Introduction

There are currently plans in the making to ban the use of lead shot in the UK^{1, 2}. Indeed a de-facto ban may occur as meat suppliers refuse to take game that have been shot with lead ammunition. There are claims that lead shot kills hundreds of thousands of wildfowl annually. While these claims about danger to wildlife have been described as inflated, lead is an extremely damaging substance to human health when in sufficient quantities, with the potential to cause neurological damage, and even defects in DNA³. Indeed lead’s effects, in particular on the mind, were observed by the Greek Nikander of Colophon in the 2nd century BC, when white lead was used to sweeten wine. Over the last half-century sources of lead in the first world have been all but removed from gasoline, paint and water piping. 

Abstract/Conclusion 

The following mathematical work is quite involved, we therefore advise the reader who does not have the time or want to expend the effort to understand the following calculations to simply read the following abstract or conclusion for a potted outline of our findings:

The conclusion is a fairly balanced one. On the one hand, game or clay pigeon shooting has the potential to distribute very significant quantities of lead (sometimes >1 tonne/ha). Additionally, according to an exponential model, this lead saturates the environment at a very significant rate, reaching an equilibrium/saturation in under one year when the relevant land has been shot upon periodically going back 20 years or more. The environment does clearly become toxified.

How to break down the issue of lead exposure for adults engaged in game or clay shooting in the field is difficult and complex. There is exposure from handling weapons and ammunition, particulate in the air, and exposure to the environment itself. 

The exponential model from a linear ordinary differential equation indicates that due to the low mass of birds, and the tendency for shot to get stuck in the gizzard, it is quite likely that a bird ingesting some pellets would experience lead poisoning. These conclusions were obviously also reached by the US Fish and Wildlife Service⁴ when they started designating steel-shot-only hunting zones for waterfowl as early as 1976⁵. Perhaps we should look to the USA. 

On the final issue of harm arising from human lead ingestion the conclusion is that due to the absence of a gizzard, and the relatively swift digestion and elimination that occurs in a human, it is very unlikely an adult would suffer lead poisoning from the ingestion of a reasonable number of lead pellets. Small children (with their small size and volume) who ingest slightly larger quantities of lead and may well experience toxicity however, particularly if the lead bodies become stuck in the digestive tract. 

The Setting

The interest in the transfer of lead in lead pipes provides us with a linear ordinary differential equation for the diffusion of lead from water pipes⁶. We will attempt to adapt this model to lead shot in a variety of environments. We will tackle three problems. 

1. Does regular use of lead shot on an area used to shoot driven game cause the ground to become toxic by scientific standards? 
2. If a bird ingests a number of lead shotgun pellets could its blood lead concentration reach toxic levels could it suffer harm or die?
3. If a human is to ingest a number of pellets, possibly on a regular basis, is it conceivable that the individual would suffer from some degree of lead poisoning? 

The Differential Equation

Transfer of lead from a solid pipe in to stationary water within the pipe can be modelled by the ordinary linear differential equation⁷: 

V dc/dt = MA (1 – c/E)       (1)

Where V is the volume of water in the pipe in m3, c is the lead concentration of this water in micrograms/litre. A is the internal area of the pipe in m3. E is the equilibrium concentration, beyond which lead ceases to be transferred from the pipe into the liquid. M is the initial mass transfer rate in micrograms per m2 per second. 

Many thanks to Vita Stembrera for providing the more intuitive form of this relationship.

The equation (1) does make some sense; the rate of change of concentration over the whole volume V is proportional to the area of the pipe and the rate of mass transfer, and decreases as the concentration approaches the equilibrium concentration or point of saturation. 

This is known as an exponential model. And indeed when it is solved it gives us the exponential equation⁶:

c= E –  (E – c₀) . exp( -AMT/VE )      (2)

For lead, from the work by Van der Leer et. al.⁶ we have that for a scenario with moderate plumbosolvency: 

M = 0.10 micrograms/m2/sec

E = 150 micrograms/litre

Part 1: Lead shot falling on one Hectare

We will adapt the scenario with a pipe to an instance where we have one hectare of land, and we consider the top 5 cm of soil. We will liken the soil to water in its properties – in the UK there is significant moisture in the topsoil in almost all seasons. 

So we have a volume: 

100m*100m*0.05m = 500m3 = 500,000 litres

From wikipedia a #6 12-bore 28g (1 oz) cartridge contains 270 pellets. The average diameter for this standard pellet is 2.59mm. Therefore the surface area of a pellet using the formula for the surface of a sphere is 2.11 e-5 m2 in scientific notation. 

