Microbial Dose Response Modeling: Past, Present, and Future

Critical Review pubs.acs.org/est

Microbial Dose Response Modeling: Past, Present, and Future Charles N. Haas* Department of Civil, Architectural & Environmental Engineering Drexel University Philadelphia, Pennsylvania 19104, United States ABSTRACT: The understanding of the risk to humans from exposure to pathogens has been firmly put into a risk assessment framework. A key element of applying this approach is the understanding of the relationship between dose and response for particular pathogens. This understanding has progressed from early use of threshold concepts (“minimal infectious dose”) thru multiple generations of models. Generation 1 models describe probability of response to exposed dose. Generation 2 models incorporate host factors (e.g., age) and/or pathogen factors (e.g., particle size of inhaled agents). Generation 3 models describe the rate at which effects develop, i.e. the epidemic curve. These (generation 1 through three models) have been developed and used in multiple contexts. Beyond Generation 3 lies an opportunity for the deep incorporation of in vivo physiological responses and the coupling of the individual host dynamics to the dynamics of spread of contagious diseases in the population. This would enable more direct extrapolation from controlled dosing studies to estimate population level effects. There remain also needs to understand broader categories of infectious agents, including pathogenic amoebae and fungi. More advanced models need to be validated against well-characterized human outbreak data.

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into the end risk assessment.3−6 This process is beyond the scope of this review. The scope of this review is on the development of modeling approaches for dose response estimation of infectious microorganisms. Contexts of Interest. Microbial dose response assessment is important for quantitative microbial risk assessment in a number of contexts. These include • Drinking water quality7,8 • Recreational water9−11 • Food safety12−15 • Biosolids16−20 • Homeland security21−23

INTRODUCTION

The role of intervention in infrastructure and sanitation to control disease predates the germ theory of disease. With the advent of the germ theory, it became clear that there was a relationship between specific diseases and specific microorganisms. The work of environmental engineers and scientists is to develop strategies for the attainment of acceptable levels of safety, recognizing (as has been true in recent decades) that absolute safety is an unattainable goal. Over the past 50 years, there has been increasing recognition of the quantitative relationship between level of risk and level of dose, that is, of a dose−response relationship. In many contexts, it is desirable to develop quantitative criteria for the levels of pathogens that provide acceptable protection against infection to people or other hosts, or to determine the risks associated with exposure to particular levels of pathogens. For this task, the field of quantitative microbial risk assessment (QMRA) was developed to parallel the process of risk assessment of chemical and physical hazards.1 Risk assessment consists of hazard assessment, followed by exposure assessment and dose response assessment. These latter two tasks are integrated into the risk characterization. The relative importance of the uncertainties in exposure assessment and dose response assessment can be determined by parametric uncertainty analysis.2 It is therefore important to determine the best model describing the dose response of a particular microbial hazard to assess the risk (and its uncertainty). The risk characterization needs to include a full assessment of uncertainties in both the dose−response and exposure assessment steps. This process often involves significant comparison among alternative models and determination of the propagation of parametric uncertainties and variabilities © 2014 American Chemical Society

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EXPERIMENTAL FRAMEWORK Regardless of the context or the microorganism, the underlying framework used for the extraction of dose response information consists of a set of experiments in which host subjects are (ideally randomly) divided into groups. Each group of subjects, comprising Ti individuals, is administered an average dose of pathogens di. At the termination of the experiment it is found that Pi subjects are “positive” for the effect under study (infection, illness, death, ...). The response is therefore quantal24 or all or none, in contrast to a graded response. Dose response data can be obtained from human trials. This was especially true many years ago. Examples of such data sets include Salmonella,25 Shigella, 26 and rotavirus.27 However, in other cases, and particularly with highly virulent pathogens, Received: Revised: Accepted: Published: 1245

September 9, 2014 December 21, 2014 December 29, 2014 December 29, 2014 DOI: 10.1021/es504422q Environ. Sci. Technol. 2015, 49, 1245−1259

Critical Review

Environmental Science & Technology

Generation 1. Generation 1 models are defined as those that delineate a functional (“dose response”) relationship between the dose administered to a group of subjects (hosts) and the probably of an individual host responding (or the expected fraction of hosts that would respond). These models are the direct descendants of the MPN assay in that they posit a functional relationship between the average dose of microorganisms administered to a set of hosts and the probability of response. However, unlike eq 1, the probability that a single organism will infect may be less than 1.0. Qualitatively this was recognized in the use of tissue culture based assays of viruses, wherein the fraction of aliquots receiving an inoculum which were “positive” (as determined by total lysis of the host cells) was used as a measure of virus concentration. In the TCID50 assay, described by Cornfield below, this was simply used as an index of potency rather than as a reference to virus “particles”: “A pool is considered sufficiently infective if 0.5 cc is capable of infecting tissue culture after at least a one-million fold dilution. The amount by which a preparation must be diluted before it loses infectivity is referred to as its titer. There is, in fact, no single dilution point at which infectivity turns abruptly to non-infectivity, and in practice the titer used is that dilution estimated to result in infectivity for 50 percent of the inoculated tissue culture tubes. The amount of virus present in an inoculum capable of infecting 50 percent of the tubes is referred to as one tissue culture infectious dose (TCID50).”36 As quantitative experiments on the infection of hosts by pathogens became more prevalent, two competing frameworks emerged described by Meynell and Stocker37 as follows: “The hypothesis of independent action postulates that the mean probability per inoculated bacterium of multiplying to cause (or help to cause) a fatal infection is independent of the number of bacteria inoculated and, for a partially resistant host, is less than unity (1> p >0)... The hypotheses of maximum and of partial synergism postulate that inoculated bacteria co-operate so that the value of p increases as the size of the dose increases.” In extremis, the hypothesis of synergism leads to a formulation of a true threshold, wherein if the dose administered is below some critical value, there is identically and precisely zero probability of adverse outcome. With careful experiments, these authors found their data to be consistent with the independent action hypothesis, and developed a framework for the underlying dose response framework. Based on this framework, and assuming that the pathogen dose was large (to avoid the effects of variability in actual numbers of organisms delivered to each host), the authors derived what is now recognized as the exponential dose response relationship (the equation below is written in more current conventional terms):

