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Systems Engineering in the Intensive Care Unit: A Model CABG Patient Post-Operative Patient Re-Awakening Time — A Legacy Paper | fentanyl


Study of Post-Operative Respiratory Recovery

Paper originally published at INCOSE 1998. Request for reproduction caused me to include the text here. Systems Engineering in the Intensive Care Unit.

In a study of the post-operative recovery of Coronary Artery Bypass Graft (CABG) patients in surgical intensive care units, it was observed that these patients tended to awaken from the effects of anesthesia at varying rates. Because the time at which patients awoke was critical to latter phases of the study, determining the factors which affected the rate of re-awakening were of primary importance. To determine the critical parameters which influenced the re-awakening time of CABG patients, a closed-form model of the patients’ cardiovascular and respiratory systems was constructed to isolate and identify the number of parameters which were thought to affect reawakening time. As a result of this analysis, a mathematical relationship was determined which described the re-awakening time of a patient as a function of several physiological parameters. These included patient body temperature, body mass, body size, and the quantity of anesthetic (fentanyl) administered during surgery.  The work discussed in this paper was performed at the University of Pennsylvania in conjunction with my Ph.D. dissertation requirements.

Key Modeling Parameters

The key input to the process was the amount of anesthesia administered during surgery, and the
key output was the time to re-awaken. Patient physiological parameters affected the reawakening
time by affecting the rate at which the metabolism processes the anesthetic. Therefore, a  mathematical representation of the patient’s rate of metabolism was required to model the
effects of the physiological parameters on reawakening time. The anesthesia, the patient
(system), and the output are represented by the block diagram of Figure 1.

Model of re-awakening CABG patient as a system represented by the parameters mass, height, body temperature and with input the fentanyl dosage and output the re-awakening time.

Figure 1. Model of re-awakening CABG patient as a system represented by the parameters mass, height, body temperature and with input the fentanyl dosage and output the re-awakening time as determined by the time to begin spontaneous breathing.

Knowing when a patient will re-awaken is important to patient care management because
patients must be weaned from post-operative mechanical ventilation. However, before weaning
can occur patients must show that they can sustain spontaneous respiratory load. This cannot happen until the patient’s body temperature has returned to normal and the effects of the anesthesia have worn off sufficiently to enable the respiratory muscles to begin operating autonomously. To facilitate patient care management, it was hypothesized that a model of a patient’s respiratory and cardiovascular systems could be constructed to provide an
estimate of when patients would begin to exhibit spontaneous respiratory response. In brief, it was hypothesized that the re-awakening time, tRA, was a function of the following parameters:

System Equation for Re-Awakening

In deriving the functional form of this model it was necessary to isolate those parameters which were thought to play some role in determining the re-awakening time. These parameters are discussed below.

Patient Age, Mass, Height

Patients varied in ages, mass, and size. Most patients were elderly, as shown in the patient age distribution of Fig. 2. However, the correlation between increased patient age and the need for the surgery is normal as CABG surgery is frequently required in individuals who have experienced many years of arterial plaque buildup.

Patient Age-Frequency Histogram

Figure 2. Patient Age-Frequency Histogram

The functional form of equation (1) indicates that patient mass can be a factor in the reawakening time. To understand the impact of this parameter on re-awakening time it is necessary to understand how changes in patient mass can result in variations in metabolization of the anesthesia. It was hypothesized that a correlation existed between the size of a patient and the patient’s mass. In average people (not abnormally obese or thin), a loose correlation appears to exist, as illustrated in Figure 3. This figure is a scatter plot of patient height versus patient mass, wherein mass varied from approximately 50 kg to 115 kg. Patient height varied between 150 cm and 210 cm. The mass and height of a patient can be related to patient body surface area. Body surface area is used as a normalizing parameter in medicine. Body surface area is incorporated as a normalizing parameter in the relationship between anesthetic dosage and re-awakening time.

Figure 3. Patient height versus mass.

Figure 3. Patient height versus mass.

Patient Body Temperature & Achieving Normothermia

Patients return from surgery cold. The reason for this reduced body temperature is a result of the surgical procedure. Because the body must be at normal temperature so that the heart and the rest of the body functions can operate optimally, reaching normal body temperature normothermia) is key to the re-awakening process. It is only after normothermia is reached that the heart pumps oxygenated blood most efficiently throughout the body. Therefore, patient temperature is a key indicator to achieving efficient cardiovascular performance.
All patients achieved normothermia before the re-awakening process commenced. The average
time to reach normal body temperature, tNORM, was approximately 2.5 hours. Figure 4 is a scatter plot of patient arrival temperature versus time to achieve normothermia. The average temperature profile is shown overlaid on this scatter plot. As can be seen in the figure, the average re-warming time of most patients was found to be in excess of 2 hours.

Figure 4. Patient temperature warming profile.

Figure 4. Patient temperature warming profile.

Deriving the Re-Awakening Model

Patients arrive from surgery dependent on a mechanical ventilator for their breathing. This results from the effect of the anesthesia: it behaves as a muscular inhibitor to ensure that the patient does not move involuntarily during surgery. As the effects of the anesthesia wear off, the patient’s respiratory performance returns with a gradual increase in the amount of air breathed by the patient as a function of time. One measure of this respiratory function is the volume of air breathed by the patient in a given minute. A qualitative relationship between the volume of air breathed in a minute (spontaneous minute volume, Vesp ) and time after surgery is shown in Figure 5.

