Chemical contaminants in food arise from a variety of sources — environmental pollution, agricultural inputs, food processing, natural occurrence in soil and water, or microbial growth on crops. Unlike biological hazards, they typically act through chronic, low-level dietary exposure rather than acute infection events, and their dose–response relationships are generally characterised using toxicological benchmark doses (BMD/BMDL) derived from animal studies or epidemiological data, rather than human challenge trials. Two broad modelling approaches are used throughout this section. For genotoxic carcinogens (arsenic, aflatoxins, acrylamide, TCDD), where no safe threshold is assumed, a linear low-dose extrapolation from the BMDL yields an incremental lifetime cancer risk (ILCR). For non-genotoxic or threshold-based effects (DON, fumonisins, OTA, ZEN), a MOE (MOE = health-based guidance value/dose) is computed; MOE < 1 indicates exceedance of the acceptable level. For some components (Cadmium), an actual dose-response exists
The eleven chemical hazards listed below were prioritised because of their relevance to dietary transitions involving animal-source food (ASF) and plant-source food (PSF) substitutions, their frequency of exceedance in dietary exposure assessments, and the availability of quantitative dose–response data suitable for use in QMRA.
Rank
Hazard
Why priority in ASF/PSF substitutions
Main food pathways to track
1
Inorganic arsenic (iAs)
Strong and consistent exceedance; sensitive to cereal/rice-based substitutions.
Rice/cereals, water, some seafood
2
Cadmium
Relevant in plant-forward substitutions; recurrent concern in biomonitoring/risk assessments.
Persistent bioaccumulative contaminants; relevant to animal fat/fish pathways in ASF scenarios.
Fish/seafood, animal fats, dairy, eggs, meat
7
DON (deoxynivalenol)
Linked to cereal-heavy diets common in PSF substitutions.
Wheat/maize/oats and cereal products
8
Fumonisins (FB1+FB2+FB3+FB4)
High in maize-dependent diets; important where maize intake increases.
Maize and maize products
9
Ochratoxin A (OTA)
Important chronic nephrotoxic concern in cereal-based and stored foods.
Cereals and cereal products, coffee, cocoa, dried vine fruits, wine, some spices
10
Zearalenone (ZEN)
Estrogenic mycotoxin relevant in cereal-rich substitution scenarios.
Maize and other cereals, cereal-based foods
11
Lead
Lower than historical levels but still important as no safe threshold.
Water, cereals, vegetables, mixed sources
General Note
We want to estimate the number of additional cases in a given population. Studies provides various kind of outputs. Given \(p(d)\) the probability of adverse effect for a given dose of a contaminant, \(p(0)\) the baseline probability of adverse effect (in the absence of contaminant, in the studied population), we have:
Additional Risk: \(AR(d) = p(d)-p(0)\),
Extra Risk is \(ER(d)= \frac{p(d)-p(0)}{1-p(0)}\). This is the probability of disease attributable to the additional exposure, conditional on not already being a case at baseline.
We assume that the ER is independent of the population. So, for a given population with \(p_{pop}(0)\), we will use:
\(\Delta = N \times (1-p_{pop}(0)) \times ER(d)\) if we have an \(ER(d)\);
\(\Delta = N \times (1-p_{pop}(0)) \times \frac{AR(d)}{1-p(0)}\) if we have an \(AR(d)\)
where \(\Delta\) is the additional number of cases.
For some situations (IQ, weight birth), other evaluations are possible (see below).
We will evaluate the additional number of cases per year, by dividing the risk over a lifetime by the period of exposure or the lifetime (EPA uses 70 years).
Also, each function will have the input in mg/kg bw.
Arsenic (OK)
Inorganic arsenic (iAs) is a naturally occurring metalloid classified as a Group 1 human carcinogen (Centre international de recherche sur le cancer 2012). Chronic dietary and drinking water exposure is associated with a broad spectrum of adverse health effects, including cancers (bladder, lung, skin, kidney), cardiovascular disease (ischaemic heart disease), diabetes, and developmental neurotoxicity and fetal growth restriction in exposed pregnancies. The dose–response models below are derived from the (2025) IRIS Toxicological Review of Inorganic Arsenic (CASRN 7440-38-2), which represents the most comprehensive and current assessment available. Each endpoint is modeled separately, as the underlying biological mechanisms and key studies differ. The input metric is chronic dietary iAs intake in µg/kg bw/day; where studies use a different dose metric (e.g. urinary arsenic, drinking water concentration), the conversion is made explicit.
Fetal, Newborn and Infant Health Outcomes
See Section 4-4 of the EPA 2025 report (starting page 4-54)
The adverse effect is a baby born weighing less than 2,500 g (5.5 lbs). In the US: the mean baby weight at birth is \(m(0) = 3,261.6 g\) (standard deviation se = 590.7g); P(0) = 8.27% of all lives US births were under 2,500 g1. From Kile et al. (2016) (used to derive the BMDL05), the estimate of β (rescaled by EPA to the normal scale) is -3.91 g decrease in birth weight per µg/L drinking water (the unit is per L of drinking water, see page 4-59). Its CI 95% is [\(\beta_{0.025} =-5.13,\beta_{0.975} =-2.7\)] g per µg/L drinking water. We estimated SEβ from the CI95% \(\left( SE_\beta = \frac{abs\left( - 5.13 - ( - 2.7) \right)}{\left( {2 \times \Phi}_{0.975} \right)} = 0.62 \right)\).
Estimate of the Extra Risk from an exposure E in µg/kg bw/day:
Conversion of the Exposure from µg/kg bw/day in “µg/L drinking water” by subtracting 0.05 µg/kg bw/d (median US dietary background dose) and dividing the result by 0.01 L/kg bw/d (mean US water consumption rate for pregnant women). \(E_{L} = \ \frac{(E - 0.05)}{0.01}\frac{\mu g}{L}\) drinking water.
Estimate of the new mean of baby weight at birth \(m(d) = m(0) + \beta E_{L}\)
Estimate of the new proportion of babies below 2,500g \(P(d) = \Phi\left( \frac{2500 - m(d)}{590.7} \right)\)
Estimate of the additional risk: \(AR = P(d) - P(0)\)
Check: for a dose of 0.237 µg/kg bw/day, the extra risk (95%) should be 5%, the additional risk 2.32%.
Code
Input: exposure in mg/kg. If available to adapt to a given country: mean BW and sd BW at birth that will provides \(p(0)\).
Output: additional risk of low birth weight (per individual).
