Mechanistic Models Fit to ED₀₀₁ Data on >40,000 Trout Exposed to Dibenzo[A,L]pyrene Indicate Mutations Do Not Drive Increased Tumor Risk

Abstract

ED₀₀₁-study data on increased liver and stomach tumor risks in >40,000 trout fed dibenzo[a,l]pyrene (DBP), one of the most potently mutagenic chemical carcinogens known, provide the greatest low-dose dose-response resolution of any experimentally induced tumor data set to date. Although multistage somatic mutation/clonal-expansion cancer theory predicts that genotoxic carcinogens increase tumor risk in linear no-threshold proportion to dose at low doses, ED₀₀₁ tumor data curiously exhibit substantial low-dose nonlinearity. To explore the role that nongenotoxic mechanisms may have played to yield such nonlinearity, the liver and stomach tumor data sets were each fit by two models that each assume a genotoxic and a nongenotoxic pathway to increased tumor risk: the stochastic 2-stage (MVK) cancer model, and a model implementing the more recent dysregulated adaptive hyperplasia (DAH) theory of tumorigenesis. MVK and DAH fits to the data sets were each excellent, but unexpectedly each MVK fit implies that DBP acts to increase tumor risk by entirely non-mutagenic mechanisms. Given that DBP is such a potent mutagen, the MVK-model fits obtained appear to be biologically implausible, whereas the DAH-model fits reflect that model’s assumption that chemical-induced tumorigenesis typically is driven by elevated repair-cell populations rather than mutations per se.

INTRODUCTION

A default linear no-threshold (LNT) approach to extrapolate increased cancer risks from environmental chemical carcinogen exposures has been applied by the U.S. Environmental Protection Agency (EPA) over the last 45 years, particularly (more recently) for carcinogens known or assumed to have a genotoxic mode of action (MOA). The LNT approach has always been based explicitly on the hypothesis that critical DNA mutations are key, rate-limiting events that can drive tumorigenesis, in accordance with the multistage somatic mutation theory of cancer (U.S. EPA 1976, 1986, 2005, 2007; Anderson et al. 1983). A stochastic two-stage/clonal-expansion version of this model was proposed originally by Armitage and Doll (1957), shortly after Watson and Crick deciphered the structure of DNA and basic elements of genetic machinery were being elucidated. This model posits that a rare, critical somatic mutation can transform a normal “stem” cell (i.e., any cell not terminally differentiated that is capable of dividing) into a premalignant cell that tends to proliferate over time, and another such mutation can transform any premalignant cell into a tumor cell. This two-stage/clonal-expansion model was later refined by Moolgavkar, Venzon, Knudson (MVK) and others (Moolgavkar and Venzon 1979; Moolgavkar and Knudson 1981; Moolgavkar 1983, 1988; Moolgavkar et al. 1988, 1989; Moolgavkar and Luebeck 1990; Leubeck et al. 1999), who obtained the exact algebraic solution that arises if this model is represented as a doubly stochastic Poisson process with parameters that are piecewise-homogeneous over any k time intervals.

According to the somatic-mutation/clonal-expansion theory, agents can increase tumor risk not only by causing mutations, but also (or solely) by increasing the number of premalignant cells that are at risk for undergoing malignant transformation by a dose-induced (or spontaneous background) mutation. It has thus become recognized that LNT cancer risk extrapolation is likely to be conservative even for genotoxic chemical carcinogens, to the extent that they happen also to increase tumor risk by inducing net premalignant-cell proliferation and/or genotoxic endpoints at tumorigenic doses, in the case that each such endpoint also happens to have a substantially nonlinear or threshold dose-response (NRC 1994; Bolt et al. 2004; O’Brien et al. 2006). However, if tumorigenesis requires one or more critical somatic mutations that can be induced in linear proportion to dose (as assumed by the somatic mutation theory of cancer and implementations of it, such as the MVK mode), then by definition a linear, mutagenic-potency coefficient in dose must dominate at very low doses. Consequently, for potently mutagenic chemical carcinogens, an LNT dose-response relationship for increased cancer risk is widely assumed to be both scientifically plausible and reasonably likely.

The ED₀₀₁ (“mega-trout”) ultra-low-dose bioassay generated detailed data on elevated liver and stomach tumor incidence observed in a total of >40,000 trout, eight groups of which were exposed to concentrations ranging from 0 to 225 ppm of dibenzo[a,l]pyrene (DBP) in food for four weeks, followed by normal diet for nine months (Bailey et al. 2009; Williams et al. 2003; 2009). This was and remains the largest single cancer bioassay study ever done, which generated the highest resolution set of tumor dose-response data, of any bioassay ever conducted using any type of experimental animal. The ED₀₀₁ study design took advantage of the fact that trout are a flexible and established model of animal carcinogenesis that can be applied efficiently, using relatively large numbers of individual animals (Dashwood et al. 1999; Reddy et al. 1999a b; Williams et al. 2003, 2009; Benninghoff et al. 2012).

