Journal of Theoretical Medicine, Vol. 6, No. 4, December 2005, 235–239 Scheduling medical procedures: how one single delay AMNON SONNENBERG†‡* and BRADFORD R. CRAIN{ †Portland VA Medical Center, Portland, OR, USA ‡Department of Medicine, Oregon Health & Science University, Portland, OR, USA {Department of Mathematics, Portland State University, Portland, OR, USA Background: Delay is a common feature of medical disease management. Delays occur becauseschedules are filled, patients forget their appointment, equipment is unavailable, or because medicaland non-medical complications interfere with the planned procedure. The aim of the present analysis isto model how one single delay can lead to multiple subsequent delays.
Methods: The consecutive stream of delays is analyzed in terms of a stochastic process comprising of arandom sum of random time periods. Any untoward event causes a procedural delay, which provides atime window of opportunity for yet another delaying event to occur.
Results: The stochastic model explains why even a single initial delay can easily lead to a multitude ofsubsequent delays. The expected overall delay is always longer than the initial delay caused by thedeferment of the initial procedure. The analysis demonstrates how in individual patients an initiallyshort delay may subsequently expand into days or weeks.
Conclusion: Because a single delay can easily burgeon into a lengthy series of multiple delays,the primary goal should be to avoid the precipitating delay at the onset.
Keywords: Health service research; Medical decision analysis; Random sums; Stochastic modeling extremely difficult to reach its elusive goals because ofthe many ensuing delays. It appears as if an initial delay in Delays are common and unfortunate features of all performing the procedure precipitates the occurrence of medical care. Most delays among hospitalized patients various other medical and non-medical events that push stem from scheduling of diagnostic test procedures [1,2].
the planned procedure further and further away.
The types of events that delay patient management relate As a typical scenario, consider a 79-year-old man who to a large variety of medical, organizational, adminis- was admitted to the hospital for post-prandial abdominal trative and technical obstacles that often render a timely pain. His serum level of total bilirubin measured 4.8 mg/dl and expeditious management difficult. Delays occur, for instance, because schedules are already filled and An abdominal ultrasound showed multiple stones in the overbooked, patients forget their appointment and need gallbladder and a dilated common bile duct of 12 mm to be rebooked, instruments become broken and unavail- diameter. The patient had been treated with low-dose able or because comorbid conditions and new compli- aspirin for cardiovascular prevention and coumadin for cations interfere with the performance of a planned atrial fibrillation. When first seen by the Gastroenterology procedure. A delay in establishing a diagnosis or initiating Consult Service on Tuesday, it was recommended to take treatment can result in severe medical consequences the patient off his anticoagulative medication and schedule [3 – 5]. Frequently, situations arise when one single delay an endoscopic retrograde cholangiography (ERC) with leads to multiple consecutive delays. It then becomes a possible stone extraction from his common bile duct on rather frustrating experience for the patient, as well as the Thursday. When tested on Tuesday morning, the patient’s physician, to appreciate the urgency of a particular INR value of 2.2 was considered still too high and the medical procedure, yet be unable to get such procedure procedure was postponed until Monday. On Friday, done. Although, the procedure may be absolutely however, the ERC endoscope broke down during a prior indicated for diagnostic confirmation or therapeutic procedure on another patient and with the second back-up resolution, nevertheless, the clinician may find it instrument still in repair, no other instrument was *Corresponding author. Email: [email protected] ISSN 1027-3662 print/ISSN 1607-8578 online q 2005 Taylor & Francis available to do the scheduled ERC on Monday. Because occurred, the probability for none, one, two, etc.