deliberate human trials are impossible to conduct. It is possible to use data from human trials and experience of the author and others have shown that dose−response relationships from animal trials are good predictors of human outcomes (e.g., during outbreaks). There are a number of such validation examples in the literature.28−30 An alternative type of data used to generate dose response information are multiple outbreaks in which for each outbreak, the average dose and the attack rate (proportion) are known. Examples of this type of analysis are the development of dose response curves for E. coli O157:H731 and Norovirus.32 Early ConceptsGeneration 0. The earliest considerations of the relationship between microorganism exposure and consequence used a concept of “infectious dose”. Under this framework, there was an apparent belief that a critical dosage existed below which there were no observable effects in host species. While the exact historical origin of the concept is unknown, this terminology dates to at least the early 1920s. For example, in a study of infection of guinea pigs by typhus, Weil and Breinl33 defined the concept as follows: “By “infecting” dose, we mean the smallest quantity of virus [sic] which just suffices to infect a guinea pig” Somewhat later, Shope34 conducted studies on the infection of mice with influenza and noted: “In certain types of experiments exact knowledge as to the minimal infectious dose of a virus is desirable. It was hoped that, with the mouse available as a test animal, accurate quantitative experiments with swine influenza virus might be possible. With this end in view virus were titrated in mice. From a group of four such experiments conducted in November of 1934, using 3 mice per dilution, it was found that mouse passage virus was active in a final dilution of 1:20 000” In mathematical terms, if the experiments are ordered by dose (d1 < d2 0 for i > z. In retrospect, Generation 0 approaches neglected two important aspects of microbial infection that are now understood. • The variation in discrete number of infectious agents in the actual dose aliquots administered to hosts • Use of a finite (often small, as in the case of Shope,34 three animals per dose) number of subjects results in potential statistical variability in response Ironically some of these issues were addressed many years earlier in the development of most probable number (MPN) assays for microorganisms, which form the basis of the Generation 1 models. Perhaps the first such rigorous statistical analysis was that of Greenwood and Yule.35 They considered that if the distribution of microorganisms in a sample was random, the probability of having zero organisms in a sample of volume V, given a density of μ (no. per unit volume) would be given by the zero term in the Poisson distribution: P(0) = 1 − exp( −μV )

p = 1 − exp( −kd)

(2)

Where k is the probability of a single organism surviving within the host and d is the dose of organisms administered. The similarities between eq 2 and eq 1 are clear, essentially differing only with regard to the survival probability (which in the MPN equation is assumed equal to 1). The dose (d) in eq 2 clearly corresponds to the product μV in eq 1. A somewhat parallel development arose from consideration of dynamics of transmission of disease via ind0oor air. Wells38 defined the concept of a “quantum of contagion” such that “When on the average 1 animal breathes 1 quantum, or the

(1)

Note that the quantity μV can be interpreted as the average “dose” of microorganisms administered to one MPN tube. Implied, but not explicitly stated in development of eq 1 is that a single organism delivered to a tube of nutrient medium in the MPN assay would have a probability of colonizing of exactly 1.0. 1246

DOI: 10.1021/es504422q Environ. Sci. Technol. 2015, 49, 1245−1259

Critical Review


binomial distribution (P2(m|j) = (j!/(m!(j − m)!))(1 − k)j−mkm). With these substitutions into eq 3, eq 2 results. Heterogeneity. The condition of the host (immune state, concomitant physiological impairments, etc.) may alter the ability to fight off microorganisms.38 Variation of physiological state of the microorganisms, for example, changes in bacterial cell envelope,46 clumping, or formation of viable nonculturable states47,48 may alter intrinsic ability of the infectious agent to survive host defenses. Therefore, the assumption of independent identical survival of agents (i.e., a single value of k and binomial variability) may be inappropriate. If there is variability in the interaction between the infectious agents and the hosts that can be characterized by a probability distribution, then one approach is to replace the simple binomial describing P2 by

total number of quanta breathed equals the total number of animals, 36.8 percent of the animals will survive”. Note that this can result from the exponential relationship 1−exp(−q) being the fraction of animals dead, when q = the quantum of contagion. In developing this concept, Wells explicitly relied on earlier work of Petrie and Morgan,39 who in experiments with on mortality of mice subcutaneously injected with pneumococcus, had dose-mortality curves that corresponded to an exponential with a median lethal dose (LD50) corresponding to roughly one organism. However, Wells recognized that the “quantum of contagion” for other organisms (via inhalation) might be substantially larger. This analysis is thoroughly consistent with eq 2 (where as k decreases, the number of organisms in such quantum increases). Wells’ approach was developed further by Riley for the case of an infective individual serving as a source of droplets of contagious organisms in a confined indoor environment.40−42 These approaches consider the source emission of quanta, the inhalation rate, the room ventilation rate and the time of exposure. While some have regarded43 the Wells-Riley approach as an alternative to dose response modeling, as the above makes clear, this is in fact not the case. They key difference is the abstraction from actual numbers of organisms to functional “quanta of contagion” in the modeling step. For the purpose of further discussion, it is useful to present the formal derivation of eq 2 which was presented by the author in 1983.44 This provides a basis for more complex types of models. Derivation of Exponential Dose Response. We consider that the process of microbial proliferation sufficient to cause an adverse effect, whether it is infection, illness or death, is the result of three steps: 1. An individual host must ingest one or more competent organisms. 2. The organisms must survive the host defenses such that they are capable of finding a site in vivo to proliferate successfully. 3. A sufficient number of proliferation sites must occur needed to cause the effect. We therefore can regard this problem as the convolution of three factors: ∞

P(d) =

P2(m|j) =

⎤

j!

(4)

where f (k) is the probability distribution (density function) for k. Since k has physical bounds of ⟨0, 1⟩, it is logical to look for a distribution bounded by those limits. Furumoto and Mickey49,50 who were interested in describing the infectivity of tobacco mosaic virus to tobacco plants proposed a beta distribution for this purpose as given by eq 5 since this distribution has support only over ⟨0, 1⟩. f (k ) =

Γ(α + β) α − 1 k (1 − k)β − 1 Γ(α)Γ(β)

(5)

The substitution of eq 5 into eq 4 and thence into eq 3 with Poisson variability and kmin = 1 can be shown44,50 to yield: P(d) = 1 −1 F1(α , α + β , −d)

(6)

The authors50 noted in passing that they tried to use a truncated normal distribution for f(x) without success. The author is unaware of any other mixing distribution that has been examined for use in eq 4. In eq 6, the function 1F1 is the confluent hypergeometric function, which, especially at the time of development of this model, was not frequently available in computer packages; currently this function is available in Maple, Mathematica, and as a third party add-in to Matlab. Therefore, the original authors derived a simplification under conditions where β “ls large”. Under these conditions eq 6 can be approximated by