The “model genotoxic” chemical and “tumor initiator” used in the ED₀₀₁ study, DBP, is one of the most potently mutagenic chemical carcinogens known (Higginbotham et al. 1993; Busby et al. 1995; Prahalad et al. 1997; Chakravarti et al. 1995; 2000; Williams et al. 2003; Mahadevan et al. 2005; Plísková et al. 2005; Mourón et al. 2006; DeMarini et al. 2011; Guttenplan et al. 2012). DBP requires metabolic activation by species-specific cytochrome P450 enzymes to generate DNA adducts, mutations, chromosome damage, and tumors (Buters et al. 2002; Schober et al. 2006; Kushman et al. 2007; Lagerqvist et al. 2008; Topinka et al. 2008; Meschini et al. 2010). From additional analyses done on stomach-tissue samples in the ED₀₀₁ study, Bailey et al. (2009) concluded that there was “clear evidence from stomach tumors that DBP-driven mutations in the Ki-ras oncogene were present even at low dose.” In Big Blue mouse cells exposed to diol-epoxide metabolites of DBP for 30 min in vitro, both DBPDNA adducts and cII mutations increased in approximate proportion to exposure concentration (Yoon et al. 2004). Such approximate proportionality was also observed between exposure-induced lacZ mutations and bulky DNA adducts measured in different tissues of groups of 25-week-old male transgenic Muta^TMMouse mice dosed daily with the DBP-related polycyclic aromatic hydrocarbon, benzo(a)pyrene (BaP) (Lemieux et al. 2011). BaP-DNA-adduct levels measured in those mice were highest in glandular stomach and liver, which Lemieux et al. (2011) pointed out were sites of first BaP contact and of primary BaP metabolism, respectively—factors also likely to have applied to the DBP that was administered in food to ED₀₀₁-study trout that exhibited exposure-induced tumors in the same two target tissues.

Because DBP is one of the most potently mutagenic carcinogens known, it is surprising that ED₀₀₁ dose-response data on liver and stomach tumors in DBP-exposed fish were reported to exhibit substantial nonlinearity at low doses (Williams et al. 2003; Bailey et al. 2009). Bailey et al. (2009) concluded that, among nine statistical/regression-type risk models explored, three models (linear probit, quadratic logit, and Ryzin-Rai) that each fit the ED₀₀₁ liver tumor data well predicted increased risks that fell increasingly below default benchmark-dose-type regulatory extrapolation assumptions with decreasing DBP dose, that low-dose tumor response was not predictable from hepatic DBP-DNA adduct biomarkers, and thus that ED₀₀₁ data provided the first experimental demonstration that EPA default LNT assumptions for genotoxic carcinogens are inherently conservative. Specifically, observed log liver-tumor likelihood and corresponding log DBP concentration were found to exhibit a substantially nonlinear (i.e., >1) slope of ∼2.3 that was significantly greater (p = 0.0007) than that of ∼1.3 exhibited between the log of the number of bulky DBP-DNA adducts measured in liver and log DBP concentration (Bailey et al. 2009). However, goodness-of-fit values for these relationships were not reported, mechanistic cancer models were not applied, and specific mechanistic hypotheses consistent with ED₀₀₁ study data were not discussed.

It is possible that, due to DBP-induced cytotoxicity, DBP may act with a dual MOA to increase tumor risk via a combination of mutagenic and nongenotoxic-promotion mechanisms. For example, DBP can potently transform cells in vitro even in the absence of detected DNA adducts (Nesnow et al. 1997; 2000). DBP can also reduce clonogenic survival by triggering cytoxicity and apoptosis, as well as trigger cell proliferation, particularly at DBP or DBP-metabolite concentrations exceeding 10 to 100 ppb in several cell types studied in vitro (Busby et al. 1995; Nesnow et al. 1997; Yoon et al. 2004; Plísková et al. 2005; Schober et al. 2006; Kushman et al. 2007; Topinka et al. 2008) and in livers of trout exposed to DBP in vivo 5 days/week for four weeks (Zipperman 1999). Tumor-response patterns observed in the ED₀₀₁ study may therefore be consistent with predictions made by a biologically based, mechanistic cancer-risk model that accounts explicitly for dual-MOA pathways of tumorigenesis.

The specific aim of the present study was to examine the dual-MOA hypothesis for DBP quantitatively, using the biologically based, mechanistic MVK cancer model that explicitly incorporates separate mutagenic and premalignant-cell-proliferation pathways to increased levels of predicted tumor risk. Specifically, the MVK model was fit to ED₀₀₁ dose-response data on liver and stomach tumor incidence in DBP-exposed trout, and relative impacts associated with best estimates of MVK parameters governing mutation and those governing premalignant-cell proliferation were calculated for each tumor endpoint.

