 Research
 Open Access
A patientspecific treatment model for Graves’ hyperthyroidism
 Balamurugan Pandiyan†^{1}Email author,
 Stephen J. Merrill†^{2},
 Flavia Di Bari^{3},
 Alessandro Antonelli^{4} and
 Salvatore Benvenga^{5}
https://doi.org/10.1186/s1297601700736
© The Author(s) 2018
 Received: 18 June 2017
 Accepted: 6 December 2017
 Published: 9 January 2018
Abstract
Background
Graves’ is disease an autoimmune disorder of the thyroid gland caused
by circulating antithyroid receptor antibodies (TRAb) in the serum. TRAb mimics the action of thyroid stimulating hormone (TSH) and stimulates the thyroid hormone receptor (TSHR), which results in hyperthyroidism (overactive thyroid gland) and goiter. Methimazole (MMI) is used for hyperthyroidism treatment for patients with Graves’ disease.
Methods
We have developed a model using a system of ordinary differential equations for hyperthyroidism treatment with MMI. The model has four state variables, namely concentration of MMI (in mg/L), concentration of free thyroxine  FT4 (in pg/mL), and concentration of TRAb (in U/mL) and the functional size of the thyroid gland (in mL) with thirteen parameters. With a treatment parameter, we simulate the timecourse of patients’ progression from hyperthyroidism to euthyroidism (normal condition). We validated the model predictions with data from four patients.
Results
When there is no MMI treatment, there is a unique asymptotically stable hyperthyroid state. After the initiation of MMI treatment, the hyperthyroid state moves towards subclinical hyperthyroidism and then euthyroidism.
Conclusion
We can use the model to describe or test and predict patient treatment schedules. More specifically, we can fit the model to individual patients’ data including loading and maintenance doses and describe the mechanism, hyperthyroidism→euthyroidism. The model can be used to predict when to discontinue the treatment based on FT4 levels within the physiological range, which in turn help maintain the remittance of euthyroidism and avoid relapses of hyperthyroidism. Basically, the model can guide with decisionmaking on oral intake of MMI based on FT4 levels.
Keywords
 Hyperthyroidism
 Methimazole
 Graves’ disease
 Thyroid receptor antibodies
Background
The incidence rate of Graves’ disease was reported to be 20−30 cases per 100,000 individuals each year [7, 8]. Sex difference plays an important role in autoimmunity especially in Graves’ disease [9] and there is high risk on women aged between 40−50 years [1, 2]. The clinical presentation of Graves’ disease includes symptoms, physical findings and hyperthyroidism. The symptoms may include anxiety, difficulty in sleeping, fatigue, weight loss, palpitations and eye swelling. The physical findings may include diffuse goiter, increased pulse pressure, tremor, warm moist palms and tachycardia. Graves’ hyperthyroidism can be categorized as overt or subclinical. Overt hyperthyroidism is defined as extremely low serum concentrations of TSH and high serum concentrations of thyroid hormones, T4 and T3 [10]. Subclinical hyperthyroidism is defined as low serum TSH, but normal serum T4 and T3 concentrations. In the United States, the prevalence of hyperthyroidism is approximately 1.2% (0.5% overt and 0.7% subclinical); the most common causes include Graves’ disease, toxic multinodular goiter and toxic adenoma [11].
The laboratory hallmark of Graves’ hyperthyroidism is finding elevated levels of serum free thyroid hormones, free T4 (FT4) and free T3 (FT3), associated with undetectable serum TSH and positivity for serum TRAb at one point in time [12]. Approximately, the normal reference range for FT4 is (7−18) pg/mL, FT3 is (23−50) ng/L and TSH (0.4−4.0) mU/L respectively [13]. Graves’ hyperthyroidism is generally treated with one of the three approaches based on patients’ choice, namely use of antithyroid drugs (thionamides) to restore euthyrodism  normal levels of thyroid hormones, destruction of thyroid via radioactive iodine or removal of thyroid via surgery. Antithyroid drugs can be used as primary treatment and can be used as pretreatment in selected patients prior to radioactive iodine therapy or prior to surgery. So, this article focuses on antithyroid drug treatment method with Methimazole (MMI), which results in the clinical progression of the patients from hyperthyroidism to euthyroidism.
