On the day of surgery, electronic white boards function as communication systems, displaying in real-time the status of cases and helping coordinate clinical work.^{1} Color coding and various icons are often used to indicate patient status (e.g., green highlighting if the patient has arrived in the holding area).^{2} Figure 1 shows an example. The information can be useful to supervising anesthesiologists,^{3} to-follow surgeons in an operating room (OR), postanesthesia care units waiting for patient arrivals,^{4} ^{,} ^{5} relief of anesthesia providers,^{2} ^{,} ^{3} ^{,} ^{6} etc.

Figure 1: Example of an electronic operating room (OR) case board (whiteboard). Displayed is a cropped screenshot from the whiteboard in use at Vanderbilt University Medical Center (left), along with the corresponding icon and color legend (right). This board shows 1 case taking longer than scheduled (red arrow) and 1 (tardy) case starting later than scheduled (right-pointing blue arrow). Actual patient names were replaced with aliases for publication purposes. The current time (denoted by the vertical red line) is 3:30 PM. In room RM 02, a laryngoscopy procedure scheduled for 1 hour and starting at 2:30 PM is taking longer than scheduled. A 1-hour bronchoscopy case that was scheduled to follow in RM 2 is available to start, as shown by the 2 icons. RM 03 is available. We chose this scenario, because inaccurate display of the mean (expected) time remaining in the ongoing case in RM 02 is not affecting rational decision making.

^{2} This is in deliberate comparison to

Table 1 , in which the vocabulary is defined explicitly. Consider 3 options with regard to potentially moving the to-follow case, depending on how time has been allocated. (A) Suppose both rooms have allocated time to 3:30 PM. Then, regardless of the estimated time remaining in the ongoing case in RM 02, the to-follow case in RM 02 will be performed in overutilized OR time. Moving the to-follow case to RM 03 would reduce the minutes of tardiness.

^{2} Therefore, the appropriate decision would be to move the case to RM 03. (B) Suppose both rooms have allocated time to 5:00 PM. Then, if the case were moved to RM 03, there would be no expected overutilized time and fewer minutes of tardiness.

^{2} The decision also would be to move the case to RM 03. (C) Suppose RM 02 has allocated time until 5:00 PM and RM 3 until 3:30 PM. If the to-follow 1-hour case were moved, it would be performed entirely in overutilized time. If the case were not moved, the overutilized time would be <1 hour, unless the time remaining in the ongoing case was as long as its originally estimated duration. Therefore, the case should remain in RM 02.

In 1999, we derived the mean expected time remaining in ongoing surgical cases.^{7} This was (appropriately) based on the 2-parameter log-normal distribution, but was limited to surgeon and procedure combinations with sufficient historical data to estimate the mean and standard deviation with little error (e.g., N ≥ 30).^{7–11} A decade later, we extended the derivations to cases with few or even no historical data.^{12} That paper focused on managerial decision making on the day of surgery, including for facilities that use times recorded on clipboards.^{12}

Among all surgical procedures, the median coefficient of variation (standard deviation divided by the mean) of case duration equals approximately 32%.^{11} Figure 2 shows a plot for a case with that coefficient of variation of 32% and a mean scheduled duration (i.e., before the patient enters the OR) of 5.0 hours. For the first few hours, each 1 hour further along in the case results in a 1-hour decline in the expected time remaining. By the scheduled end time of 5.0 hours, most (56%) of cases have finished. However, among the 44% of cases that have not ended, 1.5 hours is the mean (expected value) of the time remaining. By 6 hours, only 23% of cases are ongoing. However, for such cases, the 1.3-hour mean time remaining is similar to the 1.5-hour estimate that was made at 5 hours. By 8 hours, only 5% of cases are ongoing. However, for such cases, the 1.2-hour mean time remaining is only slightly less than estimated at 6 hours. This flat relationship between time remaining and elapsed time of ongoing cases running longer than their expected duration is a direct mathematical consequence of the 2-parameter log-normal distribution with coefficients of variations in the 10% to 50% range (i.e., precisely those of surgical procedures) (Figs. 3 and 4 ).^{11–13}

Figure 2: Time remaining in a hypothetical case with coefficient of variation of 32%. The expected (mean) duration was set at 5.0 hours before the case starts. The expected time remaining in the operating room (OR) case before exit (—) and the percentage of cases still ongoing (—) are calculated based on a 2-parameter log-normal distribution and many historical data. For each of the first several hours, the time remaining decreases linearly, but then is flat. For the Vanderbilt data, 44.0% of combinations of scheduled procedure(s), surgeon, and type of anesthetic had coefficient of variation >32%.

