Priority Queuig Models for Hospital Itesive Care Uits ad Impacts to Severe Case Patiets Matthew S. Hage, Ceter for Operatios Research i Medicie ad HealthCare Jeffrey K. Joplig, Emory Uiversity Timothy Buchma, Emory Uiversity Eva K. Lee, Ceter for Operatios Research i Medicie ad HealthCare Joural Title: AMIA... Aual Symposium proceedigs / AMIA Symposium. AMIA Symposium Volume: Volume 2013, Number 2013 Publisher: AMIA 2013-11-16, Pages 841-850 Type of Work: Article Fial Publisher PDF Permaet URL: https://pid.emory.edu/ark:/25593/s6f43 Fial published versio: https://kowledge.amia.org/ Copyright iformatio: 2013 AMIA - All rights reserved. This is a Ope Access article: verbatim copyig ad redistributio of this article are permitted i all media for ay purpose Accessed August 25, 2018 2:50 AM EDT
Priority Queuig Models for Hospital Itesive Care Uits ad Impacts to Severe Case Patiets Matthew S. Hage, MSE 1,2,4, Jeffrey K Joplig, MD, MS 5, Timothy G Buchma, MD, PhD 5, Eva K. Lee, PhD *,1,,2,3,4 1 Ceter for Operatios Research i Medicie ad HealthCare, 2 NSF I/UCRC Ceter for Health Orgaizatio Trasformatio, 3 School of Idustrial ad Systems Egieerig, 4 College of Computig, Georgia Istitute of Techology, Atlata, GA; 5 Emory Ceter for Critical Care, Emory Uiversity School of Medicie, Atlata, GA. Abstract This paper examies several differet queuig models for itesive care uits (ICU) ad the effects o wait times, utilizatio, retur rates, mortalities, ad umber of patiets served. Five separate itesive care uits at a urba hospital are aalyzed ad distributios are fitted for arrivals ad service duratios. A system-based simulatio model is built to capture all possible cases of patiet flow after ICU admissio. These iclude mortalities ad returs before ad after hospital exits. Patiets are grouped ito 9 di fferet classes that are categorized by severity ad legth of stay (LOS). Each queuig model varies by the policies that are permitted ad by the order the patiets are admitted. The first set of models does ot prioritize patiets, but examies the advatages of smoothig the operatig schedule for elective surgeries. The secod set aalyzes the differeces betwee prioritizig admissios by expected LOS or patiet severity. The last set permits early ICU discharges ad coservative ad ag gressive bumpig policies are cotrasted. It was foud that prioritizig patiets by severity cosiderably reduced delays for critical cases, but also icreased the average waitig time for all patiets. Aggressive bumpig sigificatly raised the retur ad mortality rates, but more coservative methods balace quality ad efficiecy with lowered wait times without serious cosequeces. * Correspodig author: Eva Lee, eva.lee@gatech.edu Itroductio The curret climate of critical care has a heavy challege to meet growig patiet demads while hospital capacity cotiues to shrik at a alarmig rate. Accordig to America Hospital Associatio, the umber of hospital beds has reduced by almost 25 percet i a period of 20 years. 1 Due to Certificate of Need (CON) regulatios, a average occupacy level of 85 pe rcet was required before approval to icrease capacity. 2 Sice, may hospitals had average occupacy below these rates, there was a impressio i the health care commuity that there was excess capacity. For oprofit hospitals, average rates had reached as low as 66 percet. 15 Cosequetly, available beds have cotiued to decrease across states. I April 2002, a Lewi Group survey reported 62 percet of U.S. hospitals reached or exceeded maximum operatig levels. The percetages raised to 79 percet for urba hospitals ad 82 pe rcet for level I trauma ceters. 11 The Ceter for Disease Cotrol reported the umber of aual emergecy departmet (ED) visits climbed by almost a q uarter for the decade edig i 2002. The umber of EDs reduced by 15% for the same period. 29 Settig hospital capacity by focusig o occupacy levels has led to serious circumstaces. There have bee access blocks ad substatial icreases i waitig times. 30 The relatioship betwee waitig time ad average occupacy is ot liear. At a poit, the average delay ca start to rise expoetially relative to eve small icreases i utilizatio. 16 Wait time depeds o the time betwee arrivals ad begi of service. These measures have sigificat variability, ad delays ca be cosiderably differet for idetical utilizatio levels. It is ot sufficiet to oly emphasize average occupacy levels whe evaluatig the process flow of a health care ceter. Icreasig average wait times for medical care has led to complicatios that are more sigificat tha ecoomic icetives. Poor patiet flow has bee foud to be associated with elevated mortality rates, loger legth-of-stay, ad heighteed readmissio. 4,35 Sprivulis et al. liked ED overcrowdig to a 30% relative icrease i mortality. 41 Chalfi et al. idetified delays to itesive care were correlated with loger legths of stay ad higher mortality. 3 Durig periods of stress, a decisio to admit a patiet may ot be etirely cliically drive ad urses are proe to medical errors. 24 Early discharges are more likely at high occupacy levels. The average legth of stay ca be reduced up to 16% for patiets discharged from a busy itesive care uit (ICU). 20 However, the likelihood of returig icreases substatially. 10,20,40 KC et al. foud overall bouce-back probability was 14%, but 841
