Walter Eastes, Russell M. Potter, and John G. Hadley
Owens Corning, Science and Technology Center, Granville, Ohio, USA
Originally published:
Inhalation Toxicology 12, 269280 (2000)
Copyright © 2000 Taylor & Francis
ABSTRACT
A method is presented for calculating the dissolution rate
constant of a borosilicate glass fiber in the lung, as measured
in vitro, from the oxide composition in weight percent. It is
based upon expressing the logarithm of the dissolution rate as
a linear function of the composition. It was found that the calculated
dissolution rate constant agreed with the measured value within
the variation of the measured data in a set of compositions in
which the dissolution rate constant ranged over a factor of 100.
The method was shown to provide a reasonable estimate of dissolution
over a considerably wider range of composition than what was used
to determine the parameters, such as a set of data in which the
dissolution rate constant varied over a factor of 100,000. The
dissolution rate constant may be used to estimate whether disease
would ensue following animal inhalation or intraperitoneal studies.
INTRODUCTION
The dissolution rate constant of a fiber as measured in vitro
is an important predictor of the biological effects of the fiber
(Eastes and Hadley, 1996).
This paper exhibits a reasonably accurate
and efficient method for computing this dissolution rate constant
directly from the inorganic borosilicate glass fiber oxide composition.
The dissolution rate constant k_{dis} expresses the rate at
which an extracellular fiber, especially one longer than 10 to
20 µm or too long to be effectively enveloped by alveolar
macrophages, dissolves in the mammalian lung and is thereby eliminated.
The complete elimination of the long fiber may take place by total
dissolution and removal of the dissolution products to the blood
stream, by breakage of the partially leached fiber and subsequent
macrophage mediated clearance of the fragments, or by a combination
of these mechanisms. The exact extent and contribution of each
of these mechanisms is not so important as the fact that they
both occur because of dissolution, which is captured in the
k_{dis} parameter.
It has been shown that glass fibers decrease in diameter after
intratracheal instillation in rats at the rate predicted by k_{dis}
for the same fiber composition measured in vitro
(Eastes et al., 1995).
It has also been shown that k_{dis} predicts
the biopersistence, or the disappearance of long fibers from the
rats' lungs after short term inhalation
(Bernstein et al., 1996;
Eastes and Hadley, 1995).
Finally, it was shown that k_{dis}
can predict the incidence of fibrosis and lung tumors after chronic
inhalation in rats and tumors following intraperitoneal injection
in rats through a mathematical model that also takes into account
the dose of long fibers
(Eastes and Hadley, 1994;
Eastes and Hadley, 1996).
Also, the close relationship between biopersistence and
toxicity has been demonstrated experimentally
(Bernstein, 1998).
It is clear, therefore, that an accurate estimate of k_{dis}
from composition data alone allows the prediction both of long
fiber clearance and of chronic effects associated with the fibers.
Such information is of great practical value to manufacturers
and to the regulatory community.
The dissolution rate constant k_{dis} has been measured for
a large number of different borosilicate glass fiber compositions
(Förster, 1982;
Klingholz and Steinkopf, 1982;
Leineweber, 1982;
Scholze and Conradt, 1987;
Bauer et al., 1988;
Potter and Mattson, 1991;
Christensen et al., 1994;
De Meringo et al., 1994;
Mattson, 1994a;
Thélohan and De Meringo, 1994)
and a
standardized protocol is available for performing these measurements
(Bauer et al., 1997).
A number of different protocols were used
in the references just mentioned, and they are not always quantitatively
comparable, but the results have several features in common. The
measured values range over many orders of magnitude: from below
1 ng/cm^{2}/hr for most amphibole asbestos fibers,
through single digits in these units for some refractory ceramic
fibers and thin E glass fibers that have been associated with
fibrosis and lung tumors in rats and mesothelioma in hamsters.
Double digit values have been measured for a conventional rock
wool that was associated with fibrosis but not lung tumors in
rats, triple digits (greater than 100 ng/cm^{2}/hr)
in almost all conventional insulation glass wool, to four
or five digits for some new fibers. No fiber with a k_{dis}
above 100 ng/cm^{2}/hr has produced fibrosis
or tumors in animal inhalation studies.
