Medical Application
The production of the thyroid hormone FT4 is controlled by the hypothalamus and the pituitary gland by way of a feedback system by excreting the thyroid stimulating hormone TSH in response to the level of FT4. The relation between the TSH and FT4 was found to be (by S.L. Goede):
TSH = S/exp(φ FT4)
When treating patients with a thyroid imbalance, the optimum setpoint for the feedback loop is given by:
[FT4] = ln(φ S √2)/φ, [TSH] = 1/(φ √2)
where the parameters S and φ
are obtained by fitting the model to measured TSH-FT4 combinations obtained during treatment.
Formulated like this, the numerical condition of the optimization problem is far from ideal. The sensitivity of the objective function with respect to the parameters varies widely, and the parameters have a huge range.
The model can be reformulated by choosing for the model parameters the setpoint-coordinates themselves. An additional advantage of this approach is that the calculated parameter precision applies directly to the extracted setpoint.
Here are some datasets from real patients. Try both models on Patient 2 and observe the difference.