∞

∑ ∑ P1(j|d)P2(m|j) m = k min j = m

⎡

∫ f (k)⎢⎣ m!(j − m)! (1 − k) j−m km⎥⎦dk

−α ⎛ d⎞ P(d) = 1 − ⎜1 + ⎟ β⎠ ⎝

(3)

where d = average (arithmetic mean) dose administered to the hosts; P1 (j|d) = Probability of host being exposed to exactly m organisms from that dose; P2 (m|j) = Probability of m organisms surviving from the initial j; kmin = minimum number of surviving organisms from the initial dose needed to cause the effect If the actual (integer) number of organisms to which a host is known then using the nomenclature of Haas(2002),45 a “conditional” dose response relationship can be defined, in this case P1(j|d) becomes a Kronecker delta function. The exponential model, eq 2, results from assuming one successful survivor suffices to cause the effect (kmin = 1), that there is a Poisson distribution of organisms (P1(j|d) = (dj/ j!)exp(−d)), and that each organism has an independent and identical probability (equal to k) of surviving leading to a

(7)

In this review, and in current literature, eq 6 is termed the “exact beta Poisson” equation, whereas eq 7 is termed the“beta Poisson” (or sometimes “approximate beta Poisson” equation). The first application of the exponential and beta Poisson dose response models to environmentally transmitted pathogens via the oral route was that of the author of this review.44 It was found that these models were superior at describing the results of human feeding trials to empirical approaches based on the log-normal distribution. Since that time, these models have been used widely in describing both human and nonhuman data sets with a variety of pathogens via the oral, inhalation, dermal and other routes of exposure. Examples of these applications will be discussed subsequently. 1247


Critical Review


Figure 1. Effect of β on difference between exact (eq 6) and approximate (eq 7) Models.

Figure 2. Effect of β on difference between exact (eq 10) and approximate (eq 9) models using α and N50.

1248


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Environmental Science & Technology Teunis and Havelaar51 provided extensive discussion of the differences between the exact beta-Poisson model (eq 6) and the approximation (eq 7). In particular, they note the following: • The approximation may be derived by using a gamma distribution for f(x) in eq 4. • While both the exponential (eq 2) and the approximate beta-Poisson are “scalable”, in the sense that a shift in the parameter k or β, respectively, produce simply a proportionate location shift in the dose response curve, the exact beta-Poisson (eq 6) does not have this property. This will be discussed below. • It is argued that a maximum upper limit for microbial dose−response is given by eq 2 with k = 1, that is, every organism survives to initiate an event. If this is accepted, then for certain parameter combinations, eq 7 needs to be constrained not to violate this limit. The adequacy of the approximation is shown in Figure 1. Note that (for this example of α = 0.5), as β increases, the adequacy of the approximation becomes better. Therefore, in modeling a data set, it analysis using the approximate betaPoisson reveals a small value of β, the analysis needs to be repeated with the exact model. In the application of the beta-Poisson model, it has been particularly useful to redefine the parameters such that β is replaced by N50, a median effective dose, defined by

A third possible generalization is to account for cooperative behavior among individual microorganisms. It has been established, that at least for some bacteria (e.g., ref 55), excretion of small molecular weight materials may enable “quorum sensing” to occur. While this is of importance in areas such as biofilm formation,56 gene expression,57 and sensitivity to inactivating agents,58,59 there is little information to suggest its relevance with respect to dose−response based on administered/ingested/inhaled dose. However, if evidence of such effects arose, the use of the beta-binomial distribution45 instead of the binomial for P2(m|j) would be one possible approach. Fitting Dose-Response Relationships. The objective of this paper is not to present a broad discussion on data fitting methods. However, there have been in general two approaches. In general, the data consist of a set of observations ⟨di, Pi, Ti⟩, where there are groups (the index “i” denoting group), where Ti hosts are administered an average dose (across hosts) of di microorganisms, and at the end of the experiment, Pi of the hosts are judged to be “positive” (whether by infection, illness or death). If the response probabilities between hosts in a dose group are identical and independent, this leads to a likelihood function (based on the binomial) as follows: L=

∏

Ti! f Pi (1 − fi )(Ti − Pi) Pi ! (Ti − Pi)! i

(12)

1/ α

N50 =

2

−1 β

In eq 12, f i is the predicted probability of response from a dose−response relationship evaluated at dose di, for example, the exponential or beta-Poisson. The maximum likelihood estimators of the parameters in the dose−response relationship are then those which minimize this likelihood function. Further details on the mechanics and statistics of the fitting process have been previously reviewed.60 It should be noted that fitting can also be done in a Bayesian framework, which has been found particularly useful where there are additional sources of variability, such as between strains or hosts.61,62 Indicators, Surrogates and Genetic Methods. While the focus of this review is on dose−response models for pathogens, there was been a substantial body of work (primarily associated with recreational water risk assessment) on the quantitative use of indicator organisms as predictors for risk. In early work, Fuhs63 showed that if the constancy of an indicator to pathogen ratio occurred, it is plausible to use a dose response framework for recreational water quality. What was not considered was the existence of a mixture of pathogens in any real body of water. Cabelli and colleagues64−66 conducted a series of epidemiological studies in the U.S. in which the postswimming excess likelihood of gastroenteric illness was related to the indicator organism density during swimming. This resulted in an empirical dose−response relationship which formed the basis of the U.S. water quality guidelines for primary contact waters. Recently, it has been shown that the fit of these underlying data could be explained sufficiently by an exponential dose response (in indicator dose) with a nonspecific background term.67 A number of similar studies were conducted worldwide. These have been systematically reviewed68 and confirmed the applicability of enterococci (in marine waters) and Escherichia coli as indicator organisms to estimate the risk of illness from swimming. Generation 2. A second generation of dose response models has been evolving in which the underlying dose− response parameters (of the exponential or beta-Poisson) are

(8)

N50 is the dose at which one-half of the population exposed becomes positive. With this transformation, the approximate beta-Poisson model (eq 7) can be written as −α ⎡ d(21/ α − 1) ⎤ ⎥ P(d) = 1 − ⎢1 + N50 ⎣ ⎦

(9)