Figure 3: Time remaining in a hypothetical case with coefficient of variation of 64%, twice that of

Figure 2 . The expected (mean) duration was set at 5.0 hours before the case starts. The expected time remaining in the operating room (OR) case before exit (—) and the percentage of cases still ongoing (—) are plotted as in

Figure 2 . As described in the beginning of the article, for coefficients of variation that are larger than typical for most procedure(s), the relationships between the hours since the start of the case and the time remaining are different. It is unrealistic that a 5.0-hour case would after 3.0 hours always have at least 3.0 hours to go until OR exit, and this is due to the coefficient of variation used for the figure being uncommonly large. Thus, this Figure 3 highlights that the observed behavior in

Figures 2 ,

4 , and

5 is a direct mathematical consequence of log-normal probability distributions with coefficients of variations characteristic of OR times.

^{11} ^{,} ^{12} For the Vanderbilt data, 4.7% of combinations of scheduled procedure(s), surgeon, and type of anesthetic had coefficient of variation >64%.

Figure 4: Time remaining in a hypothetical case with coefficient of variation of 20%, less than that of

Figure 2 . The expected (mean) duration was set at 5.0 hours before the case starts. The expected time remaining in the operating room (OR) case before exit (—) and the percentage of cases still ongoing (—) are plotted as in

Figure 2 . Reduction in the coefficient of variation from 32% in

Figure 2 to 20% here resulted in there being less time remaining among late running cases. For the Vanderbilt data, 82.0% of combinations of scheduled procedure(s), surgeon, and type of anesthetic had coefficient of variation >20%.

This mathematical property has a direct practical application. Although displays used on the day of surgery show the expected time remaining in cases,^{12} every commercial electronic whiteboard that we are aware of matches Figure 1 in treating the expected time remaining for cases exceeding their scheduled duration as being 0 hours (Fig. 5 ). This is, logically, incorrect statistically for all such cases (Fig. 2 ).^{12} For facilities using clipboards to manually record case progress and the estimated case completion times, what should be written for repeated viewing are not estimated times that cases will end (e.g., 5:15 PM), but rather estimates of the minutes remaining until the patient will leave the OR (e.g., 75 minutes).^{12} If a case is still ongoing 15 minutes after an estimate and closure has not yet started, the new value for the estimated time when the case will end should be calculated by adding the original estimate in minutes to the current time, not by subtracting 15 minutes from the prior estimate (see Discussion).

Figure 5: Time remaining in a hypothetical case with coefficient of variation of 0.1%. The expected (mean) duration was set at 5.0 hours before the case starts. The expected time remaining in the operating room (OR) case before exit (—) and the percentage of cases still ongoing (—) are plotted as in

Figure 2 . A coefficient of variation of 0.1% implies that when the surgeon scheduled the procedure(s), the case always takes the same amount of time, which is impossible. Notice that among cases taking longer than scheduled, the mean (expected) time remaining in the OR is zero. This is effectively what is assumed if the electronic whiteboard does not use statistical methods or historical data for modeling the time remaining in late running cases (see the beginning of the article). For the Vanderbilt data, among 1549 combinations of scheduled procedure(s), surgeon, and type of anesthetic, the smallest observed coefficient of variation was 8.3%.

