Exposure-response function

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Exposure-response function (ERF) (or exposure-response relationship) is the relationship between the exposure of a given organism, system, or (sub)population to an agent in a specific pattern during a given time and the magnitude of a continuously graded effect to that organism, system, or (sub)population.

This term has several related terms that may have slightly different meaning. Effect and response are interchangeable words. Also the word function is used instead of relationship. In Opasnet, we use the term exposure-response function (or ERF) as the generic term for different kinds of relationships. Often the exposure metric is more specifically defined in an alternative term. Two common examples:

Concentration-effect relationship: Relationship between the exposure, expressed in concentration, of a given organism, system, or (sub)population to an agent in a specific pattern during a given time and the magnitude of a continuously graded effect to that organism, system, or (sub)population. The concentration is measured at a defined site. ^[1]

Dose-response relationship: Relationship between the amount of an agent administered to, taken up by, or absorbed by an organism, system, or (sub)population and the change developed in that organism, system,or (sub)population in reaction to the agent. ^[1]<section end=glossary />

Question

What is such a representation for ERF that it fulfills the following criteria?

It is widely applicable to all kinds of agents, exposures, and responses.
A single ERF is widely applicable, within its domain, to different situations and populations.
It is mathematically clear so that impact calculations can be operationalised based on it.

Answer

An ERF is a mathematical construct describing the relationship between a response and an exposure. In the general form, it is described as a probability.

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library(OpasnetUtils)
#library(plyr)

objects.latest("Op_en2031", "initiate")
oprint(summary(EvalOutput(ERF)))

Rationale

Equations

By nature, a response to an agent occurs at individual level (with some exceptions such as agents affecting herd immunity). Thus, the individual response should be the basic unit for ERF. If individual variation is of no interest, a population ERF can simply be expressed as the average of individual ERFs.

There are several different functions that may be used. These are defined here. In all equations, these variables are used:

RR: the relative risk for unit exposure difference. Note that β = ln(RR).
E: exposure level of the individual
B: background exposure level that is considered negligible or lowest achievable. Exposures below B are not considered.
T: threshold exposure level below which no impact will occur
RR' is the relative risk for the actual exposure
I_max is the maximal relative impact
ED₅₀ is the dose that causes 50 % of the maximal impact.

NOTE! ED₅₀ parameter is given in the Threshold column, and I_max parameter is given in the ERF or Result column.

Approaches relative to background disease risk

Relative risk (RR): describes the relative risk compared with a reference exposure. The actual number of cases is calculated with the equation below. (E > T) is 1 if E is greater than T and 0 otherwise.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RR' = (e^{ln(RR) (E - max(B, T))} - 1) (E > T) + 1 }

Relative Hill: Relative Hill means an ERF function derived from Hill's plot:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle RR' = \frac{E \times I_{max}}{E + ED_{50}}}

Approaches independent of disease risk

Exposure-response slope (ERS): A linear relationship where ERS defines the slope of the exposure-response line. Typically the intercept is assumed to be 0. Response metric MUST be defined as it varies from one case to another.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = ERS \times E}

Cancer slope factor (CSF): A linear relationship between constant lifetime exposure (typically in units mg/kg/d) and lifetime probability of cancer.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(I) = CSF \times E}

Step function: Assumes that exposure at certain range is not a hazard, while exposure outside that range is. Threshold and ERF parameters are used to give the lower and upper ends of the range, respectively. The Step function includes tolerable daily intake (TDI), recommended daily intake (RDI), acceptable daily intake (ADI), no-observed-adverse-effect level (NOAEL) and related, often administrative limits. The value is TRUE (or 1) if exposure fails to meet the recommendation and FALSE (or 0) otherwise.

Pages with standardised ERF tables

If a page has a standardised ERF data table (see an example above), the data can be automatically read and used by an R code and combined with other ERF tables. Therefore it is preferable to use the standard format. It makes modelling much easier and also enables an easy way to add more endpoints to assessments, if there are interesting exposures and available ERF tables.