rose to 37.4% for early discharged patiets. 20 Higher severity patiets are associated with loger revisit stays raisig their et total legth of stay. These factors effectively reduce hospital s peak capacity, because the readmissio loads add uexpected flow related stresses. 25 Readmitted patiets have also bee foud to have higher mortality rates i additio to loger legths of stays. Sow et al. idetified mortality rates for returig patiets were 26%, three times the geeral populatio for surgical itesive care uits. 40 Readmissios from premature discharge ca icrease costs ad lead to overall worseig of medical coditios for patiets. 11 It is essetial to improve the process flow of health care ceters with motivatios that are ot purely ecoomic. The demad for itesive care is high. Gree et al. determied 90% of ICUs i New York have isufficiet capacity to provide proper medical care. 15 While ecoomics ted to favor high occupacy, 13 the quality of care does ot. This paper evaluates differet priority methods (some from literature, ad some we derive) to miimize waitig times for admissio to itesive care uits. A emphasis is placed o the severity of medical coditios rather tha exclusively focusig o market factors. The goal is to maximize the umber of patiets served while maitaiig good quality of care. Related Work I a perfect system, all patiets would arrive at the same rate ad all patiets would have the same coditio ad require idetical service time. This system would be 100% efficiet as are may automated maufacturig plats. 25 This is ot the case i the health care commuity. Patiets arrive uexpectedly with a immese diversity of coditios. Therefore, it is ecessary to optimally fit the distributios for patiet arrival ad service time. I most studies, the iter-arrival times are regarded as a egative expoetial distributio. 36 The legth of stay (LOS) ca have differet distributios for differet patiet types. 21 The fit distributios ca vary from expoetial, egative expoetial, log-ormal, or Weibull. 7,30,38,42 Kokagul et al. applied a Kolmogrov-Smirov test o five years of admissios to a teachig hospital ad foud arrivals distributed as a Poisso process ad LOS distributed as logormal. 21 Siddhartha et al. classified patiets ito emergecy ad o-emergecy care. 39 After collectig data from a emergecy departmet i Florida, patiets were grouped as emergecy care for major trauma, critical care, mior trauma, ad o-critical care cases. Noemergecy care was classified oly for primary care patiets. 53.3 percet of patiets were foud to require emergecy care ad 46.7 percet were oemergecy. The average arrival rate, service rate ad waitig time were calculated for both types. The study assumes arrivals follow a Poisso probability distributio ad service rate follows a expoetial distributio. Usig a proper priority queue disciplie, 19 it foud the average waitig time to reduce by 10 percet for all patiets. The queue gave highest priority to emergecy care patiets, because they had the larger average service time. Cha et al. utilized a more sophisticated priority queue with 9 categories of patiets. 5 Each category was classified by low, medium, or high LOS ad by low, medium, or high severity. The severity of each patiet was assessed usig criteria from Escobar et al. where admissio diagoses ad laboratory results were utilized. 12 All groups of patiets were tested with three differet priority models. The model assumed a patiet must be discharged for ew arrivals if itesive care uits are at full capacity. This is due to the iheret urgecy of itesive care. Each priority model eforced the discharge order for patiets i itesive care. The three models were based o lowest omial legth-of-stay, smallest probability of readmissio, ad lowest readmissio load. Readmissio load is defied as retur probability multiplied by average LOS for successive visits. The study results reported the readmissio load model outperformed all other priority schemes by up to 10%. Dobso et al. attempted to accurately estimate the expected umber of patiets trasferred to accommodate more critical arrivals. 9 The study did ot use a complex priority scheme compared to Cha et al. Istead, patiets were simply discharged by lowest remaiig legth of stay. A Markov model was utilized to study the effects of ICU workload o patiet bumpig. The difficulty of assigig priority to ICU admissios is to correctly idetify the severity of icomig patiets. Escobar et al. assessed the severity of each patiet by assigig the probability of mortality based o sex, age, primary coditio ad chroic ailmets. 12 16,090 ICD admissio diagoses were grouped ito 44 broad categories. Graham et al. used a simpler approach by classifyig a diagosis ito high, medium, or low risk. 14 Addig to the complicatios of accurately idetifyig patiet severity, cliicias typically write diagosis records i free-text format. There have bee 842