The principle is so important for decision making on the day of surgery that it is a prominent part of the OR management class taught by the University of Iowa.^{14} The slides for this content are online.^{a} The statistics review^{a} done by participants before the class is based heavily on being able to understand the log-normal distribution and Figure 2 .^{15} Unfortunately, we are unaware of an intuitive explanation for the mathematical relationship between the log-normal relationship and the flat relationship of the time remaining curve.^{12} Regardless, the formal explanation requires statistical sophistication beyond the background of many OR managers.^{15}

We have speculated that what is happening mechanistically is that the near-constant expected time remaining is effectively the time from the start of closing to OR exit. To try to explain the principles qualitatively, we have encouraged learners at the University of Iowa course to imagine an OR for which a microscope is used until closure starts. The event of wheeling the microscope away from the surgical field is a marker for the start of the time remaining. This explanation would be useful, because application of a single temporal fiducial for decision making results in managerial decisions with virtually no extra overutilized time versus perfect retrospective knowledge of case duration.^{2} ^{,} ^{7} However, the explanation might be wrong. As Figure 2 shows for the typical^{11} coefficient of variation, the estimated time remaining is too long to represent only the time from (1) application of the wound dressing (i.e., end of surgery) until (2) the patient exits the OR.^{16} Timestamps for these 2 epochs usually are recorded and available in most perioperative systems. The times of the start of surgical closure were not typically recorded at facilities previously studied, preventing testing of our hypothesis for the mechanistic explanation of the flat expected time remaining curve.

In the current paper, we use data from a facility with hundreds of thousands of cases and the start of closure is recorded. We test the hypothesis that, among cases taking much longer than scheduled, the mean (expected) time remaining calculated before the case starts is an unbiased estimate for the mean time from the start of closure to OR exit. If true, then this would facilitate teaching OR staff the scientific principles. In addition, facilities that have personnel in ORs who record the time of closing can use the knowledge on the day of surgery. Until closure has started, the mean time remaining for the case should not be less than the mean time from closure to OR exit (see Discussion).

METHODS
The Vanderbilt University IRB approved the study without requirement for written patient consent. Deidentified data related to procedural milestones were retrieved from the perioperative information system database at Vanderbilt University Medical Center: scheduled Current Procedural Terminology (CPT^{®} ) code, surgeon, type of anesthetic (general, regional, or monitored anesthesia care), and times of OR entrance, start of closing, and OR exit. There were 311,940 cases performed in an OR from the start of the dataset, January 2006, through November 2012. We excluded the 2.0% of cases lacking a recorded time for the start of closing.

Consider a 2-parameter log-normal distribution with relatively large (N ≥ 30) sample size.^{7–11} Our analysis applies to cases taking substantially longer than scheduled, which for the study we considered the 0.9 quantile of duration calculated before the case started.^{10} ^{,} ^{12} ^{,} ^{17} From Zhou and Dexter 1998 (Appendix 2)^{17} and reviewed in 2004 (Appendix 6),^{2} the 0.9 quantile equals:

where, for large N ,

,

, and

is the inverse of the cumulative normal distribution (i.e., 1.28).^{13} From Dexter et al. 1999 (Appendix 1)^{7} and reviewed in 2004 (Appendix 3),^{2} for common procedures with large sample sizes (e.g., N = 30), the mean (i.e., statistical “expected value”) of the time remaining equals:

This is the formula used to create Figures 2 to 5 .

For the Bayesian time remaining in cases, there is not just 1 parameter (s ) influencing the coefficient of variation, and there is no analytical expression for the expected value as in Equation (2).^{18} The reason for this is that when logarithms of OR times follow a normal distribution for a common (N ≥ 30) combination of scheduled procedure(s), surgeon, and type of anesthetic, they follow a Student t distribution for uncommon combinations (e.g., N < 10). Thus, the OR times for uncommon combinations follow the exponential of a Student t distribution. Importantly, too, there are noninteger degrees of freedom.^{10} ^{,} ^{12} Thus, although the Bayesian analysis is implemented in practice,^{10} ^{,} ^{12} for this paper we limit consideration to common procedures (i.e., where there are sample sizes N ≥ 30). This is not a limitation scientifically because to assess whether the expected value (mean) corresponds to the time from the end of surgery to OR exit, analysis would need to be limited to common procedures, anyway.