For comparison, the ED₀₀₁ data were also fit to a biologically based, mechanistic cancer risk model that differs fundamentally from the MVK model, by predicting that tumorigenic exposures, including those involving potent mutagens, typically act to increase tumor risk predominantly or virtually exclusively by increasing not mutation rates, but rather the number of cells engaged in adaptive hyperplasia (AH) in response to exposure-induced cytotoxic stress. Specifically, the “dysregulated AH” (DAH) model (Figure 1) assumes that incipient tumors arise when a stem cell that has been recruited into a (normally rare) somatically propagated AH state or program (such as that normally invoked to regenerate or repair injured or infected tissue), also contains a rare DNA mutation that blocks the normal ability of an AH-state cell to terminate AH (Bogen 2013). The DAH model is thus a one-step, state-dependent model of tumorigenesis. Because tissue repair and regeneration normally occur only sporadically throughout life, the DAH model implies that the pool of cells that most readily generate tumors is typically relatively small, but subject to periodic expansion and contraction (Bogen 2013). In contrast, the MVK model assumes that all stem cells in each tissue are susceptible to tumorigenic mutations.

FIGURE 1.

Stress-dependent tumorigenic pathways posited by the “dysregulated adaptive hyperplasia” (DAH) theory (Bogen 2013). Two types of stress (downward green “Sublethal stress” and “Cell killing” arrows) transform normal stem cells (N) to protective (P) or regenerative (R) phenotypes, respectively, epigenetically maintained by corresponding microRNA-expression profiles. A critical somatic mutation m_B or m_M (horizontal black dashed arrows) can transform an N-cell to one that is potentially prebenign (N_B) or potentially premalignant (N_M), respectively, or likewise can transform P- or R-cells to corresponding prebenign (P_B) or premalignant (P_M) proliferative cells. The critical mutation prevents normal transduction (blue “Resolution” arrows) of a tissue-specific signal required to resolve stress-induced P- or R-type adaptive hyperplasia. Conditional on surviving immune surveillance, net proliferation yields slow or more rapid clonal expansion of P_B or P_M cells, respectively. Surviving P_B clones are benign tumors. Similar to mammalian cells in tissue culture, P_M clones inevitably undergo telomere crisis, after which each surviving cell is an incipient malignant tumor cell.

Details concerning the ED₀₀₁ data that were fit, specific mathematical model representations, and fitting procedures are described in Methods. Results obtained are then presented, followed by a discussion that interprets these results and their implications for future related research.

METHODS

Bioassay Data

To explore dose-response down to a targeted 10 excess liver tumors per 10,000 animals (ED₀₀₁), a total of 40,800 Shasta strain rainbow trout fry (Oncorhynchus mykiss), initially at ∼1.5 g body weight, were administered Oregon Test Diet (OTD), or OTD mixed with 0.450, 1.27, 3.57, 10.1, 28.4, 80.0, or 225 ppm DBP for 4 weeks, then were fed control diet for 9 months (Bailey et al. 2009). The incidence of stomach and liver tumors was recorded during exposure, the 9-month post-exposure period, or sacrifice at ∼12 ±0.5 months. Based on reported body weight, the exposure period was modeled to occur during month three of the 12-month exposure period (FAO 2011). Separate enumeration of benign vs. malignant tumors was not provided (Bailey et al. 2009, Tables 1 and S1), but data from similar studies indicate that potent genotoxic polycyclic aromatic hydrocarbons such as aflatoxin B₁ and DBP can elicit predominantly malignant tumors (Reddy et al. 1999a; Williams 2012). Data from each set of four separate groups of trout that were administered the same dose were combined into one group per dose, and corresponding dose-specific chi-square tests were performed to test for consistency with binomial sampling error of tumor counts within each dose group. Only liver tumors in the 28.4-ppm dose group failed this test (p < 10⁻⁶), and a corresponding extended Fisher exact test (p < 10⁻⁵) (Baglivo et al. 1988), and for this dose group, the standard error was taken to be the empirical standard deviation (0.0759) of the four corresponding observed incidence rates.