The tables can contain the following columns (obligatory columns are in bold):

Obs (automatic)
Exposure agent (index)
Response (index)
Population (index)
Age (index)
Sex (index)
Exposure (index)
Exposure unit (index)
ER function (index)
Exposure metric (index)
Scaling (index)
Observation (hidden), containing two locations:
- Threshold
- ERF
Result
Description (description, there may be any number of description columns because they are not stored in the database)

Note! Spaces in column names will be replaced with "_" to avoid problems in the code.

**Standardised ERF tables that have been combined on this page.**
Page	Ident	Code name	Description
ERFs of environmental pollutants	Op_en5827	ERF_env2	Contains ERFs for radon, PM2.5, noise, chlorinated byproducts in drinking water, arsenic, dampness in buildings, formaldehyde, fluoride, ozone, lead, dioxin, quartz dust, asbestos.
ERF of omega-3 fatty acids	Op_en5830	ERF_omega32	Contains ERFs for Omega3 fatty acids.
ERF of methylmercury	Op_en5825	ERF_mehg2	Contains ERFs for MeHg.
ERF of dioxin	Op_en5823	ERF_diox2	Contains ERFs for dioxin TEQ.
ERFs of vitamins	Op_en6866	ERF_vit2	Contains ERF for vitamin D.
ERF of waterborne microbes	Op_en7948	ERF_micr2	Contains ERF for vitamin D.
ERF of PFAS	Op_en7974	ERF_pfas	Contains ERF for perfluorinated alkyl substances

Calculations

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# This is code Op_en2031/subgrouping on page [[Exposure-response function]]

library(OpasnetUtils)

#' subgrouping takes a subgroup column and creates new columns from that.
#' Column name and location are separated by ":" and subgroup by ";". E.g. Age: 10 years; Gender: Female
#' @param dat data.frame that contains columns Subgroup and - if type is used - Type
#' @param subg name for subgroup column

subgrouping <- function(dat, subg = "Subgroup") {
  out <- dat
  tmp <- trimws(as.character(out[[subg]]))
  tmp <- gsub(" *: *", ":", tmp)
  tmp <- gsub(" *; *", ";", tmp)
  tmp <- ifelse(tmp=="" | is.na(tmp), " : ", tmp)
  tmp <- lapply(strsplit(tmp,";"), function(x) {strsplit(x, ":")})
  out <- data.frame(dummy=rep(NA, length(tmp)))
  for(i in 1:length(tmp)) {
    for(j in 1:length(tmp[[i]])) {
      if(!tmp[[i]][[j]][1] %in% colnames(out)) {
        out[[tmp[[i]][[j]][1]]] <- NA # Create empty column if not exists
      }
      out[[tmp[[i]][[j]][1]]][i] <- tmp[[i]][[j]][2]
    }
  }
  out <- cbind(dat,out)
  out <- out[!colnames(out) %in% c(subg, "dummy"," ")]
  return(out)
}

#' prepare adjusts the data table for ovariables. Requires function subgrouping from code Op_en2031/subgrouping on page [[Exposure-response function]]
#' @param dat data.frame
#' @param type type of data that is used. Must match content in column Type
#' @param drop columns to remove
#' @return data.frame
prepare <- function(dat, type=NULL, drop=NULL) {
  out <- dat
  if(!is.null(type)) out <- out[out$Type %in% type , ]
  if(!is.null(drop)) out <- out[!colnames(out) %in% drop]
  return(subgrouping(out))
}

objects.store(subgrouping, prepare)
cat("Functions subgrouping, prepare stored.\n")

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# This is code Op_en2031/ERF2 on page [[Exposure-response function]]

library(OpasnetUtils)