The weights of pellets in grammes provided by wikipedia are not satisfactory. Clearly a #6 pellet cannot weigh 1.62g as this would mean a 28g load would only comprise around 17 pellets rather than the required 270. We therefore calculate the weight of a single pellet using the density of lead  p = 0.0113 . A #6 pellet of diameter 2.59mm has a radius of 1.295mm and therefore a volume of 9.09mm3. The resulting mass is 0.103g per pellet, and indeed, if we multiply 0.103g * 270 = 27.7g, vaguely in line with 28g load we started our calculations with. So the mass of a single pellet is 0.103g. 

On a typical pheasant drive it is reasonable to assume 8 guns take a total of twenty shots each, bearing in mind how common a miss with the second barrel is. Therefore the total number of shots that occur during the drive is 8 * 20 = 160 shots. The total number of pellets from one drive is therefore 160 * 270 = 43,200 pellets, weighing a total of 4.48 kg. 

Assuming the pheasant drive has been shot ten times per year for twenty years the total number of pellets in the topsoil is 43,200 *10 * 20 = 8.64 million, with a total weight of 896kg. The total surface area of these pellets is the area per pellet multiplied by the number of pellets:  8.64 e6 * 2.12 e-5 = 183m2. 

Going back to our differential equation, it is important to remember the result will be in micrograms of lead. 

c= E – (E – c₀) . exp( AMT/VE ) 

 We have from Van der Leer et. al.⁶  E = 150, M = 0.10, c₀ = 0,  A = 183, V =500,000 .  

Using the free online graphing tool Desmos we obtain the following plot:

What this tells us is that the equilibrium concentration comes near to being reached within around 200 days. Lead therefore diffuses and saturates the environment relatively rapidly.

Is lead at the equilibrium concentration dangerous?

Over the years the standards for lead concentration in drinking water has been gradually decreased. Starting at a maximum of 50 micrograms/l they were then reduced to 25 micrograms/litre. In 2013 the limit was reduced to 10 micrograms/l⁸. In fact, there are various substances added to drinking water in order to minimise lead concentrations⁹. 

But our above model dealt not with drinking water but instead the environment in general, and it is notoriously difficult to model how a human will interact with the natural environment. In addition there is contact with lead through the handling of lead shotgun shells, and possibly much more significantly, through the inhalation of particulate during shooting itself. All these questions are impossible to answer, so we will move on to questions two and three which investigate whether ingestion of lead shot presents a health hazard to living creatures. 

Lead Poisoning in Birds

The proposed plumbosolvency of E = 150 micrograms/litre that was used by Van der Leer et. al.⁶ does not reflect the digestive action of a human or animal. Nonetheless we will apply our exponential model to the problem of ingesting lead shot, modelling an organism as a body of water that absorbs lead from the metal pellets. 

Let us assume that five pellets are ingested. We start with a duck; a fairly typical kind of wildfowl. Assume that a duck weighs 1kg. Then as the duck is composed mainly of water it has a volume of 1 litre. Model the duck as a volume of water with a total of five pellets providing a mass transfer of lead. 

Again we use the solution to the diffusion equation: 

c= E – (E – c₀) . exp( AMT/VE ) 

And with moderate plumbosolvency E = 150, c₀=0, V=1, M=0.1 . 

The area of five pellets is 5 * 2.11 e-5 = 1.06 e-4 m2. 

We get the following plot: 

It takes longer (400-600 days) to achieve a steady state than in the previous graph showing diffusion over a hectare. However, the plumbosolvency level of 150 micrograms/litre is significantly over what is considered to be injurious to human health. Indeed the model reaches the 50 micrograms/litre level within only fifty or so days. It is therefore easy to imagine that shot stuck in the gizzard of a bird would cause significant poisoning and kill the bird well within its lifetime, and our model does not even account for any extra lead dispersal within the organism caused by the digestive process and chemicals. 