Dose-Response Models

The MVK (two-stage doubly stochastic clonal-expansion) model of cancer risk posits that a critical oncogene mutation occurring at rate m₁ causes a transition from a normal stem cell (N) to a stage-one or premalignant cell (P), and another mutation occurring at rate m₂ causes a transition of a P cell to an incipient cancer cell (M). N- and P-cells undergo processes of birth at rates b₀ and b, respectively, and loss at rates d₀ and d, respectively (via death or terminal differentiation), which may be affected by endogenous or xenobiotic exposures, with b > d (positive net proliferation) being characteristic of P-cells (Moolgavkar 1983; Moolgavkar et al. 1988; Moolgavkar and Luebeck 1990). In view of the brief periods of exposure and follow-up, it was assumed for simplicity that N is constant (as an effective time-weighted average scaling factor), b₀ ≈ d₀ (steady-state tissue maintenance) for N-cells, and the product v = N m₁ was fit to each control tumor incidence rate. Assuming that DBP acted as a pure mutagen, the MVK model was evaluated with parameters m_i = m₀(1 + q_i C(t)), (for i = 1,2), assuming m₀ = 10⁻⁶ (representing a single background mutation rate), b = g, and d = d₀, where q_i denote “potency” coefficients for DBP-induced mutations), and C(t) = DBP concentration in ppm at time t (measured in months), with C(t) = 0 if t ≤ 2 or t > 3. Assuming alternatively that DBP is a purely non-mutagenic tumor promoter that acts only to increase the net rate of proliferation of premalignant cells, the MVK model was evaluated with parameters m_i = m₀ (for i = 1,2), and b = g(1 + q₁ C(t)) exp(−q₂ C(t)) + d, and d = d₀, where q₁ and q₂ here denote potency and potency-attenuation coefficients, respectively, for DBP-induced net proliferation of P-cells, and q₂ > 0 was assumed only for liver tumors to better model the non-monotonic response pattern observed for that tumor endpoint. Fits using an MVK model parameterized to reflect a dual MOA, involving DBP-induced elevations both in mutation rates and in net rates of premalignant-cell population growth, were also explored (see Results).

To simplify DAH-model implementation, in view of the driving role of transient production of all preneoplastic (P_B and R_M) cells in proportion to the total duration of AH-generating stress that is predicted by the DAH model (see Introduction), and the fact that tumor histology information was not reported by Bailey et al. (2009) as noted above, tumors were all assumed to be malignant. In the absence of time-to-tumor data, this assumption did not clearly constrain modeling results to predict responses that differ substantially from those that could be obtained by modeling separate benign and malignant DAH pathways. The DAH model was implemented by adapting the MVK (two-stage, doubly stochastic Poisson-process) model to reflect a characteristic DAH-specific parameter structure. Specifically, the MVK-model parameters were adapted as follows: N = 1, m₁ = m₀[1 + q₁ C(t) exp(−d C(t))] {1 + 10³ q₂ Φ([ln(C(t)) − ln(μ)]/ln(σ_g))} with d = 2 q₁ (liver) or d = 0 (stomach), m₂ = 10⁻³, b = 1, d = 3/4 or 1/2 (for liver or stomach, respectively), where q_i denote hypothetical linear (i = 1) and nonlinear (i = 2) potency coefficients for DBP-induced regenerative AH, Φ = the standard normal cumulative probability distribution function (cdf), and ln = natural logarithm. Increased tumor risk due to DBP-induced AH was thus assumed to reflect a dominant pathway requiring the interaction of a mutation event, assumed to occur at a rate proportional to q₁ C(t), and the size of the regenerative AH-cell population in each target tissue, assumed to be proportional to a cumulative lognormal function of concentration C(t) with a median response at C(t) = μ and a geometric standard deviation of σ_g. The DAH model was first fit to both ED₀₀₁ data sets conditional on q₁ = 1/200 (chosen arbitrarily); additional fits to stomach-tumor data were obtained using three additional values of q₁ in the range 0 ≤ q₁ ≤ 1/10 to demonstrate that DAH-model goodness of fit is effectively independent of q₁ within this range (see Results).

Calculations and Statistical Analysis

The significance of tumor-incidence elevation observed in each exposure group compared to controls was assessed by Fisher exact test. The MVK and DAH models described were evaluated numerically by the method of Crump et al. (2005) to evaluate predicted cumulative probability R(C) of observing each tumor type at time t = 12 months after exposure to DBP concentration C during period 2 ≤ t < 3 (see Appendix). All four models were fit by inverse-variance weighted Levenberg-Marquardt minimization of a chi-square objective function comparing the fitted to observed tumor incidence rates, assuming binomial sampling errors (except as noted above for liver tumor counts in the 28.4-ppm concentration group), and associated covariances were used to calculate asymptotic chi-square goodness-of-fit statistics (Press et al. 1986). Additionally, low-dose data subsets for each tumor type were each tested for approximate linearity of dose-response using a corresponding chi-square goodness-of-fit statistic from inverse-variance-weighted linear regression.

Biologically based model predictions of increased tumor-specific risk above background were compared graphically. For this comparison, increased risk A(C) to DBP-exposed trout above background risk R(0) for each tumor type was extrapolated at low doses using each well-fitting biologically based model, assuming that R(0) arises independent of DBP exposure: i.e., assuming that A(C) = #x0005B;R(C)−R(0)]/[1−R(0)]. For the DAH model applied to stomach tumors, an exact expression for the limiting value of A(C) as C tends to zero was obtained by symbolically evaluating the (quite complicated) exact analytic expression obtained for the corresponding MVK model evaluated at time t = 12, after a total of k = 3 periods each involving constant model-parameter values, using the piecewise-recursive formula derived by Moolgavkar and Luebeck (1990) for risk R(C) conditional on k.

