Sumários

Structural model - introduction of non-linear models and examples

16 Abril 2026, 12:30 • Rafael Neto Henriques

Transitioning from linear approximations, this theoretical class introduced Non-Linear Structural Models. The session began by formally defining the mathematical characteristics that make a model non-linear and discussing why these models are essential for accurately capturing the true complexity of most physiological systems. To illustrate the different ways non-linearities manifest in biological modeling, the class explored three specific examples:

1) Enzyme Kinetics: Demonstrating non-linearity driven by interaction terms, where the rate of a reaction depends on the product of different variable concentrations.

2) Hodgkin and Huxley Action Potential Model: Highlighting systems where the model coefficients are directly dependent on the state variable itself (e.g., ion channel conductances that change dynamically based on the membrane voltage).

3) Baroreceptor Control Model: Showcasing non-linearities introduced by physiological thresholds and directional sensitivities. This was illustrated by modeling how baroreceptors exhibit non-linear firing patterns, specifically firing only when the rate of change of pressure is positive or when blood pressure drops below a defined threshold.

Advanced Numerical Methods – Runge-Kutta Implementation

15 Abril 2026, 13:30 • Rafael Neto Henriques

This session advanced the students' computational skills by introducing higher-order Runge-Kutta numerical methods for solving ordinary differential equations in their projects. The class began with a theoretical overview of the Runge-Kutta approach, explaining how it achieves greater accuracy and stability compared to the foundational Euler method. Instructors then demonstrated the application of these methods across various physiological modeling scenarios. Students learned how to implement these algorithms from scratch using custom-developed code, as well as how to leverage optimized, built-in ODE solvers available in MATLAB and Python packages to efficiently simulate their compartmental models.

Structural Models – Whole-Organ Distributed Models and Hepatic Removal

15 Abril 2026, 10:00 • Rafael Neto Henriques

Continuing with distributed modeling, this theoretical class expanded the scope from a single capillary-tissue unit to Fully Integrated Organ Models. The session addressed the physiological reality of organ heterogeneity, demonstrating how to mathematically assemble multiple capillary-tissue units in parallel, each featuring different blood flows and transit times. In its second part, the class examined the distributed model for hepatic removal of material. Students compared this spatially distributed approach - which accurately captures the progressive concentration gradient of substances as blood flows along the liver sinusoids - with its simplified lumped model counterpart. The discussion highlighted the specific scenarios where a distributed model is necessary to prevent significant estimation errors, and when a lumped approximation might still be deemed sufficient for clinical or research purposes.

Model Implementation and Numerical Troubleshooting

15 Abril 2026, 08:30 • Rafael Neto Henriques

This session was dedicated to supporting students as they began translating the physiological models from their selected research papers into functional MATLAB or Python code. Instructors provided hands-on assistance to help groups bridge the gap between theoretical mathematical descriptions and practical algorithmic implementation. A key focus of the class involved addressing specific programming and mathematical challenges raised by the students. For example it was discussed how to effectively apply numerical methods to resolve implicit root-finding problems within the dynamic systems described on one selected paper.

Structural Models – Distributed Models and the Capillary-Tissue Unit

14 Abril 2026, 11:00 • Rafael Neto Henriques

Building on the foundation of structural models, this theoretical class introduced Distributed Models. Unlike "lumped" models that assume instantaneous mixing within a space, distributed models account for spatial variations and concentration gradients across a physiological system. The session focused extensively on modeling the fundamental building block of organs: the capillary-tissue unit. Students explored the progression of these units through three levels of complexity:

1-Compartment Model: The simplest representation, where the capillary and surrounding tissue are lumped together as a single, well-mixed pool, assuming immediate equilibration.

2-Compartment Model: A more physiologically accurate approach that separates the capillary blood from the tissue space, allowing for the mathematical description of dynamic exchange and permeability between the two distinct regions.

3-Compartment Model: An advanced representation that adds a third layer of physiological fidelity, typically by further dividing the tissue into distinct interstitial and intracellular spaces (or separating the blood into plasma and red blood cells) to capture complex transport delays and binding dynamics.