No perfect curative medical treatment exists for Graves’ disease; however one can control the signs and symptoms of hyperthyroidism with the help of Methimazole (MMI). Depending upon the severity of patients’ hyperthyroidism, the starting dose of MMI is normally given between 10 to 60 mg per day [1], which can be continued for about 12−18 months until patients achieve euthyroidism. The amount of MMI can be given as a single daily oral dose or divided doses to achieve normalization of thyroid function. MMI has the benefit of onceaday administration and a reduced risk of major side effects [11]. After a single oral dose, the rate of absorption of MMI from the gut to serum happens rapidly and differently in hyperthyroid patients, so the peak serum concentration of MMI reaches within 30 to 180 min [14–16]. After the initiation of treatment, the levels of FT4 and FT3 should be monitored periodically [17] and the doses should be lowered or increased in accordance to the measured levels. In this treatment, MMI can act as an inhibitor of thyroid hormone synthesis by interfering with the action of thyroid peroxidase (TPO) enzyme in the gland. Thereby, thyroid hormone secretion gradually decreases and patients become euthyroid over time, which is the goal of treatment (see Fig. 8). The mechanism of this treatment procedure can be simply described as hyperthyroidism →euthyroidism.
Typically, higher doses of MMI is prescribed when patients are in severe hyperthyroidism; for example serum FT4 levels are greater than 3 times more than the upper reference range of FT4. If higher doses are maintained for a long time period or given when patients do not have a severe hyperthyroidism, there is a possibility for overshooting during treatment. The mechanism of overshooting can be described as hyperthyroidism →hypothyroidism. Similarly, lower doses of MMI are administered sometimes during treatment because of the variability within individuals, so there is a possibility for undershooting and that mechanism can be described as hyperthyroidism →subclinical hyperthyroidism. After stopping the MMI treatment, relapse of hyperthyroidism may occur for some patients with Graves’ hyperthyroidism. Relapse is totally independent of the type of drug and dosage administered but linked to restoration of euthyroidism [18]. Relapse mechanism can be described as euthyroidism →hyperthyroidism. Some patients exhibits exacerbation which means returning to hyperthyroidism while still on MMI treatment, as opposed to relapse (which is return hyperthyroidism after withdrawal of MMI) [19]. In this article, we explain all of the possible mechanisms involved in the Methimazole treatment through a parameter from our model.
Although the initial dose of MMI is based on FT4 levels and decided by the treating physician, the maintenance dosages of MMI are tricky and challenging during the time course of treatment. An appropriate maintenance dosage depends on frequent observation of the patient, which makes the treatment difficult in some sense and dependable on patient followup. With the model, we can test or control FT4 levels within the physiological range of (7−18) pg/mL with appropriate maintenance dosages for the entire time period of treatment. We can also predict or describe the relapse of hyperthyroidism after discontinuation of MMI treatment.
Methods
Construction of a model
We present a patientspecific treatment model to describe the effect of MMI treatment in patients with Graves’ hyperthyroidism. Basically, the model describes the clinical progression from hyperthyroidism to euthyroidism in accordance to treatment procedure. An earlier model constructed by Langenstein et al. [20] for Graves’ hyperthyroidism did not keep track of MMI treatment over time. However, their article was an inspiration and provided a base for this new work. So, we build our model with the following key assumptions and state variables.
 1.
TRAb mimics TSH and stimulates the thyroid follicular cells to grow, produce and secrete thyroid hormones [1].
 2.
After oral administration, Methimazole (MMI) absorption occurs rapidly and almost completely from gut to blood serum and its bioavailabity is estimated to be approximately 93% [21, 22].
 3.
Thyroid intakes Methimazole (MMI) from the blood serum, which in turn inactivates the functional growth of the gland [23].
 4.
 5.
 6.

x(t)= the amount of MMI (mg) per liter of blood serum at time t.

y(t)= the amount of FT4 (pg) per milliliter of blood serum at time t.

z(t)= the functional size of thyroid gland (mL) or the volume of proportion of active cells at time t.

w(t)= the amount of TRAb (U) per milliliter of blood serum at time t.

s(t)= the amount of MMI orally taken per day per liter of body volume (mg/L/day).
where x(t)≥0,y(t)≥0,z(t)>0, w(t)≥0 and the initial conditions E_{0}=(x_{0},y_{0},z_{0},w_{0}).
In Eq. (1), the first term contains the timedependent function s(t), which represents the rate of change of MMI dosing at time t. The rate of change of MMI dosing is assumed to be zero when no MMI is taken orally. The middle term, \(\frac {(k_{1}z)x}{k_{a}+x}\), represents the uptake rate of MMI by the thyroid gland with maximal saturation rate (k_{1}z). We modeled the uptake rate with MichealisMenten kinetics as supported by the literature [28]. The last term, −k_{2}x, represents the excretion or elimination rate of the drug through nonspecific mechanism.
In Eq. (2), the first term, \(\frac {(k_{3}z)w}{k_{d}+w}\) accounts for the secretion rate of free thyroxine, which we modeled through the MichaelisMenten kinetics with the maximum secretion rate as (k_{3}z). The second term, −k_{4}y, represents the elimination rate of free thyroxine from the blood.
