In Figure 2 , the results of the L-T ₄ monotherapy are illustrated for one daily intake, meaning that the simulated treatment considers one daily intake of L-T ₄. In Figure 2A , the course of the hormone concentrations over 15 days is illustrated by the continuous lines. The dashed lines represent the setpoints of the considered euthyroid generic healthy individual. Figure 2B illustrates the corresponding amount of L-T ₄ for each day. The hormone concentrations do not reach their desired setpoints. The concentration of T ₄ remains slightly higher, and the concentrations of TSH and T ₃ remain slightly lower than their respective setpoints. The concentrations of T ₄ and TSH fluctuate strongly compared to the concentration of T ₃. A high starting dosage (400 µg L-T ₄) followed by a steady-state dosage (130 µg) are optimal. The nonlinear decrease of the TSH concentrations is due to the nonlinear dependence of T ₄ and TSH, compare eqs. (S4), (S5), (S8), and (S9) in the Supplementary Material .

In Figure 3 , the simulation results of the L-T ₃/L-T ₄ combined therapy are illustrated for one daily intake. In contrast to the L-T ₄ monotherapy, the setpoint is reached for all hormone concentrations (taking into account the unavoidable daily fluctuations), no persisting offset between the hormone concentrations and their setpoints can be seen. The fluctuations of the hormone concentration of T ₃ are higher, whereas the fluctuations of T ₄ and TSH are similar compared to the L-T ₄ monotherapy. Again, high starting dosages (400 µg of L-T ₄ and 30 µg of L-T ₃) followed by steady-state dosages (121 µg of L-T ₄ and 4.7 µg of L-T ₃) are optimal.

In this section, the results of both therapies for different genetic variants are shown. In Figures 4 – 6 , we consider G _D1'=0.9G _D1, G _D1''=1.1G _D1, G _{D 1}'''=1.2G _{D 1}, respectively. The left column plots illustrate the results of the L-T ₄ monotherapy and the right column plots show the results of the L-T ₃/L-T ₄ combined therapy. As mentioned in the previous section, the adapted numerical values represent exemplarily different genotypes.

The L-T ₄ monotherapy does not restore the setpoint of healthy individuals for the case G _D1'=0.9G _D1 (compare Figures 4A, B ). There is a substantial offset visible between the T _3, T _4, and TSH concentrations and their euthyroid setpoints. This offset is higher compared to the case when no genetic variant is considered, compare Figure 2 . In turn, the L-T ₃/L-T ₄ combined therapy restores the euthyroid setpoint of healthy individuals up to some daily fluctuations.

In the case of G _D1''=1.1G _D1 (compare Figure 5 ), not only the L-T ₃/L-T ₄ combined therapy but also the L-T ₄ monotherapy reaches the euthyroid setpoint. In contrast to the L-T ₄ monotherapy, the L-T ₃/L-T ₄ combined therapy goes along with high fluctuations of T ₃ during the first days of therapy.

In the last case, where G _{D 1}'''=1.2G _{D 1}, the L-T ₄ monotherapy and the L-T ₃/L-T ₄ combined therapy lead to similar results. Both therapies reach the setpoint up to some small offset; the concentrations of T ₃ and TSH are slightly higher than their respective setpoints whereas the concentration of T ₄ remains slightly lower than the respective setpoint, which is the opposite situation to Figure 2 . In Figure 6D , one can see that after day three no L-T ₃ is needed to remain in the steady state. Therefore, the L-T ₃/L-T ₄ combined therapy reduces to an L-T ₄ monotherapy. For the sake of brevity, the results concerning the CC genotype of polymorphism rs225014 are shown in Figures 7 ,  8 of the Supplementary Material . These figures illustrate that the L-T ₃/L-T ₄ combined therapy results in better steady-state hormone concentrations, if the T ₃ setpoint is set to the upper limit of the reference range of healthy individuals. Additionally, the euthyroid setpoint is reached earlier in case of the L-T ₃/L-T ₄ combined therapy.

In this section, we show the results of the analysis regarding the frequency of medication intake. Here, we do not consider genetic variants. We compare one, two, and three daily intakes. Since Figures 2 ,  3 already show the results for one daily intake, Figures 7 , 8 complement the cases for two and three daily intakes. Table 1 summarizes the effects of each frequency of intake for both types of therapy. Exemplary, we analyze the first day and the last day of therapy, which are representative for the transient phase and the steady state.