Similarly, substitution of eq 8 into the exact form (eq 7) yields ⎛ ⎞ N P(d) = 1 −1 F1⎜α , α + 1/ α 50 , −d⎟ ⎝ ⎠ 2 −1

(10)

Using the transformed alpha−N50 parametrization, Figure 2 compares the approximate and exact solutions. Below an N50 of 10, the discrepancy becomes noticeable at an α = 0.2. There are several potential generalizations from the betaPoisson framework conceivable, although none have received widespread study. For example, the beta distribution can be generalized to more complex distributions (also over the range ). One such example is the family developed by McDonald52 which involves a power transform such that eq 5 is replaced by f (k ) =

|a|kap − 1(1 − ka)q − 1 B(p , q)

(11)

A second generalization is replacement of the Poisson distribution for P1(j|d) in eq 3 by an alternative distribution that describes nonrandom variability of microorganisms. This is motivated by a number of studies showing hyper-Poisson variability in replicate samples of microorganisms from the environment  often negative binomial.53,54 These alternative distributions can then be used to formulate a dose response relationship.45 1249


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microorganism interaction process. These can be modifications of Generation 1 models, or if explicit consideration of modulating factors are included, modifications of Generation 2 models. No present examples of dynamic modifications of Generation 2 models are known, so the focus is on discussion of modification of the Generation 1 models. A fundamental characteristic in these dynamics is the concept of incubation time. Phillipe77 defines this as “...the waiting time from exposure to an etiological agent to appearance of the [observed outcomes].” He further notes that this may be modulated by a variety of factors such as “dose and type of exposure (single or continuous), the path of the inoculum, and the features of the host, namely, age, immunity, and genetic factors.” Phenomenological Incubation Models. Apparently Sartwell78 was the first to recognize that incubation times in real populations have a probability distribution  the rationale being given was biological variability in the host. He then systematically examined a large number of human diseases (primarily data from point source outbreaks, with a small number of cases resulting from experimental trials) to assess the statistical properties of the resulting incubation time distributions.By plotting individual incubation times (on a case by case basis), he found consistency with the log-normal distribution at least by qualitative assessment on log-probability plots (with the mean and dispersion being functions of the disease). However, he did not consider alternative skewed distributions, assess the goodness of fit of his data to log-normal, or consider that the distributions (especially in the examination of outbreak data) result from convolution of a “true” dose-dependent incubation distribution with the probability distribution (e.g., Poisson) of doses received by individuals in an exposed population. Meynell79 confirmed the utility of the log-normal distribution (without examination of alternative distributions). He also observed, at least for the data sets examined, that the incubation time distribution considered for those subjects who ultimately succumb, was of the same shape regardless of the initial dose. Ercolani and Vannella80 examined the incubation time distributions for a number of bacterial infections of plants. They noted concordance with a Weibull distribution of incubation times. However, they did not report a formal goodness of fit test, nor formally compare other distributions (e.g., log-normal). Philippe77 examined Sartwell’s concept of incubation time, and his finding of log-normal behavior (which Phillippe confirmed by statistical testing), in light of concepts of microbiological multiplication in vivo being one of two processes leading to disease, the second being the pathological development of disease state. Considering the process of infective microorganism multiplication, this distributional form was mechanistically justified on the following basis: ...“Suppose that several independent cell lines are followed from generation 0 to n. Further, suppose that there is a random probability for every cell line to divide at every generation. At generation n, the distribution of cell numbers will follow a lognormal...” Therefore, on this basis the fraction of hosts (each with a cell line) that cross a critical threshold of propagated organisms will proceed to an end point disease, and this time variability is dominated by the log-normal process (if the second stage is regarded as purely deterministic). Phillippe77 note that there are multiple alternatives resulting in a log-normal and therefore

themselves regarded as functions of other conditions that describe differences in infectivity of pathogens or susceptibility of hosts. This has proceeded along several directions. It has long been known69 that for inhaled microbial aerosols, the aerosol diameter can strongly influence the infectivity of the microorganisms contained therein. It has also been known that this effect is due to respiratory anatomy of the host which influences where in the lung aerosols of particular aerodynamic sizes are likely to deposit.70,71 The more proximal deposition is to the most vulnerable features, the greater the intrinsic infectivity. In the case of inhalation of spores of Bacillus anthracis, a systematic trend in LDx (lethal dose to “x” percent in hosts) was observed. For particle diameters below 4 μm, the LDx values are constant, while above 4 μm, there was an increase in LDx with aerosol diameter.72 While each aerosol size could be fit to an individual exponential or beta-Poisson model, the quantitative incorporation of aerosol size was not attempted. Similar findings were reported for the analysis of inhalation of aerosols of Brucella suis by guinea pigs.73 An approach to quantifying the size dependency of bioaerosol infectivity was put forth by Teske et al.74 for Franciscella tularensis.The dose scaling terms in the exponential and beta-Poisson dose response models were replaced by a power law dependency in aerosol size as follows: P(d) = 1 − exp[−(A ·pd )−B d]

(13)

P(d) = 1 − [1 + d(A ·pd )−B (21/ α − 1)]−α

(14)

In these equations, pd represents the diameter of the aerosol, and A and B are the parameters characterizing size dependency. It was found that this approach was able to describe the morality behavior in Rhesus monkeys over the size range of 2.1−24 μm. It has been know that host age may influence sensitivity to infectious agents, for example in the study of smallpox (variola) virus in infant mice,75 it was found that as age increased up to 168 h, sensitivity decreased. Weir et al.76 were able to develop modifications of the exponential and beta-Poisson that provided a good description of this dependency (as an exponential function of age). The models developed were for a modified exponential P(d) = 1 − exp[−k 0exp(k1τ )d]

(15)

and for a modified beta-Poisson ⎡ ⎤−α * d P(d) = 1 − ⎢1 + (21/ α * − 1)⎥ N *50 ⎣ ⎦ with α* = α0 + α1τ N *50 = N50,0exp(N50,1τ )

(16)

The modification of the exponential and beta-Poisson for size (eqs 13 and 14), as well as the modification for host age (τ) (eqs 15 and 16) resulted from taking the basic models (eqs 2 and 7) and expressing the basic parameters (k,α,N50) in terms of another explanatory variable (τ,pd). It can be hypothesized that this may be a reasonable approach for data sets involving other modulating factors of potency. Generation 3. Generation 3 models are defined as those which incorporate some element of the dynamics of the host1250