All numerical calculations and symbolic (including algebraic, derivative, and limit) calculations were performed using Mathematica^® 9.0 software (Wolfram Research 2013). P-values <10⁻¹⁰ were indicated as being ∼0.

RESULTS

Compared to control-group incidence rates by Fisher exact tests, liver tumors were not elevated significantly at the lowest two DBP concentrations used in the ED₀₀₁ study (p ≥ 0.39), but were clearly elevated (p = 10⁻⁹) in the third-highest (3.57-ppb) DBP-exposure group that had a liver-tumor incidence rate of 0.82%, as well as in all higher dose groups (p = ∼0). Similar pairwise comparisons showed that stomach tumors were not elevated significantly at the lowest three DBP concentrations used (p ≥ 0.39), were marginally elevated (p = 0.061) in the third-highest exposure group, were significantly elevated (p = 0.020) in the forth-highest (10.1-ppb) DBP-exposure group that had a stomach-tumor incidence rate of 0.51%, as well as in all higher dose groups (p = ∼0).

Table 1 lists parameter estimates from fits to data on liver and stomach tumors in ED₀₀₁-study trout obtained using the biologically based, mechanistic MVK and DAH models, together with goodness-of-fit statistics. Neither the purely genotoxic MVK model nor the MS2 model are plausibly consistent with either set of tumor data (p = ∼0), whereas good (p = ∼0.5) and nearly identical fits to data on each tumor type were obtained using the purely nongenotoxic MVK model and the DAH model. The purely nongenotoxic MVK and DAH model fits are compared in Figure 2. Fits of the MVK model parameterized initially to reflect a dual MOA (i.e., both genotoxic and nongenotoxic pathways to increased tumor risk) converged on the pure nongenotoxic MVK fits listed in Table 1, so the latter fits represent the best MVK-model fits obtained to ED₀₀₁ data on each tumor endpoint.

TABLE 1.

MVK and DAH model fits to ED₀₀₁ study data on liver and stomach tumors in >40,000 DBP-exposed trout.

		Model parameter estimate (SE)b
Tumor type	Model typea	N m0 or p0	g or b	d	q1	q2	μ	σg	dfc	P-valuec
Liver	MVK G	0.437 (L)	0 (−)	0 (−)	1160. (L)	0 (−)	−	−	6	∼0
Stomach	MVK G	1.68 (L)	0.426 (L)	0 (−)	2.28 (L)	12.9 (L)	−	−	4	∼0
Liver	MVK N	1.10 (1.43)	0.455 (0.158)	56.9 (86.5)	0.139 (0.071)	0.00471 (0.0049)	−	−	3	0.052
Stomach	MVK N	1.14 (1.40)	0.509 (0.151)	23.9 (36.1)	0.0276 (0.0115)	−	−	−	4	0.67
Liver	DAH	5.95 (1.04)	1 (−)	3/4 (−)	1/200 (−)	4.49 (0.923)	65.0 (9.20)	3.19 (0.271)	4	0.15
Stomach	DAH	2.24 (0.287)	1 (−)	1/2 (−)	1/200 (−)	2.50 (0.560)	141 (30.2)	2.58 (0.251)	4	0.51
Stomach	DAH	2.21 (0.284)	1 (−)	1/2 (−)	1/10 (−)	0.167 (0.0270)	58.9 (8.51)	2.38 (0.285)	4	0.57
Stomach	DAH	2.23 (0.287)	1 (−)	1/2 (−)	1/40 (−)	0.626 (0.108)	78.4 (12.2)	2.39 (0.241)	4	0.52
Stomach	DAH	2.23 (0.288)	1 (−)	1/2 (−)	1/1000 (−)	6.92 (2.62)	270 (107)	2.93 (0.382)	4	0.53
Stomach	DAH	2.23 (0.289)	1 (−)	1/2 (−)	0 (−)	11.5 (5.57)	384 (194)	3.13 (0.460)	4	0.54

^aG = purely genotoxic; N = purely nongenotoxic. For the DAH model, the parameter d was pre-assigned, not optimized; two different values were inadvertently assigned for the two tumor types. See text for model-type definitions. Parameter and SE estimates listed for DAH parameter p₀ are (in order to fit these in this table) 1000-fold greater than the corresponding estimates actually calculated.

^bSE = standard error; − = parameter not applicable; (−) = SE not calculated (parameter set to a constant); L = SE >2 times the parameter-value estimate x; if SE > x, the listed model parameter is likely not identifiable (i.e., as good a fit may be possible using fewer parameters). Estimated parameter values listed each show three significant digits; assumed constants listed show infinite precision.