DRiAsLowbirth <-function(dose, nUnc =0, meanBW =3261.6, SeBW =590.7){# dose: in mg/kg bw# nUnc: return the best estimate if 0, the number of iterations otherwise# meanBW: mean BW at birth in the population (g)# SeBW: SE BW at birth in the population ExposureWater <- (dose*1000-0.05)/0.01# ug/L drinking water SE <-abs(-5.13+2.7)/(qnorm(.975)-qnorm(0.025)) Slope <-if(nUnc ==0) -3.91elsernorm(nUnc, -3.91, SE) NewMean <- meanBW + Slope * ExposureWater OldRisk <-pnorm(2500, meanBW, SeBW) NewRisk <-pnorm(2500, NewMean, SeBW)# Note: the AR is estimated directly from the p(0) in the population AR <-pmax((NewRisk-OldRisk), 0)return(AR)}# Additinal risk for a dose ot 0.237 µg/kg bw/dayDRiAsLowbirth(0.237/1000)
[1] 0.02325693
summary(DRiAsLowbirth(0.237/1000, nUnc =1E5))
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.005759 0.020599 0.023267 0.023312 0.025974 0.040647
For a dose of 0.237 µg/kg bw/day, the extra risk of low birth weight is 0.901.
See Section 4-5 of the EPA 2025 report (starting page 4-60)
The adverse effect is an infant with mild intellectual disability defined, here, as an IQ <70 points.
The IQ is assumed to follow a normal distribution with mean m(0) = 100 and se = 15. The proportion of individuals with IQ <70 is estimated by EPA to 2.5% of the population2. From EPA’s meta-analysis, the estimate of β (rescaled by EPA to the normal scale) is -0.19 decrease in IQ point per µg/L maternal urinary total As (tAs). Its CI 95% is [\(\beta_{0.025} =\)-0.39, \(\beta_{0.975} =\)+0.01] IQ point per µg/L maternal urinary tAs. We estimated SEβ from the CI95% \(\left( SE_\beta = \frac{abs\left( 0.01 - ( - 0.39) \right)}{\left( {2 \times \Phi}_{0.975} \right)} = 0.10 \right)\).
The exposure (µg/kg bw/d) need to be translated in µg/L maternal urinary tAs. El-Masri-Kenyon PBPK model ((El-Masri et al. 2018)) establishes an approximate 1:1 empirical relationship between the total urinary arsenic mg/day excretion and the mg/day oral consumption of inorganic arsenic (see EPA report, page 3-3)
Estimate of the Extra Risk form an exposure E in µg/kg bw/day:
Conversion of E from µg/kg bw/day to µg/day by multiplying E by 76 kg, the average weight of women at pregnancy.
Conversion to EL µg/L maternal urinary tAs El-Masri-Kenyon PBPK model (1:1 empirical relationship between the total urinary arsenic mg/day excretion and the mg/day oral consumption of inorganic arsenic) assuming a volume of urine of 1.8 L/day
Estimate of the new mean of IQ \(m(d) = m(0) + \beta_{0.05}E_{L}\)
Estimate of the new proportion of individuals with an IQ below 70\(P(d) = \Phi\left( \frac{70 - m(d)}{15} \right)\)
Estimate of the additional risk: \(AR = P(d) - P(0)\)
Check: for a dose of 0.315 µg/kg bw/day, the additional risk (95%) should be 0.36%.
Code
Input: exposure in µg/kg. If available to adapt to a given country: mean IQ and sd IQ that will estimate \(p(0)\).
Output: additional risk of mild intellectual disability (per individual).
DRiAsLowIQ <-function(dose, nUnc =0, meanIQ =100, SeIQ =15){# dose: in mg/kg/bw# nUnc: return the best estimate if 0, the number of iterations otherwise# meanIQ: mean IQ in the population# SeIQ: SE IQ in the population Exposure <- dose*1000*76# ug/day PK <- dose /1.8# ug/L SEIQ <-abs(-0.39-0.01)/(qnorm(.975)-qnorm(0.025)) SlopeIQ <-if(nUnc ==0) -0.19elsernorm(nUnc, -0.19, SEIQ) NewMeanIQ <- meanIQ + SlopeIQ * PK NewRiskIQ <-pnorm(70, NewMeanIQ, SeIQ) OldRiskIQ <-pnorm(70, meanIQ, SeIQ)# Note: the AR is estimated directly from the p(0) in the population AR <-pmax(NewRiskIQ-OldRiskIQ, 0)return(AR)}DRiAsLowIQ(0.315/1000)
[1] 1.196802e-07
summary(DRiAsLowIQ(0.315/1000, nUnc =1E5))
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000e+00 7.664e-08 1.198e-07 1.207e-07 1.633e-07 3.934e-07
For a dose of 0.315 µg/kg bw/day, the additional risk of mild intellectual disability is 1.2^{-7}.
Bladder cancer
Bladder cancer is one of the most robustly established cancer outcomes from chronic inorganic arsenic exposure. The EPA 2025 IRIS assessment derives a dose–response based on pooled epidemiological data from multiple high-exposure cohorts. The model is a piecewise polynomial fitted to the dose–response curve from the IRIS report, valid for intakes up to 1 µg/kg bw/day. Below 0.2 µg/kg bw/day, a linear approximation is used; above this threshold, a quadratic term is added to capture the increasing slope observed at higher doses.
These figures present Extra Risk for a lifetime (from 0 to 85 years), expressed as \(ER(d)=\frac{P(d)-P(0)}{1-P(0)}\) where P(d) is the risk of cancer for an additional dose d and P(0) is the risk of cancer. The method used is a typical lifetime analysis.
We are interested in the additional risk. We have \(AR=(1-P(0)) \times ER(d)\). In the US, using EPA data, we have \(P(0)=0.018784\). We consider, as EPA, a lifetime as 70 years.
Code
Input: iAs intake (µg/kg bw/day). If available to adapt to a given country: \(p(0)\).
Output: Bladder cancer additional risk per year
DRiAsBladder <-function(dose, P0 =0.018784, lifetime=70){# dose in mg/kg dose <- dose *1000if(any(dose >1)) warning("at least one Dose > 1 ug/kg day, so is out of domain") ER <-ifelse(dose <0.2,0.0062* dose,0.0046*dose^2+0.0053*dose) AR <- (1-P0)*ERreturn(AR/lifetime)}DRiAsBladder(0.2/1000)
[1] 1.743761e-05
For a dose of 0.200 µg/kg bw/day, the additional risk of bladder cancer per year is 1.74^{-5}.
