Mathematical Modeling of the Pituitary-Thyroid Feedback Loop: Role of a TSH-T3-Shunt and Sensitivity Analysis

Abstract

Despite significant progress in assay technology, diagnosis of functional thyroid disorders may still be a challenge, as illustrated by the vague upper limit of the reference range for serum thyrotropin (TSH). Diagnostical problems also apply to subjects affected by syndrome T, i.e., those 10% of hypothyroid patients who continue to suffer from poor quality of life despite normal TSH concentrations under substitution therapy with levothyroxine (L-T₄). In this paper, we extend a mathematical model of the pituitary-thyroid feedback loop in order to improve the understanding of thyroid hormone homeostasis. In particular, we incorporate a TSH-T₃-shunt inside the thyroid, whose existence has recently been demonstrated in several clinical studies. The resulting extended model shows good accordance with various clinical observations, such as a circadian rhythm in free peripheral triiodothyronine (FT₃). Furthermore, we perform a sensitivity analysis of the derived model, revealing the dependence of TSH and hormone concentrations on different system parameters. The results have implications for clinical interpretation of thyroid tests, e.g., in the differential diagnosis of subclinical hypothyroidism.

Keywords: TSH-T3-shunt; diagnosis; mathematical modeling; pituitary–thyroid feedback loop; sensitivity analysis; thyroid hormones.

Figure 1

Block diagram of the thyrotropic feedback control loop with an additional TSH-T₃-shunt, adapted from Ref. (1, 2). Except for G_T3, k, and G_D1, all parameters were adopted from the model in Ref. (1, 2). The parameters G_D1 and G_T3 were estimated to obtain an optimal (in a least squares sense) fit to measured in vivo FT₃-concentrations. To this end, the value of k was normalized to 1 mU/l.

Figure 2

Set of optimal (in a least-squares sense) parameters G_T3 and G_D1 when normalizing the parameter k to 1 mU/l. Due to the affine dependence of FT_3,eq (G_T3, k, G_D1) on G_T3 and G_D1, the set of optimal parameters is contained in a one-dimensional affine subspace of ${\mathbb{R}}^2$.

Figure 3

FT₃-plots $\mathrm{[}\frac{\mathit{pmol}}{l}\mathrm{]}$ over a simulation horizon of 25 days for several configurations of the TSH-T₃-Shunt. The parameters G_T3 and G_D1 are identified via least squares optimization, separately for each model configuration. (A) No shunt included. (B) Full TSH-T₃-shunt.

Figure 4

Sensitivity of T₄ w.r.t. G_T for different values of G_T. (A) $G_T=1.2\cdot {10}^{-12}\frac{\mathit{mol}}{s}$, (B) $G_T=3.375\cdot {10}^{-12}\frac{\mathit{mol}}{s}$ - nominal value, (C) $G_T=5\cdot {10}^{-12}\frac{\mathit{mol}}{s}$.

Figure 5

Stationary sensitivity of T₄ w.r.t. G_T as a G_T-dependent function. The red point indicates the nominal G_T-value from Ref. (1).

Figure 6

Sensitivity of TSH w.r.t. TRH for different values of S_S. (A) $S_S=0\frac{l}{\mathit{mU}}$, (B) $S_S=50\frac{l}{\mathit{mU}}$, (C) $S_S=100\frac{l}{\mathit{mU}}$ - nominal value, (D) $S_S=200\frac{l}{\mathit{mU}}$.

Figure 7

Plots of equilibrium T₄ and TSH levels depending on the thyroid’s secretory capacity G_T. The red point in the figures indicates the nominal G_T-value from Ref. (1). (A) Equilibrium T₄, (B) equilibrium TSH.

Figure 8

Sensitivity of FT₃ w.r.t. G_T for two versions of the HPT axis model: one incorporating the TSH-T₃-shunt and one without this extension. (A) Full TSH-T₃-shunt, (B) no shunt included.

Figure 9

Plots of the stationary sensitivity of FT₃ w.r.t. the parameter G_T as a function of the thyroid’s secretory capacity G_T. Two configurations of the model are shown: one including the TSH-T₃-shunt and one without the shunt. The red point in the Figures indicates the nominal G_T-value from Ref. (1). (A) Full TSH-T₃-shunt, (B) no shunt included.