Computational Modeling

Ordinary Differential Equations (ODE) for Modeling the Imune System on Inflamatory Response

This repository presents a computational model using Ordinary Differential Equations (ODEs) to simulate the behavior of tissue damage and immune response during an inflammatory process, such as a skin wound. The model was developed as part of an academic project for the Computational Modeling course.

Model without Immunologic Response

This model captures the basic dynamics of tissue damage, neutrophils, and pro-inflammatory cytokines without regulatory immune intervention.

We can derive Equations as follows:

$$\frac{d}{dt}TD(t) = \alpha N(t)$$ $$\frac{d}{dt} N(t)= \beta TD(t) + \gamma CH(t) -\alpha N(t)$$ $$\frac{d}{dt} CH(t) = \rho N(t) - \sigma CH(t)$$

Model with Immunologic Response

This extended model introduces the immune regulation through macrophages $M(t) $and anti-inflammatory cytokines $A(t)$.

Figure includes the immunologic response on the previous model.

$$\frac{d}{dt}TD(t) = \alpha N(t) - u_{reg} M(t)$$ $$\frac{d}{dt}N(t) = \beta TD(t) + \frac{\gamma CH(t)}{(1+\mu_{A}A(t))} -\alpha N(t)$$ $$\frac{d}{dt}CH(t) = \frac{\rho N(t)}{(1+\alpha_A A(t))} -\eta_{CH} CH(t)$$ $$\frac{d}{dt}M(t) = v N(t) -\eta_M M(t)$$ $$\frac{d}{dt}A(t) = w_{reg} M(t) -\eta_A A(t)$$

The results show that all variables grow indefinitely. Particularly, the tissue damage $TD(t)$, pro-inflammatory cytokines $CH(t)$ and neutrophils $N(t)$ follow an uncontrolled exponential pattern. This suggests that without anti-inflammatory action, the inflammatory process becomes self-reinforcing and unbounded. Conclusion: The wound never heals without immune response.

With the inclusion of regulatory terms $M(t)$ and $A(t)$, the simulations display a range of behaviors. In some configurations, both neutrophils and cytokines decay to zero, showing a resolution of inflammation. Conclusion: The immune system, when effectively modeled, is able to suppress and eventually eliminate inflammation.