# The Calculus of Patient Satisfaction

Tom Scaletta, MD FAAEM

Disclosures

December 24, 2012

### Introduction

How do calculus and patient satisfaction possibly go together? Allow me to explain. Before entering medical school, I earned an undergraduate degree in mathematics and subsequently worked as a database programmer for an actuarial firm. My favorite college math professor and early mentor knew that college was primarily a bridge for me into medicine. Yet, he often remarked that the skills I was learning in advanced calculus would never be wasted. Amazingly, he was right. After completing an emergency medicine residency, I became keenly interested in administration and currently oversee a 90,000 visit ED in the Chicago area. I now find synergy between mathematical theory and maintaining a top-tier ED that is consistently at or above the 95th percentile in patient satisfaction. In a surreal way I sometimes feel like I am channeling the wisdom he shared with me decades ago to drive process improvements in safety, cost, and satisfaction. Go figure!

Calculus, the mathematics of measuring change, operates under a recurring theme that big things are derived from little things. Achieving excellence in patient satisfaction can be approached like a calculus problem. The complex task of maximizing the patient experience in a high-volume ED requires dividing this goal into many discrete, manageable actions that can have a positive effect on patient satisfaction. Optimization of each component requires interval performance measurement and fine-tuning. Upon reassembly of all the pieces, the solution is demonstrated.

Consider the equation, B(a,b) = ∫h(x)dx, which represents the formula for the area beneath a certain curve [Figure A]. This curve could correspond to the rise and fall of patient satisfaction related to a particular aspect of service delivery over a period of time. The X axis represents the passage of time where (a) is now and (b) is the future after a discrete process change. The Y axis represents performance.

Figures A, B, and C

The Infinite Sum Theorem defines h(x) for any infinitesimally small time interval. These thin slices, denoted as ΔB, equate to h(x) Δx where Δx is the area under the curve, and are much easier to calculate [Figure B]. Subsequently, all these time intervals, when added together, will estimate the area under the curve for a certain time period (like a-b). So solving the complex problem of measuring the area under a changing curve simply requires us to sum up the areas of many thin rectangles.

That’s enough calculus for now. It’s time to segue to more practical applications.

Before we can dig into patient satisfaction, we must define it. Satisfaction is a layperson’s interpretation of the quality of health care delivery. In other words, it is based on perception of actuality, not necessarily actuality. Since each patient is unique, perceptions vary widely.

What is important to one may be meaningless to another. To add further complexity, we know a continuum exists among practitioners as to what constitutes the highest quality care.

Performance differentiators are categorized with the mnemonic, QUEST — quality, utilization, efficiency, satisfaction, and teamwork. These categories are tightly interrelated. Cost is split into resource waste (utilization) and time waste (efficiency). In the case below, every aspect of performance was great except for satisfaction. Emergency caregivers need to constantly correct patient (or family) misperceptions of what quality represents. We often try to fulfill the individual’s perspective of great service whenever feasible. Feasible implies there is no sacrifice in a more meaningful aspect of performance. For instance, curtailing test ordering is great unless it jeopardizes patient safety. Feasible also implies cost targets are met and any other limitations can be tolerated.

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