# Inconsistent Retention Function

Mithkus’s paper uses mathematical models to describe the body burden of Aluminum following the pediatric vaccination schedule. In that paper he uses a retention function based on observational data captured through an adult volunteer who was injected with Aluminum Citrate, which has a completely different solubility profile than the adjuvant we inject with vaccines.

Flarend runs experiments on white rabbits to analyze the distribution of Aluminum following administration of two kinds of Aluminum Adjuvant (Aluminum Hydroxyde (AlOH) and Aluminum Phosphate (AlPO4)). Flarend’s paper is mentioned and discussed by Mithkus who observed that the amount of Aluminum entering the blood-stream happens quite slowly (between 17% and 51% for Aluminum Hydroxide). But the paper also shows that only 6% (AlOH) to 22% (AlPO4) of the Al is ever excreted. One possible explanation for why so little is excreted is that Priest models Aluminum Citrate (in ionic form) but vaccines contain adjuvants, which have a crystalline, rather insoluble structure that are too large to be cleared by kidneys.

A natural question arises: does the mathematical model describe the experimental evidence? Below, I used Keith’s body burden accumulation (which we validated here) to show what would happen if you gave an adult 0.85 mg of Al and assumed only 17% enters the blood-stream steadily over 28 days. The amount excreted (shown in blue) is the total amount administered minus the body burden (shown in green). As you can see, Flarend measured 0.05 mg, but Priest’s model predicts about 0.09 mg, which is an 80% over-estimation of how quickly Aluminum actually leaves the body. Notice, however, that the dashed green line showing phosphate adjuvant closely matches one of the rabbits (shown as unfilled circles) and notice also that there is a great deal of variation between the phosphate rabbits (but not the hydroxide rabbits). Hence, we Priest’s model may be somewhat appropriate for the treatment of Phosphate adjuvants, but AlOH is the most common adjuvant, which is why we choose to focus on the latter.

Aside from the over-estimation of the amount excreted, the models employed by Mithkus assume a linear absorption, i.e. the 51% of AlOH is assumed to enter at a constant rate of 1.82% per day (=51%/28). In reality, the aluminum dissolves quickly and then slowly, which explains why the excretion curve flattens out over longer horizons.

To get the blue line (showing modeled excretion) to fit the black line (showing experimental excretion), I replaced the linear absorption assumption (which is a triangular distribution) with a pareto distribution, which administers the aluminum quickly in the beginning and slows down over time. The pareto distribution has only one parameter and so to fit the distribution all you have to do is to make sure that the cumulative amount absorbed after 28 days is 17%; in other words, the CDF should be 0.17 after 28 days. Therefore,

$\alpha=-log(1-0.17)/log(24*28)$

Then I took the same functional form as Priest’s Retention function and optimized the parameters in Matlab with fminsearch to get a more realistic retention function. The functional form is

$R=\gamma*t^\beta$

The optimization/ minimization procedure yields 0.8444 and -0.1002 for alpha and beta, respectively. Compare this to Priest’s values of 0.354, and -0.32. Notice how the blue line (showing modeled excretion) now nails the black line (showing the experimental evidence). So now we have something that more realistically fits the evidence and that pertains specifically to Aluminum Hydroxide used in vaccines rather than Aluminum Citrate (which would be more appropriate for dietary exposures).

The pink line shows the total amount of Aluminum that is assumed to have entered the blood. Notice how the total approaches 0.14 after 28 days, which corresponds to 17% of the 85mg administered (this is by design).

Now that we have some fresh “Priest Equations” (let’s call these the Flarend Retention functions) that look like this:

$R=0.8444t^{-0.1002}$

we can now do some interesting things like rebuild Mithkus’s MRLs, shown below. Notice how the vaccine line now consistently stays above the 1mg/kg/day Minimal Risk Level during the first 250 days of a baby’s life. At the two month visit, the exposure is about 3 times higher than the allowed body burden.

The green dashed line shows the level that would be acceptable if you used a conservative 10mg/kg/day as a level where no adverse events were detectable (NOAEL), as reported by the European Food Safety Authority (EFSA). They state:

Similarly, the lowest no-observed-adverse-effect levels (NOAELs) for effects on these endpoints were reported at 30, 27, 100, and for effects on the developing nervous system, between 10 and 42 mg aluminium/kg bw per day, respectively.

It is then typical to use these NOAEL values as Mithkus does and to divide it for uncertainty and extrapolation factors. We divide by 10 for uncertainty and an additional 3 for extrapolating from animal to human trials. Thus the MRL for a 10mg /kg/day is 0.33. To construct these MRL’s, I then used the same bio-absorption rates as Mithkus did (0.78%), meaning that 0.33mg times 0.78% makes it into the blood stream.

The dashed red line shows what the European Food Safety Authority actually recommends (1 mg/kg/week), noting that the average dietary exposure in the general population varies from 0.2 to 1.5 mg/kg/week.

Values based on the green lines or the blue lines are probably not terrible assumptions. See page 18 of ATSDR’s publication for how the MRL of 1 mg/kg/day was derived.