ERF <- Ovariable(
  "ERF",
  dependencies = data.frame(Name = c(
    "ERF_env", # [[ERF of environmental pollutants]]
    "ERF_omega3", # [[ERF_of_omega-3_fatty_acids]]
    "ERF_mehg", # [[ERF_of_methylmercury]]
    "ERF_diox", # [[ERF_of_dioxin]]
    "ERF_vit", # [[ERF_of_dioxin]]
    "ERF_micr", # [[ERF of waterborne microbes]]
    "ERF_pfas", # [[ERF of PFAS]]
    "ERFchoice", # an ovariable for choosing case-specific Responses, Exposure_units or ER_functions
    "subgrouping" # takes a subgroup column and creates new columns from that.
  ), Ident = c(
    "Op_en5827/ERF_env2",
    "Op_en5830/ERF_omega32",
    "Op_en5825/ERF_mehg2",
    "Op_en5823/ERF_diox2",
    "Op_en6866/ERF_vit2",
    "Op_en7948/ERF_micr2",
    "Op_en7974/ERF_pfas",
    "Op_en2031/ERFchoice",
    "Op_en2031/subgrouping"
  )),
  formula = function(...) {
    
    out <- OpasnetUtils::combine(
      ERF_env, 
      ERF_omega3, 
      ERF_mehg, 
      ERF_diox,
      ERF_vit,
      ERF_micr,
      ERF_pfas
    )
    out$Exposure_unit <- NULL # These are for information only and should NOT be used in calculations.
    out$Exposure <- NULL

    if("Iter" %in% colnames(out@output)) {
      out@output <- fillna(out@output, marginals = "Iter")} # To make sure that deterministic ERFs work properly.
    out <- unkeep(out * ERFchoice, cols=c("ERFchoiceSource", "ERFchoiceResult")) #technical nuisance removed
    marginals <- colnames(out@output)[out@marginal]
    coln <- colnames(out@output)
    out@output <- subgrouping(out@output, subg = "Subgroup")
    out@marginal <- colnames(out@output) %in% union(marginals, setdiff(colnames(out@output), coln))
    
    return(out)                         
  }
)

objects.store(ERF)
cat("Ovariable ERF stored.\n")

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# This is code Op_en2031/initiate on page [[Exposure-response function]]

library(OpasnetUtils)

#' subgrouping takes a subgroup column and creates new columns from that.
#' Column name and location are separated by ":" and subgroup by ";". E.g. Age: 10 years; Gender: Female
#' @param dat data.frame that contains columns Subgroup and - if type is used - Type
#' @param subg name for subgroup column

subgrouping <- function(dat, subg = "Subgroup") {
  out <- dat
  tmp <- trimws(as.character(out[[subg]]))
  tmp <- gsub(" *: *", ":", tmp)
  tmp <- gsub(" *; *", ";", tmp)
  tmp <- ifelse(tmp=="" | is.na(tmp), " : ", tmp)
  tmp <- lapply(strsplit(tmp,";"), function(x) {strsplit(x, ":")})
  out <- data.frame(dummy=rep(NA, length(tmp)))
  for(i in 1:length(tmp)) {
    for(j in 1:length(tmp[[i]])) {
      if(!tmp[[i]][[j]][1] %in% colnames(out)) {
        out[[tmp[[i]][[j]][1]]] <- NA # Create empty column if not exists
      }
      out[[tmp[[i]][[j]][1]]][i] <- tmp[[i]][[j]][2]
    }
  }
  out <- cbind(dat,out)
  out <- out[!colnames(out) %in% c(subg, "dummy"," ")]
  return(out)
}

ERF <- Ovariable(
  "ERF",
  dependencies = data.frame(Name = c(
    "ERF_env", # [[ERF of environmental pollutants]]
    "ERF_omega3", # [[ERF_of_omega-3_fatty_acids]]
    "ERF_mehg", # [[ERF_of_methylmercury]]
    "ERF_diox", # [[ERF_of_dioxin]]
    "ERF_vit", # [[ERF_of_dioxin]]
    "ERF_micr", # [[ERF of waterborne microbes]]
    "ERFchoice" # an ovariable for choosing case-specific Responses, Exposure_units or ER_functions
  ), Ident = c(
    "Op_en5827/initiate",
    "Op_en5830/initiate",
    "Op_en5825/initiate",
    "Op_en5823/initiate",
    "Op_en6866/initiate",
    "Op_en7948/ERF_micr",
    "Op_en2031/ERFchoice"
  )),
  formula = function(...) {
    
