========================================================================== SCIENCE JOURNALING CLUB :: ROMAN LEAD DOSE-RESPONSE CHAIN ========================================================================== seed = 20250106 numpy 2.4.2 Club reconstruction. Emissions taken from McConnell et al. (2025); everything between emissions and IQ is recomputed here from scratch. ========================================================================== STAGE 0 :: THE BOX ========================================================================== domain -10E to 40E, 30N to 65N area A 1.439e+13 m^2 = 14.392 million km^2 removal velocity v_d 1.50 cm/s = 4.734e+05 m/yr scavenging volume v_d * A = 6.813e+18 m^3/yr at steady state all emitted lead deposits inside the box, so C = E/(v_d A) ========================================================================== THE CHAIN AT CENTRAL ASSUMPTIONS ========================================================================== 1. emissions E = 3.50 kt Pb/yr 2. air lead C = 5.1375e-04 ug/m^3 = 0.514 ng/m^3 3. blood lead rise dBLL = 2.391 ug/dL Neolithic baseline = 1.00 ug/dL total child blood lead = 3.391 ug/dL 4. IQ deficit at total BLL = -4.905 points IQ deficit at baseline = -2.298 points ROMAN-ATTRIBUTABLE LOSS = 2.607 IQ points published for comparison: air 0.49-1.04 ng/m^3 (domain means), dBLL about 2.4 ug/dL, total about 3.4 ug/dL, decrement 2.5-3 IQ points. ========================================================================== THE PAPER'S TWO EMISSION SCENARIOS, RUN THROUGH OUR BOX ========================================================================== scenario E kt/yr C ng/m3 dBLL BLL tot IQ lost -------------------------------------------------------------------------- 1 Rio Tinto dominant 4.20 0.617 2.453 3.453 2.653 2 distributed sources 3.00 0.440 2.340 3.340 2.568 club midpoint 3.50 0.514 2.391 3.391 2.607 ========================================================================== VALIDATION :: DOES OUR IQ REGRESSION REPRODUCE THEIR PUBLISHED CHECKS? ========================================================================== US children 1976-1980, GM 15.2 ug/dL ours 9.23 published 9.2 diff 0.03 US children 2011-2016, GM 0.8 ug/dL ours 1.95 published 1.9 diff 0.05 The regression is reproduced to within rounding, so any disagreement downstream is in the atmospheric box, not in the epidemiology. ========================================================================== STEEPNESS :: MARGINAL IQ COST OF ONE MORE MICROGRAM PER DECILITRE ========================================================================== d(IQ)/d(BLL) = slope / (BLL + 1), so the curve is steepest at zero dose. BLL ug/dL IQ deficit marginal pts vs first ug -------------------------------------------------------------------------- 0.00 -0.000 3.3150 1.000 0.50 -1.344 2.2100 0.667 1.00 -2.298 1.6575 0.500 2.00 -3.642 1.1050 0.333 3.39 -4.904 0.7551 0.228 5.00 -5.940 0.5525 0.167 10.00 -7.949 0.3014 0.091 15.20 -9.232 0.2046 0.062 25.00 -10.801 0.1275 0.038 40.00 -12.310 0.0809 0.024 The first ug/dL costs 3.315 IQ points. The fortieth costs 0.0809, a factor of 41.0 less. Roman children sat on the steep part. That is the whole reason a blood lead level a modern clinic would call unremarkable does this much damage. ========================================================================== THE FLATNESS OF THE AIR-TO-BLOOD STEP ========================================================================== dBLL = exp(1.932) * C^0.140 = 6.9033 * C^0.140 A tenfold change in air lead changes blood lead by 10^0.140 = 1.3804x, i.e. 38.0 percent. C ug/m^3 C ng/m^3 dBLL ug/dL IQ lost -------------------------------------------------------------------------- 1.0e-06 0.001 0.9978 1.342 1.0e-05 0.01 1.3774 1.737 1.0e-04 0.1 1.9013 2.215 5.1e-04 0.5134 2.3907 2.607 1.0e-03 1 2.6246 2.779 1.0e-02 10 3.6229 3.427 1.0e-01 100 5.0010 4.153 1.0e+00 1000 6.9033 4.950 At effectively pristine air (1 pg/m^3) the regression still returns 0.998 ug/dL, which lands on the Neolithic tooth-enamel figure of about 1.0 ug/dL. That is either a good sign or a coincidence, and we cannot tell which from outside the data. ========================================================================== SWEEP :: THE CHAIN ACROSS THE PLAUSIBLE EMISSION RANGE ========================================================================== E kt/yr C ng/m^3 dBLL BLL total IQ lost vs CDC 3.5 -------------------------------------------------------------------------- 1.0 0.147 2.006 3.006 2.303 0.86 1.5 0.220 2.123 3.123 2.399 0.89 2.0 0.294 2.211 3.211 2.468 0.92 2.5 0.367 2.281 3.281 2.523 0.94 3.0 0.440 2.340 3.340 2.568 0.95 3.5 0.514 2.391 3.391 2.607 0.97 4.0 0.587 2.436 3.436 2.641 0.98 4.2 0.617 2.453 3.453 2.653 0.99 5.0 0.734 2.513 3.513 2.698 1.00 6.0 0.881 2.578 3.578 2.745 1.02 8.0 1.174 2.684 3.684 2.821 1.05 12.0 1.761 2.841 3.841 2.930 1.10 Emissions span a factor of 12 across that table. The IQ answer spans a factor of 1.27. Two compressive steps in series flatten a twelvefold input into a 1.27-fold output. ========================================================================== SENSITIVITY :: ONE ASSUMPTION AT A TIME, EVERYTHING ELSE HELD CENTRAL ========================================================================== assumption range carried low high span % of base -------------------------------------------------------------------------- air-to-blood exponent b 0.140 +/- 0.054 1.938 3.413 1.475 56.6 IQ regression slope -3.315 +/- 0.6155 2.123 3.091 0.968 37.1 Neolithic baseline BLL 0.5 - 1.5 ug/dL 2.225 3.160 0.935 35.9 air-to-blood intercept a 1.932 +/- 0.181 2.294 2.947 0.653 25.1 removal velocity v_d 0.5 - 3.0 cm/s 2.436 2.894 0.459 17.6 emission rate E, wide 2.0 - 6.0 kt/yr 2.468 2.745 0.277 10.6 domain area A 1.0 - 2.0 e13 m^2 2.525 2.700 0.175 6.7 emission rate E 3.0 - 4.2 kt/yr 2.568 2.653 0.085 3.3 central answer = 2.607 IQ points DOMINANT ASSUMPTION: air-to-blood exponent b (0.140 +/- 0.054), moving the answer by 1.475 points, which is 57 percent of the central value. The atmospheric assumptions (v_d, A) are the ones the club had the most freedom to choose, and they matter least. The epidemiological transfer functions, which we copied verbatim and cannot adjust, matter most. ========================================================================== MONTE CARLO :: ALL ASSUMPTIONS VARIED TOGETHER, 200000 SEEDED DRAWS ========================================================================== quantity p2.5 p16 p50 p84 p97.5 -------------------------------------------------------------------------- air lead ng/m^3 0.231 0.338 0.629 1.161 1.736 blood lead rise ug/dL 1.012 1.581 2.468 3.815 5.794 total blood lead ug/dL 1.884 2.544 3.486 4.854 6.846 IQ points lost 1.167 1.778 2.634 3.743 5.093 median IQ loss 2.634 points 68% interval 1.778 to 3.743 95% interval 1.167 to 5.093 P(loss > 1 point) 98.9 percent P(loss > 2 points) 76.1 percent P(loss in 2.5 - 3.0) 19.3 percent P(total BLL > CDC 3.5) 49.5 percent The 95% band is wide because we let the two regression coefficients wander over their published standard errors. That is honest. It also means our interval is not the paper's interval, which comes from a different and much more careful error propagation. ========================================================================== CONTEXT :: WHERE 3.4 ug/dL SITS AMONG NUMBERS WE CAN CHECK ========================================================================== population / reference BLL IQ deficit vs Roman -------------------------------------------------------------------------- US children 2011-2016 (NHANES GM) 0.80 1.95 0.24 Neolithic Europe (tooth enamel) 1.00 2.30 0.29 Roman Empire, Pax Romana (this model) 3.39 4.90 1.00 CDC blood lead reference value, 2021 3.50 4.99 1.03 CDC level of concern, 1991-2012 10.00 7.95 2.95 US children 1976-1980 (NHANES GM) 15.20 9.23 4.48 Roman child, total blood lead 3.39 ug/dL CDC reference value since 2021 3.50 ug/dL ratio 0.969 A Roman child at the empire-wide average would today sit just under the threshold at which an American clinic opens a case file. Leaded-petrol-era US deficit 9.23 points Post-phase-out US deficit 1.95 points Recovered by removing tetraethyl lead from petrol: 7.28 points Roman airborne burden 2.61 points Roman burden as a share of the American petrol burden: 36 percent ========================================================================== INVERSION :: WHAT WOULD IT TAKE TO REACH THE LEADED-PETROL ERA? ========================================================================== target total blood lead 15.2 ug/dL required air lead 172.7 ug/m^3 = 1.727e+05 ng/m^3 required emission rate 1.177e+06 kt Pb/yr that is 3.362e+05 times the Roman rate. The exponent of 0.14 is doing all of that work. Because blood lead responds to the 0.14 power of air lead, matching twentieth-century American children on the airborne route alone would need an absurd emission rate. Which is the point: their lead was not mostly airborne. ========================================================================== SCHEMATIC EMISSION HISTORY USED BY THE INTERACTIVE COMPANION ========================================================================== Shape taken from the narrative published in the paper and its 2018 predecessor; absolute scaling is ours, pinned so the Pax Romana plateau sits at the club's 3.5 kt/yr midpoint. This is NOT digitised data. year E kt/yr nbPb pg/g BLL tot IQ lost event -------------------------------------------------------------------------- 500 BCE 0.05 0.573 2.319 1.679 Iron Age background; Greek silver at Laurion beginning 400 BCE 0.10 0.596 2.453 1.811 Athenian silver coinage at full output 300 BCE 0.20 0.641 2.602 1.950 Carthaginian working of Iberian ores 206 BCE 0.70 0.870 2.909 2.221 Rome takes Iberia from Carthage 150 BCE 1.40 1.190 3.103 2.382 Republican mining expansion in Hispania 120 BCE 1.80 1.373 3.178 2.442 Republican maximum, late 2nd century BCE 91 BCE 1.00 1.007 3.006 2.303 Social War 80 BCE 0.60 0.824 2.868 2.186 Sullan civil wars; output near background 49 BCE 0.40 0.733 2.765 2.097 Caesar crosses the Rubicon 15 BCE 1.20 1.099 3.058 2.346 Augustan settlement; sharp rise begins 0 CE 2.40 1.647 3.268 2.513 Empire consolidating 50 CE 3.20 2.013 3.361 2.584 Julio-Claudian and Flavian output 100 CE 3.60 2.196 3.400 2.614 Trajanic peak 150 CE 3.50 2.150 3.391 2.607 Antonine plateau 165 CE 3.40 2.104 3.381 2.600 Antonine Plague arrives 180 CE 1.90 1.419 3.195 2.456 output falls with the plague 250 CE 0.80 0.916 2.945 2.252 third-century crisis 400 CE 0.50 0.779 2.821 2.146 late Empire 600 CE 0.20 0.641 2.602 1.950 post-Roman minimum ice calibration: nbPb(pg/g) = 0.4571 * E(kt/yr), background 0.55 pg/g ========================================================================== FIGURE DATA :: EXACT NUMBERS PLOTTED IN THE ARTICLE ========================================================================== Figure 2a, air-to-blood transfer function, C in ng/m^3 -> dBLL ug/dL 0.010 1.377 0.030 1.606 0.100 1.901 0.300 2.217 0.513 2.390 1.000 2.625 3.000 3.061 10.000 3.623 30.000 4.225 100.000 5.001 300.000 5.832 Figure 2b, blood-to-IQ transfer function, BLL ug/dL -> deficit points 0.00 -0.000 1.00 -2.298 2.00 -3.642 3.39 -4.904 5.00 -5.940 8.00 -7.284 12.00 -8.503 15.20 -9.232 20.00 -10.093 30.00 -11.384 40.00 -12.310 Figure 3, sensitivity spans in IQ points (sorted, widest first) air-to-blood exponent b 1.938 3.413 span 1.475 IQ regression slope 2.123 3.091 span 0.968 Neolithic baseline BLL 2.225 3.160 span 0.935 air-to-blood intercept a 2.294 2.947 span 0.653 removal velocity v_d 2.436 2.894 span 0.459 emission rate E, wide 2.468 2.745 span 0.277 domain area A 2.525 2.700 span 0.175 emission rate E 2.568 2.653 span 0.085 Figure 4, blood lead ladder US children 2011-2016 (NHANES GM) 0.80 deficit 1.95 (post-phase-out) Neolithic Europe (tooth enamel) 1.00 deficit 2.30 (pre-metallurgy baseline) Roman Empire, Pax Romana (this model) 3.39 deficit 4.90 (club output) CDC blood lead reference value, 2021 3.50 deficit 4.99 (triggers case management) CDC level of concern, 1991-2012 10.00 deficit 7.95 (withdrawn as a threshold) US children 1976-1980 (NHANES GM) 15.20 deficit 9.23 (leaded-petrol peak) Figure 1, chain compression factors emissions 3.0 -> 4.2 kt/yr is a factor of 1.400 air 0.440 -> 0.617 ng/m^3, factor 1.400 blood 2.340 -> 2.453 ug/dL, factor 1.048 IQ 2.568 -> 2.653 points, factor 1.033 ========================================================================== END ==========================================================================