The ability of k_{dis} to predict biological effects reliably
depends critically on measuring it in a way that is relevant to
the dissolution that occurs for long fibers in the lung. Not only
is it important that the composition and pH of the simulated lung
fluid be the same as in the lung in those components that affect
fiber dissolution, but also it is important that the dissolution
products be removed as quickly in vitro as they appear to be in
the lung. Otherwise the dissolution products may build up in vitro
to the point at which they affect the measured dissolution rate
in ways that do not happen in the lung. To avoid this problem,
the typical invitro dissolution measurement apparatus causes
the simulated lung fluid to flow through a loose assembly of fibers
at a rate high enough to remove the dissolution products so that
the pH and measured dissolution rate constant is not affected
significantly
(Mattson, 1994b).
In this way, k_{dis} values
are obtained that predict reasonably well the dissolution of long
fibers in the lung.
The dissolution rate of a fiber is proportional to its surface
area and is thus a zero order reaction
(Scholze, 1988).
The parameter
k_{dis} in this reaction is a property of the material in
the fiber, and does not depend on its size or shape. Thus k_{dis}
depends on the fiber composition, conventionally expressed
as the weight percent of oxides. One may expect, therefore, to
be able to calculate k_{dis} to a reasonable approximation
directly from the oxide composition. (The k_{dis} also depends,
but to a much lesser extent for most fibers, on the glass structure
that is affected by the fiber cooling rate and thus by the method
of fiberizing and the fiber diameter
(Potter and Mattson, 1991;
Scholze, 1988).
These effects are much smaller than the influence
of composition on k_{dis}; usually they are so small as to
be unobservable
(Scholze, 1988);
and they may be ingnored for the present purposes.)
The next section describes the theory for calculating k_{dis}
from composition, and the following one shows the application
to a large set of compositions with measured dissolution rate.
In the last section, this method is contrasted with other ways
of estimating fiber biodurability.
THEORY
The dissolution of glass fibers in nearly neutral water solutions occurs by the reaction of a water molecule or some part thereof with a part of the glass structure containing a cation, here called the "dissolving species", replacing it with the water species. Since the dissolution products in the lung appear to be rapidly removed from the vicinity of the fiber, the reaction is most likely not diffusion limited, and the reaction with water is the rate limiting step. Although it does not apply strictly because it models isolated molecules, chemical reaction rate theory (Eyring et al., 1944) is useful here if it is applied to the species that are dissolving, neglecting the lesser influence of more distant atoms. The reaction rate is given by
(1) 
where k_{dis} is the reaction rate, A and
are
constants for the given reaction, k is the Boltzmann constant,
and T is the absolute temperature, here fixed at body temperature,
and ln is the natural logarithm. Equation (1) is the logarithmic
form of the familiar Arrhenius equation. The constant A is related
to the frequency of collisions among the reacting species, which
would be approximately the same for every dissolving species in
the glass fiber. The constant is the free energy required
to create the transition state involving both the dissolving species
and the water species, starting from the original glass in water.
The thus depends on the dissolving species, with the
average given by
(2) 
where is the for dissolving species
i present at mole fraction X_{i}, and the sum is over all
the dissolving species in the glass structure.
Thus it is seen that ln k_{dis} is linear in the mole fractions
X_{i} and therefore so is log k_{dis}, the logarithm to
the base 10. Equations (1) and (2) may be combined and new constants
P_{i} defined merely for convenience so that
(3) 
where W_{i} is the weight percent of oxide i and the sum
is over all n oxides including SiO_{2}. Equation
(3) invokes the approximation that the oxide weight percents W_{i}
are proportional to the mole fractions of the corresponding
dissolving species, which could be justified only over relatively
narrow ranges of composition. Indeed, the nature of the dissolving
species might well be different for widely different glass compositions.
Thus rock wool fiber compositions, for example, require a different
set of coefficients P_{i} than the ones for borosilicate glass
fibers exhibited in this paper.
In order to compute k_{dis} for a glass fiber composition
given by the oxide weight percent W_{i}, for i = 1, 2, ...
n, the coefficients P_{i} in Eq. (3) are all that are needed.
These may be determined most easily by fitting measured k_{dis}
values to the W_{i} for a range of compositions. The
best fit P_{i} in the sense of minimum chi squared is desired,
but estimates of their precision and the quality of the fit are
also important.