The data were sorted by scheduled procedure(s), then by surgeon, then by category of anesthetic (general, regional, or other), then by date, and then by start time. The category of anesthetic and inclusion of surgeon matches that shown to be predictive of case duration by Strum et al.^{19} This produced a time series for each combination of procedure(s) and surgeon. For each combination with more than 30 cases, the first 30 cases were used to calculate the 0.9 quantile, based on the 2-parameter log-normal distribution (Equation 1). That 0.9 quantile was compared with the duration of the 31st case. This was a check that, as reported at multiple previous facilities,^{10} ^{,} ^{17} ^{,} ^{20} the probability was 10% that the duration of the 31st case exceeds the 0.9 quantile. This condition was in no way assured because Equation (1) assumes a 2-parameter log-normal distribution (see Appendix of Ref. ^{11} ).^{21} ^{,} ^{22} The mean (expected value) of the time remaining in a case was calculated, conditional on the duration exceeding the 0.9 quantile, using Equation (2). If at the time of the 0.9 quantile closing had not yet started for the 31st case, the pairwise difference was calculated between (a) the observed time from closure to OR exit and (b) the calculated (Equation 2) mean (expected) time remaining. The process was repeated comparing the 2nd to 31st case with the 32nd case, and so forth, until there were no more cases for the combination. This moving window process was used because it represents how forecasts are used in practical implementation.^{10} ^{,} ^{12} The process was repeated using each of the other combinations of scheduled procedure(s), surgeon, and anesthetic. There were N = 3962 pairwise differences of 1549 combinations. The overall mean ± SEM was calculated among these 3962 differences.

We performed 3 sensitivity analyses. First, the preceding steps were repeated based on the scheduled procedure(s) and surgeon (N = 4274 pairwise differences of 1613 combinations). Second, the steps were repeated only with the scheduled procedure(s) (N = 4320 pairwise differences of 1624 procedures). Third, we created a positive control. Figure 3 shows that when a large (e.g., >50%) coefficient of variation is used, the log-normal distribution does not have the properties of Figure 2 . The steps were repeated using broad categories of procedure. For all cases with a single scheduled procedure, the Current Classifications Software for Services and Procedures category (CCS)^{b} was determined based on the scheduled CPT. The mean was taken of the durations for each CCS. For all cases with more than 1 scheduled procedure, the CCS used was that associated with the longest individual CPT mean. This approach follows that described by Strum et al.^{8} for CPTs. Using the CCS with the longest individual median duration resulted in the identical choice of CCS for all cases of the scheduled procedure(s). There were N = 17,068 pairwise differences created of 166 categories. Since the categories included many different CPT procedures, the large mixture of durations pooled resulted in more categories having coefficients of variation exceeding 50% than for the preceding 3 analyses (47.6% vs 11.6%, 12.2%, and 12.1%, respectively).

RESULTS
Classifying cases by scheduled procedure(s), surgeon, and category of anesthesia, the mean pairwise difference between the observed time from closure to OR exit and the mean (expected) time remaining in the case was 0.2 ± 0.4 minutes. Classifying cases by scheduled procedure(s) and surgeon, the mean pairwise difference was 0.7 ± 0.4 minutes. Classifying by scheduled procedure(s), the mean pairwise difference was 0.6 ± 0.4 minutes. In contrast, classifying by broad category alone (i.e., CCS), the mean pairwise difference was large, 30.2 ± 0.4 minutes (see preceding paragraph). Thus, our hypothesis was supported.

For each of the 3 methods of classification, the 0.9 quantile was exceeded by 10.2% ± 0.01% of cases, each with at least 101,337 cases (i.e., bias was +0.2% ± 0.01%). This (expected) predictive ability of the 2-parameter log-normal has been reported previously in multiple studies.^{10} ^{,} ^{17} ^{,} ^{20}

DISCUSSION
Our results show that if a case is taking longer than the expected (scheduled) duration, and closure has not yet started, electronic whiteboard displays should not show that the time remaining in the case is less than the mean time from closure to OR exit (i.e., not like in Figure 5 ). Similarly, if closure has started, the expected time remaining that is displayed should not be longer than the mean time for closure. For decision making on the day of surgery, if whiteboards are used, presenting such information with face validity is of organizational value, since otherwise trust may be eroded in the reliability of the predictions. For example, staff may lose confidence in the predictive accuracy of the system if the pancreaticoduodenectomy in OR 1 ended after 2 hours due to nonresectable disease and the patient has been extubated, but the whiteboard continues to display the Bayesian expected 3 hours remaining to complete the case. From a practical perspective, a lookup table is created periodically using historical data for each combination of scheduled procedure(s), surgeon, and type of anesthetic.^{12} The data then are used to update the time remaining for cases that have exceeded their scheduled case durations. A limitation is that although our results are reasonable overall for all specialties of a multidisciplinary surgical suite, there may be exceptions that we simply do not have the sample size to detect and study.