the two physicians performing ERC left to attend a two- day conference, the next available time slot for ERC was P(N ¼ 1) ¼ qp, P(N ¼ 2) ¼ qp 2, etc. with the corre- only on the following Friday. In the meantime, the patients sponding geometric probability mass function (pmf): elevated serum bilirubin and alkaline phosphatase trended down and it was hypothesized that the initial cholestasis had stemmed from passed gallstone. Rather than wait for the ERC, the patient underwent an uneventful laparo- The expected overall number of secondary events E[N ] scopic cholecystectomy. An intra-operative cholangio- gram again revealed a dilated common bile duct with asuspected gallstone lodged above the papilla. The ERC was eventually done on the following Friday, that is, seventeen days after the initial hospital admission and The scenario from above represents only one example of many similar clinical instances, where the cumulativeoccurrence of several unpredicted events can result in an extensive overall delay. Why does this happen? The aim of the present article is to describe this process and analyze the mechanisms underlying its occurrence. The analysis is not concerned with waiting queues in general or patient flow through medical systems, but focused solely on how one delaying event prepares the ground for additionalsubsequent delays. A stochastic model is developed to provide estimates for expected delays in a large set of Since in general, var[N ] ¼ E[N 2] 2 (E[N ])2, the two var½NŠ ¼ E½NðN 2 1Þ þ NŠ 2 ðE½NŠÞ2 In the present context, the term “event” refers to anincident which causes delay. The term “delay” refers to the excess time needed to accomplish a medical task. Rarely does one delaying event come alone, as frequently the first The expected overall delay E[D ] corresponds to the event lends to the occurrence of a second, third, etc.
expected number of delays E[N ] multiplied by the subsequent delaying event. The initial delay caused by the expected length of the individual delays E[d first event opens up a time window, during which another delay, running its course uninterrupted by any subsequent delaying event can occur. Let d be the average length of event, is expected to be twice as long as the preceding time of a delay. During the time window of length d, the probability for a second delay arises. The second delay provides a new time window for yet another delaying event. A delaying event can occur anytime within the window of opportunity provided by the previous delay.
Because on the average, the next event will occur in the The variance of the overall delay var[D ] is calculated middle of the previous delay, each new event adds d/2 to according to the general formula for the variance of a sum the overall delay. The overall delay D equals var½DŠ ¼ var ½d1 þ d2 þ d3 þ . . . þ dNŠ ¼ var ½NdiŠ where the individual delays di are assumed to be independent and identically and uniformly distributed over an interval [0,d ] with an expected value E[di] ¼ d/2 Substituting with the terms from above, equation (7) and a variance var[di] ¼ d 2/12 [6]. Once the first initiating delay has occurred, the probability for the occurrence ofany secondary delaying event is p and for its non- occurrence q ¼ 1 2 p. The aim of the following analysis is to develop an estimate for the expected length of theoverall delay and its variance. The number of secondary The stochastic model of consecutive delays was delaying events N is random. After the primary delay has (with errors of less than 5%) was obtained between the predicted and the simulated values of the overall delay Dand its standard deviation SD.
Using equation (6) given in the methods, an event probability of p ¼ 50% and a delay of d ¼ 3 days result in an overall expected delay of D ¼ 4.5 days. Similarly,a higher event probability of p ¼ 90% results in an overall expected delay of D ¼ 16.5 days. As the event probability p increases, the overall delay D becomes longer andlonger. With a probability p < 1 close to certainty that each new delay will provide sufficient time for yet another Expected overall delay as function of probability and lengths event to occur and cause another delay, the overall length of D stretches out to infinity. Figure 1 shows the overalldelay as function of delay probability and length.
of figure 2, the gray area represents the confidence interval Not every patient, however, will necessarily experience between the mean overall delay plus two standard the overall delay D. First, the chain of events delaying the deviations. As the two upper graphs demonstrate, even a procedure may stop at any given point in time, when due short initial delay of 1 – 3 days can easily expand into to their random nature, no new events occur. Second, since a protracted overall delay, especially, if the tendency for the lengths of individual delays fluctuate around an repeat delays creeps above 50%. The lower two graphs average value d, some events may cause much shorter or demonstrate that, in dealing with long individual delays, much longer delays than d. The standard deviation of the the expected overall delay is markedly increased even with overall delay is given by equation (8) in the methods, its low underlying risks of delay. As a general rule, the overall result being easily calculated on a spreadsheet or hand- delay must always be expected to turn out longer than the held calculator. In the first example from above, p ¼ 50%, d ¼ 3 days, D ¼ 4.5 days and the standard deviation of theexpected delay is calculated as SD ¼ 2.3 days. The 95%confidence interval of the expected delay ranges betweenD ^ 1.96 SD, that is, from 0 to 9.0 days. In the second example, p ¼ 90%, d ¼ 3 days and D ¼ 16.5 days. Thestandard deviation of the expected delay is SD ¼ 14.5 Sometimes, it can become a rather frustrating experience days. Hence, the 95% confidence interval for the expected for a physician managing patients to pursue an obvious delay of 16.5 days ranges between 16.5 ^ 1.96 £ 14.5 diagnostic or therapeutic goal, but then encounter a seemingly endless number of obstacles that push Figure 2 serves as a general guide to estimate the a planned intervention further away into the future and expected overall delay for a large set of possible scenarios.