An important practical consideration in applying (3) to a variety
of glass fiber compositions is to deal with extra oxides, those
present in small quantities for which corresponding coefficients
P_{i} are not available. It is a poor approximation to simply
ignore these extra oxides, because the weight percents W_{i}
for which coefficients are available may sum to significantly
less than 100% and the predicted k_{dis} will be biased low.
A better method, and the one used here, is to normalize each composition
to 100% in those oxides for which coefficients are available.
This method is equivalent to treating the extra oxides as a weighted
average of the oxides for which coefficients are known.
It is interesting and useful that the theory developed here, although
tacitly assuming that all components of the glass fiber dissolve
at the same rate, works well in practice even in many cases in
which some components leach out of the fibers much more rapidly
than others and leached layers form on the fibers. This behavior
has been observed both in vitro and in vivo. The fact that a single
dissolution rate constant can fit the observed dissolution data
reasonably well even in these cases suggests that these leached
layers are not dense enough to hinder significantly the movement
of the dissolving species away from the fiber or significantly
change the environment in which the fibers dissolve.
RESULTS
An excellent data set of measured k_{dis} with the composition
of each fiber is available
(Potter and Mattson, 1991;
Mattson, 1994a)
for over 90 fiber compositions obtained by the same protocol.
A tab delimited text file containing these data are available by
clicking here.
The range of composition covered by these data is shown for the
six major oxides in Figure 1 and for six minor oxides in Figure
2. In these diagrams, there is one panel for each oxide, and it
has a small red square marking the weight percent of that oxide in
each fiber. It is seen in Figures 1 and 2 that the oxides are
distributed somewhat unevenly. For some oxides like ZnO
the data are so sparse that it would not be possible to obtain
a reliable coefficient for it. Other oxides, like TiO_{2}
and Fe_{2}O_{3} vary over such a small
range in weight percent that the measured k_{dis} would not
be expected to be sensitive to these variations. The data for
K_{2}O have both of these problems, and a coefficient
was not determined for it, either. For the remaining eight oxides,
it is possible to choose a range in each oxide that the available
data cover reasonably well. The other oxides, for which the data
are not sufficient to determine reliable coefficients, are limited
to a small enough weight percent that the oxide is not expected
to impact the dissolution rate significantly. These ranges are
indicated by horizontal lines in the middle of each panel. There
are 62 fiber compositions in the data set that fall within the
ranges indicated by the horizontal lines in Figures 1 and 2, and
these are fit to Eq. (3) to determine the coefficients P_{i}.
Figure 1. Composition range of the available measured dissolution rate data in six major oxides in weight percent. Each small red square shows the weight percent of the corresponding oxide in one measured glass composition. The data displayed here and used throughout this paper, including the composition in all oxides and the measured dissolution rate, are available as a tab delimited text file by clicking here.
Figure 2. Composition range of the available measured dissolution rate data in six minor oxides in weight percent. Each small red square shows the weight percent of the corresponding oxide in one measured glass composition. The data displayed here and used throughout this paper, including the composition in all oxides and the measured dissolution rate, are available as a tab delimited text file by clicking here.
Not only the measured dissolution rate for the compositions but
also an estimate of the standard error of the measured log k_{dis}
is needed in order to minimize chi squared. Since 13
of the compositions in the data set were measured more than once,
the pooled standard deviation of these replicate measurements
was used as an estimate of the standard error of every datum.
The pooled standard deviation of log k_{dis} was 0.1158.
The coefficients P_{i} determined by minimizing chi squared
using the method of singular value decomposition
(Press et al., 1992)
along with the standard error of each coefficient and various
statistics are shown in Table 1. It may be seen that the coefficients
for CaO, MgO, Na_{2}O,
B_{2}O_{3}, and BaO are similar in magnitude
to one another, taking into account the standard error of each,
and the coefficient of Al_{2}O_{3} is approximately
twice as large and negative. These results mean that
Al_{2}O_{3} "strongly decreases dissolution rate by a
factor approximately two times greater than those of the [former]
group"
(Potter and Mattson, 1991),
which was noted when these
measurements were published originally.