Critical Review

Environmental Science & Technology (like in most empirical kinetic studies), the observation of rate cannot be used absent other information to conclusively demonstrate mechanism. Nishiura critically reviewed the basis for a log-normal distribution for incubation time.81He concluded that there are a number of right skewed distributions that should be considered as candidates, but that alternative distributions have rarely been compared for goodness of fit. Further, he pointed out the difficulty of interpreting outbreak onset curves by incubation time unless the exposure period is short. This difficulty was demonstrated in an attempt to identify the incubation time distribution from the Milwaukee Cryptosporidium outbreak.82 To do this, it was necessary to convolute the incubation time distribution with an unknown exposure distribution.83 It was found that, while good fits to the time course of cases could be obtained, the distributional forms of the exposure distribution and incubation time distribution could not be uniquely determined solely from the epidemic curve. It is our hypothesis that information derived from dose− response-time studies using Generation 2 approaches could be used to reduce model uncertainty in describing epidemic curves. More recently, Lee et al. did a systematic review of a number of viral outbreaks (Norovirus, Rotavirus, Caliciviruses, Astrovirus) to develop incubation time curves.84 They found close estimates of the median incubation time (and also good fits) of log-normal, Erlang, gamma and Weibull distributions. Birth Death Models. In one sense, phenomenological incubation models can be consider as “macro” descriptors in that they attempt to depict disease progression in a population of hosts. The birth-death modeling is a complementary “micro” approach in which the progression of the pathogen in an individual host is being described. The basic framework of Birth-Death models for microbial infection were described by Armitage et al.:85 “the birth-death model assumes that the outcome is determined by successive random events which continually operate as long as even one organism remains viable in its host. Thus, in each small interval of time dt after inoculation, each organism still extant has a probability λdt of dividing, a probability μdt of dying, and a probability 1 − (λ − μ)dt of doing neither. When μ > λ, the host must invariably survive if the dose is substantially less than the final number of organisms causing the response” There are several important aspects of this framework that are important to note. First, is that there is an assumption that the birth and death parameters (μ and λ) are themselves constant and identical among microorganisms and hosts. Second, is that an end point (disease) results when the multiplication in vivo surpasses a particular number. Much of the development of birth death modeling of microbial infection has followed this set of assumptions, and the evolution will be discussed prior to returning to these assumptions as a basis for future work. Williams86 developed the distribution for the incubation times based on such a birth-death model. He noted that this is a dynamic version which has an underlying exponential dose− response distribution. The following relationships predict such a distribution.

f (τ ) =

⎛ 1 ⎞ ⎛ ⎛ τ ⎞⎞ exp⎜ − τ − e−τ ⎟I1⎜2 δ0exp⎜ − ⎟ ⎟ ⎠ ⎝ ⎝ 2⎠⎠ exp(δ0) − 1 ⎝ 2 δ0

(17)

where ⎛N⎞ τ = (μ − λ)t − ln⎜ ⎟ ⎝ν⎠

(18)

In eqs 17 and 18, the symbols are defined as follows: • A dimensionless initial average mean dose: δ0 = (d/ν) • A dimensionless ratio of growth and death parameters: ν = (μ/(μ − λ)) The function I1 is the modified Bessel function of the first kind, and N is the assumed in vivo microbial body burden beyond which the end point occurs. Williams noted that the form of this distribution was skewed right (such as also occurs with the log-normal distribution). However, he did not test the adequacy of this model to fit experimental data, nor compare this with prior approaches such as Sartwell’s.78 Shortly thereafter, Williams and Meynell87 examined data on in vivo bacterial counts during infection. They noted a limitation in the above framework: “discrepancies exist, the most serious being that the basic model predicts that if inoculated organisms succeed in increasing even slightly, they are virtually certain to increase to an infinite extent’. This would mean that every infection either ends almost as soon as it starts or is fatal, whereas in practice organisms often increase to a limited extent so that the host falls ill and then recovers.” They then proposed that the birth rate (μ) and death rate (λ) could be functions of either time since exposure, or current in vivo counts, and they demonstrated that this concurred with actual measurements of microorganisms in hosts during the course of infection. Morgan and Watts88 tested the Williams model against data reported by Sartwell78 and found that there was poor identifiability of parameters, although the model provided good fit. In particular, the time to effect data only provided good estimates of the difference between birth and death rates, μ − λ, and also the estimates of the threshold burden to effect (N), appeared implausible. They concluded that collection of additional data beyond time to effect would be necessary for a definitive assessment of this issue. Following the 2001 U.S. anthrax terrorism incident,89 there were attempts to apply a birth-death time to response model to the dynamics of development of cases. Anthrax is contracted inhallationally by respiring spores of Bacillus anthracis. The spores are then taken up by lung macrophage, where they germinate into vegetative cells and multiply.90 The multiplication results in the formation of bacterial toxins whose effect causes the disease state. Brookmeyer91 developed a model for the development of inhalation anthrax based on competitive processes of spore germination versus clearance (natural physiological processes removing foreign particles from the lungs). The underlying assumption was that once germination occurred, the time to disease occurrence would be fast such that a critical burden of microorganisms and toxins would be achieved. Based on this approach, the following model (which is a time dependent version of an exponential dose−response) was developed: 1251


Critical Review

Environmental Science & Technology ⎡ − dμ ⎤ g p(d , t ) = 1 − exp⎢ (1 − e−(μg + λs)t )⎥ ⎢⎣ μg + λs ⎥⎦

response-time experimental data. Subsequent work has shown a good fit to other experimental test data (e.g., refs 73,93−95). In a general sense, these classes of models can be written as an extension of eq 21. In more general terms, the extension of the exponential dose response model,

(19)

In eq 19, μg is the spore germination rate, and λs is the spore clearance rate. Note that at infinite time, this simplifies to an exponential dose response relationship (eq 2) where k = (μg/ (μg + λs)) . Wilkening23 modified the Brookmeyer91 model by convoluting the probability of spore germination with a second probability distribution of outgrowth of the vegetative microorganisms to reach a critical “threshold” at which infection would occur. Using this, he derived the following equation for the (expected) body burden as a function of initial dose and time, where μv is the growth rate of the vegetative cells. N (d , t ) =

μg d μg + ks

[e μv − exp( −(μg + ks)t )]

p(d , t ) = 1 − exp[−dk 0G(t ; θ )]

In the extension of the beta-Poisson model, ⎡ ⎢ p(d , t ) = 1 − ⎢1 + ⎣

(20)