In Eq. (3), the first term, \(\left (\frac {w}{z}N\right)\), represents the growth rate of the functional thyroid gland in the presence of TRAb in the blood wherein N indicates the maximal growth ratio. The second term, −k_{6}zx, represents the inactivation rate of thyroid functional size by the MMI treatment.
In Eq. (4), the first term, k_{7}, represents the maximum production rate of TRAb due to autoimmune response. The middle term, \(\frac {k_{7}x}{k_{b}+x}\), represents the inhibition rate of TRAb due to MMI dosing. We further assume the the maximum inhibition rate is the maximum production rate of TRAb. The maximum inhibition rate will be reached asymptotically as the concentration of MMI dosing increases in the blood. The last term, −k_{8}w, represents the natural decay rate of TRAb.
The form of this model uses the idea of the functional size of the thyroid that was developed in our first article [26]. To be more precise, the rate of change of functional size (as opposed to thyroid size) over time was described as growth rate minus inactivation rate of functional size. The growth rate of functional size was modeled by a fraction of TSH concentration to the functional size when TSH is available in the blood serum. When TSH is not available, the growth rate decreases at physiologically standard constant rate N. By assumption (1), we replace TSH by TRAb (see Eq. (3)). The inactivation rate of functional size was influenced by the interaction of MMI concentration and the functional size [25].
Stability analysis
We first analyze the stability of steady state for untreated Graves’ hyperthyroidism patients and subsequently analyze the stability of steady state for patients treated with MMI. The following remark describes the steady state for untreated patients:
Remark 1
Theorem 1
When s(t)=0 and x_{0}=0, model (1)(4) with all parameters positive has a unique steady state E_{1} (hyperthyroid state) in the hyperplane x=0, which is asymptotically stable.
Proof
\(a_{1}=\frac {k_{1} k_{7}}{k_{8} N k_{a}}+\frac {k_{5} k_{8} N^{2}}{k_{7}}+k_{2}+k_{4}+k_{8} \)
\(a_{2}=\frac {k_{8} N k_{a} \left (k_{5} k_{8}^{2} N^{2}+\left (k_{2}+k_{4}\right) k_{8} \left (k_{5} N^{2}+k_{7}\right)+k_{2} k_{4} k_{7}\right)+k_{1} k_{7} \left (k_{5} k_{8} N^{2}+k_{7} \left (k_{4}+k_{8}\right)\right)}{k_{7} k_{8} N k_{a}}\)
\(a_{3}=\frac {k_{8} N k_{a} \left (\left (k_{2}+k_{4}\right) k_{5} k_{8} N^{2}+k_{2} k_{4} \left (k_{5} N^{2}+k_{7}\right)\right)+k_{1} k_{7} \left (k_{5} k_{8} N^{2}+k_{4} \left (k_{5} N^{2}+k_{7}\right)\right)}{k_{7} N k_{a}}\)
\(a_{4}=\frac {k_{4} k_{5} k_{8} N \left (k_{2} k_{8} N k_{a}+k_{1} k_{7}\right)}{k_{7} k_{a}}\)
This completes the proof. □
Remark 2
We denote the euthyroid steady state as E_{2}=(x_{2},y_{2},z_{2},w_{2}). The analytical description of this state is not possible. So, we present the qualitative analysis of the existence of the euthyroid state and its stability.
Theorem 2
Proof
Equation (9) is the quadratic equation for x_{2}, which has a positive root and a negative root for (c−k_{1}z_{2}−k_{2}k_{ a })<0 or (>0) with all parameters are positive. By Descarte’s Rule of Signs, we see there is a sign change for (9) so there exists a root x_{2} in the positive orthant.
which also lies in the positive orthant for z_{2}>0 and w_{2}>0. Therefore, there exists a unique euthyroid state E_{2}=(x_{2},y_{2},z_{2},w_{2}) in the positive orthant when c>0.
Suppose the denominator of (17) is positive, we have \(\frac {\partial x_{2}}{\partial c}\) greater than zero, then \(\frac {\partial w_{2}}{\partial c}, \frac {\partial z_{2}}{\partial c}\) and \(\frac {\partial y_{2}}{\partial c}\) are all negative, which implies w_{2}, z_{2}, y_{2} are all decreasing functions of c. □
Theorem 3
The euthyroid state E_{2}=(x_{2},y_{2},z_{2},w_{2}) is locally asymptotically stable in (1)(4) when s(t)=c>0 and w=Nz.