Figure 5. Evolution of spontaneous minute volume over time.

Figure 5. Evolution of spontaneous minute volume over time.

The re-awakening time was defined by the author as the ability to breathe a steady volume of
air, Vesp, of not less than 1 L/min with 80% confidence. The confidence was determined by monitoring Vesp over a 10 minute interval. If it was found that 80% of all Vesp measurements
exceeded 1 L/min, then the patient was said to be re-awakening. This criterion was selected by the author as an empirical threshold to affirm the reawakening of a patient. Qualitatively, this is illustrated in Figure 6.

Figure 6. Illustration of re-awakening time.

Figure 6. Illustration of re-awakening time.

It was found that once patients began breathing at approximately 1 L/min the volume of air they
continued to breathe spontaneously increased monotonically until they were able to breathe
completely on their own, defined in Figure 5 as breathing between 6 and 10 L/min (depending on body mass). The confidence value of 80% was selected as a threshold for measurements to ensure that the re-awakening of patients was indeed genuine and not spurious or an artifact of a
procedure that may have stimulated the patient to breathe spontaneously. The setting of the threshold on this parameter is of itself a subject of some study. The generalized mathematical relationship between fentanyl dosage and re-awakening time was determined to be most highly correlated to normalized fentanyl dosage, ƒn. This parameter is calculated by computing the ratio of the fentanyl dosage to average patient body surface area. Rollings [Rollings 1984] provides an expression for body surface area, BSA, as a function of patient mass and height:

Equation 2, Body Surface Area


BSA is the patient body surface area measured in square-meters;

m is the mass of the patient, measured in kilograms; and,

h is the patient height, measured in centimeters.

Analysis revealed that a more definitive relationship between anesthetic dosage and time to
reach 1 L/min was achieved by relating this parameter to both the patient body mass and body
surface area, as opposed to body mass alone. Body surface area was selected as an additional
normalizing parameter because the equation defining body mass incorporates an expression
relating to patient height. Patient height affects the size of the body torso, which ultimately affects the size of the lungs and their volume, as well as the cardiovascular system. The relationship between lung volume and body surface area motivates the argument that larger, heavier patients need more anesthetic than smaller, lighter patients. Hence, the patient body mass and surface area (BSA) are used to normalize the dosage. The normalizing parameter, ƒn, is defined as follows:

fentanyl dosing

The regression curve defining the re-awakening time is as follows [1]:

Re-awakening time

A second-order regression was chosen as this represented the lowest order expression with an adequately high correlation coefficient. A plot of the raw data, the mean time to reach 1 L/min., and the curve of equation (4) (solid line) is provided in Figure 7. The upper and lower curves that bound the mean of Figure 7 also define the 1-s standard deviation about the mean value, or 64 minutes. Coincidentally, these bounding curves also define the 80% confidence region for measurements. The plot of Figure 7 provides and indication of when the patient will re-awaken after surgery. Having an estimate of re-awakening time can provide critical care staff with a marker for the time from arrival until the time at which patients will begin breathing on their own. This knowledge provides critical care staff with both a guide for
managing the patients within their care, and as a measuring device with which to determine whether a patient’s progress is normal.

Figure 7. Model to estimate time for patient to begin breathing 1 L/min of air, in comparison with measurements.

Figure 7. Model to estimate time for patient to begin breathing 1 L/min of air, in comparison with measurements.


The model of re-awakening time derived here provides a simple and useful tool for estimating patient re-awakening time. This knowledge is vital for the critical care physician as it facilitates the patient care management process, enabling ICU staff to anticipate when and how patients should be evolving during the normal recovery process. It was generally observed that the larger the fentanyl dosage, the longer the time interval until a patient exhibited spontaneous respiratory function. Why? One simple reason is that larger fentanyl doses result in patients who tend to re-awaken more slowly. Patients given heavier doses tended not to re-awaken for perhaps 3-4 hours, and then their respiratory systems responded more slowly. As a result, these patients required ventilatory support for extended periods of time. However, wide variations existed which made the coupling between fentanyl dosage and re-awakening time less significant. What might cause these wide variations? There are a number of reasons. The most obvious is that patients metabolic rates differ, resulting in some patients being able to metabolize
the drug more quickly. Ultimately, by treating the patient as a system of processes, it can be shown that even as complex as the human body is, simple mathematical relationships can be developed to model the outcome of the effects of anesthetic on human respiratory and cardiovascular behavior, and that these simple models can provide a useful tool for physicians to estimate and monitor the expected respiratory responses of patients recuperating from the effects of surgery.


[Zaleski 1996] Zaleski, John R. Modeling Post-Operative Respiratory State in Coronary
Artery Bypass Graft Patients: A Methodology for Weaning Patients from Post-Operative Mechanical Ventilation. Ph.D. Dissertation, The University of Pennsylvania, 1996.
[Rollings 1984] Rollings, Robert C. Facts and Formulas. Nashville, TN: By the author, 1984.


Posted in Mathematical Modeling and Simulation, Medical Device Integration / Connected Medical Devices, Syndromic Surveillance.

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