λ (d , t ) =

exp(a0t + a1) exp((b0/t) + b1) exp(c0tc2 + c1) (d0/t) + d1

(

∂p(d , t ) ∂t

(23)

)

1 − p(d , t )

then it can be shown

(24)

(25) 94

for the exponential model (eq 22) that

k N = 0 g (t ; θ ) d k

(26)

where g(t;θ) is the corresponding density function to G(t;θ), i.e., g(t;θ) = ((∂G(t;θ))/(∂t)). For the beta-Poisson at low dose, it can also be shown that ⎡ α(21/ α − 1) ⎤ N ⎥ g (t ; θ ) =⎢ d N50k ⎦ ⎣

(27)

Both eqs 26 and 27 have a common form, predicting that a log−log plot will have a slope of unity, that is, ln((N/d)) = ln[g(t;θ)] + C. Hence if body burden versus time is available, and if g(t;θ) is available, the consistency of this model can be checked. This was tested94 using dose−response-time data on response of monkeys to inhaled aerosols of Francisella tularensis, where the function G was found to be Weibull. Future Work. There is remaining work to do in the development and validation of Generation 3 models. Additional confirmatory assessments of models and comparisons of approaches with real time dependent response data, coupled with in vivo measurements are desirable. One big question remains unanswered−what is the appropriate metric of the in vivo loading for particular organisms that characterizes “switchover” to the infected/ diseased/fatal state? The birth-death models have been developed using an underlying framework that once a body burden exceeds a “threshold”, the “switchover” will occur. The models of Huang21,93,94 are consistent with a proportionality between the body burden and the rate of “switchover”. These are not the only two possibilities−for some organisms, it may be the integration of cumulative numbers, such as ∫ Ndt would be best associated with the kinetics of the conversion happening, perhaps where the generation of toxin is responsible for pathology. These questions lead to the formulation of new classes of models, which we term “Beyond Generation 3”, discussed in the next section.

Table 1. Dependencies of Exponential and Beta-Poisson Dose Response Parameters on Time (From Huang21)

exponential exponential-reciprocal exponential-power inverse time

(21/ α

λ(d , t ) = kN

Where d and k are as defined in the exponential model, and G(t) is a monotonically increasing function that describes the likelihood that at least one sustaining colony would arise. This function is bounded by G(0) = 0 and G(∞) = 1. The approach of Gart presaged the work of Huang,21,93,94 who explicitly modified the exponential (eq 2) and betaPoisson (eq 7) models with a time dependent function. Gart’s approach suggests that the time dependency could usefully be modeled by making the exponential dose response parameter, k, time dependent. In the beta-Poisson model, with two parameters, it might be that both α and N50 are time dependent. Huang21 did an exploratory data analysis of a number of data sets where onset to end point was measured (in animals) following controlled dosing. For the data sets described by the beta-Poisson model (eq 7), it was found that α was not time dependent. The time dependencies for the exponential k and the beta-Poisson N50 parameters were found to fit one of the classes of models in Table 1. Applying these class of models to a variety of microorganisms (Yersina pestis, Bacillus anthracis, Francisella tularensis, and Mycobacterium tuberculosis) provided a good fit to dose−

functional form G(t)

N50 G(t ; θ )

⎤−α ⎥ − 1)⎥ ⎦

If the hazard rate function is assumed to be proportional to the instantaneous body burden of infectious organisms, that is,

(21)

functional descriptor (for k or N50)

d

where G(t;θ) is formally a cumulative probability distribution, with θ being a set of parameters of that distribution. With these definitions, a hazard rate function can be defined as94

This approach provided as good a fit as an empirical log-normal incubation model. Extending Dose Response Models to Include Explicit Time To Effect. Gart92 developed a time to effect model in which he considered that each original organism would colonize independently, and looked at the probability that one or more colonies would persist until time t (and having done so, cause an adverse effect). Using this approach, he arose at the follow equation, which can be viewed as an extension of the exponential model (eq 2). p(t , d) = 1 − exp[−kdG(t )]

(22)

1252


Critical Review

Environmental Science & Technology Beyond Generation 3. We reserve the term “Beyond Generation 3” for models that incorporate some aspects of quasimechanistic modeling of in vivo physiological processes. There are several threads of recent research that highlight elements that might be combined in future models of this category. By incorporating more mechanistic detail, it may be possible to describe dynamics, response to multiple doses, impacts of host susceptibility, etc. without resort to more empirical formulations (such as incubation time distributions). One overarching approach that has emerged is the “key events dose response framework” (KEDRF) developed by a working group organized by ILSI.96 In this approach, the following tasks are undertaken: • Describe the pathway of key events occurring between initial exposure and the end point of concern • Characterize the kinetics and dynamics of each of the key events • Assess what events appear to be control points or determinative in influencing the kinetics This approach could be useful in driving research needs or assessing the existence of biomarkers of intermediate effects. The application of this framework to microbial dose response was illustrated by the assessment of the relationship between dose of ingested Listeria monocytogenes in a pregnant mother and mortality of the fetus.97 The key events identified for this outcome were: • Pathogen survival in the upper GI tract • Establishment in the gut and attachment and take up by epithelium • Escape from phagosomes and transfer to phagocytes • Trans-placental transfer • Growth in fetus More recently, Roser et al.98 have developed a conceptual key events framework to the assessment of ear infections caused by Pseudomonas aeruginosa. The events identified in this case were: • Transport (via advection and motility) of microorganisms toward swimmers • Approach and reversible attachment to the epidermis • Colonization of follicles with pre-existing abrasions • Infection Based on this, a systematic program to gather data for the quantification of each step was proposed. As of this point, a full quantitative realization of the KEDRF, or models of such complexity have not been realized. However, models incorporating various aspects of host physiology in response to particular pathogens have been developed, which will be discussed below. For a particular pathogen-host-effect constellation, the integration of features from these sorts of models (and others to be developed) would need to occur. Deposition and Colonization. The first step in any microbial-host interaction is transport to a suitable site where multiplication (colonization) could occur. In the case of inhalation anthrax, this is a process of deposition to the lung alveoli where multiplication can occur. It was shown that the deposition process can be modeled as a Markov Chain−which essentially results in a first order partitioning between inhaled concentration and airway concentration in the alveoli.99 Once delivered to susceptible tissues in the alveoli, multiplication in the lung macrophage can be described by growth kinetic models.100