Lung Cancer
Lung cancer risk from inorganic arsenic is well established epidemiologically and forms part of the EPA 2025 IRIS combined cancer assessment. Like bladder cancer, the dose–response is derived from occupational and environmental cohort studies, and uses a piecewise polynomial structure. The model is valid for chronic iAs intakes up to 1 µg/kg bw/day. At low doses (below 0.2 µg/kg bw/day) the relationship is approximately linear; above this level a quadratic term captures the observed dose–response curvature.
Similarly, we are interested in the additional risk. We have \(AR=(1-P(0)) \times ER(d)\). In the US, using EPA data, we have \(P(0)=0.056978\). We consider a life expectancy of 70 years.
Code
Input: iAs intake (µg/kg bw/day). If available to adapt to a given country: \(p(0)\).
Output: Lung cancer additional risk per year
DRiAsLung <-function(dose, P0 =0.056978, lifetime=70){# dose in mg/kg bw/day dose <- dose *1000if(any(dose >1)) warning("at least one Dose > 1 ug/kg day, so is out of domain") ER <-ifelse(dose <0.2, 0.0078* dose, 0.0025*dose^2+0.0077*dose) AR <- (1-P0)*ERreturn(AR/lifetime)}DRiAsLung(.200/1000)
[1] 2.209366e-05
For a dose of 0.200 µg/kg bw/day, the additional risk of lung cancer is per year 2.21^{-5}%.
IHD
Ischaemic heart disease (IHD) risk from inorganic arsenic is derived from the EPA 2025 IRIS assessment, fitting a polynomial to the dose-response curves provided. The model applies for exposures up to 1 µg/kg bw/day.
Similarly, we are interested in the additional risk. We have \(AR=(1-P(0)) \times ER(d)\). In the US, using EPA data, we have \(P(0)=0.398464\). We consider a life expectancy of 70 years.
DRiAsIHD <-function(dose, P0 =0.398464, lifetime =70){# dose in mg/kg bw/day dose <- dose *1000if(any(dose >1)) warning("at least one Dose > 1 ug/kg day, so is out of domain") ER <-0.0187*dose^2+0.0878*dose AR <- (1-P0)*ERreturn(AR/lifetime)}DRiAsIHD(1/1000)
[1] 0.0009151941
For a dose of 1 µg/kg bw/day, the additiannal risk of IHD per year is 9.15^{-4}.
Diabetes
Epidemiological evidence links chronic inorganic arsenic exposure to an increased risk of type 2 diabetes, likely through mechanisms involving oxidative stress, impaired insulin secretion, and disruption of glucose homeostasis. The EPA 2025 IRIS assessment provides a dose–response curve for diabetes risk, which is fitted here as a polynomial. As with the cardiovascular endpoint, the model applies for exposures up to 1 µg/kg bw/day and returns an excess risk estimate.
Similarly, we are interested in the additional risk. We have \(AR=(1-P(0)) \times ER(d)\). In the US, using EPA data, we have \(P(0)=0.398247\). We consider a life expectancy of 70 years.
DRiAsDiabete <-function(dose, P0 =0.398247, lifetime =70){# dose in mg/kg bw/day dose <- dose *1000if(any(dose >1)) warning("at least one Dose > 1 ug/kg day, so is out of domain") ER <-0.0205*dose^2+0.1039*dose AR <- (1-P0)*ERreturn(AR/lifetime)}DRiAsDiabete(1/1000)
[1] 0.001069401
For a dose of 1 µg/kg bw/day, the additional risk of Diabetes per year is 0.00107.
Cadmium
Cadmium (Cd) is a non-essential heavy metal that accumulates in the body, particularly in the kidneys and bones, with a biological half-life of 10–30 years. It is classified as a Group 1 human carcinogen (IARC 2012, lung cancer from inhalation), though the primary concern from dietary exposure is nephrotoxicity and, at lower doses, bone demineralisation leading to osteoporosis. EFSA (2009, (EFSA Panel on Contaminants in the Food Chain (CONTAM) 2011)) established a tolerable weekly intake (TWI) of 2.5 µg/kg bw/week, based on a benchmark dose for tubular kidney damage. Dietary cadmium exposure is driven mainly by cereals, vegetables, and root crops — food groups that increase in consumption when diets shift toward plant-source foods — making cadmium a priority contaminant in ASF/PSF substitution scenarios. Because the relationship between dietary intake and internal dose (urinary cadmium) is mediated by absorption efficiency, body stores, and renal accumulation over decades, a toxicokinetic (PBPK) model is ultimately needed to translate dietary exposure into kidney burden. The dose–response below is expressed as a function of urinary cadmium concentration (µg/g creatinine) as the internal dose metric, following Pouillot, Santillana Farakos, and Van Doren (2022).
Osteoporosis
Pouillot, Santillana Farakos, and Van Doren (2022) evaluated the risk of osteoporosis as a function of Urinary Cadmium (ug/g creat) in the US population.
We can evaluate a polynomial from the data from this graph. We have, for the black line (50th quantile in the uncertainty dimension), for Urinary cadmium < 2 ug/g creat. Limiting to a degree 3, we have:
We estimate the annual incidence of chronic kidney disease stages 4 and 5 that is specifically attributable to cadmium exposure, for an individual of a given age. GFR is considered as normally distributed with a mean function of age. The probability of falling below the clinical thresholds for CKD stage 5 (GFR < 15) and stage 4 (15 ≤ GFR < 30) can be computed analytically with the normal CDF at both age and age plus one year.
The function first converts a dietary cadmium intake into an internal dose (urinary cadmium concentration) using the same Amzal et al. (2009) model. It then constructs a stochastic baseline GFR distribution drawn from a Swedish reference population, which is age-corrected in two steps: first back-adjusted to age 40 as a common reference point, then projected forward to the individual’s current age and age plus one year. Below age 40, no age-related GFR decline is applied.
From this baseline, two parallel GFR distributions are maintained — one without cadmium exposure and one where GFR is reduced proportionally to the internal cadmium dose, following a dose-response relationship from (Åkesson et al. 2005). The difference between these two probabilities gives a one-year incidence rate, computed separately for the exposed and unexposed distributions.
The cadmium-attributable incidence is then simply the excess incidence in the exposed group over the background incidence due to aging alone, for each CKD stage.
Code
Input: dose (in mg/kg bw/day), type (“ckd4” or “ckd5”), p(0): the prevalence of type in the population of age in the population of interest. If NULL, will consider the curent population, age: the age, stoc: if TRUE, will provide iter values, if FALSE , will provide the mean of iter values.
Output: Additional risk of Chronic Kidney Disease for the people of the given age, for one year.
