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Model Predictive Controller to Regulate Cortisol Levels in Individuals With Adrenal Insufficiency

This paper proposes a Model Predictive Controller (MPC) integrated with a novel mathematical model of the hypothalamus-pituitary-adrenal (HPA) axis to serve as a virtual assistant for physicians, enabling the optimization of cortisol replacement therapy for patients with primary or secondary adrenal insufficiency while rigorously adhering to dosage constraints.

Original authors: Renuka Joshi, Nayana Saha, Vittal Srinivasan, Stanislaw H. Zak, Cary N. Mariash

Published 2026-07-13

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CC BY 4.0

Original authors: Renuka Joshi, Nayana Saha, Vittal Srinivasan, Stanislaw H. Zak, Cary N. Mariash

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine your body has a tiny, super-smart thermostat inside your brain that controls a special "stress hormone" called cortisol. This hormone is like the body's emergency fuel; it helps you handle stress, keep your energy up, and fight off infections. In a healthy person, this thermostat works on a natural daily rhythm, pumping out more fuel in the morning to wake you up and less at night so you can sleep.

But for people with a condition called Adrenal Insufficiency (AI), this system is broken. It's like the furnace in their house has stopped working. They can't make their own cortisol, so they have to get it from a pill (hydrocortisone) to survive. The big problem? Figuring out exactly how much to take and when is tricky. If they take too little, they feel weak and sick. If they take too much, it can be dangerous.

Right now, doctors usually play it safe with a "fixed schedule." They might say, "Take 12 mg at 8:00 am, 4 mg at 2:00 pm, and 4 mg at 8:00 pm." The authors of this paper ran computer simulations to see how well this fixed schedule works. They found that while it does get the hormone levels into the right ballpark, it's not perfect. It's like driving a car with cruise control set to a single speed; it gets you there, but it doesn't adjust if the road gets steep or flat.

The New Idea: A "Virtual Co-Pilot"

To do better, the researchers built a new, super-detailed mathematical map of how the body's stress system works. This map includes the daily rhythm (the circadian clock) and how the body naturally tries to stop making cortisol when there's too much of it.

Using this map, they designed a "Model Predictive Controller" (MPC). Think of this MPC as a smart, virtual co-pilot for the doctor. Instead of just guessing a fixed schedule, this co-pilot constantly looks ahead. It asks: "If I give this dose now, what will the hormone levels look like in one hour? Two hours? Will they be too high or too low?"

The co-pilot then calculates the perfect dose to keep the levels just right, but it has strict rules it must follow:

No negative doses: You can't take "minus" pills.
No overdose: It won't suggest a dose that's too high (the limit is set so the patient doesn't get sick from too much cortisol).
Real-world pills: Since you can't buy a 5.45 mg pill at the pharmacy, the co-pilot rounds the numbers to the nearest 5 mg (like 5 mg, 10 mg, or 15 mg) and sticks to the usual times: 8:00 am, 2:00 pm, and 8:00 pm.

What the Simulations Showed

The team tested this virtual co-pilot on two different types of broken stress systems:

Primary AI: The adrenal glands (the furnace) are totally broken.
Secondary AI: The brain's thermostat (the hypothalamus and pituitary) is broken, so it never tells the glands to work.

In their computer simulations, the MPC was able to guide the hormone levels to match a "target" perfectly. Whether the target was a steady line or a wavy line that mimics the natural morning-to-night rhythm, the co-pilot adjusted the doses to follow it closely.

For example, when they told the system to aim for a cortisol level of 10 µg/dL or 15 µg/dL, the controller figured out the exact mix of pills needed to get there without overshooting. Even when they tried to mimic a changing day with a wavy "sine wave" target, the controller adapted, though it was slightly delayed (a tiny "phase lag")—which is normal for how the body reacts.

The Bottom Line

The paper doesn't claim this is a cure or that it's been tested on real people yet. Instead, the authors show that in their simulations, this smart, rule-following virtual assistant can do a much better job than the old "fixed schedule" method. It proves that it's possible to use math to create a personalized, optimal plan for taking cortisol.

The researchers suggest that the next step would be to tweak the math so it fits specific patients and then test it on real data. But for now, this study shows that a smart, virtual co-pilot could one day help doctors prescribe the perfect dose of cortisol, keeping patients safe and healthy.

Technical Summary: Model Predictive Controller to Regulate Cortisol Levels in Individuals With Adrenal Insufficiency

Problem Statement
Adrenal insufficiency (AI), or hypocortisolism, is a condition characterized by the inability of the adrenal glands to produce sufficient cortisol, a hormone critical for regulating stress, metabolism, and immune responses. AI manifests in two primary forms: primary AI (adrenal disorder) and secondary AI (hypothalamic-pituitary disorder). While AI cannot be cured, it is managed through cortisol replacement therapy, typically involving fixed doses of hydrocortisone (HC). However, standard fixed-dose strategies are often suboptimal; they fail to account for the complex, non-linear dynamics of the hypothalamus-pituitary-adrenal (HPA) axis, including endogenous circadian rhythms and feedback loops. Furthermore, both overdosing and underdosing pose significant health risks, with studies indicating that higher replacement therapy correlates with increased mortality. The challenge lies in developing a treatment strategy that rigorously adheres to dosage constraints while dynamically regulating cortisol levels within a healthy range.