Proof
Therefore, the origin is locally asymptotically stable for (19)(22) and thus euthyroid state is locally asymptotically stable for (1)(4). This completes the proof. □
Numerical simulations
Data
We have 90 Graves’ hyperthyroidism patients data in which we have information for steady state levels of free thyroxine (FT4), free triiodothyronine (FT3), thyroid receptor stimulating antibodies (TRAb), thyroid peroxidase antibodies (TPOAb), thyroglobulin antibodies (TgAb), Methimazole (MMI) loading and maintenance dosage levels and age (Additional file 1). Serum TRAb is a biomarker for Graves’ disease, however it is not a criterion for the dose of antithyroid drugs [11]. So, antithyroid drug (MMI) is prescribed based on solely FT4 and FT3 levels. When data was collected, FT4 and FT3 measured in SI units (pmol/L) that we converted FT4 to the conventional unit (in terms of pg/mL) by using the factor 0.7769 and converted FT3 to the conventional unit (in terms of ng/L) by using the factor 0.651 for our purposes [30]. Since FT4 data was collected from different labs across Sicily, Itlay, the normal reference range adopted by these labs was different. So, we have center the FT4 data for commonly used clinical reference range (7−18) pg/mL by using the formula \(y=\frac {(187)*(xLFT4)}{(UFT4LFT4)}+7\), where x is a actual measurement of FT4 value (in pg/mL), LFT4 and UFT4 are lower and upper reference range values of FT4 used in the labs (in pg/mL). The same idea for the formula was used in our previous articles [26, 31]. For some patients in our data, TRAb was not measured in the first visit due to high levels of FT4 and that indicated the 1st evidence of hyperthyroidism. Just note that TSH were undetectable for many of our patients at the first clinical visit.
Summary statistics of FT3, FT4, TRAb, TPOAb and TGAb for 1st clinical visit of 90 patients
FT3:  FT4:  TRAb:  TPOAb:  TGAb:  

(23  50) ng/L  (7  18) pg/mL  (0  1) U/mL  (< 100) U/mL  (< 100) U/mL  
Mean  77.5  32.6  16.05  793.5  795.2 
Median  70.9  31.2  11.4  263.5  138.5 
Mean variability  77.5±37.5  32.6±11.3  16.05±22.5  793.5±1822.2  795.2±3760 
Initial conditions
We can calculate s(t) and k_{2} from MMI dosing information but parameters k_{1} and k_{ a } are difficult to estimate due to high variability in patients’ thyroid gland MMI uptake and MMI distribution in blood and body volume. So, we consider the functional size as an hidden compartment and assume z_{0}=30 mL for hyperthyroidism patients. With this assumption, we estimate parameter values k_{1} and k_{ a } in next section. Therefore, our initial conditions for numerical simulation is of the form E_{0}=(x_{0},y_{0},30,w_{0}).
Parameter estimates
Parameter Estimates, Description, Units and Source
Parameter  Description  Value  Source  Units 

k _{1}  Relative maximum uptake rate of MMI  8.374×10^{−3}  Literature [28]  mg/(mL∗Lday) 
k _{2}  Elimination rate of MMI  3.3271  Estimated  1/day 
k _{ a }  MichealisMenten constant for half maximal uptake rate of MMI  0.358068  Simulation  mg/L 
k _{3}  Relative maximum secretion rate of FT4  0.119  Calculation  pg/(mL^{2}day) 
k _{ d }  MichealisMenten constant for half maximal secretion rate of FT4  0.05  Simulation  U/mL 
k _{4}  Elimination rate of FT4  0.099021  Estimated  1/day 
k _{5}  Relative growth rate of the functional size of thyroid gland  1×10^{6}  Simulation  mL^{3}/(U∗day) 
N  Maximal growth ratio  0.833  Calculation  U/(mL^{2}) 
k _{6}  Thyroid inactivation rate constant  0.001  Simulation  mL/mg∗day 
k _{7}  Maximal production rate of TRAb  0.875  Calculation  U/(mL∗day) 
k _{ b }  Inhibition rate of TRAb  1.5  Simulation  mg/L 
k _{8}  Elimination rate of TRAb  0.035  Estimated  1/day 
The parameter k_{4} is the excretion rate of FT4 which assumed to be exponential due to a nonspecific mechanism so it can be calculated as \(k_{4}=\frac {ln(2)}{7 \text {days}}=0.099021~\text {1/day}\).
In Eq. (3), we determine the value of N from the initial concentration of TRAb (w_{0}) and the functional size of the thyroid gland (z_{0}) at the hyperthyroid state or from the patients data. Again, the equilibrium argument was used to estimate the parameter \(N=\frac {w_{0}}{z_{0}}\) when the MMI treatment was not given. The parameters k_{5} and k_{6} were assumed to be one as found through the simulation.