DRiAsCKD <-function(dose, type ="ckd4", p0 =NULL, age =60, iter =1E6, stoc =FALSE, seed =NULL){# dose in mg/kg bw/day Cd_ug_kg_bw_day <- dose *1000if(!is.null(seed)) set.seed(seed) dose <-PBPKCadmium(Cd_ug_kg_bw_day, age = age, iter = iter)# GFR for Swedish population (mean age 49.6 years)# for age 40 and below, assuming a 0.8 reduction per year gfr_baseline <-rnorm(iter, mean =107, sd =19) -0.8* (-9.6)if(age >=40){ gfr_age <- gfr_baseline -0.8* (age-40) gfr_age_plus1 <- gfr_baseline -0.8* (age+1-40) } else { gfr_age <- gfr_age_plus1 <- gfr_baseline }# From Akesson, 2005: 0.078 reduction above 1 ug/g creat gfr_cd_age <- gfr_age*(1-0.078*(dose-1)) gfr_cd_age_plus1 <- gfr_age_plus1*(1-0.078*(dose-1))if(type =="ckd4") { lim =30 } elseif(type =="cdk5"){ lim =15 } elsestop("wrong type") prob_ckd_age <-pnorm(lim, gfr_age, sd =19, lower.tail =TRUE) prob_ckd_age_plus1 <-pnorm(lim, gfr_age_plus1, sd =19, lower.tail =TRUE) inc_rate_ckd_age <- prob_ckd_age_plus1 - prob_ckd_age# with Cd exposure prob_ckd_age_cd <-pnorm(lim, gfr_cd_age, sd =19, lower.tail =TRUE) prob_ckd_age_cd_plus1 <-pnorm(lim, gfr_cd_age_plus1, sd =19, lower.tail =TRUE) inc_rate_ckd_age_cd <- prob_ckd_age_cd_plus1 - prob_ckd_age_cd# CKD5 probability only attributable to Cd exposure inc_rate_ckd_cd_only <- inc_rate_ckd_age_cd - inc_rate_ckd_age# Provide AR if no if(is.null(p0)){ Result <- inc_rate_ckd_cd_only } else { Result <- inc_rate_ckd_cd_only / (1-prob_ckd_age_plus1) }if(stoc) return(Result)return(mean(Result))}DRiAsCKD(1.5/1000)
[1] 0.0003221741
Methylmercury
For the dose–response assessment of developmental neurotoxicity, the critical endpoint is IQ loss in children from prenatal MeHg exposure. The dose–response relationship is linear, using maternal hair mercury as the biomarker of exposure (Axelrad et al. 2007):
\[\Delta IQ = -\beta \times E_{Hair_{MeHg}}\]
with \(\beta = 0.18\) IQ points lost per µg/g maternal hair Hg, derived from a Bayesian integration of three prospective epidemiological studies (New Zealand, Seychelles Islands, Faroe Islands). The dietary dose is converted to hair mercury using:
\[E_{Hair_{MeHg}} = 250 \times E_{Blood_{MeHg}}\]
and
\[E_{Blood_{MeHg}} = \frac{dose \times A \times f \times bw}{b \times V}\]
\(E_{Blood_{MeHg}}\) : concentration of mercury in blood (μg/L)
dose : daily dietary mercury intake (μg/kg bw per day)
A : absorption factor (0.95)
f : the absorbed fraction distributed to the blood (0.05)
bw : body weight (65 kg for a pregnant woman)
b = elimination rate constant (ln(2) / half-life in blood = 0.014 per day)
V = blood volume (9 % x bw in a pregnant female).
The population-level risk of intellectual disability (IQ \(<\) 70) is estimated assuming IQ follows a normal distribution (mean = 100, SD = 15):
Input: the dose in µg/kg bw/day, mean IQ and sd IQ in the population (note: will be used to evaluate p(0)), bw: body weight of the mother.
Output: Additional risk of mild intellectual disability
# IQ dose-response: linear model (Axelrad et al. 2007 / FDA 2014)DRMeHgIQ <-function(dose, meanIQ =100, sdIQ =15, bw=65) {# dose in mg/kg bw/day dose_ugkg_day <- dose *1000# Convert dietary dose to hair Hg (µg/g) blood_Hg_ug_L <- (dose_ugkg_day *0.95*0.05) /(0.014*0.09) hair_Hg <- blood_Hg_ug_L *250/1000# Slope: 0.18 IQ points per µg/g hair (Axelrad et al. 2007) beta <-0.18 delta_IQ <--beta * hair_Hg# Excess proportion with IQ < 70 (intellectual disability)# Assumes IQ ~ Normal(100, 15) new_mean <-100+ delta_IQ P_MMR <-pnorm(70, new_mean, sdIQ) P_MMR_0 <-pnorm(70, meanIQ, sdIQ) AR <- P_MMR - P_MMR_0return(AR)}DRMeHgIQ(0.229/1000)
[1] 0.001434987
Lead
Lead (Pb) is a toxic heavy metal with no known biological function. It is classified as a probable human carcinogen (IARC Group 2A) and is particularly harmful to the developing nervous system. There is no established safe level of lead exposure — JECFA, EFSA, and EPA all conclude that the dose–response for neurodevelopmental effects is linear with no threshold. The primary concern from dietary exposure in children is IQ loss and an increased prevalence of mild intellectual disability (IQ < 70). Lead accumulates in bone and is released back into circulation during periods of bone resorption (pregnancy, menopause, ageing), creating a complex toxicokinetic relationship between dietary intake and blood lead concentration. The dose–response framework used here follows Carrington et al. (2019/05/01/) and EFSA (2013): dietary intake is converted to blood lead concentration using a linear pharmacokinetic coefficient, and blood lead is then linked to IQ shift using a linear slope, yielding a shift in the population distribution of IQ scores and the resulting excess prevalence of mild intellectual disability.Not ethat the relationship is non-linear. The linear slope is a simplification.
The mean IQ in the US population is distributed as a normal (100, sd = 15) IQ. The threshold for mental retardation is 70. So, the proportion of infants with mental retardation in the absence of Lead is \[\Phi\left( \frac{70 - 100}{15} \right) = 2.28\%\]
The loss of IQ per µg/dl is 1/1.8 IQ/(µg/dl) with an upper bound of 1/1.2 IQ/(µg/dl) (EFSA) or 0.48 IQ/(µg/dl) (lower bound: 0, upper bound 1.19 IQ/(µg/dl)) (Carrington et al, 2015).