successful attempts to use machie learig ad atural laguage techiques to correctly associate otes with hierarchical codes, such as SNOMED- CT ad ICD-9. 6,8,32,33,34,37 However, these methods have bee foud to have cosiderably lower performace i data poor cases. 37 More successful results were attaied whe a large volume of cliical reports, laboratory results ad follow-up reports were available. Regardig strategies for aalyzig ICU workflow, Cha et al. oly prioritized patiets by how they were bumped from the ICU rather tha admitted. Discharges were eforced by attemptig to miimize readmissio load accordig to several factors, icludig retur probability ad LOS. Dobso et al. also prioritized patiet trasfers from the ICU. 9 They were ordered accordig to their remaiig legth of stay. Both of these studies used sophisticated priority schemes, but were ot etirely realistic. Patiets were automatically admitted whe requestig ICU etry by bumpig lower priority patiets. However, i healthcare settigs it is ot ucommo for average wait times for a ICU to exceed 4 hours, 28 ad bumpig patiets ca cause sigificat medical complicatios. 10,20,40 Methods ad Computatioal Desig Data preparatio 32,531 medical records were retrieved from a large urba hospital over a o e year period from March 2010 to April 2011. Each record icluded the patiet s id, registratio umber, diagosis, ad etrace ad exit times of each reserved room durig the etire hospital stay. Five separate itesive care uits were aalyzed for this study: Cardiovascular (CV) Surgery, Neurosurgery, Medical, Neurosciece, ad Surgical. Sice the distributio of LOS may vary amog differet patiet types, 21 the Iput Aalyzer i Rockwell Area 13.5 was used to fit the LOS distributio for each ICU. Of 5,465 hospital ICU visits, 813 cotaied a missig etry. 14.8 percet of records icluded the time a patiet exits a ICU room without the time of etry. These offedig records were temporarily removed to calculate the LOS distributios for each ICU uit (Table 1). The fitted distributios were the used to sample etrace times for the records with missig etries. ICU LOS distributio CV Surgery 1 + LOGN(84.8, 115) Neurosurgery 5 + LOGN(55.8, 75.7) Medical 2 + LOGN(54.4, 53.9) Neurosciece 4 + LOGN(56.6, 85.4) Surgical 7 + LOGN(67.6, 101) Table 1. Legth of stay distributio 843 ICU Emergecy arrival distributio CV Surgery GAMM(9.37, 0.948) Neurosurgery EXPO(13.8) Medical GAMM(8.79, 0.937) Neurosciece WEIB(35.9, 1.06) Surgical WEIB(8.86, 0.984) Table 2. Emergecy arrival distributio Arrival rates were calculated after all hospital ICU visits cotaied complete records for etry ad exit. Full lists of etrace times were geerated, ad distributios were fitted from iterarrival times for each ICU. Arrivals were separated by emergecy ad scheduled surgery admissios (Tables 2, 3). Other statistics were calculated to help idetify the process flow of patiets through the system. These icluded retur rates after a p atiet leaves a ICU, after a patiet leaves the hospital, ad after a patiet is forcibly bumped from a ICU. Mortality rates were determied for patiets eterig a ICU for iitial ad retur visits (Table 4). ICU Scheduled arrival Distributio CV Surgery 493 * BETA(0.554, 2.51) Neurosurger WEIB(23.2, 0.695) Medical 2 + LOGN(2.98e+003, 3.97e+004) Neurosciec WEIB(68.9, 0.883) Surgical GAMM(43.7, 0.715) Table 3. Scheduled arrival distributio ICU P(R r) P(R e) P(M) P(R t) P(M R) CV Surgery 0.068 0.039 0.077 0.118 0.226 S Neurosurgery 0.034 0.069 0.172 0.111 0.288 Medical 0.073 0.113 0.160 0.056 0.213 Neurosciece 0.039 0.069 0.195 0.250 0.407 Surgical 0.061 0.059 0.075 0.257 0.159 Table 4. Probabilities for ICU returs ad mortality. Retur probability from room P(R r), retur probability from hospital exit P(R e), mortality probability P(M), retur probability after early discharge P(R t), ad mortality probability after retur P(M R) Natural laguage processig of cliical diagosis records The medical records obtaied were ot comprehesive eough to coduct a full cotextual aalysis. I particular, the diagoses from patiet records received did ot cotai stadardized codes, such as ICD-9. They were free text etries ragig oly up to 54 characters at maximum. This limits text aalysis for each record to a few words at most, but it is useful to test the applicability of atural laguage processig whe the cotet is very miimal.