Methodology
The authors propose a novel mathematical framework combining advanced modeling with Model Predictive Control (MPC) to serve as a virtual assistant for physicians.

Mathematical Modeling of the HPA Axis:
The study constructs a non-linear differential equation model of the HPA axis that integrates three key components:
- The structural model by Andersen et al. (2013), which accounts for glucocorticoid and mineralocorticoid receptors and gland size limits.
- The endogenous circadian rhythm function $C(t)$ from Bangsgaard and Ottesen (2017), which models the diurnal variation of hormone release.
- Strong negative feedback mechanisms from cortisol to Corticotropin-Releasing Hormone (CRH), as suggested by Sriram et al. (2012), to accurately simulate AI conditions.
  The model defines three state variables: CRH concentration ( $x_1$ ), ACTH concentration ( $x_2$ ), and cortisol concentration ( $x_3$ ). An input variable $u$ represents the exogenous cortisol replacement therapy. The model is parameterized separately for primary and secondary AI, with specific values for stimulus rates and depletion rates derived from literature.
Open-Loop Validation:
Before implementing control, the model was validated using an open-loop simulation of standard clinical practice. Fixed daily doses of hydrocortisone (20 mg and 25 mg), divided into three administrations (60% morning, 20% afternoon, 20% evening), were applied to the model. These simulations confirmed that while fixed dosing could elevate cortisol to normal ranges, it resulted in non-optimal trajectories and did not fully replicate the physiological nuances of the HPA axis.
Model Predictive Control (MPC) Design:
To achieve optimal regulation, the authors designed an MPC controller.
- Linearization and Discretization: The non-linear HPA model was linearized around an equilibrium state (where $u=0$ ) and discretized using a zero-order hold with a sampling time of 1 hour.
- Controller Configuration: The MPC was configured with a prediction horizon ( $N_p$ ) of 10 hours and a control horizon ( $N_c$ ) of 4 hours. It utilizes a Kalman filter to estimate unmeasured states (CRH and ACTH) based on measured cortisol levels.
- Constraints: The controller enforces strict constraints: input doses must be non-negative and capped at 15 mg per dose, with a daily limit of 30 mg. To align with clinical reality, the calculated optimal doses are quantized to the nearest 5 mg increment.
- Objective: The controller minimizes a quadratic cost function to drive the cortisol concentration toward a reference trajectory (constant or sinusoidal) while respecting the input constraints.

Key Contributions
The paper outlines five specific contributions:

The development of a novel HPA axis model that explicitly incorporates both circadian and ultradian rhythms.
The simulation of two distinct AI conditions: primary and secondary adrenal insufficiency.
The introduction of an input variable $u$ to mathematically model cortisol replacement therapy.
The verification of the proposed model against published patient data for primary AI.
The construction of an MPC-based virtual assistant capable of generating optimal, constrained cortisol replacement strategies for both AI types.

Results
Simulation results demonstrate the efficacy of the proposed approach:

Open-Loop: Fixed-dose simulations showed that while cortisol levels could be brought into the normal range, the strategy was not optimal and resulted in spikes that varied by day.
Closed-Loop (MPC): The MPC successfully regulated cortisol levels for both primary and secondary AI across various reference trajectories (constant 10 µg/dL, constant 15 µg/dL, and a sinusoidal profile mimicking diurnal variation).
- For Primary AI, the controller effectively suppressed elevated CRH and ACTH levels via negative feedback while tracking the target cortisol levels.
- For Secondary AI, the controller maintained suppressed CRH and ACTH levels (consistent with the pathology) and successfully tracked the reference cortisol trajectory.
- The MPC demonstrated the ability to adapt dosing patterns to changing reference demands, with only a slight physiological phase lag observed between the reference and the administered cortisol.
Constraint Handling: The controller successfully adhered to the imposed constraints (non-negative, max 15 mg/dose, max 30 mg/day) and quantized the output to clinically available doses (multiples of 5 mg).

Significance and Claims
The paper claims that the proposed MPC-based virtual assistant offers a rigorous method for addressing the constraints inherent in cortisol replacement therapy, which fixed-dose strategies cannot achieve. By integrating a physiologically accurate model with an optimization-based controller, the system can generate treatment strategies that maintain cortisol within a healthy range while minimizing the risks of over- or under-dosing. The authors position this work as a step toward personalized medicine, where a physician's decision-making is supported by a tool capable of calculating optimal, patient-specific dosing schedules. The study concludes that while the current model is validated against existing data, future work will focus on adapting model parameters to specific individuals and potentially integrating machine learning and advanced nonlinear observers to further refine predictive accuracy and explore interactions with other endocrine axes, such as the hypothalamus-pituitary-thyroid (HPT) axis.

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