Mathematics and Clinical Concepts, Part I: The Need for Formal Methods and Theories

Clinical care is a complex activity, and the systems designed to support it must manage that complexity.   Building systems to assist with patient care requires converting real-world messiness into something computers can manipulate, which comes down to, at some point, 0s and 1s.   Obviously, this is not currently a straightforward process.   The difficulties inherent in transferring clinical concepts to computing systems become obvious when we look at EHR systems.   Clinical care is a process-oriented, data-based activity.   Systems that support clinical care must support processes, participants, and data. Currently, only the latter has significant support.

At present EHR system design is an ad-hoc sort of process.  We have standards that describe the data that EHRs should hold and the minimum number of functions they should provide.  High-level features, such as workflows and interactions between healthcare professionals, have no agreed upon representation. The same is true for simple entities such as problem lists and medication lists.  Currently, both are usually captured as simple text strings in relational databases.   Is this ideal?  Is a problem list merely a list of SNOMED terms or should it be able to interact with the other data in the EHR?  Our idea of a problem list is based on the two-dimensional notion of a paper list, but can it be, should it be, more?

Ultimately, creating an EHR or other clinical information system requires the creation of abstractions and models of real-world entities.   The goal of this series of posts is to present discrete mathematics as a tool for developing computing-friendly abstractions and models of clinical care processes and data.

Discrete versus continuous mathematics
Susanna Epp, in her book, Discrete Mathematics with Applications, Fourth Edition, states (1):

Discrete mathematics describes processes that consist of a sequence of individual steps.  This contrasts with calculus which describes processes that change in a continuous fashion.

Discovering discrete math was a revelation to me, even though I had encountered some areas of it previously.  After all, who hasn’t heard of set theory or propositional logic?  The problem for me was that I looked at them simply as subjects from math classes, not as tools for modeling clinical concepts.   Like every other healthcare professional, I‘ve had experience with calculus and functions that operate on real numbers (think decimal points and rates, decays, acceleration).   Of course, there are plenty of real numbers in clinical care, but the care process itself is discrete.      A medication may be divided into a fractional amount (say breaking a 25 mg tab in half), but the administration of that medication is all or none–either the patients takes it or he does not, a visit occurs or it does not, a lab test is done or it is not.

In their book, Modeling Business Processes: A Petri Net-Oriented Approach, van der Aalst and Stahl, discuss discrete, dynamic systems, which they go on to model using Petri nets (based on graph theory)(2).  The authors demonstrate quite well that a variety of discrete, dynamic systems can be modeled using Petri nets–why not complex clinical systems?

The need for formal methods and theories
In my last post, I spoke about the need for formal methods and theories in developing clinical systems.   If such theories and methods existed, it would be much easier to compare and contrast different approaches to building systems.   Without them, all solutions are local and not readily comparable with competing solutions.  We have no model, abstract or concrete (i.e., a prototype), for an ideal EHR; therefore, how can we know when we have built one?

Thomas Kuhn, in The Structure of Scientific Revolutions, describes the situation that existed in the field of optics before Newton provided the concepts that allowed each competing group of researchers to see how its work compared to the others (3).

At various times all these schools made significant contributions to the body of concepts, phenomena, and techniques from which Newton drew the first nearly uniformly accepted paradigm for physical optics. Any definition of the scientist that excludes at least the more creative members of these various schools will exclude their modern successors as well. Those men were scientists. Yet anyone examining a survey of physical optics before Newton may well conclude that, though the field’s practitioners were scientists, the net result of their activity was something less than science. Being able to take no common body of belief for granted, each writer on physical optics felt forced to build his field anew from its foundations. In doing so, his choice of supporting observation and experiment was relatively free, for there was no standard set of methods or of phenomena that every optical writer felt forced to employ and explain. Under these circumstances, the dialogue of the resulting books was often directed as much to the members of other schools as it was to nature. That pattern is not unfamiliar in a number of creative fields today, nor is it incompatible with significant discovery and invention. It is not, however, the pattern of development that physical optics acquired after Newton and that other natural sciences make familiar today.

This aptly describes the state of clinical information systems research and development today.

The goal for this series of posts
The first step in producing formal methods and theories is finding a way to describe mathematically key concepts of the clinical area under investigation.  Of course, this is easier said than done.  My aim in this series of posts is to present, using clinical examples, the areas of discrete mathematics that I think can be used to describe and model clinical concepts.   I will discuss selected topics from the following domains: propositional and predicate logic, sets, functions, relations, and graphs.  More time will be spent on graphs than other areas because of its value in modeling processes.

I will try to keep each post to 1500 words or less, so they are not too long.  When posts are longer than 1500 words, I will split them and publish the first half on Monday and the second on Wednesday.     See you on Wednesday…

  1. Epp SS. Discrete Mathematics with Application, Fourth Edition. Belmont, CA: Thomson-Brooks/Cole; 2010
  2. van der Aalst W, Stahl C. Modeling Business Processes: A Petri Net-oriented Approach. Cambridge, MA: MIT; 2011
  3. Kuhn TS. The Structure of Scientific Revolutions, Second Edition, Enlarged. Chicago: University of Chicago; 1970

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