Due to the difficulty of uiquely matchig a patiet s diagosis with miimal cotet ad o-restricted etries, the goal is to istead classify the severity of a patiet s coditio based o these free text etries. Severity is the calculated by idetifyig key words show to have high prevalece i cases of mortality. Of 2,950 diagoses, 486 r esulted i mortality. The cliical terms used i mortality cases were treated with higher severity. A list of words was geerated from all diagosis records. Aother list was produced oly from the mortality records. NLTK, a atural laguage processig toolkit for Pytho, was used to tokeize the words i each list. 31 It was importat to oly iclude words i the Eglish dictioary ad remove ay commo stop words. Wordlist is a corpus icluded i NLTK that cotais 234,943 uique Eglish words, ad the Eglish Stopwords corpus cotais 127 u ique words. These corpora facilitate more sigificat words to be idetified i diagosis records, but may medical terms may be improperly excluded. It is possible that commo words used by cliicias are ot icluded i the stadard Eglish dictioary provided by the NLTK library. SNOMED-CT is a s tadardized referece that cotais millios of medical cocepts developed by the America Pathologists ad the Uited Kigdom s Natioal Health Service. 25 The July 2011 release cotaied 988,921 uique medical terms. We use this release to augmet the list of Eglish words provided by the NLTK corpus. SNOMED-CT was tokeized ad stop words were removed usig the NLTK library. SNOMED-CT was foud to cotai 94,581 uique words ad whe combied with the Wordlist corpus, the uio created a j oit corpus of 304,760 uique words. This added 69,817 medical words facilitatig more cotet for aalysis. With oly utilizig the Wordlist corpus, 6,008 words were matched from diagosis records. The joit SNOMED-CT Wordlist corpus matched 6,535 words icreasig the data size by 8.7 percet. I atural laguage processig, oe of the challeges is to ot treat words differetly that have idetical roots or map to the same stem. The words walkig, walker, ad walked all map to the stem walk. Stemmig is a p rocess that reduces iflected or derived words to their appropriate root. I this study, the Lacaster stemmer provided by the NLTK toolkit was used. 683 u ique words were foud from the diagosis records, ad 231 uique words were foud by diagosis mortality records. With applyig the Lacaster stemmer, the uique words were reduced to 635 ad 222 respectively. The frequecies of each uique word were the calculated. A severity score could the be calculated by utilizig the word frequecy distributio for all diagosis records ad the distributio for mortality records. The TF-IDF score is a weight used i iformatio retrieval. It measures the importace of a t erm i a documet, but it is offset by the frequecy the term appears i the etire corpus. The method i this study is ot exactly idetical to iformatio retrieval, ad there are may possible variats of the TF-IDF calculatio. 27 The importace of the term is measured by the frequecy it appears i all mortality records. It is offset by the frequecy it appears i all diagosis records. Therefore, a higher score will be give to a term that occurs ofte i mortality records but ot ofte i all diagosis records. Istead of summig the TF-IDF score for each term, the scores are averaged. This way more beig terms ca reduce the severity of the diagosis. The TF-IDF scores for each diagosis record were calculated as N idft = log dft tf idftd, = tftd, idft Score( q, d) = ave ( tf idf ) t q td, where N is the umber of words i the diagosis records that were matched with the NLTK Wordlist corpus ad the SNOMED-CT corpus, df t is the term frequecy i all diagosis records, d is the set of words i mortality records, t is the set of words i the curret diagosis record, tf t,d is the term frequecy i mortality records, ad q is the set of words i the curret diagosis record that exist i d. Patiets were clustered ito ie differet groups similar to the study by Cha et al. 5 Each group has three possible levels for severity ad three possible levels for LOS. LOS level is divided ito three rages by service hours (h). Groups are allocated by LOS < 25h, 25h < LOS < 57h, ad LOS > 57h. This resulted i a equal amout of records for each LOS level. After the TF-IDF score was calculated for each diagosis record, severity groups were clustered usig the K-Meas algorithm. 26 After K-Meas clusterig, the rages for TF-IDF scores for each severity group were TF-IDF < 0.07, 0.07 < TF-IDF < 0.19, ad TF-IDF > 0.21. The mortality rate for records i each severity group accurately reflected the average TF-IDF score. The lower severity groups both had mortality rates at roughly 15 percet. Seve percet of etries were classified with highest 844