keep the physician from reaching his/her goal. There In the four graphs, the probability values of delays are seems to be a continued interference by a slew of minor varied between 0 and 100%, while the average lengths of and often trivial events that sidetrack the entire work-up, the individual delays are varied between 1, 3, 5 and 10 complicate the medical pursuit and lead down a lengthy days. Shorter delays may occur, for instance, when key and convoluted path, before the medical goal that has been medical personnel are not available, instruments break so clearly discernible from the onset is finally achieved.
down or abnormal laboratory values are encountered.
Such delays are a common feature among hospitalized Mid-sized delays are frequently associated with patients patients and are similarly encountered in the health care harboring comorbid conditions or with organizational systems from different countries [9 – 12].
obstacles that limit the number of time slots available for In the present analysis, a model is developed to describe procedures. Long delays are most likely encountered in the stochastic process that underlies the recurrence of instances of complex medical procedures or surgical medical and non-medical events delaying medical interventions that are difficult to schedule and that require procedures. In essence, any untoward event is modeled sophisticated equipment or the interplay of multiple to cause a procedural delay, which provides a time window medical subspecialties. As a general rule, all types of of opportunity for yet another delaying event to occur.
delays tend to be longer in the elderly as opposed to The consecutive stream of delays is thus analyzed in terms younger patients. Although, the initial delay stems from of a random sum of random time periods. In this model, deferment of a medical procedure, subsequent delays independent events are characterized by a similar rate may be caused occasionally by events outside medicine of occurrence and a similar average length of delay. As in and unrelated to the underlying disease. In each graph any mathematical model, these assumptions simplify Expected length of the overall delay in relation to the probability of recurrent individual delays. In the four graphs, the mean lengths of the individual delays are varied between 1, 3, 5 and 10 days. The gray area represents the confidence interval between the mean overall delay plus twostandard deviations.
the reality of clinical medicine and they do not always healthcare system, delays in scheduling and appointments, represent the entire complexity encountered in the time spent in awaiting areas to be seen by a physician, management of actual patients. To derive the estimates nurse or technician and waiting times spent before surgery, from above, several simplifying assumptions have to be X-ray or other medical procedures. A large body of made. The delays are assumed to be independent of each stochastic models and administrative instruments has been other and equally uniformly distributed. However, actual developed to analyze and manage patient flow within delays in clinical practice may be distributed according to medical systems. Queuing theory has been utilized to a lognormal, exponential or some other statistical estimate lengths of waiting lines and waiting times distribution. If two delays occur consecutively, the in patient scheduling and hospital operations [13 – 16].
preceding first delay is assumed to become superseded The instruments of theory of constraints have been used to by the subsequent second delay. In reality, some delays identify components of delays and how to eliminate them could be additive. Consecutive delaying events may be in order to improve system performance [17 – 19].
linked causally and not only temporally. The lengths of In contrast with such administrative goals to streamline consecutive delays may depend on each other or prolong an overburdened medical system, the present model was over time as they accumulate in the individual patient.
aimed to illustrate how the management of an individual In spite of its simplifications, however, the model provides patient becomes derailed by a series of short delays that useful insights into the occurrence of delays associated can accumulate into one major delay. The analysis has with medical procedures. The stochastic model explains been focused on the perspective of a physician struggling how even a short initial delay can easily result in a lengthy to expedite the patient’s work-up rather than the overall delay. It also demonstrates why in some patients perspective of an administrator trying to improve the the overall delay may extend over days or even weeks.
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