^cdf = degrees of freedom for chi-square goodness of fit test associated with the indicated p-value.

FIGURE 2.

Nongenotoxic or “pure promoter” MVK model (blue curves) and DAH model (balck curves) fit to ED₀₀₁ “mega-trout” study data (open points) on the incidence of liver tumors (solid curves) and stomach tumors (dashed curves) in Shasta strain trout fry fed different concentrations of DBP for four weeks, then followed on control diet for nine months before sacrifice. Error bars denote ±1 SE.

Figure 3 plots tumor- and model-specific increased risks predicted by the nongenotoxic (pure-promoter) MVK model (blue curves), and by the DAH model (black curves). In this figure, the fit to liver-tumor data shown (blue dashed curve) assumed a value of 0.005 for the DAH-model parameter q₁, which specifies the potency with which DBP is assumed to increase the critical DAH-mutation rate. To obtain the four DAH-model fits to stomach-tumor data shown (as black dotted curves) in Figure 3, four different corresponding values of q₁ were assumed (including the value q₁ = 0.005) as indicated in the figure. At very low DBP concentrations (< ∼1 ppm) administered for four weeks, the fits to liver- and to stomach-tumor data that were made using the purely nongenotoxic MVK model are ∼550-fold and ∼120-fold greater, respectively, than those made using the DAH model conditional on q₁ = 0.005. Figure 3 illustrates how the magnitude of this MVK-to-DAH ratio of predicted, model-specific increased risk is directly and linearly proportional to the assumed value of the DAH-model parameter q₁.

FIGURE 3.

Each of the risk models plotted in Figure 2 in relation to the tumor data is re-plotted here with the Y-axis re-expressed as increased tumor risk over each corresponding background risk estimated to occur in the absence of DBP exposure. The DAH-model fits each involve a parameter q₁ that defines assumed mutagenic potency. A q₁ value of 0.005 was assumed for the DAH-model fit to liver-tumor data (blue dashed curve). Four different values of q₁ (including 0.005, as indicated) were used to obtain four corresponding fits to stomach-tumor data (dotted black curves).

As values of maximum DBP concentration C = Max[C(t)] approach 0, the terms involving exp and Φ in the DAH-model expression given in Methods clearly reduce to 1 and 0, respectively. Conditional on this simplification, analysis (as described in Methods) of the DAH model yielded the following corresponding exact expression for increased risk A(C) as a function of C in the limit as C→0, as a function of model parameters m₀ and q₁:

$$\begin{array}{c}\text{A}(\text{C})=\left\{\left(r_1/2000\right)-\text{ln}\left[\left(r_2+r_1\hspace{0.17em}\text{X}\right)/\left(r_2+r_1{\hspace{0.17em}\text{X}}^{9/10}\right)\right]\right\}{m}_0{q}_1\hspace{0.17em}\text{C}\\ =\sim 0.16011\hspace{0.17em}{m}_0\hspace{0.17em}{q}_1\hspace{0.17em}\text{C},\end{array}$$ (1)

where r₁ = R−c, r₂ = R+c, $\text{R}=\sqrt{253001}$, c = 499, and X = exp(R/100) (see Appendix). As illustrated in Table 1 for DAH fits to the stomach tumor data, when DAH-model parameters are re-optimized using different values of q₁ in the range 0 ≤ q₁ ≤ 1/10, all the resulting fits are good and virtually identical within the range of the data. Each such fit involves a virtually identical value of m₀, which essentially fits the background tumor rate. Optimizing the two lognormal parameters contained in the DAH model thus generates the good fit achieved to all the non-background data for each tumor type. Numerical evaluations of the DAH model that was fit to stomach-tumor data show that for all C ≤ 0.1 ppm, Equation 1 is accurate to within ∼2.5% for all q₁ ≤ 1/10 and to within ∼0.5% for all q₁ ≤ 1/40. Because models found to fit well to data on both tumor endpoints (Table 1) are structurally similar, analogous results are expected concerning how well Equation 1 approximates DAH-model fits to data on liver tumors, conditional on similar ranges for the values of q₁ and C. Consequently, the DAH model applied essentially combines a lognormal dose-response model that dominates at higher doses, together with a “hockey-stick”-type model that dominates at much lower doses, over which the limiting linear “hockey-stick” dose-response slope is determined entirely by the mutagenic-potency parameter q₁ (Figure 3).

Stepwise chi-square goodness-of-fit tests for consistency with dose-response linearity, each performed on a tumor-specific dose-response data subset consisting of the k lowest-concentration data points, showed that the liver-tumor data subset, including the 3.57-ppm exposure group exhibiting a 0.82% rate of liver-tumor incidence, was found to be only marginally consistent with linearity (k = 4, p = 0.054); all larger subsets of the concentration-ordered liver-tumor data (i.e., those including the 10.1-ppm exposure group exhibiting a 5.5% rate of liver-tumor incidence) were clearly inconsistent with linearity (k ≥ 5, p < 10⁻⁸). Dose-response linearity was thus determined to be plausible only for rates less than 2% of that observed for liver tumors in the highest DBP-concentration group studied.

