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MBW:Interaction Between the Thyroid and Pituitary Glands

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Executive Summary

This paper uses a Bernoulli differential equation of the second order to model the interaction between the thyroid and pituitary glands. Using the chain rule the author converts the nonlinear differential equation into a first order linear differential equation. The method of integrating factors is then used to solve the differential equation.

This molecular model describes the relationship between the levels of TSH[1] (a hormone created by the pituitary gland) and T3/T4[2] (hormones produced by the thyroid). Thus, this paper analyzes the biological pituitary-thyroid system by modeling the hormones involved.


In a recent paper [1], A mathematical model of pituitary–thyroid interaction to provide an insight into the nature of the thyrotropin–thyroid hormone relationship, by Melvin Khee-Shing Leow, a simplified model for understanding the process of the control relation between hormones produced in the pituitary and thyroid glands is discussed.

The model, while ignoring some aspects of the pituitary-thyroid system, still demonstrates important aspects of the relation between thyroid and pituitary hormones. The simplifications made are due to both the aspects of the system which are harder to measure, and in order to create a model that is understandable to biologists and physicians. In addition, the model focuses on the direct relation between thyroid and pituitary hormones instead of a process over time. The net result of the model is a better understanding of the basic relation between thyroid and thyroid stimulating hormones.

Biological System

The described four hormone system

There are several hormones under consideration. The pituitary gland controls the output of hormones of several other glands, but it itself is under control by the hypothalamus. The hypothalamus creates thryotropin releasing hormone (TRH), which makes the pituitary thyrotrophs (the part of the pituitary gland dealing with the thyroid) create thyroid-stimulating hormone (TSH). TSH then controls the level of output of thyroid hormones, L-3,5,3′-triiodothyronine (T3) and L-3,5,3′,5′-tetraiodothyronine (T4). The T3 hormone acts upon receptors in the hypothalamus and pituitary glands to limit the output of TRH and TSH. Thus, we have a situation where an elevated amount of T3 decreases the levels of TRH and TSH, which in turn limits the production of T4 and T3, and so there is a self-balancing system in healthy humans. In a situation with an overactive thyroid, hyperthyroidism, the levels of TSH and TRH will be reduced, while an underactive thyroid, hypothyroidism, will stimulate production of TSH and TRH.

While T3 is to some extent created by the thyroid, the majority of it (80%) is created by the transformation of T4 to T3. Thus, while T3 is the active hormone, in most cases either one can be measured to find the level of free thyroid hormones. So, one may choose which to use in the model, and due to clinical practice T4 is chosen as the variable of measurement. (For the other cases, where T3 is required, the model can be adjusted accordingly).

The simplified two hormone system

The contribution of TRH is neglected in the model. This is due in part to the difficulties in measuring it; any model including TRH would have large errors in its parameters due to the uncertainty in measuring levels of it. In addition, some studies with euthyroidism (normally behaving, healthy thyroid), hypothyroidism and hyperthyroidism, that showed that although there were measurable changes in T4 and TSH levels, TRH levels did not significantly change. However, levels of TSH are much easier to measure and more sensitive to change. Hence, in this model, only the relation between TSH and T4 levels are considered. To the right is a diagram with the simplified model.


The variables of interest:

H: levels of TSH

T: levels of T4 (or T3 if desired, or the total of both).

k: a proportionality constant

\alpha : stimulatory factor of the thyroid (the amount of T that the thyroid produces depends on the current level of T, which is bundled in alpha, and the level of H, which multiplies alpha)

-{\frac  {dH}{dT}}=kH(T+\alpha H)

Essentially, this differential equation says that the negative of the rate at which the level of H changes with respect to the current level of T is proportional to the current level of H times the existing level of T plus the T that is currently being produced. Note that H is always decreasing with increasing T in this model.

The model becomes more complex when we regard alpha. Experiments have shown that the rate at which T is produced decreases as T increases. This results in an assumption of exponential decrease.

-{\frac  {d\alpha }{dT}}=\beta \alpha

\alpha =\alpha _{0}e^{{-\beta T}}

β: decay constant.