severity ad were foud to have a mortality rate of 46 percet. LOS distributios were the fitted for each severity group (Table 5). C S # TF- P(M s) P(M s) Expressio 1 31,42 IDF 0.026 0.143 / 0.852 1 + LOGN(72,87.4) 2 0831 0.11 0.152 0.897 4 + LOGN(49.7, 60.3) 3 174 0.293 0.466 2.755 6 + LOGN(23.7, 19.4) Table 5. Severity group results from K-Meas clusterig. Severity group C S, average TF-IDF score for severity group, mortality rate for severity group P(M s), ratio betwee group mortality rate ad average After patiets were successfully grouped ito ie separate classes, multiple statistics were calculated for later use by priority models i the simulatio model. These icluded average iitial LOS, retur rate ad average retur LOS (Table 6). C P C LOS C S # LOS Piit P(R P ) P(R P ) LOS Pret 1 1 1 1 19.468 0.075 0.950 41.971 2 1 2 1 19.606 0.074 0.928 28.543 3 1 3 4 19.720 0.065 0.822 44.027 4 2 1 1 39.491 0.076 0.960 54.448 5 2 2 9 38.179 0.041 0.520 50.014 6 2 3 2 34.405 0.091 1.147 44.280 7 3 1 2 159.712 0.124 1.562 134.017 8 3 2 9 149.481 0.041 0.520 182.770 9 3 3 6 110.284 0.167 2.103 124.621 Table 6. Patiet group results after clusterig. Patiet group C P, LOS group C LOS, severity group C S, average iitial legth of stay for patiet group LOS Piit, retur rate for patiet group P(R P), ratio betwee group retur rate ad average retur rate P(RP)/μ R, ad average retur legth of stay for patiet group LOS Pret Simulatio Model A simulatio model was built usig Rockwell Area 13.5 to aid i the developmet ad evaluatio of the process flow of five itesive care uits. A separate submodel was created for each ICU: CV Surgery, Neurosurgery, Medical, Neurosciece ad Surgical. Each submodel had both scheduled ad emergecy arrivals. Scheduled arrivals were direct trasfers after a appoited operatio or surgical procedure, ad emergecy arrivals were uexpected admissios. The iter-arrival distributios were fitted usig the Area Iput Aalyzer for both cases (Tables 2, 3). ad patiet workflow, ad their iter-depedeices o patiet care ad resources. After a patiet departs a itesive care uit, they are trasferred to a itermediate care room before dismissal. The patiet may retur to a ICU after trasfer to a itermediate room, ad they may also retur after exitig the hospital. The distributios for LOS i itermediate rooms after ICU discharges were fitted with Iput Aalyzer. Distributios were also calculated for duratios betwee patiet hospital exits ad subsequet ICU returs (Table 7). Locatio Expressio Itermediate Room LOGN(157, 294) before ICU retur Itermediate Room WEIB(76.4, 0.697) before hospital exit Outside hospital 67 + 8.82e+003 * BETA(0.467, before ICU retur 2.15) Table 7. Itermediate room ad hospital exit distributios Estimated probabilities from hospital records were utilized i the simulatio model. The retur ad mortality rates were separately calculated for each ICU (Table 4). The Retur module i our computerized model captures all possibilities for returs ad exits. It also icludes mortality cases where patiets do ot survive their ICU stay. The simulatio model tests six differet queuig methods ad each is executed i Rockwell Area for a period of 90 days with te replicatios. The results reported for each queuig model are averages over all replicatios. ICU Resource Allocatio The goal of this system is to aggressively test the process flow of the hospital uder heavy coditios. The give umbers of beds were approximated for each ICU accordig to a M/M/s queuig model. The model assumes there are s idetical servers with ulimited waitig room capacity. Service duratio follows a expoetial distributio while arrivals occur at a costat rate accordig to a Poisso process. Give the umber of servers s, average arrival rate λ, ad average service time 1/μ, the mea waitig time i the queue W q ca be calculated uder the M/M/s model: 18 Differet umbers of beds were allocated ad a separate queue was desigated for each ICU. The full computermodel cotais scheduled ad emergecy arrivals for all five ICUs. Further, each ICU is modeled i detail, icludig service, queues, cliical 845
p W q D = L / λ = 1 s 1 = 0 ρ = λ / sµ λ p0 (1 s)! µ = λ p0 ( s) s s s! µ 1 s 1 s s+ 1 ( ρs) ρ s p0 = + ρ < 1 = 0! s!( s ρs) ICU λ λ s µ s r s e CV Surgery 0.125 0.013 0.011 18 16 Neurosurgery 0.105 0.034 0.015 20 11 Medical 0.122 0.002 0.017 14 11 Neurosciece 0.042 0.014 0.015 7 6 Surgical 0.140 0.032 0.013 20 15 Table 8. M/M/s Queueig Model parameters for each ICU. Arrival rate (patiets/hour) λ, Arrival rate from scheduled surgeries (patiets/hour) λ s, service rate (patiets/hour) µ, umber of beds i the hospitals s r, umber of beds i simulatio model s e q ρ Lq = p 1 ρ p where L q is the mea umber of patiets i the queue, p D is the probability that a arrival will experiece a delay for service, ρ is the average utilizatio for the queuig system, ad s is the umber of servers. I the 2001 US Natioal Hospital Ambulatory Medical Care Survey (NHAMCS), the average waitig time for a ICU bed reported was approximately 4.1 hours. 