“Since these dose response models use blood lead concentrations as the dose metric, evaluation of dietary lead exposure requires quantification of the relationship between dietary lead exposure and blood lead concentrations. The present analysis used a linear slope with the range identified by World Health Organization (2000) of 0.05–0.16 μg/dL per μg/day of lead exposure. ((Carrington et al. 2019/05/01/))” […]”
The link from the diet to the lead in blood use a coefficient of 0.105 (CI inf: 0.052; 0.16) (µg/dl)/(µg/day)
Code
Input: dose in mg/day (warning; not /kg bw)
Output: additional risk of Mild Intellectual Disability, per year. Exposure is adssumed to last 5 years (Carrington et al. 2019/05/01/)
The aditioanl risk of Mild Intellectual Disability from the ingestion of 3.5 µg/day lead is 0.01%.
Acrylamide
Acrylamide is a genotoxic carcinogen classified as “probably carcinogenic to humans” (Group 2A, IARC). EFSA Panel on Contaminants in the Food Chain (CONTAM) (2015) conducted a margin of exposure (MOE) assessment based on rodent cancer data. The most sensitive endpoint is incidence of Harderian gland adenomas and adenocarcinomas in male B6C3F1 (BMD10 = 0.17 mg/kg bw/day).
EFSA consider that an MOE of 10,000 would be considered as safe, found MOE of 50-425. Since the calculated MOE values are all substantially lower than 10 000, the CONTAM Panel concluded that the MOEs across surveys and age groups indicate a concern with respect to neoplastic effects.
EPA (2010) developed an Oral Slope Factor of \(8.3 \times 10^{-1}\) per mg/kg-day (endocrine, reproductive tumor). For non cancer, the RfD was \(2 \times 10^{-3}\) mg/kg-day.
Aflatoxins are genotoxic carcinogens (IARC Group 1). The cancer potency approach follows IARC (2002) and WHO/JECFA (2016), using a linear dose–response relationship between dietary aflatoxin B1 (AFB1) equivalent intake and hepatocellular carcinoma (HCC) risk. Potency factors (expressed relative to AFB1) are used to convert the mixture to AFB1-equivalents: B1 = 1, B2 = 0.1, G1 = 0.1, G2 = 0.01, M1 = 0.01 (JECFA 2016).
The cancer potency slope factor differs between hepatitis B virus (HBV) positive and negative populations:
HBV-negative: \(SF = 0.017\) cases/year/100,000 population per ng AFB1-eq/kg bw/day
HBV-positive: \(SF = 0.269\) cases/year/100,000 population per ng AFB1-eq/kg bw/day
\[Risk_{HCC} = SF \times dose_{AFB1\text{-}eq}\]
Input: aflatoxin intake in ng AFB1-equivalents/kg bw/day
Dioxins and dioxin-like PCBs (DL-PCBs) are persistent organic pollutants assessed using WHO toxic equivalency factors (TEF) to convert the mixture to TCDD-equivalents (TEQ). EFSA Panel on Contaminants in the Food Chain (CONTAM), Knutsen, Alexander, et al. (2018) and ECHA/EFSA (2018) provide the current European risk assessment framework. The critical effect is semen quality (sperm concentration) from human evidence.
(EFSA Panel on Contaminants in the Food Chain (CONTAM), Knutsen, Alexander, et al. 2018) established a TWI (Tolerable Weekly Intake) of 2 pg WHO-TEQ/kg bw/week, based on a BMDL5 of 14.1 pg TEQ/kg bw/week for semen quality. The dose–response uses a benchmark approach: excess risk is proportional to exceedance above the TWI, linearised at low doses.
For cancer (TCDD is IARC Group 1), a linearised slope factor from US EPA (2010) (Note: from EPA 2003 external review) is used: \[ILCR = SF_{cancer} \times dose_{TEQ}\] with \(SF = 1.0 \times 10^{6}\) (mg TEQ/kg bw/day)\(^{-1}\) for all cancers combined when the target risk range \(10^{-5}-10^{-7}\).
Input: chronic dioxin + DL-PCB intake in mg WHO-TEQ/kg bw/day
Output: incremental lifetime cancer risk per year, assuming a lifetime of 70 years.
Deoxynivalenol (DON, vomitoxin) is a trichothecene mycotoxin produced by Fusarium spp. The primary adverse effect in humans is gastrointestinal — nausea, vomiting, diarrhea — with an acute threshold. (EFSA Panel on Contaminants in the Food Chain (CONTAM), Knutsen, Alexander, Barregård, Bignami, Brüschweiler, Ceccatelli, Cottrill, Dinovi, Grasl-Kraupp, et al. 2017) established a TDI of 1 µg/kg bw/day for the sum of DON, 3-Ac-DON, 15-Ac-DON and DON-3-glucoside based on a BMDL05 of 0.11 mg/kg bw per day for reduced body weight gain, and an acute reference dose (ARfD) of 8 µg/kg bw based on a NOAEL for vomiting in humans of 26 µg DON/kg bw per a single eating occasion.
There is no established in-vivo genotoxic or carcinogenic effect; the dose–response is therefore threshold-based.
Fumonisins are mycotoxins produced by Fusarium verticillioides and related species. FB1 is the most abundant and toxic; the mixture is assessed as FB1+FB2+FB3 equivalents (FB4 is minor). EFSA Panel on Contaminants in the Food Chain (CONTAM), Knutsen, Barregård, et al. (2018) established a TDI of 1 µg/kg bw/day for the sum FB1+FB2+FB3, based on a BMDL10 of 0.1 mg/kg bw/day for hepatotoxicity in rats. See also EFSA Panel on Contaminants in the Food Chain (CONTAM) (2014).
The MOE approach is therefore used.
Input: dietary fumonisin intake (mg/kg bw/day, sum FB1+FB2+FB3)
Ochratoxin A is a nephrotoxic mycotoxin produced by Aspergillus and Penicillium spp. (EFSA Panel on Contaminants in the Food Chain (CONTAM) et al. 2020) established a BMDL10 for nephrotoxicity (tubular kidney damage) of 4.73 µg/kg bw/week in pigs. OTA is classified as possibly carcinogenic (IARC Group 2B). EFSA (2020) used a MOE approach for the carcinogenic endpoint rather than a linear slope, as the mode of action may be non-genotoxic.
For cancer, an upper-bound MOE approach:
\[MOE = \frac{BMDL_{10,cancer}}{dose_{OTA}}\]
with \(BMDL_{10} = 4.73\) µg/kg bw/week (EFSA Panel on Contaminants in the Food Chain (CONTAM) et al. 2020). Pending elucidation of the MoAs for the genotoxicity/carcinogenicity of OTA, the EFSA Panel concluded that an MOE of 10,000 to the neoplastic reference point is warranted for the risk characterisation of OTA, albeit being the most conservative approach until the MoAs are clarified.
