The workshop participants are researchers in areas of mathematics, engineering, physics, statistics, and physiology mastering traditional methodologies used within these fields, but find themselves challenged to use and combine these methodologies to get better understanding of the complex dynamics observed in clinical data. Part of the purpose of this workshop was to help these groups to communicate, and to develop a common body of problem-solving skills. The overall goal is to use sophisticated mathematical techniques to predict and analyze dynamics observed in complex physiological systems.
The participants at this workshop interacted in a mutually supportive role supplementing each other knowledge and expertise. Clinicians provide the source of problems and opportunities to test modeling approaches. In this subject area there are some mathematical models that are so complex that it is difficult to identify the parameters. Other models are too simplistic to have any real descriptive value. One of the main goals of this workshop is to find "middle-level models" that are realistic and meaningful, and that will give rise to sub-models that can be applied as clinical tests. These tests should use data that can be collected clinically. What is most desirable are non-invasive tests that can be performed on humans.
The main focus of this workshop was to understand problems behind cardiovascular and respiratory regulation. Particular focus problems included modeling and analysis of data for heart rate regulation in health and disease and understanding how space-flight can impact control of heart rate and blood pressure in astronauts post-space flight. Heart rate dynamics were discussed both using methods from time series analysis and also from modeling perspectives. While some of the feedback dynamics may be understood and can be explained using differential equations models there are still questions arising in how these models can be adapted to account for multiscale fractal dimensions, which seem to be an important quantity that varies between health and disease. One topic of discussion was how to merge these approaches using Bayesian statistical methods and stochastic differential equations.
Another illustrative example of the sort of feedback loops that arise in medical situations is the glucose-insulin control in the human body. People suffering from diabetes do not have a working control loop. Researchers would like to construct an external control that will modify the dysfunctional control loop in the body of a diabetic patient.
Similar problems occur in sleep apnea which is now understood to be a consequence of delay and elevated controller gains in the feedback system that controls respiration. The levels of oxygen and carbon dioxide are sensed at two sites in the body but it takes time for blood gases which are exchanged in the lungs to reach the sensory sites. This results in the delay in the feedback loop.
A variety of mathematical tools are used to analyze the situtations that have been described. Control theory is used to devise feedback loops; Kalman filters are a useful tool in this process. Optimization theory is used for parameter estimation. Ordinary differential equations and delay differential equations are used for modeling. Numerical methods translate the theoretical ideas into data. Fast Fourier transforms and, more recently, wavelets are used to break signals up into rudimentary frequencies. Statistics are particularly useful for distinguishing different groups of patients under study.
The physiological problems studied at this workshop provide bases for models that represent real-world applications. Part of the goal of the workshop is to generate discussions between clinicians and mathematicians, so that clinicians can understand and benefit from the modeling approach. This give-and-take is resulting in new advances both in theory and applications.