In Eq. (4), we determine the maximal production rate of TRAb at the hyperthyroid state of patient. More precisely, the parameter estimate of k_{7}=k_{8}w_{0} in the absence of MMI treatment. The parameter k_{8} was the death rate of TRAb, which we assume to be exponential in the model and that was calculated as \(k_{8}=\frac {ln(2)}{20~\text {days}}=0.035~ \text {1/day}\). The parameter k_{ b } was the halfmaximal inhibition concentration of TRAb (MichaliesMenten constant) that can be estimated with fmincon procedure with lower bound 3 and upper bound 12.
Simulation of the model
Model validation
Individual patient parameter values
Parameter  Patient# 20  Patient# 31  Patient# 55  Patient# 70 

k _{1}  8.374×10^{−3}  8.374×10^{−3}  8.374×10^{−3}  8.374×10^{−3} 
k _{2}  3.3271  3.3271  3.3271  3.3271 
k _{ a }  0.358068  0.358068  0.358068  0.358068 
k _{3}  0.085  0.08975  0.08992  0.11784 
k _{ d }  0.067  0.07  0.081  0.075 
k _{4}  0.099021  0.099021  0.099021  0.099021 
k _{5}  1×10^{6}  1×10^{6}  1×10^{6}  1×10^{6} 
N  0.250  0.058  0.207  0.293 
k _{6}  0.001  0.001  0.001  0.001 
k _{7}  0.26  0.061  0.22  0.308 
k _{ b }  4.95  11.8  4.09  3.15 
k _{8}  0.035  0.035  0.035  0.035 
We will first explain patient #20 time course data and then simulate the model in accordance with MMI treatment. In November 2010, this patient was diagnosed with Graves’ hyperthyroidism. At the first clinical visit, patient’s FT4 and TRAb levels were 25.63 pg/mL and 7.5 U/mL respectively with respect to normal reference ranges of FT4, (7  18) pg/mL and TRAb, (0  1) U/mL. At this visit, this patient was put under MMI treatment. A loading dosage was started with 6 tablets/day and asked to decrease tablets consumption by one stepwise every month. So, patient #20 took 6 tablets/day (i.e., 30 mg/day) for first month, 5 tablets/day (i.e., 25 mg/day) for second month and 4 tablets/day (i.e., 20 mg/day) for third month. After third month, patient followed up for checkup and the dosage was lowered to 2 tablets/day (i.e., 10 mg/day) for next three months because FT4 levels were 17.7 pg/mL (close to upper normal reference limit) and TRAb levels were not measured at this visit. After six months, patient visited the clinic and FT4 levels were measured 13.8 pg/mL (but not TRAb) and the dosage was further lowered to 1 tablet/day and prescribed for another six months. Now, after twelve months or 1 year, patient’s FT4 and TRAb was 13.14 pg/mL and 0.7 U/mL so the dosage level continued with same amount 1 tablet/day for another six months. Finally, after 1 year and six months, FT4 and TRAb was 12.46 pg/mL and 0.6 U/mL; therefore MMI dosage was stopped.
As before, we first explain patient #31 time course data and then proceed with model validation. At the first visit, this patient’s FT4 and TRAb levels were 26.43 pg/mL and 1.74 U/mL respectively, which indicates Graves’ hyperthyroidism in accordance with normal reference range. So, this patient was put under MMI treatment. The treatment initiated with loading dose of 4 tablets/day for 1st month, lowered stepwise to 1 tablet/day. That is, 3 tablets/day for 2nd month, 2 tablets/day for 3rd month and 1 tablet/day for 4th month. After 4 months taking MMI, patient’s followup revealed FT4 levels were 20 pg/mL and TRAb was not measured at that time. The treatment was continued with 2 tablets/day for 1 month and then 1 tablet/day for 23 months. In a subsequent followup, MMI treatment was discontinued because FT4 and TRAb levels were 11.5 pg/mL and 0.45 U/mL respectively. Patient #31 followedup for check up after 2 years of no treatment, FT4 and TRAb levels were 11.36 pg/mL and 0.43 U/mL normal.
As before, we first explain patient #70 time course data and then validate the model. At the first clinical visit, this patient’s FT4 and TRAb levels were 35.04 pg/mL and 8.8 U/mL respectively, which indicates Graves’ hyperthyroidism in accordance with normal reference range. So, this patient was put under MMI treatment at this visit. MMI dosage was prescribed as 6 tablets/day for first month, 5 tablets/day for second month, 4 tablets/day for third month, 3 tablets/day for fourth month, 2 tablets/day for fifth month and 1 tablet/day from sixth month to tenth month (that is to say four months of maintenance dosage). After tenth month, patient’s FT4 and TRAb were 9.3 pg/mL and 1.3 U/mL, so MMI treatment was continued with 1 tablet/day for another seven months. So, after seventeen months, patient followed up for checkup and now FT4 and TRAb were 8.97 pg/mL and 1.4 U/mL. Since FT4 measurement was closer to the lower reference range, MMI treatment was stopped for six months after this visit but asked to follow up on the thyroid status. Then, after six months, patient’s FT4 and TRAb levels were 31.2 pg/mL and 4.7 U/mL respectively so relapse occurred. A second course of MMI treatment started again with 4 tablets/day for 1 month, 2 tablets/day for three months and then 1 tablet/day for three months. In the followup of after seven months, patient’s FT4 and TRAb levels were 9.87 pg/mL and 0.9 U/mL respectively. MMI treatment continued with lower dosage 1 tablet/day for another six months and then patient FT4 levels remained as 9.87 pg/mL but TRAb levels decreased to 0.3 U/mL. Finally, treatment was stopped.