Input: chronic OTA intake (µg/kg bw/day)
Output: Hazard Quotient (nephrotoxicity) and MOE (cancer)
DROTA <-function(dose_mgkg_day) { dose_ugkg_day <- dose_mgkg_day *1000 BMDL10_cancer <-4.73/7# µg/kg bw/week MOE_cancer <- BMDL10_cancer / dose_ugkg_day # MOE > 10,000 considered of low concern for genotoxic carcinogens (EFSA)return(MOE_cancer)}# Example: 0.015 µg/kg bw/day (typical European adult)DROTA(0.015/1000)
Biological hazards — bacteria, viruses, parasites, and their toxins — are the leading cause of foodborne illness globally. Unlike chemical contaminants, which exert effects through chronic exposure, biological hazards typically act through a single ingestion event, and the dose–response relationship describes the probability of infection or illness following ingestion of a given number of pathogens or toxin units. Two families of models dominate microbial QMRA: the exponential model (\(P = 1 - e^{-rd}\)), which assumes a Poisson-distributed dose and a constant per-organism probability \(r\) of initiating infection, and the beta-Poisson model (\(P = 1 - (1 + d/\beta)^{-\alpha}\)), which relaxes the assumption of homogeneous host susceptibility by allowing \(r\) to vary according to a beta distribution. The beta-Poisson model generally provides a better fit to human challenge data when substantial inter-individual variability in susceptibility is present.
A key feature of microbial dose–response models is that they are derived almost exclusively from human volunteer feeding studies or, where these are unavailable, from outbreak back-calculation. Both approaches have limitations: volunteer studies use laboratory-adapted strains in healthy adults and may underestimate real-world risk, while outbreak-derived estimates are subject to substantial uncertainty in dose reconstruction. Where possible, the most current and best-supported models from the QMRA Wiki (qmrawiki.org) and peer-reviewed literature are used here.
The nineteen biological hazards listed below were prioritised for their burden of disease, their relevance to ASF/PSF dietary substitutions, and the availability of quantitative dose–response data.
Rank
Microbiological hazard
Why priority in ASF/PSF substitutions
Main food pathways to track
1
Salmonella spp.
High burden across settings; relevant in both ASF and PSF chains.
Meat-borne cestodes; T. solium has high severity potential.
Undercooked pork/beef
17
Trichinella spp.
Relevant in pork/wild game contexts.
Undercooked pork, wild boar, game meats
18
Yersinia enterocolitica
Mostly pork-associated, context-dependent.
Pork and cross-contaminated foods
19
Vibrio spp.
Relevant in coastal/raw shellfish consumption contexts.
Raw/undercooked shellfish and seafood
20
Echinococcus spp.
Lower frequency but severe outcomes; region-specific.
Contaminated produce/environment
Salmonella spp.
Salmonella is one of the most common causes of foodborne bacterial gastroenteritis worldwide, responsible for an estimated 93.8 million cases and 155,000 deaths annually. Non-typhoidal Salmonella (NTS) causes self-limiting gastroenteritis in most individuals, but can progress to bacteraemia and invasive disease, particularly in young children, the elderly, and the immunocompromised. The two most clinically important serotypes are S. Enteritidis (primarily linked to eggs and poultry) and S. Typhimurium (broader host range, including pork and beef). In the context of ASF/PSF dietary substitutions, Salmonella is a priority hazard because it is prevalent across both poultry-derived ASF pathways and plant-source foods such as sprouts, sesame/tahini, and spices.
The most classical dose-response model is the one from FAO/WHO (2002). It is a beta-Poisson dose–response model fitted to human outbreak data, with parameters \(\alpha = 0.1324\) and \(\beta = 51.45\). It predicts the probability of illness (not just infection) following ingestion of a given dose, making it directly applicable to disease burden estimation.
Multiple pathotypes of diarrheagenic E. coli (DEC) cause foodborne illness. The best characterised dose–response is for enterotoxigenic E. coli (ETEC), based on human feeding trials. The QMRA Wiki recommends a beta-Poisson model fit to pooled DEC challenge data.
For ETEC (most relevant foodborne pathotype), using the beta-Poisson approximate form:
Campylobacter dose–response is based on human feeding studies (Black et al. 1988) and analysed by P. F. Teunis and Havelaar (2000) using a beta-Poisson model. This is the standard model used in EU and international QMRA for Campylobacter in poultry.
Listeria monocytogenes is a Gram-positive, facultatively intracellular bacterium that causes listeriosis — a rare but severe foodborne illness with a case-fatality rate of 20–30%. Unlike most bacterial pathogens, L. monocytogenes can grow at refrigeration temperatures, making it particularly hazardous in ready-to-eat (RTE) chilled foods. Clinical manifestations range from mild febrile gastroenteritis in healthy individuals to invasive listeriosis (septicaemia, meningitis, encephalitis) in pregnant women, neonates, the elderly, and the immunocompromised. In ASF/PSF substitution scenarios, Listeria is a concern wherever chilled RTE foods — including plant-based analogues — are processed or stored without adequate temperature control. The dose–response is strongly population-dependent: risk in the elderly and immunocompromised is orders of magnitude higher than in healthy adults, which makes sub-population stratification essential for accurate risk characterisation.
The latest dose-response model was published in Pouillot et al. (2024) . Because we don’t know the virulence of the strains or their distribution, we will use the EFSA model (marginal over virulence).
The input is the average (Poisson distribution) dose per serving (CFU/serving). The output is the probability of invasive listeriosis for the population. Note that a sub-population can be specified.
Code
library(doseresponsemodels)DRListeria <-function(dose, population =0, Poisson =TRUE) {DRQuick(dose, model="EFSA", population = population, Poisson = Poisson)}
STEC (EHEC, incl. O157/non-O157)
Shiga toxin-producing E. coli (STEC) can cause severe haemorrhagic colitis and haemolytic uraemic syndrome (HUS). The dose–response is characterised by a very low infectious dose. Strachan et al. (2005/08/15/) fitted a beta-Poisson model to E. coli O157:H7 human challenge data:
For E. coli O157:H7 , we use the beta-Poisson approximate form:
For the more severe HUS endpoint (mainly in children <5 and elderly), a conditional probability of HUS given illness of approximately 3–8% is applied (Mead and Griffin 1998).
Alternatively, for infants <15 year old, we could use the dose-response from Perrin et al. (2015):
\(P(illness)=1-exp(-r \times dose)\) with \(r(age)=r_0 \times exp(k \times age)\) and \(1<age<15\), \(k=-0.38\), \(r_0=log_{10}(-2.33)\).
