A mathematical model of fiber carcinogenicity and fibrosis in inhalation and intraperitoneal experiments in rats


A hypothesis is presented that predicts the incidence of tumors and fibrosis in rats exposed to various types of rapidly dissolving fibers in an inhalation study or in an intraperitoneal (IP) injection experiment, for which the response to durable fibers has been determined. The model takes into account the fiber diameter and the dissolution rate of fibers longer than 20 µm in the lung, and it predicts the measured tumor and fibrosis incidence to within approximately the precision of the measurements.

The basic concept of the model is that a rapidly dissolving long fiber has the same response in an animal bioassay as a much smaller dose of a durable fiber. Long, durable fibers are considered to have special significance since no effective mechanism is known by which these fibers may be removed. In particular, the hypothesis is that the effective dose of a dissolving long fiber scales as the residence time of that fiber in the extracellular fluid. For example, a certain dose of a fiber that dissolves in one year acts like half that dose of a fiber that requires two years to dissolve. The residence time of a fiber is estimated directly from the average fiber diameter, its density, and the fiber dissolution rate as measured in simulated lung fluid at neutral pH.

The incidence of fibrosis in a recent series of chronic inhalation tests at the Research and Consulting Company (RCC) in Geneva, Switzerlend, is predicted well by the mathematical model. The observed lung tumor rates in these studies are consistent with this model. The model also predicts the incidence of mesothelioma in the IP model of Pott and colleagues.

The model allows one to predict, for an inhalation or IP experiment, what residence time and dissolution rate is required for an acceptably small tumorogenic or fibrotic response to a given fiber dose. For an inhalation test in rats at the maximum tolerated dose (MTD), such as the ones completed at RCC, the model suggests that less than 10% incidence of fibrosis would be obtained at the maximum tolerated dose of 1 µm diameter fibers if the dissolution rate were greater than 80 ng/cm2/hr. The dissolution rate that would give no detectable lung tumors in such an inhalation test in rats is much smaller. Thus a fiber with a dissolution rate of 100 ng/cm2/hr has an insignificant chance of producing either fibrosis or tumors by inhalation in rats even at the maximum tolerated dose used in the RCC study.


There has been intense interest recently in determining what properties of certain fibers are responsible for producing respiratory disease in humans and in laboratory animals. The disease potential has long been linked to the dose, dimension, and durability of the fibers (Pott and Friedrichs, 1972; Stanton and Wrench, 1972). The various types of asbestos fibers, which are associated with mesothelioma, lung cancer, and fibrosis when inhaled, differ in each of these properties from insulation wool glass fibers, for example, which are not associated with these diseases when inhaled. Certainly the airborne fiber concentrations of asbestos to which workers were exposed many years ago (Liddell, 1991), were hundreds or thousands of times higher than that experienced by workers manufacturing or installing insulation glass fibers (Hesterberg and Hart, 1994; Jacob et al., 1992; 1993). The diameters of asbestos and insulation glass fibers are also markedly different, with asbestos fibers typically 0.1 to 0.2 µm or thinner, whereas airborne glass fibers typically average around 1 µm (Hesterberg and Hart, 1994), which is considerably less than the average in the product itself (Christensen et al., 1993).

Another property in which asbestos fibers and insulation wool glass fibers differ greatly, and the subject of this paper, is the dissolution rate of the fibers in the extracellular lung fluid. The dissolution rate of fibers can be measured in vitro in simulated lung fluid and the physical chemistry of the process is reasonably well understood (Potter and Mattson, 1991; Leineweber, 1984; Scholze, 1988). The dissolution rate constants measured in vitro depend significantly on the measurement conditions, and it is important to measure them in a way that is relevant to dissolution in the lung (Mattson, 1994a; 1994b). Therefore the in-vitro measurement methods used to determine the dissolution parameters given in this paper were tested by comparing the in-vitro results to glass fibers recovered from rat lungs at various times following intratracheal instillation. It was found that long fibers, for example, those 20 µm or longer, dissolve at the same rate in the lung as in vitro (Eastes et al., 1995). It was further found that the rate of disappearance of long (> 20 µm) glass, rock, and slag wool fibers was predicted by the dissolution rate measured in vitro by these methods (Eastes and Hadley, 1995). Short glass and asbestos fibers, on the other hand, do not appear to dissolve in the lung, but are cleared efficiently by macrophage-mediated physical action after inhalation (Eastes and Hadley, 1995).

