1. Introduction
In recent years, the mathematical modeling of human thyroid hormone homeostasis via the hypothalamic–pituitary–thyroid feedback loop has received an increasing amount of attention. Starting from early phenomenological models, more precise models have been developed based on molecular and pharmacokinetic data, see, e.g., Ref. (1–3, 4–6) for recent surveys on existing modeling approaches. These mathematical models can give important insight into the functionality of the hypothalamic–pituitary–thyroid axis and can be used to simulate the dynamic behavior of thyroidal hormone concentrations under different (euthyroid and non-euthyroid) conditions, and sometimes also for clinical decision-making (7). Furthermore, in Ref. (8), a method is proposed to compute personalized euthyroid setpoints that can be used for individualized diagnosis and treatment of thyroid diseases. While this static model is appealing due to its simplicity (only two parameter values have to be estimated), it does not consider any dynamic phenomena in the HPT axis, which are, however, of great importance for a deepened understanding of the HPT axis and ultimately the development of personalized optimal medication strategies. Another drawback is the absence of any consideration of T3, which has been shown to be significant not only as a key actor in the hypothalamic–pituitary–thyroid feedback loop (4, 9) but also in maintaining a good quality of life (5).
The main objective of this paper is an improved mathematical modeling of the HPT axis in order to obtain a more detailed understanding of the dynamic phenomena occurring in thyroid hormone homeostasis. In particular, as a first contribution, we extend the model originally developed in Ref. (1, 2) in order to incorporate new insights obtained through several recent clinical studies. In particular, we incorporate a direct TSH–T3 path inside the thyroid, accounting for the central T3 production by the thyroid. Existence of such a TSH–T3-shunt was hypothesized and demonstrated in several experiments and clinical observations (10–15). In Ref. (10), it was shown that L–T4-treated athyreotic patients exhibit decreased FT3 concentrations despite normal free thyroxine (FT4) levels, which would not be the case if peripheral FT3 was mainly produced by deiodination of peripheral FT4. Furthermore, the sum activity of step-up deiodinases (GD) is positively correlated with the TSH concentration (11) and with the thyroidal volume (12) and significantly decreases after thyroidectomy. These observations suggest that besides the peripheral T4/T3 conversion, also TSH-stimulated deiodinases inside the thyroid contribute to the total T3 production. In our work, we show that the extended model including such a TSH–T3-shunt is in good accordance with various clinical observations. For example, we show that the FT3 concentration shows a clear circadian pattern, as was observed in vivo in Ref. (16). Notably, this is not the case in the previous model, which did not include the TSH–T3-shunt.
As a second main contribution of this paper, we perform a sensitivity analysis of the derived model. Loosely speaking, the (first-order) sensitivities are a measure for how “sensitive” certain system states (i.e., TSH or hormone concentrations) are with respect to changes in certain parameters (such as, e.g., the thyroid’s secretory capacity GT). These sensitivities reveal structural insight into the functionality of the hypothalamic–pituitary–thyroid axis and can provide explanations for certain clinical observations. For example, we show that the sensitivity of TSH with respect to GT is much higher for low values of GT (i.e., in hypothyroidism) than for high values of GT (i.e., in hyperthyroidism). This fact can be used to explain why in clinical practice, TSH concentrations may significantly vary beyond the upper limit of the reference range despite normal thyroid function.
The remainder of this paper is structured as follows. Section 2 presents the extended mathematical model and discusses the identification of the required (additional) parameters. In Section 3, we show simulation results of the derived model and discuss the observed properties (such as the existence of a circadian rhythm in FT3 concentrations). A sensitivity analysis of TSH, FT4, and FT3 concentrations with respect to different parameters is performed in Section 4. Finally, we conclude the paper in Section 5.