In general, concerning both types of therapy, an increasing number of medication intakes reduces the fluctuations of all hormone concentrations. In particular, we focus on the concentrations of T ₃. Regarding the L-T ₄ monotherapy (compare Figures 2 , 7 ), the T ₃ concentrations do not fluctuate and smoothly increase until the steady state is reached.

In turn, the T ₃ concentrations fluctuate substantially concerning the L-T ₃/L-T ₄ combined therapy (see Figures 3 ,  8 ). These fluctuations are higher in the transient phase and smaller in the steady state. The strongest fluctuations occur for one daily intake.

Note that the small variations of the L-T ₄ dosage (after the steady state has been reached) regarding three daily medication intakes (compare Figure 8D ) are due to numerical issues and not due to medical reasons. We need to solve a highly nonlinear optimization problem where the solver can be stuck in local minima.

As mentioned in the introduction, one difficulty of the current treatment strategies is the trial-and-error process in order to find the correct individual dosage. The here presented procedure represents the first step to improve this (potentially unnecessary) trial-and-error process. For a generic hypothyroid patient, the daily dosages can directly be adopted from the simulation results of the MPC. In contrast to (18), the MPC does not only compute an optimal steady-state dosage but also optimal dosages regarding the transient phase of the therapy.

Furthermore, the usage of an MPC provides the opportunity to adapt the dosage much earlier, if necessary. In Figures 2 , 3 , one can see that the steady-state hormone concentrations are reached within the first week of the start of the therapy. Therefore, if the resulting thyroid hormone replacement strategy does not yield to the desired outcome, one can adapt the dosage(s) already after one or two weeks, as it is also suggested in (18). It is not necessary that the adaptation is made after 4-6 weeks, as it is currently recommended (1).

Another interesting aspect is the tendency that high dosages at the beginning of the therapy followed by lower dosages are optimal. In the past, some experts suggested that the treatment strategy should start with low dosages and should be continued by increasing the dosages only slowly (sometimes designated as “start low go slow” policy), especially for elderly patients (1, 30). However, this strategy seems outdated for cardiac asymptomatic patients (31). The results presented here underline once again the advantageous effects of high starting dosages: the setpoint is reached much faster with this policy than by following the “start low go slow” policy. These considerations apply especially to myxoedema coma, the most severe complication of hypothyroidism. The question of optimal substitution treatment remains controversial (32), and the modality of optimum treatment continues to be uncertain, mainly due to the low number of clinical studies and inherent difficulties in performing controlled trials. The population at risk from myxoedema coma predominantly includes elderly subjects and patients suffering from cardiovascular disease, thereby suggesting that the dosage should be escalated slowly. However, several case series and small studies reported it to be beneficial to start with a single high-dose intravenous bolus of 500 µg and to continue with a maintenance dose of 50 to 100 µg daily (33, 34).

One important long-term perspective of this work is to translate these results into clinical practice. Before this translation can be realized, two issues must be considered. First, the optimal dosages determined by means of the MPC may not be available in form of a commercial product. Second, the medication dosages should not be adapted every day. As visible in Figure 2B , the results of the MPC suggest that the (generic) patient should take in 400 µg of L-T ₄ for the first four days. In the following days, the patient should take in several different dosages before a constant dosage of approximately 130 µg of L-T ₄ should be taken in. In clinical practice, one might simply prescribe 400 µg of L-T ₄ for the first four days. Subsequently, one could directly prescribe the closest commercially available dosage to 130 µg of L-T ₄. Following this procedure, one only adapts the dosage once and additionally only uses dosages that are available in form of a commercial product. An additional issue must be considered regarding the prescription of L-T ₃. Currently, the commercially available L-T ₃ dosages are remarkably higher than the dosages that are recommended in clinical guidelines and that are determined in this work. For example, in Germany, the available L-T ₃ medications such as Thybon^® 20 or the combined L-T ₃/L-T ₄ preparation Prothyrid^® contain 20 µg or 10 \mu\g of L-T₃, L-T ₃ respectively. These dosages are approximately four times or two times higher compared to the optimal dosages determined by means of the MPC in this work. Therefore, the implementation of our work (and of the mentioned guidelines) in clinical practice imperatively requires the availability of pharmaceutical preparations containing lower dosages of L-T ₃.