Question 3: Lead Shot Digested by a Human

Assume that a person ingests five lead pellets and assume a human has a weight of 100kg and therefore, being mostly of water, a volume of 100 litres. The only variable we need to change from the above model for a bird is therefore volume. We get the following result: 

But a human doesn’t have a crop or a gizzard. We do admittedly have an appendix but when something becomes stuck in the appendix this is considered an unlucky occurrence, not to be expected. From our graph lead levels do not reach significance for hundreds of days, based on this model, and with the human digestive system fully processing intake within a typical 24 hour time frame, it seems inconceivable that a human would suffer lead poisoning from ingestion of a reasonable number of lead pellets. However, there is the issue of the action of the digestive system to take in to account. Lead does react (albeit slowly) with hydrochloric and sulphuric acid, which are found in the human stomach. Nonetheless, checking pubmed.gov for case reports of adult lead poisoning from pellets or rounds returns no results. 

Exceptions to the above might include when small children¹⁰ ingest larger lead bodies or a great number of pellets; children have a much smaller body weight and volume than  an adult. Another compounding issue is if a lead body was to get stuck in the digestive system¹¹.

Conclusion 

The conclusion is a fairly balanced one. On the one hand, game or clay pigeon shooting has the potential to distribute very significant quantities of lead (sometimes >1 tonne/ha). Additionally, this lead saturates the environment at a very significant rate, reaching an equilibrium/saturation in under one year when the relevant land has been shot upon periodically going back 20 years or more. The environment does clearly become toxified.

How to break down the issue of lead exposure for adults engaged in game or clay shooting in the field is difficult and complex. There is exposure from handling weapons and ammunition, particulate in the air, and exposure to the environment itself. 

The exponential model from a linear ordinary differential equation indicates that due to the low mass of birds, and the tendency for shot to get stuck in the gizzard, it is extremely likely that a bird ingesting some pellets would experience lead poisoning. These conclusions were obviously also reached by the US Fish and Wildlife Service⁴ when they started designating steel-shot-only hunting zones for waterfowl as early as 1976⁵. Perhaps we should look to the USA. 

On the final issue of harm arising from human lead ingestion the conclusion is that due to the absence of a gizzard, and the relatively swift digestion and elimination that occurs in a human, it is very unlikely an adult would suffer lead poisoning from the ingestion of a reasonable number of lead pellets. Small children (with their small size and volume) who ingest slightly larger quantities of lead may well experience toxicity however, particularly if the lead bodies become stuck in the digestive tract. 

References:

1. Colley T. ‘No excuses’: 2024 phase out of lead shot possible with regulatory intervention, says report. Available at: https://www.endsreport.com/article/1799479/no-excuses-2024-phase-lead-shot-possible-regulatory-intervention-says-report.

2. Wildlife and Countryside Link. Report for parliamentary group concludes that ban on lead shot can be achieved by 2024. Available at: https://www.wcl.org.uk/ban-on-lead-shot-can-be-achieved-by-2024.asp#:~:text=The%20consultation%20proposals%20include%20a,in%20force%20by%20late%202024.

3. Johnson FM. The genetic effects of environmental lead. Available at: https://www.sciencedirect.com/science/article/abs/pii/S138357429700032X?via%3Dihub.

4. U.S. Fish & Wildlife Service. U.S. Fish & Wildlife Service. Available at: https://www.fws.gov/.

5. Middleton H. THE AMERICAN HUNTER STILL BALKS AT SWITCHING FROM LEAD TO STEEL SHOT. Available at: https://vault.si.com/vault/1984/11/05/the-american-hunter-still-balks-at-switching-from-lead-to-steel-shot.

6. Van Der Leer D, Weatherill NP, Sharp RJ, Hayes CR. Modelling the diffusion of lead into drinking water. Appl Math Model. 2002;26:681-699.

7. Vita Stembrera. Mass transfer differential equation. 

8. Cardew PT. Plumbosolvency Control in Soft Waters. Available at: https://www.deepdyve.com/lp/wiley/plumbosolvency-control-in-soft-waters-F7hbmVD8R4.

9. Hayes CR, Croft TN. Optimisation of plumbosolvency control using
computational modelling techniques: a demonstration
project for the Government of Alberta, working with the
City of Calgary and EPCOR (Edmonton). Available at: https://open.alberta.ca/dataset/c249d0ea-e494-4218-9b4a-300cb666afb5/resource/31bdfcc0-9a90-406b-8d22-2b0cf5bbbc7b/download/2014-optimisationplumbosolvency-feb2014a.pdf.

10. Treble G, Thompson T. Elevated blood lead levels resulting from the ingestion of air rifle pellets. Available at: https://pubmed.ncbi.nlm.nih.gov/12220020/.

11. Rozier B, Liebelt E. Lead Pellet Ingestion in 3 Children: Another Source for Lead Toxicity. . 2019.