MMSC special topic in mathematical physiology
Optimising IVF drug dosing
Report © 2026 George Hildick-Smith, shared under CC BY-NC-ND 4.0. Third party images and quotations remain the property of their owners.
Overview
In in vitro fertilisation (IVF), hormone injections stimulate the ovaries to produce several mature eggs. Here we build a simple model of the key hormones, FSH, LH and oestradiol that are used in the ovulation cycle. Together with the number of maturing follicles, using Hill functions for the feedback between them we can create a system of ordinary differential equations and numerically simulate them.
On top of the model sits an optimisation problem: choose the dose schedule that gives the most mature follicles at egg retrieval for the least drug, without the dose or the LH level reaching unsafe values. Restricting doses to realistic sizes and times turns it into an integer programme that can be searched directly.
What I found
- In this model the LH level can never exceed a fixed ceiling however much FSH is given, a useful check when fitting it to real data, since the ceiling has to sit above the ovulation threshold.
- The best injection times are always the latest ones allowed, because the drug may wear off. That reduces the problem to choosing the dose sizes.
- In the worked example, the best schedule is one unit on day 11 and half a unit on each of days 12 and 13, giving about 2.17 mature follicles at low drug cost.
