Complete Scientific Theories in IINTS-AF¶

This document summarizes the physiological and mathematical mechanisms currently implemented or exposed for pre-clinical simulation in the IINTS-AF Digital Twin SDK. It is an implementation reference for EUCYS-style explanation, not clinical validation.

The simulation engine includes the AdvancedMetabolicModel, an extended 18-state differential-equation model derived from the Bergman-style core and expanded with research-oriented metabolic stress states.

1. The Bergman Minimal Model¶

Foundation: The core system relies on the classical Bergman minimal model (1989) which uses a 3-compartment design to track Plasma Glucose (\(G\)), Plasma Insulin (\(I\)), and Insulin Action (\(X\)).

Mathematics:

\[ \frac{dG}{dt} = -X \cdot G + p_1 \cdot G_b + R_a \]

\[ \frac{dX}{dt} = -p_2 \cdot X + p_3 \cdot \max(I - I_b, 0) \]

Here, \(p_1\) represents glucose effectiveness (GEZI) and \(p_3\) represents insulin sensitivity.

Practical Impact: Determines the fundamental balance between how fast glucose falls due to the body's natural metabolism versus the action of administered insulin.

2. Multi-Compartment Gastric Emptying (Dalla Man)¶

Foundation: The Dalla Man (2006) absorption model.

Mathematics: Rather than assuming immediate absorption, carbohydrates flow through three delayed compartments:

\[ \frac{dQ_{sto1}}{dt} = -k_{gri} \cdot Q_{sto1} + D_{carbs} \]

\[ \frac{dQ_{sto2}}{dt} = k_{gri} \cdot Q_{sto1} - k_{emp} \cdot Q_{sto2} \]

\[ \frac{dQ_{gut}}{dt} = k_{emp} \cdot Q_{sto2} - k_{abs} \cdot Q_{gut} \]

Practical Impact: Prevents ingested carbohydrates from instantly entering the blood. It models the physiological delay of gastric emptying, resulting in a realistic glucose spike 45-90 minutes after eating.

3. Subcutaneous Insulin Absorption Kinetics¶

Foundation: Hovorka / Dalla Man pharmacokinetics.

Mathematics: Tracks the physiological lag (\(\tau\)) between pump delivery (\(u_{ins}\)) and plasma appearance via two subcutaneous compartments:

\[ \frac{dS_1}{dt} = u_{ins} - k_a \cdot S_1 \]

\[ \frac{dS_2}{dt} = k_a \cdot S_1 - k_a \cdot S_2 \]

\[ R_{a, I} = k_a \cdot S_2 \]

Practical Impact: Controls the dangerous "Insulin On Board" (IOB) tail. Pumped insulin does not work immediately but only reaches its peak effect in the tissue after 60-90 minutes.

4. Lipotoxicity & Free Fatty Acid (FFA) Dynamics¶

Foundation: Advanced pathophysiology of insulin resistance.

Mathematics: Insulin inhibits lipolysis. Without insulin, FFA (\(F\)) rises and down-regulates insulin sensitivity (\(p_3\)):

\[ \frac{dF}{dt} = l_0 \cdot e^{-l_1 I} - k_f F \]

\[ p_{3, eff} = p_3 \times \frac{0.4}{\max(0.4, F)} \]

Practical Impact: Models a strong insulin-resistance pressure when an insulin pump fails or is occluded for hours, so later correction boluses may become less effective in the simulation.

5. Ketogenesis & Diabetic Ketoacidosis (DKA)¶

Foundation: The ketone-production cascade under insulin deficiency.

Mathematics: Ketone (\(K\)) production is driven by extreme FFA levels and near-zero insulin:

\[ \frac{dK}{dt} = k_0 \cdot F \cdot e^{-k_1 I} - k_2 K \]

Practical Impact: Allows the simulator and audit tools to flag DKA-risk trajectories during severe pump failure or acute illness scenarios.

6. Hypoglycemia-Associated Autonomic Failure (HAAF)¶

Foundation: Cryer's theory of defective counter-regulation.

Mathematics: Tracks "hypo-memory". Past hypos suppress future adrenaline rescue mechanisms:

\[ \Delta_{hypo} = \max(0, 70-G) \]

\[ \frac{dHAAF}{dt} = k_{build} \cdot \Delta_{hypo} \cdot (1-HAAF) - k_{decay} \cdot HAAF \]

Practical Impact: Mimics clinical reality: patients who experienced a deep hypoglycemic event during the night are physiologically much more vulnerable to another, deeper hypo the next day because their counter-regulatory adrenaline response is depleted.

7. Circadian Rhythms & Dawn Phenomenon¶

Foundation: Chronobiology and counter-regulatory morning hormones.

Mathematics: Applies a continuous sinusoidal wave to Endogenous Glucose Production (EGP), peaking around 05:00 AM:

\[ \text{Circadian} = 1.0 + A_{circ} \cdot \cos\left(\frac{2\pi}{24} (t_{hours} - 5)\right) \]

Practical Impact: Creates the infamous "Dawn Phenomenon" where patients experience unexplained, severe high blood sugars early in the morning despite not eating anything.

