiency of particle deposition in the lung. The ranges of EA that we obtained were generally consistent with published values obtained from rats exposed to a variety of aerosols, including aluminosilicate (4,36,37), DEP (38,39) [note that 66% of lung deposition was assumed to be in the alveolar region (39)], Ga203 (4,40,41), PuO2 (42), U 08 and U02 (43), and an 198Au labeled aerosol (44). We included monodisperse particles, characterized by their aerodynamic diameters (36,37) as well as polydisperse aerosols characterized by their MMADs (4,(38)(39)(40)(41)(42)(43). As shown in Figure (10). Thus, we conclude that the values of EA that we assigned to the various experimental groups should not have unduly influenced the behavior of our model predictions.
The validity of MM-like kinetic models is considered from three perspectives. First, reasonable agreement is to be expected between model predictions and observed lung burdens when the same data are used to estimate the clearance-related parameters and also to evaluate goodness of fit (as for Sb2O3 in Fig. 1, for PTT in Fig. 3,4, for CB in Fig. 6, and for DEP in Fig. 7). Second, all of the pairs of kmax and mI/2 that were used to define clearance rate coefficients for the four dusts were obtained from the elimination phases of inhalation experiments after exposure had been terminated. Yet, we observed excellent agreement between observed lung burdens and model predictions during the accumulation phases of all experiments as well (Fig. 1,5,8,9). Third, more validity can be given to a model in which parameters estimated from one study successfully predict lung burdens from another, as was the case for PTT (Fig. Volume 105, Number 5, May 1997 * Environmental Health Perspectives offer avenues for explaining the behaviors observed. However, in the absence of particularly rich and varied data resources, questions of accuracy tend to plague such highly parameterized models. Thus, when data are sparse, simple empirical models offer advantages for the accurate prediction of kinetic behavior, even if the reasons for such behavior remain elusive. Certainly our model (defined by Equations 2 and 3) falls into this latter category because it contains only a single compartment and two clear-5) and DEP (Fig. 8,9). Such agreement of model predictions across studies for two different dusts strengthens the notion that MM-like clearance behavior may be generally applicable to insoluble particles.
We suspect that the parsimony achieved by applying MM-like kinetics to alveolar clearance relates to its underlying biological plausibility. Table 4 compares the main characteristics of traditional MM enzyme kinetics to the analogous process that we postulate to be operative in the lung. The key feature leading to the mathematical forms of both relationships rests in the inherent feedback systems, which are related to dissociation of enzyme-substrate complex on the one hand and to redistribution of phagocytized particles (associated with the deaths of AMs) on the other. Although the concept of particle redistribution has long been recognized and has often been included in retention models (10,20,(22)(23)(24)(25), we are apparently the first to recognize the parallel between redistribution and dissociation of the enzyme-substrate process and, thereby, to justify the mathematical simplicity of MM kinetics in this context.
The MM-like kinetics that underlie our model also lend substance to the notion of lung overload, where it is postulated that the clearance rate slows only after some critical threshold burden has been exceeded. From Equation 1 it is easily shown that when m << ml/2, linear (or first-order) kinetics prevail and the alveolar clearance rate coefficient k is approximately equal to kmax. Thus, the threshold burden is defined by the condition where m << m112.
Previously (27), we estimated threshold burdens of 0.11, 0.16, 0.40, and 0.46 mg for Sb2O3, PTT, DEP, and polyvinyl chloride powder, respectively (whose values of mI/2 were estimated to be 0.69, 0.97, 2.49, and 2.90 mg, respectively). This implies that lung overload occurs when the lung burden exceeds about 16% of m1/2. The good fits of all experimental data to our model underscore the point that dust overloading is probably a manifestation of nonlinear kinetics of the MM type according to the relationship given in Equation 1. Our model can also be interpreted in the context of particle sequestration, where it has been postulated that clearance stops completely for some portion of the lung burden (15,16). Yu et al. (35) argued that particle sequestration can be explained in practice by the effect of slowed alveolar clearance rather than a complete breakdown of the clearance process, given that animals have a finite lifetime. Our model offers a similar explanation in the sense that alveolar clearance, under Equation 1, is mass-dependent and, in the extreme, will  Articles * Yu and Rappaport lead to radically diminished clearance rate coefficients. Thus, although conceptual differences remain in defining particle sequestration, in practice there is not much dif& ference between our approach and that of Yu et al. (35). We also note that the most consistent bias observed between predicted and observed burdens under our model involved the underestimation of burdens of DEP ( Fig. 9-1 1) during late stages of the experiments (times greater than about 1 year), where sequestration would be expected to play a more prominent role. We conclude that MM-like clearance kinetics provide a reasonable basis for a general model describing the retention of insoluble particles in the lungs of F344 rats. We encourage others to investigate the suitability of our model for describing retention of other aerosols in the rat lung as well as for describing the behavior of insoluble dusts deposited in the lungs of other species, including humans.