Input: dose in CFU/serving
Output: probability of illness
DRSTEC <-function(dose) {1- (1+ dose /2.2183)^(-0.0571)}DRSTECPerrin <-function(dose, age){if(any(age<1| age >15)) warnings("DR developed for 1-15 year old individuals") k <--0.38# CI95% [-0.40; -0.36] r_0 <-10^-2.33# CI95% -2.56, -2.12] r_age <- r_0 *exp(k * age)return(1-exp(-r_age*dose))}DRSTEC(c(1e2, 1e3, 1e4))
[1] 0.1964486 0.2946530 0.3814821
DRSTECPerrin(c(1e2, 1e3, 1e4), 10)
[1] 0.01040904 0.09934766 0.64878602
Norovirus
Norovirus is the leading cause of acute viral gastroenteritis worldwide, responsible for an estimated 685 million cases and 212,000 deaths annually, predominantly in children under 5 years in low-income countries (Bartsch et al. 2016). It is a non-enveloped single-stranded RNA virus with an extremely low infectious dose, high environmental stability, and the ability to spread efficiently through contaminated food, water, and person-to-person contact. In the food chain, norovirus is particularly associated with raw shellfish (oysters concentrate the virus from contaminated water), ready-to-eat produce, and food handler contamination. It is highly relevant in PSF dietary scenarios, as fresh produce and minimally processed plant foods are frequent vehicles. A key feature of norovirus epidemiology is the existence of a fraction of the population that are fully susceptible (lacking the FUT2-encoded histo-blood group antigen receptor), while others are genetically resistant — a heterogeneity that motivates the fractional Poisson dose–response model.
Multiple dose-response models were developed in the literature. However, Schmidt (2015) found that simple models, such as the fractional model of (M. J. Messner, Berger, and Nappier 2014) were not more justified than more complex ones with regards to the available data. We therefore use the fractional Poisson model: \[P_{inf}(Dose, P) = P \times \left(1 - e^{-Dose}\right)\]
It can be read as: a proportion P are fully susceptible to norovirus and will get sick as soon as 1 virion is present in the serving, while the norovirus particles are distributed according to a Poisson distribution. M. J. Messner, Berger, and Nappier (2014) estimated P to 0.722.
Input: average dose in a serving (Poisson distribution)
Cryptosporidium is a protozoan parasite responsible for cryptosporidiosis, a self-limiting but potentially severe diarrhoeal illness. It is one of the most important waterborne pathogens globally and a leading cause of diarrhoea in children under 5 in low- and middle-income countries (LMICs). Two species are most relevant in human disease: C. parvum (zoonotic, associated with cattle and contaminated water/produce) and C. hominis (anthroponotic, restricted to humans). The oocysts are the environmentally resistant infective stage, shed in faeces and capable of surviving months in moist environments; they are highly resistant to chlorine disinfection but susceptible to UV treatment, making water treatment a critical control point. In the context of ASF/PSF dietary transitions, Cryptosporidium is particularly relevant where fresh produce irrigation uses untreated surface water.
We will use an exponential model with \(r = 4.19 \times 10^{-3}\) from DuPont et al. (1995), reanalysed by Michael J. Messner, Chappell, and Okhuysen (2001) . Note that heterogeneity has been found from one strain to the other (Peter F. M. Teunis, Chappell, and Okhuysen 2002) but the value above, obtained from the Iowa isolate, can be considered a central estimate.
Input: ingested oocysts (average dose of a Poisson distribution)
Anisakis spp. cause anisakiasis through ingestion of live larvae in raw or undercooked marine fish. The dose–response for Anisakis is driven by a single larva being sufficient to cause disease. Each larva can penetrate the gastric mucosa independently.
A simple single-hit exponential model is appropriate, with \(r \approx 1\) (i.e., near-certain infection if a live larva is ingested), adjusted for the probability that larvae survive food processing. The key uncertainty is the fraction of larvae that are alive at consumption.
\[P(anisakiasis) = 1 - e^{-r \times N_{live}}\]
with \(r \approx 0.5-1.0\) ((EFSA Panel on Biological Hazards (BIOHAZ) 2010): near 1 for gastric anisakiasis given live larva ingestion), and \(N_{live}\) the number of live larvae per serving.
Input: number of live Anisakis larvae per serving
Output: probability of anisakiasis
DRAnisakis <-function(N_live_larvae, r =1.0) {# r ~ 1: each live larva has ~100% chance of causing disease if swallowed# EFSA 2010: gastric anisakiasis from a single larva is essentially certain P_anisakiasis <-1-exp(-r * N_live_larvae)return(c(P_anisakiasis = P_anisakiasis))}# Example: 0.01 live larvae per serving (low-contamination raw fish)DRAnisakis(0.01)
P_anisakiasis
0.009950166
DRAnisakis(1)
P_anisakiasis
0.6321206
Bacillus cereus
There are no established dose-response curves for B. cereus. (EFSA 2016)
Cereulide: Agata (1995) data reanalysed by (European Food Safety Authority (EFSA) et al. 2026) found a BMDL10 of 4.2 µg/kg bw (reduced to 3.08 µg/kg bw) (emesis in Asian house shrews), leading to an ARfD or 0.014 µg/kg bw.
Clostridium perfringens
Clostridium perfringens causes food poisoning through the production of enterotoxin (CPE) in the gut during sporulation. Disease requires ingestion of large numbers of vegetative cells (\(>10^6\) CFU/serving). (Golden et al. 2009) proposed an exponential model based on human volunteer studies and outbreak data:
\[P(illness) = 1 - e^{-r \times dose}\]
with \(r = 1.8266 \times 10^{-11}\)(Lee et al. 2016). The very low \(r\) reflects the high infectious dose requirement.
Input: dose in CFU/serving
Output: probability of illness
DRCperfringens <-function(dose, r =1.8266E-11) {# Exponential model, Haas et al. 1999 / QMRA Wiki P_illness <-1-exp(-r * dose)return(c(P_illness = P_illness))}DRCperfringens(c(1e6, 1e7, 1e8))
Staphylococcus aureus food poisoning is caused by preformed heat-stable enterotoxins (SE). Illness is toxin-mediated, not infection-dependent. The dose–response is characterised by a threshold for staphylococcal enterotoxin A (SEA), the most potent type. Evenson et al. (1988) suggest a minimum emetic dose of 100–200 ng SEA per person.
[@heidingerQuantitativeMicrobialRisk2009] used a value of 84 ng SEA to triger an illness.