As before, we first explain patient #55 time course data and proceed to model validation. At the first clinical visit, this patient’s FT4 and TRAb levels were 27.01 pg/mL and 6.22 U/mL respectively, which indicates Graves’ hyperthyroidism in accordance with normal reference range. So, this patient was put under MMI treatment at this visit. MMI dosage was prescribed as 3 tablets/day for first month, 2 tablets/day for second month and 1 tablets/day for third month. After third month during followup, patient’s FT4 levels measured as 10.03 pg/mL and TRAb not measured, so MMI treatment was continued with 1 tablet/day for another six months. After nine months, patient’s FT4 and TRAb levels were 8.99 pg/mL and 1.17 U/mL respectively. Since FT4 measurement was closer to the low normal reference limit, MMI treatment was stopped. During followup after 23 months of no treatment, patient’s FT4 and TRAb levels were 40.28 pg/mL and 2.46 U/mL respectively and therefore relapse occurred. A second course of MMI treatment started with loading dose 4 tablets/day for first month, 3 tablets/day for second month and 2 tablets/day for third month. After third month during followup, patient’s FT4 levels were 12.41 pg/mL and TRAb levels not measured. Treatment was continued with lower dosage 1 tablet/day for another five months and then FT4 levels were measured as 10.79 pg/mL and TRAb not measured. Treatment continued again for another 11 months with lower dosage of 1 tablet/day. In a subsequent followup, patient’s FT4 levels and TRAb levels were 11.36 pg/mL and 0.7 U/mL respectively.
Results and discussions
When the treatment parameter s(t)=c=0 and the initial concentration of MMI x_{0}=0, there is a unique steady state corresponding to hyperthyroidism, which is asymptotically stable. After the initiation of MMI treatment, that is s(t)=c>0 and x_{0}>0, the hyperthyroidism steady state is moving on the monotonic decreasing function w=Nz towards subclinical hyperthyroidism and then euthyroidism. Once patients achieve euthyrodism, the maintenance dosage is prescribed to maintain FT4 values within the normal reference range. The value of treatment parameter c is determined in accordance with loading or maintenance dosages, treatment time period and body volume of patients. The initial concentration of MMI is estimated from the loading dosage of MMI.
One way to use the model is to run an experiment on clinical dosing amount and schedules. For example, by keeping the dosing MMI amount constant (say 5 mg) throughout the treatment period and varying the dosing schedule, say every 4 h for the 1st 130 days and/or every 8 h for the 2nd 130 days. We can actually check to see if that assumption ever results in hypothyroidism. Next, by keeping the dosing schedule constant say every day for 360 days, we can vary the dosing amount from high doses (say 30 mg) to low doses (say 5 mg) each day. After reaching the lowest dosage amount, restart the dosage amount from high to low doses again. In this way, we can actually check to see if that assumption results in subclinical hyperthyroidism, euthyroidism or hypothyroidism. Another way to use to model is to find the response rate of thyroid gland to different concentrations of MMI in the blood serum. Also, the model can be extended to account for cellular autoimmune responses, absorption rates from the gut to serum and drug toxicity.
Conclusions
For Graves’ hyperthyroidism patients, we developed a patientspecific treatment model to describe the time course of FT4 levels after administration of Methimazole (MMI) and to control FT4 levels within the physiological range (7−18) pg/mL with an appropriate maintenance dosage schedule. Our primary interest is to explain the natural history of Graves’ hyperthyroidism MMI treatment, restore FT4 levels with an appropriate dosage amount and provide a better dosage schedule through the model. The model takes the form of a system of ordinary differential equations with four state variables, namely concentration of MMI (x) at time t, concentration of free thyroxine (y) at time t, concentration of TRAb (w) at time t and the functional size of thyroid gland (z) at time t. The model has total of thirteen positive parameters in which we fix twelve parameters for all runs except a treatment parameter s(t)=c. All fixed twelve parameters are determined from patients’ data and literature (see Table 3). The treatment parameter value c varies between patients, which in fact depends on amount of MMI orally taken, number of days of loading or maintenance dosages given, and body volume of the patient.