Interaction with Host Physiological Systems. Detailed modeling reflecting the interaction between the pathogen and host physiological systems have started to appear. Perhaps the simplest case is the description of skin pathogens. A risk assessment model of Staphylococcus aureus on skin was developed.101 As a first step, a growth model was developed of the following form: dN = −k1N exp( −k 2t ) + k 3N (Nmax − N ) dt

(28)

in eq 28, the first term represents a declining decay rate, and the second term represents a growth term of logistic form with a maximum population density (Nmax). This equation is evaluated from the initial condition, @t = 0,N = d. The parameters (k’s) were calibrated from human data. This was then used with an area under the curve model (AUC) and the risk of skin infection was found to be an exponential using AUC as a metric: p(t ) =

∫0

t

N (x)dx

(29)

102

Pujol et al. developed an elaboration of a birth death model in which they envision in vivo interaction of infectious microorganisms (P) and “immune particles” (I). The density of immune particles is governed by nonspecific increase, as well as induction, and the interaction between immune particles and microorganisms results in the inactivation of both. They then formulated a Markov Chain model, in which transition of a given host from state (P,I) is described by a matrix of probabilities as depicted by the graph in Figure 3.

Figure 3. Transition State Matrix for Pujol102 Model. P = in vivo pathogens, I = in vivo “immune particles”. Movement from P to P + 1 indicates an increase in number of pathogens by one, and movement from I to I + 1 indicates an increase in number of “immune particles” by one. αp: arrival rate of new pathogens (from dosing); αI: non specific increase rate in new immune particles; λI: pathogen proportional induction rate of I; θP: pathogen reproduction rate; γI: natural death rate of immune particles; PIδP: loss rate of pathogens due to specific interaction with immune particles; PIδI: loss rate of immune particles due to specific interaction with pathogens.

In this figure, the parameters are defined as follows: This model was then used to describe dose response, hosts in which the pathogen number increased uncontrollably were deemed “positive”. Fits as good as that of Generation 1 models were obtained, however this class of models has a greater number of adjustable parameters. One of the findings of this model is the important effect of dose rate (αp) on effect. 1253


Critical Review


Figure 4. Schematic flow diagram of SEIR disease transmission model.

or use of strains expressing different proteins (e.g., florescence proteins, antibiotic resistance markers). With advances in molecular biology, the spectrum of methods potentially useable for this application will undoubtedly increase. Coupling with Transmission Models. The development of dose response modeling up until Generation 3 focused on more detailed elucidation of what happens in an individual host. However, individual hosts exist in a population, and the consequences of exposure to a population of hosts must consider the interaction between hosts  which is particularly important when the infectious agent is contagious. This approach uses the “microparasite” framework as devised by Anderson and May107 and as such do not include pathogen multiplication and change in life cycle outside the host. These factors may be important for some agents. However, as noted,107 ‘The division into microparasites and macroparasites, whether made on biological or mathematical grounds, is necessarily a rough one. The distinction essentially corresponds to the extremes of a continuum.” A number of infectious diseases of interest in food, water and airborne exposures are contagious−i.e., some individuals previously exposed can transmit the agent to new individuals (who were not exposed to an initial source). These agents include the viruses, such as norovirus, bacteria, such as E. coli O157:H7 and protozoa, such as Cryptosporidium. Classical mathematical epidemiology107,108 describes members of a population (of hosts) as being in one of several classes: Susceptible (S) These are individuals who are capable of becoming infected by the pathogen of interest.. Exposed (E) These are individuals who will become infectious. The corresponds to the proportion, “p” of those who have experienced a dose.. Infectious (I) These are individuals, symptomatic or not, in whom the course of infection has proceeded to the point where they can spread disease to others−i.e., they are contagious.. Recovered (R) These are individuals who are no longer contagious, and are no longer capable of becoming infected. Elaborations of the basic model, include a rate at which these individuals lose immunity and once again become susceptible. Individuals may move from one class to another in accordance with the flow diagram shown in Figure 4. This SEIR model can be described by a set of differential equations107 as follows:

Clearly the expansion of models of this class to incorporate more sophisticated aspects of pathogen interaction with the host immune system offer promise. In particular, if certain properties of the host immune system are determined to be relatively pathogen independent, the ability to generalize across diverse pathogens will be increased. Promise of New Methods and Approaches. The availability of new methods of experimentation coupled with detailed mathematical modeling will allow further development of beyond Generation 3 models. This could allow the full quantification of elements of the KEDRF. This could mirror the development of computational toxicology in chemical risk assessment103 in which detailed physiological and metabolic information is used to assess toxicity of differing organic stressors. The potential is that common features and parameters of particular in vivo processes may be similar across different pathogens, thereby allowing more rapid assessment of the dose response of new infectious agents. A pioneering study was conducted by Grant et al.104 In this study, mice were injected intravenously with Salmonella enterica and postinfection colonization of the blood, liver and spleen (the principal organs found to accumulate the pathogen) determined. They were able to fit a compartment model of the following form to the dynamics of bacterial numbers (after a bolus dose)in the three organs: dnL = μL nL + n0ηLexp[−t(ηL + ηs)] dt dnS = μS nS + n0ηS exp[−t(ηL + ηs)] dt

(30)

In this equation, μL and μS are the net growth rates in the liver and spleen, and ηL and ηS are the transfer rates from the blood to the liver and spleen. The term n0 exp[−t(ηL + ηs)] represents the blood concentration at time t (note there is no growth or death assumed in the blood). This was found to provide a good description of the in vivo dynamics of the pathogen. These authors also coinfected mice with “bar coded” strains of the same S. enterica. In this approach, short segments of non coding DNA are added into non coding regions of the base DNA of the parent strain. The “bar coded” strains could then be individually monitored by quantitative polymerase chain reaction (qPCR) during the course of the infection. This enabled the stochastic nature of the early infection process to be discerned, and the organ-specific birth and death rates of the pathogens to be estimated. Kaiser et al.105 used similarly tagged strains to study infection of Salmonella typhimurium in mice−with a focus on the initial infection of the cecal lymph nodes. A continuous time Markov Chain process was defined to model the number of microorganisms in these lymph nodes assuming a constant arrival rate from the gut, a division rate in the lymph node and a clearance rate from the lymph node. Experimental data from infection with the tagged Salmonella was then used to fit the parameters of this model−and good correspondence between the fitted models and observations could be obtained. There are other such “tagging” methods that can be employed. These include106 use of nonreplicating plasmids, 1254