28 I this study, the average arrival rate ad service duratio were determied for each itesive care uit. Usig the M/M/s model, the average wait times were calculated with the give umber of beds for each ICU (Table 8). Parameters i the simulatio model are determied empirically so as to match the hospital statistics for ICU admissio delay to accurately evaluate the beefits for differet test settigs. Usig the M/M/s model, performace measures were calculated for each ICU for differet levels of bed availability. Sice the M/M/s assumptio of expoetial service times ca lead to uderestimatig actual cogestio, 17 the umber of beds selected by the simulatio model were associated with mea waitig times betwee 1.8 D p 3.2 hours (Table 9): CV Surgery (16), Neurosurgery (11), Medical (11), Neurosciece (6), Surgical (15) CV Surgery Neurosurgery Medical Neuroscie Surgical S W q s W q s W q s W q s W q 11 705.54 8 52.162 8 60.122 3 267.6 11 413.78 1 81 8 12 51.580 9 14.948 9 15.611 4 23.19 12 42.430 8 13 18.190 10 5.684 10 5.834 5 5.606 13 15.265 14 7.972 11 2.333 * 11 2.396 * 6 1.531 * 14 6.725 15 3.755 12 0.972 12 1.006 7 0.417 * * 15 3.170 * 16 1.809 * 13 0.400 13 0.419 16 1.525 17 0.871 14 0.160 14 0.171 ** 17 0.733 18 0.414 ** 15 0.062 18 0.348 16 0.023 19 0.161 17 0.008 20 0.073 ** 18 0.003 19 0.001 20 0.000 ** Table 9. Estimated wait times for each ICU usig M/M/s Queueig Model. Number of beds s, average wait time (hours) W q. * W q for s used by simulatio model ** W q for s used by the hospital Classificatio of severity group After a patiet arrives at the hospital i the simulatio model, they are classified ito oe of ie differet groups based o their severity score ad LOS. The LOS is geerated from the distributio for the requested ICU. There are prior values for the percetage of patiets i each severity group. However, the LOS distributios are slightly differet for each severity group (Table 5). For example, it is rare to fid a patiet with high severity ad high LOS. It would ot be etirely accurate to assig the severity group based oly o prior probabilities. Therefore, a p osterior probability is calculated by multiplyig the prior probability with the likelihood give a patiet s LOS: P(C s LOS) = P(C s)p(los C s ) P(LOS) P(LOS) = P(C s )P(LOS C s ) sεs P(LOS C s ) = P(LOS; μ s, σ s ) (l LOS μ s ) 2 2σ s 2 1 P(LOS; μ s, σ s ) = LOS σ s 2π e where C s is the severity group class, LOS is the sampled value for legth of stay from the ICU distributio, p(c s LOS) is the posterior probability of 846
belogig to C s give the LOS, p(c s ) is the prior probability of belogig to C s, p(los C s ) is the likelihood of observig the LOS give C s, μ s ad σ s are parameters of the log-ormal distributio for C s. The severity group is assiged to the admitted patiet based o the calculated posterior probabilities for each class. Each group has a set of mortality rates determiig whether the patiet will die durig their stay i the ICU (Table 5). Maagig Artificial Variability There is substatial atural variability i hospital admissios through the emergecy departmet, but there is also artificial variability. I this study, we foud that 28.2% of etries were admitted to a ICU from elective surgeries. If adjusted for patiet volume, scheduled surgical admissios ca vary eve more tha through the Emergecy Departmet (ED). 23 This ca have reciprocal effects where high surgical volumes ca delay operatios ad icrease waitig times for a available room. Operatios ca be cacelled due to a shortage of ICU beds. I this study, the distributio is calculated for iterarrival times to each ICU from scheduled ad uscheduled admissios. A Passive model is first tested that uses o priority scheme ad factors atural ad artificial variability of arrivals. Each model reports the total patiets served, severe patiets admitted, average waitig times, utilizatio rate, retur rate, ad mortalities. The Smooth Model is similar to the Passive Model, except it uses a ideal surgery schedule where there is o artificial variability. This is to help determie the effects the surgery schedule has o the hospital process flow. The average time betwee arrivals is calculated for scheduled admissios for each ICU (Table 8). Istead of usig the fitted distributios for scheduled admissios, patiets arrive at times equidistat from each other for each ICU. The Smooth Model is ot realistic, because eve operatio times ca vary i ideal cases where elective surgeries are scheduled at efficiet times. It is oly used for evaluatio purposes. All subsequet models utilize fitted distributios for scheduled admissios, but test differet priority methods for admittig ad bumpig patiets. Priority models Typically, a queue admits etries o a first-comefirst-serve (FCFS) basis. However, priority queues allow differet classes to be treated differetly. Without preemptio, higher class items ca jump ahead of others withi the queue. However, service caot be iterrupted for ay items i process. I a preemptive priority class, higher class items ca discotiue other items curretly i service. 