Dose-response linearity for DBP-induced stomach tumors was likewise determined to be rejected only when the third-highest dose group (4.6% of which developed stomach tumors) was included (k = 6, p = 0.00015). Dose-response linearity was thus determined to be plausible only for rates less than ∼6% of that observed for stomach tumors in the highest DBP-concentration group studied.

DISCUSSION AND CONCLUSIONS

Results from the present study show that the two biologically based, mechanistic models of tumorigenesis (MVK and DAH) examined each provide good fits to liver- and stomach-tumor incidence data from the ED₀₀₁ study. Surprisingly and counter-intuitively, however, the good MVK model fits obtained that are listed in Table 1 all assume a purely nongenotoxic MOA, in each case with zero predicted contribution from exposure-induced mutations by one of the most potently mutagenic carcinogens known. The nongenotoxic MVK model applied expresses increased net proliferation of premalignant cells as a linear function of DBP concentration. Nevertheless, predictions made by this model were found to be consistent with a lognormal pattern of observed responses in relation to DBP concentration that is exhibited by the DAH model that was fit to the same tumor data.

The observation that nongenotoxic MVK-model fits listed in Table 1 are each statistically consistent with the data to which these models were fit, while corresponding genotoxic MVK-model fits are not plausibly consistent with those data, does not prove that DBP-induced mutations played no role in elevating tumor incidence in the ED₀₀₁ study. As noted in Results, a low-dose subset of each tumor-specific data set is clearly statistically consistent with an LNT dose-response. Even if none of the data were determined to be statistically consistent with low-dose linearity, from a statistical standpoint, it is a well-recognized practical impossibility to disprove that an LNT dose-response does not actually occur at sufficiently low doses for any endpoint, particularly if it occurs spontaneously at a non-negligible rate, even using tens of thousands of experimental animals as done in the ED₀₀₁ study. However, as also noted in Results, each subset of ED₀₀₁-study data found to be consistent statistically with low-dose linearity was also found to be rather limited, representing very small fractions—less than about 2% and 6%—of the maximum rates of incidence observed for liver and stomach tumors, respectively. These are rather small upper statistical bounds on the plausible contribution of low-dose linearity to tumor rates increased by exposure to DBP, one of the most potently mutagenic carcinogens known.

Forward mutations are potently induced by DBP, and are readily detected in animal cells in vitro and in vivo as approximately linear functions of exposure, even at concentrations that are at most marginally cytotoxic (see Introduction). Therefore, if the somatic-mutation/MVK model of cancer is valid, MVK-model fits obtained to ED₀₀₁-study tumor data should have involved estimated increases above background mutation rates, and corresponding low-dose dose-response linearity over response ranges that extend beyond merely marginal elevations above background tumor rates. This expectation is clearly inconsistent with the modeling results obtained for liver tumors discussed, and possibly also with those for stomach tumors, even after addressing plausible bounds on statistical uncertainty that pertains to the lowest portions of each observed tumor dose-response pattern. Thus, while MVK-modeling results were obtained that are statistically consistent with ED₀₀₁-study tumor data, the results do not appear to be biologically plausible because they imply that DBP-induced mutations played implausibly small or even no role in tumorigenic effects that were potently elicited by DBP exposure.

Like the MVK model, the mechanistic DAH model also assumes that mutation plays a critical, rate-limiting role in tumorigenesis. However, the DAH model places no lower bound on the assumed positive magnitude of this role. If the single critical mutation posited by the DAH model for each tumor type is typically extremely rare, then this model predicts that elevated tumor incidence after sustained chemical exposure can arise primarily (or solely) by an associated increase in the number of cells recruited into AH-cell populations that, according to this model, most typically are the source of exposure-induced tumors. The DAH model thus provides one example of a mechanistic, biologically based model of tumori-genesis that happens to provide good fits to ED₀₀₁ tumor data, based on modeling assumptions that are consistent with data concerning different toxic cellular effects of DBP. However, the fact that good DAH-model fits were obtained, of course, may instead simply coincidentally reflect the approximate lognormal pattern that happens to be exhibited by both tumor types that were elevated by DBP in the ED₀₀₁ study.