There is high variability among Graves’ hyperthyroid patients, making each patient unique so treatment with MMI is typically challenging. Right loading, maintenance dosage amount and schedule would help the patients control their free thyroxine levels within the normal reference range over time. Our model can be used for each patient to generate dosage amount and schedule which in turn may help prevent undershooting, overshooting and relapses. Basically, the model can predict when to discontinue the treatment which helps with decisionmaking on the amount of oral intake of MMI over time. With appropriate MMI dosage schedule, the clinical progression from hyperthyroidism → euthyroidism can be achieved effectively. We validated the model with data from four Graves’ hyperthyroidism patients. To measure the quality of prediction, we have calculated root mean square error for all of these four patients which can be crosschecked with the reference interval. The nonnormalized root mean square error interval has been obtained for FT4 and TRAb using our patients’ data.
Declarations
Acknowledgements
We would like to thank the reviewers for their comments and suggestions to improve this manuscript.
Funding
Not applicable.
Availability of data and materials
All data is not explicitly included in the article. However, authors can share the data upon request.
Authors’ contributions
P and M conceptualized, developed and analyzed the model for Graves’ hyperthyroidism. P wrote the manuscript. DB, A and B collected patients medical data with the ethical approval from Italian hospitals and contributed the data for model validation. They shared valuable insights on autoimmunity, Graves’ disease and hyperthyroidism. All authors read and approved the manuscript.
Ethics approval and consent to participate
This study involved the secondary analysis of deidentified data collected for another purpose. All data are RETROSPECTIVE and presented in an AGGREGATED fashion. So, no IRB approval was required. Patient numbers used in the model validation section of the article are spreadsheet numbers from data, which can not be used to identify the patients name, family name or other specific details.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
References
 Burch HB, Cooper DS. Management of Graves Disease: A Review. JAMA. 2015; 314(23):2544–54.View ArticlePubMedGoogle Scholar
 Weetman AP. Grave’s disease 18352002. Horm Res. 2003; 59 Suppl 1:114–8.PubMedGoogle Scholar
 De Leo S, Lee SY, Braverman LE. Hyperthyroidism. Lancet. 2016; 388:906–918.View ArticlePubMedPubMed CentralGoogle Scholar
 Dong YH, Fu DG. Autoimmune thyroid disease: mechanism, genetics and current knowledge. Eur Rev Med Pharmacol Sci. 2014; 18(23):3611–8.PubMedGoogle Scholar
 Lee HJ, Li CW, Hammerstad SS, Stefan M, Tomer Y. Immunogenetics of autoimmune thyroid diseases: A comprehensive review. J Autoimmun. 2015; 64:82–90.View ArticlePubMedPubMed CentralGoogle Scholar
 Neumann S, Place RF, Krieger CC, Gershengorn MC. Future Prospects for the Treatment of Graves’ Hyperthyroidism and Eye Disease. Horm Metab Res. 2015; 47(10):789–96.View ArticlePubMedGoogle Scholar
 Nystrom HF, Jansson S, Berg G. Incidence rate and clinical features of hyperthyroidism in a longterm iodine sufficient area of Sweden (Gothenburg) 20032005. Clin Endocrinol (Oxf). 2013; 78(5):768–76.View ArticleGoogle Scholar
 Brent GA. Clinical practice. Graves’ disease. N Engl J Med. 2008; 358(24):2594–605.View ArticlePubMedGoogle Scholar
 Merrill SJ, Mu Y. Thyroid autoimmunity as a window to autoimmunity: An explanation for sex differences in the prevalence of thyroid autoimmunity. J Theor Biol. 2015; 375:95–100.View ArticlePubMedGoogle Scholar
 Tegler L, Gillquist J, Lindvall R, Almqvist S. Secretion rates of thyroxine, triiodothyronine, and reverse triiodothyronine in man during surgery. Acta Endocrinol. 1982; 101(2):193–8.PubMedGoogle Scholar
 Ross DS, Burch HB, Cooper DS, Greenlee MC, Laurberg P, Maia AL, Rivkees SA, Samuels M, Sosa JA, Stan MN, Walter MA. 2016 American Thyroid Association Guidelines for Diagnosis and Management of Hyperthyroidism and Other Causes of Thyrotoxicosis. Thyroid. 2016; 26(10):1343–421.View ArticlePubMedGoogle Scholar
