Alex describes his experience of taking lithium and his search for information on its pharmacokinetic properties. He wanted to calculate his peak serum concentration and find out his approximate lithium half-life using the exponential decay formula. However, he found that there was no semi-log plot in the literature and no mention of the dual half-life of lithium. He searched for papers on simulations of the pharmacokinetic curve and found references to multi-compartmental models. Eventually, he found a chapter on lithium pharmacokinetics in a book and learned that the two half-lives he observed corresponded to the alpha and beta phases of lithium removal from blood vessels after peak concentration, which could be described using multicompartment models.

When I first started to take lithium, I wanted to calculate my peak serum concentration using my 12-hour sample value. I knew it was going to be a very rough estimation, but for me, doing the calculation made me feel less anxious about toxicity.

My plan was to use the exponential decay formula 1 to find the peak concentration (assuming the peak occurred around five hours). I wanted to find out, given I knew my serum creatinine levels, what my approximate lithium half-life was. I went looking for data in the lithium pharmacokinetic literature to figure this out.

This is how I came across the Hunter paper and the graph within (The one I have placed in the Appendix and in the A4 section above). The first thing I noticed was that there was no semi-log plot. Coming from a physics background, I found this strange. The exponential decay function is easily linearisable 2. By making the graph linear it is much easier to see where the data deviates from the algebraic function that is supposed to model it.

No matter, I thought. I would linearise the graph myself. What I found both surprised and confused me. Lithium seemed to have two elimination half-lives. See the semi-log plot below of the subject represented by the filled circles in Figure 1 of Hunter

The pharmacokinetic curves of all six subjects clearly showed this dual half-life property. I went searching for any mention of this dual half-life property in the pharmacokinetic literature. I couldn’t find anything. I found more data, but no semi-log plots and no mention of a dual elimination half-life of lithium. Further, all the guidelines and papers on optimal lithium serum concentration had no mention of it either (I don’t think any of the references I have included in the appendix of my Lancet Psychiatry article mentions anything approaching a dual half-life of lithium).

At the time I was very sick, and my brain just wasn’t functioning properly. I eventually settled on a stable dose of lithium and forgot about the dual half-lives – I put it down to the kidneys working harder or something after peak concentration (I know very little about how the kidneys filter out lithium). My focus was entirely on getting better.

When I was thinking about how physicists could help psychiatric researchers, I couldn’t think of a concise explanation. It wasn’t one thing in my head. It was a bunch of little aspects, little pieces, missing from many of the psychiatric papers I had read, that aren’t missing from the majority of physics papers I have seen.

My best idea was to give an example. My brain is still not in great shape. But it is working much better since my time on lithium. I was determined to get to the bottom of the mysterious double half-life of lithium.

My first thought was to look for papers on simulations of the pharmacokinetic curve. From my experience in physics, whenever there are linear trends in data, this strongly suggests it can be modelled. The first time around, I couldn’t find anything other than very rudimentary patient therapeutic dose predictions from 12-hour levels (usually some simple multiplication or division involving creatinine clearance).

This time, however, I did start to find papers on lithium pharmacokinetic simulation. I found it odd that none of the previous experimental papers I found seemed to reference these studies – even if it was only as a “future prospects” addition in the conclusion.

In the abstracts of the simulation papers, a term kept coming up: multi-compartmental model. I was unfamiliar with it. When looking up what the term meant, I saw this plot on the Wikipedia page on multicompartment models in pharmacokinetics.

The diagram looked very similar to the semi-log plot Hunter data. I now had scientific terms to search for. Eventually, in the second edition of the book “Applications in clinical pharmacokinetics” , I found a chapter on lithium pharmacokinetics. Figure 17-1 caught my attention

It was apparent that the two half-lives in the Hunter data corresponded to the alpha and beta phases of lithium removal from blood vessels after peak concentration. Further, quoting from the text:

After oral administration, lithium concentrations follow a complex concentration/time curve that is best described using multicompartment models”

This statement was followed by five references. The 1977 paper by Amdisen was the most interesting. In the inset of Figure 3, the distribution and elimination phases were shown, though crudely, with experimental data (compare linear dotted lines with red and blue lines from my Hunter graph earlier on). In Figure 11, Amdisen used a two-compartment simulation model, first outlined in Caldwell , to show that it aligned well with the data of one of the subjects. The Hunter graph discrepancy now made sense.

## References

1.
Bauer LA. Chapter 17. Lithium. In: Bauer LA, editor. Applied Clinical Pharmacokinetics, 2e [Internet]. New York, NY: The McGraw-Hill Companies; 2008 [cited 2023 Feb 7]. Available from: https://accesspharmacy.mhmedical.com/content.aspx?aid=3518187
1.
Caldwell HC, Westlake WJ, Connor SM, Flanagan T. A Pharmacokinetic Analysis of Lithium Carbonate Absorption from Several Formulations in Man. The Journal of Clinical Pharmacology and New Drugs [Internet]. 1971 [cited 2023 Feb 7];11(5):349–56. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1177/009127007101100505
1.
Amdisen A. Serum Level Monitoring and Clinical Pharmacokinetics of Lithium. Clin Pharmacokinet [Internet]. 1977 Apr 1 [cited 2023 Feb 7];2(2):73–92. Available from: https://doi.org/10.2165/00003088-197702020-00001
1.
Hunter R. Steady-state pharmacokinetics of lithium carbonate in healthy subjects. British Journal of Clinical Pharmacology [Internet]. 1988 [cited 2022 Nov 25];25(3):375–80. Available from: https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1365-2125.1988.tb03316.x
1. C(t) = C_0e^{-\lambda t} where: C(t)  is the serum lithium concentration at time t , C_0  is the peak serum concentration (assuming peak is at t=0 ) and the decay constant  \lambda = \ln (2)/t_{1/2}  where t_{1/2}  is the half-life.
2. To put the exponential decay function into linear form, y = mx + c , take the natural logarithm of both sides i.e. \ln (C(t)) = \ln ( C_0) e^{-\lambda t}  simplify using logarithm rules: \ln(C(t)) = \ln(C_0) -\lambda t . The half-life can be found through the gradient m =- \lambda . Essentially, any straight line that can be seen in the Hunter semi-log plot (after peak) represents a half-life of lithium.