At very low DBP concentrations (less than about 1 ppm) administered for four weeks, the fits to liver- and to stomach-tumor data obtained using the purely nongenotoxic MVK model were determined to be ∼550-fold and ∼120-fold greater, respectively, than those obtained using the biologically based, mechanistic DAH model conditional on a value of 0.005 for the DAH-model parameter q₁ (Figure 3). The magnitudes of these relative differences in predicted increased risk are effectively arbitrary, because Equation 1 (see Results) implies that these relative differences in predicted increased risk are linearly proportional to the value of the q₁ parameter used to implement the DAH model, over a fairly large q₁-range bounded by zero. As noted in Results, the DAH model thus exhibits a dose-response pattern that combines an “arbitrary-slope hockey-stick” at low doses, with a lognormal S-shaped response at higher doses. Thus when DAH-model fits to each tumor set are conditioned on different values of q₁, the goodness of fit remains virtually unaffected, but the “slope” of the low-dose hockey-stick part of the fit is re-scaled in proportion to q₁. When the Y-axis is log-transformed as in Figure 3, this re-scaling of slope appears as a vertical shift of the low-dose portions of the fitted (dotted black) curves (i.e., the portions at which DBP concentration is less than ∼1 ppm), which each have unit slope on a log scale.

The ED₀₀₁ “mega-trout” bioassay offers the greatest statistical power currently available to resolve different low-dose dose-response relationships for tumors induced by a genotoxic chemical carcinogen. Substantial (>10-fold) improvement in statistical power in this regard is unlikely to arise for years or perhaps decades. Consequently, a direct experimental test between low-dose dose-response predictions made, for example, by the DAH vs. the MVK model using animal bioassay data, is not feasible either currently or perhaps even within the next decade. At this point, such a comparison can be made only by generating additional experimental data and by improving theory concerning fundamental molecular mechanisms of tumorigenesis. Such developments thus continue to be the long-recognized key to improved risk assessment for environmental chemical carcinogens (Albert 1994).

APPENDIX

All MVK and DAH models fit were implemented numerically as the solution for predicted cumulative likelihood R(t) that a malignant cell appears by time t = 12 months for a general MVK model expressed in terms of the parameters v(t), m₂, b(t), and d, which were defined in Methods in fundamentally different ways for each model considered, as indicated where applicable as functions of time t. Crump et al. (2005) showed that a general solution for R(t) = 1 − C(t, t), can be evaluated by numerically solving the following linked system of ordinary differential equations involving three auxiliary variables (A, B, and C) each a function of starting time s and ending time t, with 0 ≤ s ≤ t:

$$\text{dA}(s,t)/\text{d}t=b(t-s){\text{A}}^2(s,t)+d-\left[b(t-s)+d-m_2\right]\text{A}(s,t)$$ (A-1)

$$\begin{array}{c}\text{dB}(s,t)/\text{d}t=2b(t-s)\text{A}(s,t)\text{B}(s,t)+d\\ -\left[b(t-s)+d-m_2\right]\text{A}(s,t)\end{array}$$ (A-2)

$$\text{dC}(s,t)/\text{d}t=v(t-s)\text{C}(s,t)[\text{A}(s,t)-1],$$ (A-3)

with initial conditions A(0, t) = C(0, t) =1 and B(0, t) = −m₂. For piecewise-constant parameters, R(t) can also be solved analytically (see below).

From the general analytic solution by Moolgavkar and Luebeck (1990) defining R(t) for an MVK model with piecewise-constant parameters during intervals 0 ≤ t < 2, 2 ≤ t < 3, and t ≥ 3, the exact value of R(t) for the DAH model described in Methods with parameters v = ∼m₀(1 + q₁ C) for very small maximum DBP concentration C, m₂, b, and d is given by

$$\begin{array}{c}\text{R}(t)=1-\text{exp}[-\left(m_0/b\right)\{\text{A}+\text{Fln}\left({\text{S}}_2/{\text{S}}_3\right)\\ +\text{ln}\left[{\text{B}\hspace{0.17em}\text{S}}_3/\left(2{\text{r}\hspace{0.17em}\text{S}}_2\right)\right]\}],\end{array}$$ (A-4)

where A = (h−r)F/2, h = b−d−m₂, r = [(b+d+m₂) − 4 b d]^1/2, F = 1 + q₁ C, B = (r+h) + (r−h)X^t, X = exp(r), S_i = r + b − d − m₂ + (r − b + d + m₂)X^t−i, t = 12, i = 1,2, and ln denotes natural logarithm. Mathematica 9.0^® symbolically calculated the derivative of increased risk A(C), defined in Methods, with respect to C in the limit as C→0 as

$$\begin{array}{c}{\text{Lim}}_{\text{C}\to 0}\left[\text{dA}(\text{C})/\text{dC}\right]=\{[b-d-m_2-\text{r}\\ +2\text{ln}\left({\text{S}}_2/{\text{S}}_3\right)]/(2\hspace{0.17em}b)\}{m}_0{q}_1,\end{array}$$ (A-5)

which, using definitions given above and multiplying by C, yields Equation 1 (in Results) after substituting values of m₀, m₂, b, and d defined in Methods for the DAH model.