It has long been believed that long fibers are the most biologically active, probably because their aerodynamic behavior allows fibers with much greater lengths than a macrophage can totally ingest, to enter the lower lung (Timbrell, 1976). Efficient macrophage-mediated clearance of these long fibers therefore cannot occur. If the long fibers are durable, they will accumulate if the exposure continues (Davis, 1994). Persistent long fibers can then result in incomplete or "frustrated" phagocytosis with leakage of macrophage contents, leading to chronic inflammation (Holt, 1987). The ability of a fiber to dissolve rapidly in the extracellular fluid would appear therefore to be an important means of reducing the dose of the most biologically active fibers to the lung parenchyma or to the pleura (Boffetta, 1994).

The model described here to predict the development of fibrosis or tumors is based on the hypothesis that a rapidly dissolving fiber acts like a much smaller dose of a durable fiber. A durable fiber is here considered to be one that does not dissolve during the lifetime of the species of interest, or about two years for the rat. For example, a given dose of a fiber that dissolves in one year is assumed to act like half that dose of a fiber that requires two years or more to dissolve.

The next section describes more precisely the nature of this mathematical model and the following one presents tests of its accuracy for an inhalation study and for intraperitoneal experiments in rats. The last section discusses a number of implications of the model.


When animals are exposed to various doses of durable fibers, it is found that the tumor incidence is a function of the dose of the appropriate size of fiber (Stanton and Wrench, 1972). Accordingly, the starting point for this analysis is the assumption that the incidence of disease following administration of durable fibers is given by a function f(X), where X is the dose according to some convention. For example, a logit form for f(X) has been used to describe the mesothelioma incidence after intraperitoneal (IP) injection in rats (Pott et al., 1990a), as well as both lung tumors and fibrosis following inhalation in rats (Eastes and Hadley, 1994). However, in contrast to these previous treatments, the present work establishes the dose-response function f(X) directly from the observed durable fiber results without the assumption of an analytical functional form.

The hypothesis proposed here is that the dose-response relation for durable fibers, f(X), holds also for rapidly dissolving fibers, if the dose is adjusted by the fraction of the species lifetime that these long fibers remain in the lung. That is, the disease incidence f for dissolving fibers becomes

where the adjustment factor is defined as
the ratio of tD, the time a fiber of diameter D remains in the lung, to the lifetime of the animal tL. For the rat studies used here to test this hypothesis, the lifetime is taken to be two years.

The lifetime of a fiber tD can be estimated from the rate law for fiber dissolution found in vitro and confirmed for fibers 20 µm or longer in vivo as well (Eastes et al., 1995). It has been found (Leineweber, 1984; Mattson, 1994a; Potter and Mattson, 1991; Scholze, 1988) that a wide variety of vitreous fibers dissolving in simulated lung fluid at nearly neutral pH, at a flow rate high enough to simulate the rapid removal of dissolution products in the lung, decrease in diameter at a constant rate given by

where kdis is the dissolution rate constant and is the fiber density. It follows then that a fiber with initial diameter D decreases to zero diameter in a time
If the administered fibers do not all have the same diameter, then it is the average lifetime tD that is required in Eq. (4). Since the fiber lifetime is proportional to the fiber diameter, the average fiber diameter is used in Eq. (4). Thus the diameter D should be understood to be the number weighted, arithmetic mean fiber diameter for a collection of fibers with a distribution of diameters.

Equation (1) along with Eqs. (2) and (4) provide a mathematical model to predict the incidence of disease at a given dose of long fibers X that dissolve at the rate given by the dissolution rate constant kdis. The particular function f in Eq. (1) depends on the disease (fibrosis, lung cancer, or mesothelioma) and on the route of administration (inhalation or injection) in the particular bioassay being used, as well as on the units in which the dose is expressed.