 Menconi F, Marcocci C, Marino M. Diagnosis and classification of Graves’ disease. Autoimmun Rev. 2014; 13(45):398–402.View ArticlePubMedGoogle Scholar
 Baloch Z, Carayon P, ConteDevolx B, Demers LM, FeldtRasmussen U, Henry JF, LiVosli VA, NiccoliSire P, John R, Ruf J, Smyth PP, Spencer CA, Stockigt JR. Laboratory medicine practice guidelines. Laboratory support for the diagnosis and monitoring of thyroid disease. Thyroid. 2003; 13(1):3–126.View ArticlePubMedGoogle Scholar
 Fumarola A, Di Fiore A, Dainelli M, Grani G, Calvanese A. Medical treatment of hyperthyroidism: state of the art. Exp Clin Endocrinol Diabetes. 2010; 118(10):678–84.View ArticlePubMedGoogle Scholar
 Skellern GG, Stenlake JB, Williams WD, McLarty DG. Plasma concentrations of methimazole, a metabolite of carbimazole, in hyperthyroid patients. Br J Clin Pharmacol. 1974; 1(3):265–9.View ArticlePubMedPubMed CentralGoogle Scholar
 Okamura Y, Shigemasa C, Tatsuhara T. Pharmacokinetics of methimazole in normal subjects and hyperthyroid patients. Endocrinol Jpn. 1986; 33(5):605–15.View ArticlePubMedGoogle Scholar
 John M, Sundrarajan R, Gomadam SS. Antithyroid drugs in pediatric Graves’ disease. Indian J Endocrinol Metab. 2015; 19(3):340–6.View ArticlePubMedPubMed CentralGoogle Scholar
 Laurberg P. Remission of Graves’ disease during antithyroid drug therapy. Time to reconsider the mechanism?Eur J Endocrinol. 2006; 155(6):783–6.View ArticlePubMedGoogle Scholar
 Prout TE, Asper SP. Exacerbation of hyperthyroidism by methimazole during iodide therapy. AMA Arch Intern Med. 1958; 101(2):364–8.View ArticlePubMedGoogle Scholar
 Langenstein C, Schork D, Badenhoop K, Herrmann E. Relapse prediction in Graves disease: Towards mathematical modeling of clinical, immune and genetic markers. Rev Endocr Metab Disord. 2016; 17(4):571–81.View ArticlePubMedGoogle Scholar
 Pittman J, Beschi R, Smitherman T. Methimazole: its absorption and excretion in man and tissue distribution in rats. J Clin Endocrinol Metab. 1971; 33(2):182–5.View ArticlePubMedGoogle Scholar
 Jansson R, Lindström B, Dahlberg P. Pharmacokinetic properties and bioavailability of methimazole. Clin Pharmacokinet. 1985; 10(5):443–50.View ArticlePubMedGoogle Scholar
 Kim H, Lee TH, Hwang YS, Bang MA, Kim KH, Suh JM, Chung HK, Yu DY, Lee KK, Kwon OY, et al.Methimazole as an antioxidant and immunomodulator in thyroid cells: mechanisms involving interferon γ signaling and h2o2 scavenging. Mol Pharmacol. 2001; 60(5):972–80.View ArticlePubMedGoogle Scholar
 Johansen K, Andersen AN, Kampmann J, Hansen JM, Mortensen H. Excretion of methimazole in human milk. Eur J Clin Pharmacol. 1982; 23(4):339–41.View ArticlePubMedGoogle Scholar
 Andrade VA, Gross JL, Maia AL. The effect of methimazole pretreatment on the efficacy of radioactive iodine therapy in graves hyperthyroidism: oneyear followup of a prospective, randomized study. J Clin Endocrinol Metab. 2001; 86(8):3488–93.PubMedGoogle Scholar
 Pandiyan B, Merrill SJ, Benvenga S. A patientspecific model of the negativefeedback control of the hypothalamuspituitarythyroid (HPT) axis in autoimmune (Hashimoto’s) thyroiditis. Math Med Biol. 2014; 31(3):226–58.View ArticlePubMedGoogle Scholar
 Jansson R, Dahlberg PA, Johansson H, LindstrÖm B. Intrathyroidal concentrations of methimazole in patients with graves disease. J Clin Endocrinol Metab. 1983; 57(1):129–32.View ArticlePubMedGoogle Scholar
 Huang G. [Relation between the dosage of methimazole and its intrathyroidal concentrations]. Zhonghua Yi Xue Za Zhi. 1991; 71(6):301–3.PubMedGoogle Scholar
 EdelsteinKeshet L. Mathematical Models in Biology.Philadelphia: Society for Industrial and Applied Mathematics (SIAM); 2005.View ArticleGoogle Scholar
 Online Conversion Calculator. http://unitslab.com/node/121. Accessed 19 May 2017.
 Pandiyan B, Merrill SJ, Benvenga S. A homoclinic orbit in a patientspecific model of hashimotos thyroiditis. Diff Equ Dyn Syst. 2016;:1–18. https://doi.org/10.1007/s1259101603355.