This gives us

-{\frac  {dH}{dT}}=kHT+kH^{2}\alpha _{0}e^{{-\beta T}}

We can simplify this by using v=H^{{-1}}. First, we relate {\frac  {dv}{dT}} and {\frac  {dH}{dT}}:

{\frac  {dv}{dT}}={\frac  {dv}{dH}}{\frac  {dH}{dT}}=-{\frac  1{H^{2}}}{\frac  {dH}{dT}}

And so,

H^{2}({\frac  {dv}{dT}})=kHT+kH^{2}\alpha _{0}e^{{-\beta T}}

From which we get,

{\frac  {dv}{dT}}-kTv=k\alpha _{0}e^{{-\beta T}}

a first order linear differential equation. In order to solve this, we use an integrating factor,


Multiplying both sides by I,

e^{{-kT^{2}/2}}({\frac  {dv}{dT}}-kTv)=k\alpha _{0}e^{{-\beta T}}e^{{-kT^{2}/2}}

{\frac  d{dT}}(ve^{{-kT^{2}/2}})=k\alpha _{0}e^{{-(\beta T+kT^{2}/2)}}

Here we may integrate both sides with respect to T, from 0 to T_{{final}}

ve^{{-kT^{2}/2}}]_{0}^{{T_{f}}}=k\alpha _{0}\int _{0}^{{T_{f}}}e^{{-(\beta T+kT^{2}/2)}}\,{\mathrm  {d}}T

=k\alpha _{0}[{\frac  {-e^{{-(\beta T+kT^{2}/2)}}}{\beta +kT}}]_{0}^{{T_{f}}}

Note: The integration above is given in the paper by the author. This summarizer has been unable to confirm its validity. (Please see the wiki page about a correction to this integral.)

This, replacing Tfinal with T for simplicity of notation, and using H0 as the initial value of H, gives us,

{\frac  {e^{{-kT^{2}/2}}}{H}}-{\frac  1H}_{0}=k\alpha _{0}[{\frac  1\beta }-{\frac  {e^{{-(\beta T+kT^{2}/2)}}}{\beta +kT}}]

From here, we use basic algebraic manipulation to obtain

H={\frac  {H_{0}\beta (\beta +kT)}{e^{{kT^{2}/2}}(\beta +kT)(\beta +kH_{0}\alpha _{0})-k\beta H_{0}\alpha _{0}e^{{-\beta T}}}}

Limits and simplifications

As we stated before, H_{0} is the initial value of H, thus when T = 0, we should obtain H = H_{0}

H={\frac  {H_{0}\beta ^{2}}{\beta (\beta +kH_{0}\alpha _{0})-k\beta H_{0}\alpha _{0}}}={\frac  {H_{0}\beta ^{2}}{\beta ^{2}}}=H_{0}

This corresponds to the maximum level of TSH. Taking T -> Infinity, we find that H approaches zero (the more T we have, the less we want produced).

\lim _{{t\to \infty }}{H(T)}=0

since the e^{{k(T^{2})/2}} term grows quite large very quickly, while the other exponent term e^{{-\beta T}} goes to zero.

The overriding factor in these equations is exp(k(T^2)/2). By dividing the equation through by (β + kT), and assuming that k β H0 α0 exp(-β T)/(β+kT) decreases rapidly to zero as T increases (it is much less than the term with exp[k(T^2)/2] ), we obtain the approximation,

H\approx {\frac  {H_{0}\beta }{e^{{kT^{2}/2}}(\beta +kH_{0}\alpha _{0})}}

Noting that many of the terms are constants, we can simply look at

TSH\propto {\frac  {1}{e^{{k(freeT4)^{2}/2}}}}

Thus, we can look at an approximation of the curve which is much simpler to compute and analyze than H itself. One can explicitly show that in the simplified equation, {\frac  {d^{2}H}{dT^{2}}}=0 when T=({\frac  1k})^{{1/2}} (In the full equation it is quite difficult to calculate the second derivative, and an explicit solution for T cannot be found for when the second derivative is equal to zero.) The plot below is of the approximate equation, as well as the point of inflection (in order to create a plot, H0 and k were both taken to be equal to one). The article provides some evidence that the full equation would have approximately the same sigmoidal configuration.

