Optimal Control Methods to Analyze Acute Inflammation

By Karthika Swamy Cohen

An individual or animal responds to acute stress such as infection or trauma with an acute inflammatory response, involving a cascade of events directed by cells and molecules in the body that locate invading pathogens or damaged tissue and then send a signal, recruiting and instigating other cells and effector molecules. These cells then set off a response to eliminate the offending agents or pathogens, restoring the body to health. 

However, if the inflammatory response is too weak, the body may succumb to the inflammation, and if it’s too strong, it may cause tissue damage leading to persistent inflammation. Either can cause organ failure and death. 

In these cases, intervention is needed to restore normal homeostasis in the organism. But intervention strategies are difficult to design without a complete understanding of the complex immune response.

This necessitates tools that can give insight into the inflammatory response and better forecast the effectiveness of therapies. 

At a minisymposium at the SIAM Conference on Control and Its Applications taking place in Pittsburgh, Pa. this week, Judy Day of the University of Tennessee, Knoxville described mathematical techniques that can be used to address these challenges in the field of immunology.

Day studies complex inflammatory response dynamics with nonlinear mathematical models. The canonical and highly nonlinear dynamical systems being used to analyze these problems are very compact, making the system easy to study, Day explained. 

These models typically exhibit tristability or three stable states. This means there are three possible states the model could reach given the initial parameter set: a healthy stable state where the cells respond to the insult (or pathogen) and resolve back to the normal state; a state where the cell can eliminate the pathogen but isn't able to resolve to the normal state; and a third state where it isn’t able to resolve to the normal state and is also unable to eliminate the pathogen.

Day described three strategies that her group tested on virtual patient populations (for lack of volunteers for a real clinical trial). The virtual clinical trial involved 1000 virtual patients and accounted for heterogeneity as would be seen in a normal patient population

The first strategy is an optimal control study, which involves comparisons of different types of objective functions, while focusing on general strategy. The second approach uses model predictive control, which combines predictions of the real system at a future state based on a mathematical model; measurements are then derived from the system in order to compute a control move to help optimize the desired outcome for a specific process variable. The third method is model free control, which has the novel feature that the two outputs that must be driven are sensorless. This is overcome by assigning suitable reference trajectories to two other outputs with sensors.

Static therapy was modeled after standard sepsis treatment uniformly applied to all patients. Day's group then kept track of how the different strategies performed overall on the virtual population, also monitoring any harm caused to patients.

The objective was to reduce damage and pathogen levels as well as minimize the dosage it takes to get to that healthy state. Prescribed constraints in the dosing amounts and duration prevent the system from exceeding toxicity levels. 

The model proved to be an effective method to investigate and compare different control strategies to determine therapeutic intervention strategies for a difficult biomedical problem. 

Click here for more coverage of the 2017 SIAM Annual Meeting.

Karthika Swamy Cohen is the managing editor of SIAM News.