16 I this study, both preemptive ad o-preemptive models were tested to aalyze the process flow of itesive care uits. Four differet priority models were evaluated i our simulatio model. Specifically, we derive ad test models that both restrict ad allow bumpig while factorig the cosequet mortality ad retur rates. Greedy: The greedy method 39 gives patiets with highest LOS the greatest priority. Usig queuig theory, Siddharta et al showed that admittig patiets with larger LOS before others lowered the overall average wait time. 39 The Greedy model is o-preemptive where bumpig of patiets is ot permitted i ay case. Higher priority patiets i the queue are ot permitted to iterrupt lower priority patiets i service. Hybrid: The hybrid method admits patiets based o their severity ad their LOS. A patiet i the highest severity group will be admitted first, but patiets i the lower severity groups will be ordered accordig to their average LOS. The Hybrid model is also a o-preemptive method. It factors admissio ot oly o efficiecy, but also o the severity of the patiet s coditio. The ext two priority models are both preemptive. They allow the service of lower priority patiets to be iterrupted if a higher priority patiet is admitted. Severity (Coservative) Bumpig: The Coservative Bumpig model is idetical to the Hybrid model i the order patiets are placed i the queue. However, a severe patiet (C s = 3) i the queue ca bump a osevere patiet (C s < 3) from service. No-severe patiets caot bump ay patiets from service. Nosevere patiets are bumped by lowest remaiig legth of stay plus the associated readmissio load: LOS = LOS + P( R ) LOS tot rem P Pret where LOS rem is the remaiig service time, P(R P ) is the average retur rate for the patiet group, ad LOS Pret is the average service time for returs for the patiet group (Table 6), LOS tot is the estimated total service time. The readmissio load is the product of 847
retur probability times retur LOS, which is calculated usig a similar method to the study by Cha et al. 5 Aggressive Bumpig: Severe patiets ca still ot be discharged from the ICU while i service. However, o-severe patiets will be bumped whe ay type of patiet requests admissio to the ICU. Patiets are discharged i the same order as the Coservative Bumpig model. Aggressive Bumpig is similar to the method used by Cha et al., except severe patiets are restricted from ICU trasfer before completio of service. If a patiet is bumped while i service, they will have a higher retur rate as foud i our hospital trasfer records data (Table 4). Subsequetly, the retured patiets also have a higher mortality rate. All four differet priority models are tested to determie the effects o waitig time, retur rate ad mortality. Results Table 10 reports the results for all six queuig models. Without eforcig ay priorities for admissio, the Passive Model reported higher average waitig time i the queue (4.5 hours) ad fewer total patiets served (1,024). The utilizatio rate was also 4% higher tha ay other model. The Smooth Model also does ot eforce priorities, but arrivals from elective surgeries occur at a costat rate. The hospital oly schedules surgeries Moday through Friday ad operatig hours ca vary sigificatly. The Smooth Model is a ideal case that removes all variatio from scheduled surgery arrivals. It gave impressive results whe compared to the Passive Model at 2.5 hours for average waitig time ad 1,035 for total patiets served. This raised the amout of patiets as well as lowerig delays. This showed reducig artificial variability is beeficial if it is possible to eforce a more regimeted surgery schedule. variability utilizig the fitted distributios for scheduled surgery arrivals. The Greedy model oly prioritizes patiets by their expected LOS. It was able to serve 1,043 patiets at a average waitig time of 2.95 hours. This model could ot capitalize o the beefits of uiform patiet arrivals as with the Smooth Model, but it was able to report better performace measures tha the Passive Model. The average wait time for the Greedy Model was 0.4 hours higher tha the Smooth Model, most likely due to temporary bottleecks from variatio i arrivals. The Greedy model focuses o efficiecy rather tha patiet severity. The Hybrid Model prioritizes severe patiets above all others. No-severe patiets are prioritized by expected LOS idetical to the Greedy Model. The Hybrid Model served 1,033 patiets at 3.6 hours average waitig time. These are weaker results, but the average waitig time for severe patiets was oly 1.4 hours compared to 3.8 hours i the Greedy Model. The Hybrid Model also had the lowest retur rate at 16.4%. The Coservative Bumpig ad Severity Bumpig models reported results with substatial differeces. Both preemptive queuig models prioritize patiets by severity idetical to the Hybrid Model. The Coservative model ca oly bump less severe (C s < 3) patiets from service whe the most severe (C s = 3) request ICU admissio. The Aggressive Model bumps less severe patiets from service for ay patiet requestig admissio. The Coservative Bumpig model served 1,038 patiets ad oly bumped a average of 7.8 from service. The average waitig time was 0.8 hour for severe patiets ad 2.7 hours for all patiets. The mortality rate