Most practicing physicians are familiar with the prescription of the dosages regarding an L-T ₄ monotherapy. However, they are potentially not familiar with the prescription of L-T ₃ making the treatment with an L-T ₃/L-T ₄ combined therapy challenging. Possible guidelines for physicians are given by the European Thyroid Association (13). Four alternative formulas are given, but no preference regarding one formula is mentioned. Consequently, the treating physician decides the formula so far. We provide assistance to this decision by comparing the formulas of (13) to our simulation results. The formulas in (13) are based on a prior treatment with L-T ₄ that normalizes the thyroid hormone concentrations, which would be in our case around 130 \mu \g, compare Figure 2 . The resulting dosages for the L-T ₃/L-T ₄ combined therapy of the four different formulas are given in Table 2 . The dosages regarding an L-T ₃/L-T ₄ combined therapy determined by means of the MPC are approximately 121 µg of L-T ₄ and 4.7 of L-T ₃. By comparing these dosages to the ones given in Table 2 , one can conclude that the simulation results fit best to formula B2. This suggests that from the four different methods proposed in (13), formula B2 seems to be the best strategy.

So far, we only considered a generic hypothyroid patient. A matter of ongoing research deals with a reliable individualization of the mathematical model. When this is the case, the treatment strategies obtained by applying the MPC will be optimal individual strategies. This would clearly pave the way into the era of personalized medicine. In a first clinical trial exploiting the advantages of mathematical modeling to determine optimal dosages, the importance of an individualized model is pointed out (18). However, the individualization of our complex model is currently not straight-forward, one problem in this context is that some model parameters cannot be identified uniquely based on only few individual hormone measurements (15, 20). This issue could potentially be resolved using more frequently measured (dynamic) hormone data as available in (18), compare the discussions in (15) and (20), which is subject of future work.

Interestingly, clinical studies such as (35) document that athyroid patients that are treated with an L-T ₄ monotherapy have higher T ₄ but similar T ₃ concentrations compared to the patients’ prethyroidectomy concentrations. Higher T ₄ concentrations in the case of the L-T ₄ monotherapy are also visible in the simulations of the mathematical model, compare Figure 2 . However, the simulations show slightly decreased concentrations of T ₃ compared to the concentrations of healthy individuals, see Figure 2 . This can be explained by the design of the MPC (compare Section S2 of the Supplementary Material ). The objective is to reach simultaneously the euthyroid steady-state concentrations of T ₃, T ₄ and TSH leading to a trade-off between too high T ₄ and too low T ₃ concentrations. If one chooses a cost function that penalizes solely the deviation of T ₃ from its steady state, the result is a T ₃ conncentration similar to the setpoint of healthy individuals and a higher concentration of T ₄ than the one observed in Figure 2 as reported in (35). For the sake of brevity, the corresponding simulation results are illustrated in Section S3.2, Figure 3 of the Supplementary Material. Additionally, that section contains simulation results with cost functions that penalize mainly the deviations of T ₄ or TSH from their euthyroid setpoints. These results illustrate that the course of the hormone concentrations depends on the tuning of the weighting matrix Q, compare Figures 4 and 5 of the Supplementary Material .

Note that the authors in (16) underline the importance of high serum T ₃ concentrations for a successful outcome of the L-T ₃/L-T ₄ combined therapy. This objective could be easily met by adapting the cost function of the MPC so that the difference between the measured hormone concentrations and the upper limit of the reference range is penalized. In addition, for patients affected by differentiated thyroid cancer a cost function that solely penalizes the T ₃ concentrations can be benefical, since for these patients the free T₃ (FT ₃) concentratios are particulary important for the relief of symptoms, compare (36).