dS = −βSI dt

(31)

dE = βSI − γE dt

(32)

dI = γE − δI dt

(33)

dR = δI dt

(34) DOI: 10.1021/es504422q Environ. Sci. Technol. 2015, 49, 1245−1259

Critical Review

Environmental Science & Technology This assumes homogeneously mixed classes, no birth/ immigration or death/emigration (note that (dS/dt) + (dE/ dt) + (dI/dt) + (dR/dt) = 0), and no return of the recovered to the susceptible class. β is the force of infection resulting from the contagious mixing between susceptible and infectious populations, γ is the rate of conversion to the infectious state− essentially the reciprocal of the mean of the incubation time, and δ is the rate of recovery of infectiousness−essentially the reciprocal of the mean of the duration of infectiousness. Particularly with prolonged time, it is possible that some of the recovered population may once again become susceptible (i.e., R → S) which would require additional rate terms in eqs 31 and 34. It is noteworthy that this classical formulation fails to include primary exposure via contact with the contamination (water, air, etc.) itself. The extension of this model to incorporate such primary exposures would directly couple with the dose− response (and extensions) discussed above. This will be discussed below. In such cases, it may be important to also include models of the fate and transport of the pathogen in the environmental reservoirs. Examples include the modeling of the outbreak of cholera in Haiti following earthquake recovery109 as well as viruses and protozoans where there is a recirculating exposure between wastewater and water in a population.110,111 Parenthetically, β is a lumped parameter that must embody the dose exchanged during a contact event as well as the dose− response function. The term γ is clearly directly related to incubation time. The formulation in eqs 32 and 33 imply an exponential incubation time distribution, which is clearly inconsistent with much of the prior data, for example Sartwell.78 Eisenberg et al.110 were apparently the first to modify the conversion of susceptible to exposed individuals for the direct acquisition of infection via a dose dependent term. In their approach, eqs 31 and 32 get modified as follows, where ḋ is the dose rate (dose per unit time) experienced by the individuals. dS = −βSI − αd ̇ dt

approaches, incubation times frequently are much different than this. At least two approaches to this issue exist and further comparison is needed: 1 Eisenberg and colleagues112,113 have expanded the exposed state to multiple stages, for example, E1, E2, etc. This results in a gamma distribution of the overall incubation time, with a rate of conversion between stages,for example, E1 → E2 equal to γ1, γ2, etc. with all γi equal. Using this approach, the transmission model remains a series of ordinary differential equations. If the homogeneous population assumption is relaxed, either by spatial, age, or other susceptibility factors, then the overall transmission model assumes a more complex structure. 2 A formal incubation time distribution can be used, either empirically such as the use of the log-normal,78 or from that estimated from “Generation 2” Models. In this case, the transmission models result in a coupled set of integro-differential equations.83 Research Needs. There clearly remains the opportunity to test the framework(s) of dose response models against new varieties of microorganisms. In particular, the following classes of pathogens are under-represented in the knowledge base: • Helminths are of importance in management of biosolids and reuse of wastewaters−especially in developing countries.114−116 However, there is only a dose− response curve available for Ascaris,117 although QMRA is being recognized as an important tool for risk management internationally.118 • Fungal pathogens have not received extensive attention. Cryptococcus is emerging as a human infection that may be associated with water.119,120 A variety of fungal pathogens have been found to be important potential pathogens via indoor environmental exposures.121−125 Dose−response information is nonexistent for these agents. • While there are robust dose−response models available for Giardia and Cryptosporidium,2,62,126−128 there is increasing recognition of the importance of other protozoans.129 In particular water-borne or water associated cases (frequently fatal) have occurred with Naegleria130−134 and Acanthamoeba.135−139 Dose−response information for these agents is needed.. • The 2014 outbreak of Ebola in West Africa140,141 have highlighted the need for dose response information for this organism, as well as other less studied pathogens. The efficient determine of parametric and model uncertainty, especially with sparse data, needs more attention. In addition, more deliberate comparisons between animal derived dose response relationships and human illnesses (e.g., during outbreaks) would bolster the utility of animal models. Dose−response models to date have focused on single pathogens. However, in realistic situations (e.g., contamination of water by sewage) the possibility for concomitant exposure to multiple pathogens are exist. Whether there can be interactions in the responses is not known. There have been a number of outbreaks involving multiple pathogens in both water142−145 and food.146,147 Both controlled multiorganism dosing experiments, and retrospective analyses of outbreaks with good exposure data would be important to develop a theory of exposures to multiple agents.

(35)

dE = βSI + αd ̇ − γE (36) dt The term αḋ accounts for the acquisition of the exposure (leading to infectious state). This form may be obtained by differentiating either the exponential (eq 2) or beta-Poisson (eq 7) with respect to time and taking the limit at low dose. This is equivalent to assuming that the proportion of successfully exposed individuals is small. This overall approach to incorporate microorganisms explicitly into transmission models is in a state of development. Some specific areas for further development are • When the dose per exposure is not small (i.e., it is above the linear region in an exponential or beta-Poisson formulation), the term α in eqs 35 and 36 need to be revised. This is particularly the case in repeated or ongoing exposures−in such cases the transmission models will need to be coupled to other models of the “Beyond Generation 3” category. • The use of a constant, γ for the conversion rate between exposed and infectious is equivalent to assuming an exponential incubation time distribution. As noted above, and particularly in the discussion of “Generation 2” 1255


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While Generation 1 models have been validated by showing consistency with actual disease oubreaks, there have been no studies of actual disease outbreaks that have well characterized exposure. Generation 1 models would be predictive of the overall attack rate, while Generation 2 models would be predictive of the distribution of time of onset−a more powerful test. The advancement of Generation 3 and beyond models would need to rely on the detailed studies of in vivo pathogen dynamics and interactions with the immune system. A coupling of the tools of microbial risk assessment with the growing field of systems biology148 would be a fruitful approach.copy

■

AUTHOR INFORMATION

Corresponding Author

*Phone: 215 895 2283; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Work in this area is the result of a long period of collaboration and research with multiple lines of support. My long time collaborators, Joan Rose (Michigan State University) and Charles Gerba (University of Arizona) have been incredibly important. I have been fortunate to have many many students whose intellectual interaction have been influential in shaping my ideas. For some of the work developing dose-response models, funding was received by AWWARF, ILSI, the USEPA and Department of Homeland Security, and the American Cleaning Council.

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REFERENCES

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