8. Physiological Renal Glucose Clearance (RGC)¶

Foundation: Kidney filtration physics.

Mathematics: When glucose exceeds the Renal Threshold (~180 mg/dL), kidneys excrete it. Modeled via a softplus function to prevent stiff-ODE crashes:

\[ RGC = c_{renal} \cdot 10 \cdot \ln\left(1 + \exp\left(\frac{G - 180}{10}\right)\right) \]

Practical Impact: Acts as the body's natural safety valve. Without this mathematical sink, simulated glucose levels in a patient suffering from pump failure would rise to infinity.

9. Exercise Physiology & Stress¶

Foundation: Metabolic shifts during physical exertion.

Mathematics: Exercise intensity (\(E\)) increases insulin sensitivity (\(p_3\)) and drives insulin-independent muscle uptake:

\[ p_{3,eff} = p_3 \cdot (1 + 2E) \]

\[ \text{Uptake}_{muscle} = E \cdot 0.005 \cdot G \]

Practical Impact: Creates exercise-driven downward glucose pressure for testing whether algorithms reduce insulin early enough during exertion-induced hypoglycemia risk.

10. Residual Beta-Cell Autoimmune Decay¶

Foundation: The T1D "Honeymoon Phase".

Mathematics: The residual healthy Beta-cell mass fraction (\(\beta\)) undergoes exponential autoimmune decay:

\[ \frac{d\beta}{dt} = -\alpha \cdot \beta \]

\[ \text{Secretion}_{endo} = \beta \cdot \gamma \cdot \max(G - h, 0) \]

Practical Impact: Allows researchers to benchmark algorithms over multi-year lifespans, testing how well an AI adapts as the patient slowly shifts from a mild "Honeymoon" phase to a brittle, 100% dependent diabetic.

11. Exogenous Glucagon Kinetics¶

Foundation: Emergency hormonal rescue pharmacokinetics.

Mathematics: Simulates glucagon transport (\(\Gamma\)) causing direct hepatic glycogen release:

\[ \frac{dY_1}{dt} = u_{gluc} - \frac{Y_1}{\tau} \quad \rightarrow \quad \frac{d\Gamma}{dt} = \frac{Y_2}{\tau \cdot V_{gluc}} - k_e \Gamma \]

Practical Impact: Enables pre-clinical testing of bi-hormonal pump logic that can deliver both insulin and glucagon in low-glucose-risk scenarios.

12. Multi-Macronutrient Gastric Emptying¶

Foundation: Advanced meal composition (Fat & Protein).

Mathematics: Fat (\(Q_{fat}\)) exponentially delays gastric emptying (\(k_{emp}\)). Protein (\(Q_{prot}\)) triggers slow gluconeogenesis:

\[ k_{emp,eff} = \frac{1}{\tau_{meal}} \cdot e^{-0.02 \cdot Q_{fat}} \]

\[ R_{a,prot} = 0.5 \cdot k_{prot} \cdot Q_{prot} \]

Practical Impact: Simulates the dangerous "Pizza Paradox", where delayed fat and protein absorption causes unexpected and massive hyperglycemic spikes up to 6 hours after a meal.

13. Cannula Degradation & Lipohypertrophy¶

Foundation: Mechanical tissue resistance & inflammation.

Mathematics: Subcutaneous absorption (\(k_a\)) degrades linearly by up to 30% after wearing the pump for 48 hours (2880 mins):

\[ k_{a,eff} = k_a \cdot \left(1 - 0.3 \cdot \max\left(0, \frac{t - 2880}{2880}\right)\right) \]

Practical Impact: Penalizes algorithms in long-term endurance tests. After wearing the infusion set for 3 days, tissue inflammation occurs, causing insulin to be absorbed at an increasingly slower and erratic rate.

14. Menstrual Cycle Hormonal Drifts¶

Foundation: Female biology and cyclical resistance.

Mathematics: A 28-day low-frequency sinusoidal wave alters insulin sensitivity (\(p_3\)), peaking in resistance during the Luteal phase:

\[ p_{3,eff} = p_3 \cdot \left(1 - 0.25 \cdot \sin\left(\frac{2\pi \cdot t}{28 \cdot 24 \cdot 60}\right)\right) \]

Practical Impact: Tests whether an algorithm is adaptive enough to handle the subtle, multi-week hormonal resistance drifts (like PMS) that heavily impact insulin requirements in female patients.

15. Acute Illness & Cytokine Resistance¶

Foundation: Immune system stress response.

Mathematics: A sickness severity factor (\(\zeta\)) increases basal glucose production (\(G_b\)) while reducing tissue sensitivity:

\[ Gb_{eff} = Gb \cdot (1 + 0.8 \cdot \zeta) \]

\[ p_{3,eff} = p_3 \cdot (1 - 0.5 \cdot \zeta) \]

Practical Impact: Designed as a hyperglycemia stress test. Illness scenarios can increase simulated insulin requirements, so algorithms must adapt without violating safety limits.

Conclusion¶

Together, these mechanisms form a cohesive 18-state pre-clinical Digital Twin for stress-testing insulin algorithms under controlled, documented assumptions.