As mentioned in the previous section, the L-T ₄ monotherapy can only approximately restore the setpoint of healthy individuals (with some remaining offset), whereas the L-T ₃/L-T ₄ combined therapy can restore this setpoint. In the case of the L-T ₄ monotherapy, the missing endogenously produced T ₄ is simply replaced by L-T ₄. T ₃ is synthesized out of T ₄ in peripheral organs and in the thyroid mainly by D1 and by D2. The peripheral T ₃ production is restored by an L-T ₄ monotherapy, since the serum T ₄ is replaced by L-T ₄. In turn, the production of T ₃ in thyroid cells (describing the TSH-T ₃ shunt) cannot be restored by L-T ₄, because L-T ₄ only enters the pituitary-thyroid feedback loop in the periphery and not within the thyroid. Consequently, the L-T ₄ dosages increase, until the best trade-off between a too low concentration of T ₃ and a too high concentration of T ₄ is found.

Regarding the L-T ₃/L-T ₄ combined therapy, the missing endogenously produced T ₄ and the subsequent peripheral T ₃ production can again be replaced by L-T ₄. In contrast to the L-T ₄ monotherapy, the production of T ₃ in thyroid cells is compensated through the intake of L-T ₃. Since only a small fraction of T ₃ is synthesized within thyroid cells (15), the necessary L-T ₃ dosages are rather low.

As explained above, the main reason for the resulting different treatment outcome of an L-T ₄ monotherapy compared to an L-T ₃/L-T ₄ combined therapy is the fact that the T ₃ production also takes place in thyroid cells (the so-called TSH-T ₃ shunt, compare (15)). When this intrathyroidal T ₃ production is not considered in the mathematical model, the final treatment outcome between the two types of therapy is the same. In particular, in (14), where no direct T ₃ synthesis based on TSH is considered in the model (also referred to as TSH-T ₃ shunt), the authors conclude that the L-T ₃/L-T ₄ combined therapy is not superior to the L-T ₄ monotherapy, because the same hormone concentrations are reached and because of undesired fluctuations of the T ₃ concentrations in the case of the L-T ₃/L-T ₄ combined therapy.

Since the difference in the outcome of the two therapies is rather small (compare Figures 2 , 3 ), it might seem questionable whether an L-T ₃/L-T ₄ combined therapy should be recommended in practice. However, when it is known that the patient is affected by an AA or CC genotype in polymorphism rs2235544 in DIO1, the difference between the L-T ₄ monotherapy and the L-T ₃/L-T ₄ combined therapy in the simulations of the model can be substantial. On the one hand, when patients have an AA genotype (leading to lower activity of D1, here exemplary considered by G _D1'=0.9G _D1) they might profit considerably from an L-T ₃/L-T ₄ combined therapy, compare Figure 4 . In this case, less T ₃ is produced out of T ₄ in peripheral organs and in the thyroid gland. The lack of endogenously produced T ₃ cannot be compensated by means of the L-T ₄ monotherapy. Again, the controller increases the L-T ₄ dosages further in order to reach the best trade-off between a too low T ₃ concentration and a too high T ₄ concentration. In turn, the L-T ₃/L-T ₄ combined therapy restores the setpoint of healthy individuals by means of higher L-T ₃ dosages. The difference to the case without genetic variants is an increased offset between the T ₃, T ₄ and TSH concentrations and their respective setpoints regarding the L-T ₄ monotherapy and increased T ₃ fluctuations regarding the L-T ₃/L-T ₄ combined therapy. On the other hand, when the patient has a CC genotype leading to a higher activity of D1 (here considered by G _D1 ^′​′=1.1G _D1), there is no difference in the outcome of both therapies. There is enough endogenously produced T ₃ such that the setpoint of healthy individuals is reached even for the L-T ₄ monotherapy. The intrathyroidal production of T ₃ (that cannot be replaced by an L-T ₄ monotherapy) is compensated by the increased D1 activity. In contrast to the L-T ₄ monotherapy, the L-T ₃/L-T ₄ combined therapy goes along with high fluctuations of T ₃ during the first days of therapy. Furthermore, the simulation results for G _{D 1} ^{′​′​′}=1.2G _{D 1} (which is another simple exemplary modeling of two C-allele) show that both therapies reach the setpoint up to some small offset; the concentrations of T ₃ and TSH are slightly higher than their setpoints whereas the concentration of T ₄ remains slightly lower than the respective setpoint. Here, the intrathyroidal production of T ₃ is overcompensated by the increased activity of D1, explaining the opposite results of Figure 2 (where a too low concentration of T ₃ and a too high concentration of T ₄ is observed). Therefore, an additional intake of L-T ₃ would lead to even higher concentrations of T ₃ which are not desired. Consequently, even though two medication inputs are available, it is impossible to reach the euthyroid setpoint. In Figure 6D , one can see that after day three no L-T ₃ is needed to remain at the steady state. Therefore, the L-T ₃/L-T ₄ combined therapy reduces to an L-T ₄ monotherapy. If one additionally prescribes L-T ₃ after day three, the T ₃ concentrations would be even higher which potentially leads to symptoms of hyperthyroidism.