One feature of this model is that it has no adjustable parameters. Once the response f(X) to asbestos or to other durable fibers is known for a particular bioassay, then the response to any other fiber type is predicted from the dissolution rate constant, a property of the fiber composition measured in vitro. The implication of this model is that a lifetime exposure to a rapidly dissolving fiber acts like less than a lifetime exposure to a durable fiber. Since such less than lifetime exposures to durable fibers were also included in the inhalation studies, this feature provides a convenient way to test the model.

It remains to establish the meaning of the incidence f and the dose X. For lung cancer or mesothelioma, f is the fraction of animals diagnosed with this disease during the study. For fibrosis, a number of different indicators of the condition could be used. Here f is taken to be the fraction of animals that are found to have at least minimal fibrosis, defined as Wagner Grade 4 or above (McConnell et al., 1984).

The dose X should be the total number of "relevant" fibers to which the animal is exposed throughout the study. For a chronic inhalation study, it is considered here to be the total number of long fibers that are inhaled. This dose is not the same as the fiber lung burden at any time because the lung burden is the equilibrium value between the number of fibers continually inhaled and those continually removed by macrophage-mediated physical clearance, or by dissolution. Therefore, the total number of respirable, long fibers in the aerosol, multiplied by the average air volume inhaled by the animal while exposed to the fibers, will be used here to characterize the total dose X. Respirable fibers are those that can be inhaled into the deep lung of the rat and are generally those less than 1 µm in diameter (Bernstein et al., 1995). The length of fibers that is relevant to lung disease is less clear, but it likely that fibers short enough to be engulfed and transported by alveolar macrophages are not associated with lung disease in the absense of significant overload (Davis, 1994). For the sake of definiteness, fibers longer than 20 µm were chosen in what follows as a surrogate for long fibers (Kuschner, 1987). Also, since the length distributions of the synthetic vitreous fibers in these studies were similar, the results were also similar whether 5 µm, 10 µm, or 20 µm was chosen as the minimum length of a long fiber. Thus the definition of relevant fibers used here is the number of aerosol fibers less than 1 µm in diameter and simultaneously greater than 20 µm in length.

For the intraperitoneal injection studies, the concept of dose is uncertain. It is clear that the injection of foreign materials into the sterile serosal cavity represents a very nonphysiological exposure. Not only are large quantities of materials injected as a bolus, but there is no anatomical filtration of fiber sizes, such as occurs after inhalation. Also, while there appears to be a biological basis for the unique aspects of long fibers in the lung, it is not clear if this is also relevant to the peritoneal or pleural cavities. For example, recent data have indicated that primarily the short fibers translocate to the pleural cavity following inhalation (Gelzleichter et al., 1995). Given this uncertainty, the model was tested simply by using the dose reported by the study authors, which was essentially fibers less than 2 µm in diameter and greater than 5 µm long (Pott et al., 1990a).


The hypothesis described in the previous section was tested by applying it to three different endpoints in rats: fibrosis and lung cancer after inhalation and mesothelioma following intraperitoneal injection. The inhalation studies were sponsored by the Thermal Insulation Manufacturers Association (TIMA) and were performed by the Research and Consulting Company (RCC) in Geneva (Hesterberg et al., 1993; Mast et al., 1993; McConnell et al., 1994). The intraperitoneal injection studies are those of Pott and coworkers (Pott et al., 1990a; 1990b; Pott, 1991).

In the RCC inhalation studies, separate groups of male Fischer 344 rats were exposed to different types of synthetic vitreous insulation wool fibers at different concentrations, to chrysotile, and to crocidolite asbestos for 6 hours per day, 5 days per week. Typically 3 to 6 rats were sacrificed at a time, at intervals from 3 months to over 2 years. The flow-through, nose only inhalation apparatus has been previously described (Bernstein et al., 1994).

The fiber types studied at RCC, against which this model was tested, along with their properties, are listed in Table 1. The dissolution constants kdis given in Table 1 were measured in vitro (Mattson, 1992; 1994b; Potter and Mattson, 1991), except for the asbestos and RCF 1, which are measurements of fibers with compositions similar to those used at RCC. The average diameters of the fibers in Table 1 were taken from published reports (Hesterberg et al., 1993; Mast et al., 1993; McConnell et al., 1994). The densities of the fibers were measured (Christensen et al., 1993; Mattson, 1992).