Clostridium botulinum produces botulinum neurotoxin (BoNT), one of the most toxic substances known. The dose–response is extremely steep — the human lethal dose is estimated at ~1–2 ng toxin/kg bw (LD50 ≈ 1–2 ng/kg bw by ingestion; (Arnon et al. 2001)). Even sub-lethal doses cause severe paralysis.
For foodborne botulism risk assessment, a single-hit exponential model is used based on the LD50 for type A toxin (the most common in foodborne outbreaks):
\[P(intoxication) = 1 - e^{-r \times dose}\]
with \(r = \ln(2) / LD_{50}\), where \(LD_{50} = 1000\) ng/person (ingestion route, conservative estimate; (Arnon et al. 2001).
Input: dose of BoNT-A in ng/serving
Output: probability of intoxication (clinical botulism)
Giardia duodenalis (syn. G. lamblia, G. intestinalis) dose–response is based on human volunteer studies (Rendtorff 1954) reanalysed by QMRA Wiki using a exponential model:
Input: dose in cysts/serving or per drinking event
Output: probability of infection
DRGiardia <-function(dose, r =0.0199) {# Exponential model, Teunis et al. 2002 / QMRA Wiki P_inf <-1-exp(-r * dose)return(c(r = r, P_infection = P_inf))}DRGiardia(c(1, 10, 100))
r P_infection1 P_infection2 P_infection3
0.0199000 0.0197033 0.1804501 0.8633046
Cyclospora cayetanensis
Cyclospora cayetanensis dose–response data from human studies are very limited. No formal human volunteer dose–response study exists. The available data come from outbreak investigations.
Taenia spp. (T. solium / T. saginata)
Taenia spp. infection risk from meat consumption depends on ingestion of viable cysticerci. The dose–response for both T. solium (pork) and T. saginata (beef) is modelled as a single-hit exponential, as each viable cysticercus can establish an intestinal tapeworm infection.
Based on experimental data from animal studies and human epidemiology, \(r \approx 0.5\)–\(1.0\) per viable cysticercus (near-certain establishment per cysticercus). For T. solium, the additional severe outcome of neurocysticercosis (NCC) arises from egg ingestion (faecal-oral), not from meat consumption.
Input: number of viable cysticerci per serving
Output: probability of intestinal taeniasis
DRTaenia <-function(N_cysticerci, r =1.0) { P_taeniasis <-1-exp(-r * N_cysticerci)return(P_taeniasis)}DRTaenia(0.01)
[1] 0.009950166
DRTaenia(0.001)
[1] 0.0009995002
Trichinella spp.
Trichinella infection results from ingestion of viable muscle larvae in undercooked pork or game. Like Taenia, the dose–response follows a single-hit exponential model, as each larva can establish infection:
\[P(infection) = 1 - e^{-r \times N_{larvae}}\]
Teunis et al. (2012) fitted \(r = 0.01\) (on average. Large variability) from human outbreak data, reflecting that not every larva successfully establishes (partial immunity, digestion losses). Clinical trichinellosis (myositis) typically requires ingestion of \(\geq\) 70 larvae; severe disease \(\geq\) 300 larvae.
Input: number of viable Trichinella larvae per serving
Two key pathogenic Vibrio species are relevant in food safety: V. parahaemolyticus (raw shellfish) and V. vulnificus (high fatality in immunocompromised). Beta-Poisson models have been developed by USFDA (2005) for V. parahaemolyticus.
Echinococcus granulosus (cystic echinococcosis) and E. multilocularis (alveolar echinococcosis) cause severe larval cyst disease following accidental ingestion of eggs from contaminated produce, water, or dog fur. No formal human dose–response study exists.
Approach for Nutrition
Nutrition studies on cohorts provide a Relative Risk (RR) and its confidence interval (95% CI) extracted from the literature. As an example: Diabetes RR: 1.19 (CI: 1.04 ; 1.37) per 100 g red meat per day (Pan et al. 2011). It is assumed that:
The RR established per 100g can be intrapolated (log linearly) to values lower than <100g substitution;
The uncertainty around the RR is assumed to follow a lognormal distribution.
Example:
# Lower bound of the RR for 100g low <-1.04# Upper bound of the RR for 100gup <-1.37RR_beta_low <-log(low)/100RR_beta_high <-log(up)/100# The mean is then (mu <- (RR_beta_low+RR_beta_high)/2)
[1] 0.001770157
# The standard deviation is estimated from the bound(sigma <- (RR_beta_high-RR_beta_low)/(qnorm(0.975,0,1)-qnorm(0.025,0,1)))
Then, incidence (number of cases per 100,000 individuals) for the given illness in the given population is collected from the literature par age/sex category. Example:
The RR application is unclear. The RRs are defined at the population level for average consumption changes, but the model has individual-level consumption data. Whether it applies the RR at the individual level and aggregates, or applies it directly to population incidence, matters a lot for the result. These are not equivalent approaches.
Diabetes incidence vs. incidence relevant to diet. Diabete incidence includes all diabetes cases regardless of whether they are diet-attributable. Applying a dietary RR to total incidence implicitly assumes the entire disease burden is modifiable by this dietary change, which may overestimate the effect.
We want: \(C_{avoided} = C_{\text{REF}} - C_{\text{ALT}}\)
Cases under each scenario: for each individual \(i\), we compute their expected risk under each diet:
where \(x_{REF,i}\) and \(x_{ALT,i}\) are individual \(i\)’s red meat consumption (g/day) under the reference and alternative diet, and \(RR(x)\) is the dose-response function relative to zero (or some reference) consumption.
For diabetes, we assume that the RR is log-linear, so:
\(RR(x) = exp(\beta \times x)\) where \(\beta = log(1.12) / 100\), meaning a 100g/day increase multiplies risk by 1.12.
These formulas make the assumption that \(R_{baseline,i}\) are similar for all \(i\). It could be necessary to evaluate this per sub-groups of individuals saring tyhe same baselline risk. What is \(R_{baseline}\)?
Baseline_risk is the risk at zero (or reference) consumption — not total diabetes incidence. We would back-calculate it from diabetes incidence:
This ensures the model reproduces the observed diabete incidence under the current diet, and then projects what happens under the alternative. This way we are not assuming all diabete cases are diet-attributable — we are simply anchoring the model to observed incidence and computing the marginal change.
This is internally consistent, uses the individual-level data the model already has, and avoids the overestimation problem entirely — because the baseline risk is calibrated to reproduce observed incidence rather than applied naively to all diabete cases.
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Footnotes
Note: if the birth weight did follow a normal distribution, we would have P(0) = 9.9%.↩︎
Note: if the IQ did follow a normal distribution, we would expect P(0) = 2.3%.↩︎