In conclusion, our results suggest that in case of genotype AA regarding polymorphism rs2235544 of DIO1, an L-T ₃/L-T ₄ combined therapy can be beneficial, whereas an L-T ₄ monotherapy seems to be better suited (with respect to the achieved thyroid hormone concentration) in case of genotype CC concerning polymorphism rs2235544 of DIO1. However, clinical studies do not document a better response (in terms of symptoms) to an L-T ₃/L-T ₄ combined therapy when patients are affected by this polymorphism (26). This difference to our result could be explained by the hypotheses mentioned in (16) stating that the success of the L-T ₃/L-T ₄ combined therapy strongly depends on whether the T ₃ concentrations are brought to the upper reference range and on the residual thyroid function. Furthermore, the simulations of polymorphism rs225014 reveal that the L-T ₃/L-T ₄ combined therapy is better to treat hypothyroidism in terms of the treated steady-state hormone concentrations, if the T ₃ setpoint is set to the upper limit of the reference range of healthy individuals, compare Figures 7 , 8 in the Supplementary Material . This result is in line with the study (26), where the authors document a better response to the L-T ₃/L-T ₄ combined therapy for patients affected by the considered genotype. Once again, these results underline the relevance of the hypothesis that the L-T ₃/L-T ₄ combined therapy is successful if the T ₃ concentrations are brought to the upper reference range of healthy individuals because the L-T ₃/L-T ₄ combined therapy does not yield to an improved outcome in terms of the treated steady-state hormone concentrations, if the T ₃ setpoint remains unchanged. In any case, the benefit of the L-T ₃/L-T ₄ combined therapy might depend strongly on genetic variants. Such genetic variants could play a crucial role in explaining why some studies document an advantage of the L-T ₃/L-T ₄ combined therapy over the L-T ₄ monotherapy and others not as also suggested in (13).

Before going into detail regarding the optimal L-T ₃ intake frequency in order to reach stable T ₃ concentrations, we briefly want to comment on the impact of the L-T ₄ intake frequency on the course of the T ₄ concentrations. These are known to be stable in the case of an L-T ₄ monotherapy for one daily intake (13). The simulations of the L-T ₄ monotherapy emphasize this observation, compare Figures 2 ,  7 as well as Table 1 . Two or three daily medication intakes lead to reduced fluctuations of T ₄ as shown in Table 1 . However, since many patients feel well with one daily intake of L-T ₄, there is potentially no need in increasing the intake frequency although the fluctuations would decrease. Furthermore, the fluctuations of the T ₄ concentrations are similar concerning the L-T ₃/L-T ₄ combined therapy. This is due to the similar dosages of L-T ₄ regarding the L-T ₄ monotherapy and the L-T ₃/L-T ₄ combined therapy. In addition, the fluctuations of the T ₄ concentrations are similar in the transient phase and in the steady state, compare ΔT ₄ in Table 1 for one daily intake. The long half-life of approximately one week leads to stable hormone concentrations, such that the fluctuations are not impacted considerably by the different medication dosages occurring in the transient phase and in the steady state.

Considering the T ₃ concentrations in Figures 2 and 7 as well as in Table 1 , one can see that the L-T ₄ monotherapy leads to stable T ₃ concentrations, even in the case of one daily intake. Regarding the L-T ₃/L-T ₄ combined therapy, the frequency of the intake of L-T ₃ influences substantially the fluctuations of the T ₃ concentrations, compare Figures 3 , 8 , as well as Table 1 . As expected, a higher intake frequency of L-T ₃ results in lower fluctuations of the T ₃ concentrations. However, a higher intake frequency is also less convenient for patients, especially since these therapies are often life-long therapies. Given the numbers reported in Table 1 , a good trade-off could be to consider two daily intakes in case of the L-T ₃/L-T ₄ combined therapy.