Oxide  Coefficient  Standard Error 
SiO_{2}  0.01198  0.00285 
Al_{2}O_{3}  0.21410  0.01102 
CaO  0.10806  0.01119 
MgO  0.13761  0.01262 
Na_{2}O  0.09386  0.00867 
B_{2}O_{3}  0.14669  0.00908 
BaO  0.06921  0.03095 
F  0.11867  0.06134 
R^{2}  0.96 

Significance  0.97 

Degrees of Freedom  54 

The statistic R^{2} is the fraction of the variance of the
data that is accounted for by the prediction. The fact that R^{2}
is large here, over 95%, indicates that the simple Eq.
(3) captures nearly all of the important effects on k_{dis}
for this range of fiber compositions. The degrees of freedom is
the number of data used for the fit minus the number of coefficients
determined by the fit, here 8.
The significance of the chi squared statistic is the probablity
that the extent to which the measured data deviate from Eq. (3)
could be the result of random variations with the estimated standard
error. The fact that the significance is over 95% here indicates
that Eq. (3) explains the measured data to well within the variations
in the measured data themselves.
In addition to the statistical measures, another way to assess
the quality of the prediction is to plot the calculated value
against the measured value of k_{dis} as in Figure 3. It is
seen here that the calculated value agrees with the measured one
reasonably well over the entire factor of 100 in dissolution rate
spanned by the data. Furthermore, Figure 3 does not show any tendency
for the predicted k_{dis} to deviate more at the extremes
of k_{dis} represented in the data.
Figure 3. Dissolution rate constant k_{dis} calculated
by Eq. (3) compared to the measured value for the data used to
determine the coefficients P_{i}.
The fact that the deviation of the predicted from the measured
k_{dis} is fairly uniform over a wide range of k_{dis}
and composition and the fact the Eq. (3) is based on the chemistry
of the dissolution process, suggests that this equation with the
coefficients just determined might provide a reasonable approximation
somewhat outside the range over which the coefficients were fit.
That this is so is seen in Figure 4 in which the calculated k_{dis}
is plotted against the measured values for all of the
compositions in the original data. In Figure 4, the compositions
that were used in the fit to determine the P_{i} are shown
as small squares just as in Figure 3, whereas the other measurements,
not used in the fit, are shown as X. It may be seen in Figure
4 that the coefficients of Table 1 provide a reasonable estimate
of k_{dis} over a factor of 100,000, much wider than that
of the data used to determine the coefficients. This fact does
not necessarily imply that one could develop coefficients that
cover a wider range of composition equally well by using all of
the data. When that is done, the accuracy of the prediction over
the restricted data (the small squares in Figures 3 and 4) becomes
worse.
Figure 4. Dissolution rate constant k_{dis} calculated
by Eq. (3) compared to the measured value for the complete set
of measured values. The data used to determine the coefficients
are denoted by small red squares,
whereas the other data, covering
a wider range of composition and k_{dis}, are denoted by a
blue X.
The agreement is not unreasonable in most cases, even over the
wider range.
DISCUSSION
The method of calculating glass properties by expressing them
as linear functions of weight or mole fraction with coefficients
determined by fitting to measurements has been used in glass technology
for over a century (Huggins and Sun, 1943). The theory outlined
here suggests that, for fiber dissolution, the logarithm of k_{dis}
is the appropriate quantity to approximate as a linear
function of composition over a fairly wide range of composition,
and this possibility was verified in practice.
Another approximation related to k_{dis} has appeared in European
regulatory affairs. The "Carcinogenicity Index" K_{I},
proposed in the German TRGS 905
(BMA, 1995),
is given as
(5) 
with C_{i} = 1 for Na_{2}O, K_{2}O,
CaO, MgO,
B_{2}O_{3}, and BaO, and C_{i} = 2
for Al_{2}O_{3}. It may be seen that these coefficients
C_{i} are approximately proportional to those in Table 1,
although they are not the same since Eq. (4) is not normalized
to all oxides and does not contain SiO_{2} The values
of the coefficients in Eq. (4) are seen to be identical to the
statement of Potter and Mattson, 1991, noted previously.