Our results match previous reports that, before a case starts, statistical methods can reliably be used to assist in conflict checking for resources, filling holes in the OR schedule, preventing holes in the schedule, etc.^{2} ^{,} ^{10} ^{,} ^{23} ^{,} ^{24} These OR management decisions are important economically and involve the shortest or longest time that a case might take, not the mean, and thus proportional variability needs to be included, requiring use a Bayesian approach.^{2} ^{,} ^{10} For example, use of the longest time is important when 2 surgeons need an institution’s only surgical robot and the start time of the later starting surgeon needs to be determined.^{2} Use of the shortest time is important when one surgeon is scheduled to follow another surgeon in the same OR. If the potential conflict is not addressed in the first example, the second surgeon might not be able to start his case if the other case using the robot runs longer than the scheduled time. In the second example, if the surgeon plans his day to arrive in the suite based on the scheduled start time of his case, there will be underutilized OR time if the first case finishes early.

Finally, our results suggest that when teaching application of the log-normal distribution for case duration prediction (Figs. 2 and 4 ), the time for closure to OR exit can be used as a reasonable mechanistic explanation. Table 1 gives an example of application of the principle, adapted from Figure 1 (e.g., with matching cues), and with the time estimates of Figure 2 .^{25} ^{,} ^{26}

Table 1: Example of Operating Room Teaching Lesson Created by Matching Cues from Vanderbilt

Figure 1 and Applying Principles from

Figure 2 RECUSE NOTE
Franklin Dexter is Statistical Editor and the Section Editor of Economics, Education, and Policy for the Journal. The manuscript was handled by Dr. Steven L. Shafer, Editor-in-Chief, and Dr. Dexter was not involved in any way with the editorial process or decision.

DISCLOSURES
Name: Vikram Tiwari, PhD.

Contribution: This author helped conduct the study.

Attestation: This author has approved the final manuscript.

Conflicts of Interest: The author has no conflicts of interest to declare.

Name: Franklin Dexter, MD, PhD.

Contribution: This author helped design the study, conduct the study, analyze the data, and write the manuscript.

Attestation: This author has approved the final manuscript.

Conflicts of Interest: The University of Iowa sets up statistical analyses for hospitals using their operating room information systems. Income from the Division’s consulting work is used to fund Division research. FD has tenure and receives no funds personally, including honoraria, other than his salary and allowable expense reimbursements from the University of Iowa. He and his family have no financial holdings in any company related to his work, other than indirectly through mutual funds for retirement.

Name: Brian S. Rothman, MD.

Contribution: This author helped conduct the study.

Attestation: This author has approved the final manuscript.

Conflicts of Interest: The author has no conflicts of interest to declare.

Name: Jesse M. Ehrenfeld, MD, MPH.

Contribution: This author helped conduct the study.

Attestation: This author has approved the final manuscript.

Conflicts of Interest: The author has no conflicts of interest to declare.

Name: Richard H. Epstein, MD, CPHIMS.

Contribution: This author helped design the study, conduct the study, analyze the data, and write the manuscript. This author is the archival author.

Attestation: This author has approved the final manuscript.

Conflicts of Interest: The author has no conflicts of interest to declare.

FOOTNOTES

a See end of the “Decision-making on the day of surgery” lecture. Available at: www.FranklinDexter.net/education.htm . The “Statistics for Anesthesia” review also is available at that web page. Accessed February 17, 2013. Cited Here

b Clinical Classifications Software for Services and Procedures. Available at: http://www.hcup-us.ahrq.gov/toolssoftware/ccs_svcsproc/ccssvcproc.jsp . Accessed February 18, 2013. Cited Here

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