    out <- OpasnetUtils::combine(
      ERF_env, 
      ERF_omega3, 
      ERF_mehg, 
      ERF_diox,
      ERF_vit,
      ERF_micr
    )
    out@marginal[colnames(out@output) == "Exposure_unit"] <- FALSE
    if("Iter" %in% colnames(out@output)) {
      out@output <- fillna(out@output, marginals = "Iter")} # To make sure that deterministic ERFs work properly.
    out <- unkeep(out * ERFchoice, cols=c("ERFchoiceSource", "ERFchoiceResult")) #technical nuisance removed
    marginals <- colnames(out@output)[out@marginal]
    coln <- colnames(out@output)
    out@output <- subgrouping(out@output, subg = "Subgroup")
    out@marginal <- colnames(out@output) %in% union(marginals, setdiff(colnames(out@output), coln))
    
    return(out)                         
  }
)

threshold <- Ovariable(
  "threshold",
  dependencies = data.frame(Name = c(
    "threshold_env", # [[ERF of environmental pollutants]]
    "threshold_omega3", # [[ERF_of_omega-3_fatty_acids]]
    "threshold_mehg", # [[ERF_of_methylmercury]]
    "threshold_diox", # [[ERF_of_dioxin]]
    "threshold_vit", # [[ERFs of vitamins]]
    "threshold_micr", # [[ERF of waterborne microbes]]
    "ERFchoice" # an ovariable for choosing case-specific Responses, Exposure_units or ER_functions
  ), Ident = c(
    "Op_en5827/initiate",
    "Op_en5830/initiate",
    "Op_en5825/initiate",
    "Op_en5823/initiate",
    "Op_en6866/initiate",
    "Op_en7948/ERF_micr",
    "Op_en2031/ERFchoice"
  )),
  formula = function(...) {
    
    out <- OpasnetUtils::combine(
      threshold_env, 
      threshold_omega3, 
      threshold_mehg, 
      threshold_diox,
      threshold_vit,
      threshold_micr
    )
    
    out@marginal[colnames(out@output) == "Exposure_unit"] <- FALSE
    if("Iter" %in% colnames(out@output)) {
      out@output <- fillna(out@output, marginals = "Iter")} # To make sure that deterministic ERFs work properly.
    out <- unkeep(out * ERFchoice, cols=c("ERFchoiceSource", "ERFchoiceResult")) #technical nuisance removed
    marginals <- colnames(out@output)[out@marginal]
    coln <- colnames(out@output)
    out@output <- subgrouping(out@output, subg = "Subgroup")
    out@marginal <- colnames(out@output) %in% union(marginals, setdiff(colnames(out@output), coln))
    
    return(out)                         
  }
)

objects.store(ERF, threshold, subgrouping)
cat("Ovariables ERF, threshold and function subgrouping stored. Page: Op_en2031, code: initiate.\n")

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# This is code Op_en2031/ERFchoice on page [[Exposure-response function]]

library(OpasnetUtils)

ERFchoice <- Ovariable(
  "ERFchoice",
  dependencies = data.frame(Name="exposure"),
  formula = function(...) {
    out <- cbind(unique(exposure@output["Exposure_agent"]),Result=1)
    return(out)
  }
)

objects.store(ERFchoice)
cat("Object ERFchoice stored.\n")

Technical issues

In Opasnet, the use of term Exposure-response function (ERF) is recommended. The pages about ERFs should be named: "ERF of <agent> on <endpoint> in <population>."
ERFs are typically variables, and they should be categorised to Category:Exposure-response functions
In the definition of a variable, an ERF data is described as a t2b table under subheading Data. The actual ovariable that is used in models is defined in an "initiate" code under subheading Calculations.

References

↑ ^1.0 ^1.1 IPCS_WHO

Weihsueh A. Chiu and Wout Slob. (2015) A Unified Probabilistic Framework for Dose–Response Assessment of Human Health Effects. Environ Health Perspect; doi:10.1289/ehp.1409385

[IPCS_WHO-1] 1.0 ^1.1 IPCS_WHO

[1]

Exposure-response function

Contents

Question

Answer

Rationale

Equations

Pages with standardised ERF tables

Calculations

Technical issues

See also

References

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