It is seen by comparing Eq. (4) with Eq. (3) that log k_{dis}
should be approximately linear in K_{I}. That this is so is
shown in Figure 5, which plots the measured k_{dis} on a logarithmic
scale against K_{I} for the set of borosilicate glass fibers
used previously. The R^{2} statistic for the linear fit of
K_{I} to log k_{dis}, shown in Table 2, is large, indicating
a good fit. For these compositions, K_{I} is a reasonably
good predictor of k_{dis}, which is why it is a good predictor
of tumors following IP injection, as proposed in TRGS 905.
Equation (3) proposed here is a better predictor of k_{dis}
than K_{I}, over the same range of composition, however, as
seen by comparing Figure 5 with Figure 3. This comparison may
be made more precise by use of the standard error of the fit,
shown in Table 2 for the linear fit of K_{I} to
log k_{dis}
and for the calculated k_{dis} of Eq. (3). The improvement
in the standard error for the calculated k_{dis} by Eq. (3)
over that by K_{I} is found to be significant at the 1% level.
Statistic  Calculated k_{dis}  K_{I} 
R^{2}  0.96  0.91 
Standard error  0.095  0.132 
Figure 5. "Carcinogenicity Index" K_{I} plotted
against the measured dissolution rate constant k_{dis} for
a series of glass compositions. The correlation is not so good
as that of the method of this paper, shown for the same glass
compositions in Figure 3.
It is seen then that Eq. (3), along with the coefficients given
in Table 1, provide a reasonably accurate method for estimating
the dissolution rate constant k_{dis} for a borosilicate glass
fiber if its composition in oxide weight percent is known.
A computer program that runs in a web page is available to perform this
calculation for any given borosilicate glass composition.
This program may be started by clicking here.
The range of composition over which the compute estimate is accurate is shown
in Figures 1 and 2.
From the dissolution rate constant, it is
then feasible to estimate whether disease would ensue following
animal inhalation or intraperitoneal studies, as shown previously
(Eastes and Hadley, 1996).
The ability to determine k_{dis} simply from the composition
of a glass fiber should be useful both to fiber manufacturers
and to researchers who are interested in the dissolution rate
of glass fibers. Additionally, it provides a tool by which interested
regulatory bodies may monitor the dissolution rate of insulation
glass wools simply by monitoring their composition without expensive
and timeconsuming animal tests.
So far, this work has focused exclusively on k_{dis} as the
measure of dissolution rate in vitro and in vivo because it is
a conventional parameter. However, dissolution rate, especially
when expressed in the conventional units of
ng/cm^{2}/hr,
does not relate in any intuitive way to the clearance
of fibers from the lung. To make the connection between dissolution
and fiber removal more obvious, the authors recommend that the
parameter dissolution time t_{dis} in units of
days/µm be used instead of k_{dis}. The relation
between these two measures of dissolution
(Eastes and Hadley, 1996) is
(6) 
where is the density of the fiber. In physical terms, t_{dis}
in days/µm is the time required to dissolve a 1
µm diameter fiber. It would be the time to eliminate this
fiber from the lung if dissolution were the only process operating
on the fiber. Of course there are other mechanisms besides dissolution
that serve to remove fibers, and dissolution mostly affects the
long, extracellular fibers in any case. For these reasons, t_{dis}
cannot be equated to the clearance time, but rather indicates
the time for one of the clearance mechanisms, long fiber dissolution,
to happen. The parameter t_{dis} should not be confused with
the half life often used to describe particle clearance. The t_{dis}
is appropriate for a zero order fiber dissolution reaction,
whereas a half life is appropriate for a first order clearance
mechanism
(Scholze, 1988).
A comparison of t_{dis} with the equivalent value of k_{dis}
is given in Table 3 for a typical glass fiber density. One
important result of expressing dissolution as t_{dis} rather
than k_{dis} is that it illustrates the unlikely biological
relevance for inhaled fibers of increasing k_{dis} much above
100 ng/cm^{2}/hr. At this rate, long fibers
would undergo complete dissolution and hence removal more rapidly
than macrophagemediated clearance generally removes inhaled particles
or short fibers. Indeed, Table 3 shows that many of the large
k_{dis} values exhibited in Figures 3 and 4 have dissolution
times less than a few days, which is unlikely to have a biological
impact in the lung.
t_{dis} [days/µm]  k_{dis} [ng/cm^{2}/hr] 
5  1000 
10  500 
25  200 
50  100 
100  50 
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