Approx HTeqn Graph.jpg

Application to various states

The normal state: as T4 rises, TSH quickly decreases, as one can quickly see from the graph and the simplified equation. This in turn will decrease the creation of T4. While the model is not built for patterns over time, the net result (as less T4 is produced and the current level of T4 decreases through natural processes) is that the level of T4 returns to normal. Likewise, a decrease in T4 is compensated for by an increase in TSH. In Graves’ disease, a thyrotoxic state causing hyperthyroidism, a circulating TSH receptor autoantibody (TSHRAb) stimulates the thyroid more than TSH normally would. The net effect in the model is that the amount of T4 influencing TSH, normally the current level of T plus new T created as a result of stimulation by H,

T=T_{a}+\alpha H



where g is a constant greater than zero that depends on the type of TSHRAb present. Using the approximated version of the model, we have

H\approx {\frac  {C}{e^{{k[T_{a}^{2}+2T_{a}*g*(TSHRAb)^{2}+g^{2}*(TSHRAb)^{4}]}}}}

These previous equations are in line with some of the reported situations that have occurred due to Graves’ disease. In order to treat Graves’ disease, large doses of Anti-Thyroid Drugs (ATD) are required at the start to be clinically effective, due to the large amount of T4 being produced by the thyroid. This would then allow the free T4 to decline to normal levels.

A more encompassing model

One important observation is necessary to improve the model. When levels of TSH fall extremely low due to hyperthyroidism, there exists a lag time between when thyroid hormones reach normal levels and when TSH levels rise significantly. This lag can be on the order of weeks or months after the initial hyperthyroid returns to normal (euthyroid), or even hypothyroid. After levels of TSH do rise, they act normally. This lag time for TSH to react (both in cases of hyperthyroidism and hypothyroidism) implies that the model should be modified accordingly. While the improved model is not included in the article, some results of it are. Any treatment of hyperthyroidism or hypothyroidism should account for the fact that the doses should be adjusted carefully to allow both T4 and TSH levels to approach the euthyroid equilibrium, as different initial values will lead to different paths for T4 vs. TSH levels.

Analysis of the model

The model used here was quite limited for the sake of simplicity. However, it achieved its goal that “readily observable, easily measured, widely familiar variables” were used to explain “some common phenomena of T4-TSH interactions known to every experienced clinician.” (page 285) In addition, for cases where T3 and T4 are not quite as substitutable for each other, it can be readily modified. While there are some deficiencies in any model that does not take into account all possible variables, this mathematical model does give some further insight into the process at hand. The type of relationship between T4 and TSH is more fully known than simply knowing that they are inversely related. The specific “inverse exponential-power relationship” found in the model “might hint at a molecular process in which a chain of multistep ligand-receptor binding kinetics are facilitated tremendously by the changes in molecular conformation of the receptor surface as a ligand approaches that amplify each subsequent step.” (p. 285 - 286)

The desired effect of this model is that a better understanding of the mathematical relation between T4 and TSH will lead to a better understanding of the underlying biological process by biologists, and of what is happening with patients by clinicians. A more complete understanding of the biological system requires both a biological description of the system, and mathematical models that describe and correctly predict the system.

Recent Literature

Narelle et. al., 2013[3] cite the paper of discussion as they explain their findings regarding the complexity behind the relationship between TSH and free T4. In Narelle et. al., 2013 it is emphasized that the TSH-T4 relationship is key to understanding and diagnosing thyroid disease. This study confirms that sex and age are important factors in the function of the hypothalamo-pituitary-thyroid axis. As a result, the findings of this study contradicts the notion that the log TSH and T4 can be represented by a linear mathematical model. Instead, evidence suggests that the relationship depends on sex and age and can be better represented by two overlapping negative sigmoid curves.


  1. Leow, M., 2007. A mathematical model of pituitary–thyroid interaction to provide an insight into the nature of the thyrotropin–thyroid hormone relationship. Journal of Theoretical Biology 248, 275-287.

A link to the paper, as of 3/31/2011, can be found here

See Also

MBW:Correcting the analysis for Interaction Between the Thyroid and Pituitary Glands