was oly raised by 0.4 percet compared to the Hybrid Model. The Aggressive Model served 1,051 patiets bumpig 93 patiets with a average waitig time of 1.1 hours. The retur rate icreased by 1.0 percet ad it r eported the highest mortality rate for ay model at 8.8%. It is clear that bumpig ca prove to be beeficial but oly i heavily restricted cases. Priority queuig models were tested with artificial Model Priority Order # Patiets P(R) P(M) B W q W qs Util Passive 1,024 0.173 0.078 0 4.515 4.556 0.693 Smooth 1,035 0.169 0.075 0 2.560 2.610 0.670 Greedy 7,8,9,4,5,6,3,2,1 1,043 0.169 0.074 0 2.946 3.840 0.660 Hybrid 9,6,3,7,8,4,5,2,1 1,033 0.164 0.075 0 3.562 1.411 0.655 Severity Bumpig 9,6,3,7,8,4,5,2,1 1,038 0.165 0.079 7.9 2.768 0.847 0.652 Aggressive Bumpig 9,6,3,7,8,4,5,2,1 1,051 0.174 0.088 93.4 1.062 0.961 0.666 Table 10. Priority Queuig Model Results. Priority order for patiet groups, total patiets served, retur rate P(R), mortality probability P(M), umber of bumped patiets B, average waitig time i the queue for all patiets W q, average waitig time i the queue for severe patiets Wqs, average utilizatio rate 848
Coclusio Healthcare ceters that focus o operatig at highest efficiecy may cosequetly sacrifice the quality of care. By evaluatig several differet priority methods, the ICU system-based simulatio model helps idetify the costs of prioritizig by severity rather tha efficiecy. Severe priority methods do raise overall waitig times ad lower the amout of patiets served, but added beefits reduce further medical complicatios. Shorter wait times for severe patiets result i lower retur ad mortality rates. Severe priority methods ca show substatial ehacemets by coservatively allowig bumpig policies. Permittig early discharges with severe priority models resulted i wait times close to the most efficiet models. However, without firm restrictios, bumpig ca sigificatly raise the mortality ad retur rates. There are several potetial future research studies that ca be coducted with appropriate types of data. Our approach is applicable to other hospital data streams, for example, ICD diagosis codes, patiet resource eeds, ad hospital utilizatio status. Specifically, it would be beeficial to accurately categorize the diagosis for each patiet usig idividual ICD diagosis codes. This would help determie if a p atiet retur was due to a early discharge or because of a etirely ew coditio. Further, i our earlier readmissio work 43, hospital resource usage ad utilizatio iformatio were employed to help predict patiet readmissio characteristics ad the impact o patiet eeds ad quality of care. The studied hospital has five distictly specialized itesive care uits. There may be evets where the requested ICU is full ad a patiet is diverted to a ICU of a d ifferet specialty. 22 It would be advatageous to examie the implicatios regardig permitted diversios for associated coditios. A aalysis could be coducted whether patiets beefit from diversios to ICUs of differet specialties rather tha remaiig i the queue for the desired locatio. Patiet admissios ca also be evaluated more globally. If estimated wait times were available for each hospital, the costs ca be cosidered for redirectig patiets to aother hospital. The studied hospital herei has a s ister medical ceter at a locatio about six miles away. It would be iterestig to review records for cases where patiets were blocked access ad directed to this alterative locatio. A future study will aalyze these cases ad determie if trasfer times were lower tha estimated wait times for direct admissio. Eve i circumstaces where total wait time were reduced by diversio, complicatios ca result from the additioal trasit time. Optimizig patiet flow i healthcare settigs is a challegig balace betwee maagig efficiecy ad maitaiig quality of care. Hospitals ca become more proficiet ad resourceful i daily operatios by cotiuig to build system models that attempt to idetify ad ivestigate all sigificat iterdepedet factors. Ackowledgemet The study is partially supported by a grat from the Natioal Sciece Foudatio. Refereces 1. America Hospital Associatio. 2000. Hospital Statistics 2000. Chicago, Ill.: America Hospital Associatio. 2. Brecher, C., ad Spiezio, S. Privatizatio ad Public Hospitals: Choosig Wisely for New York City. New York: Twetieth Cetury Fud, 1995. Prit. 3. Chalfi, D. B., S. Trzeciak, et al. (2007). "Impact of delayed trasfer of critically ill p atiets from the emergecy departmet to the itesive care uit." Crit Care Med 35(6): 1477-83. 4. Cha, A., G. Aredts, et al. (2008). "Causes of costraits to patiet flow i emergecy departmets: a compariso betwee staff perceptios ad fidigs from the Patiet Flow Study." Emerg Med Australas 20(3): 234-40. 5. Cha, Carri W, Nicholas Bambos, ad Gabriel J Escobar. Maximizig Throughput of Hospital Itesive Care Uits with Patiet Readmissios. workig paper (2010): 1-41 6. Chie-Hsig Che, ad Chug-Chia Hsu. (2009). "Idexig ICD-9 codes for free-textual cliical diagosis records by a ew esemble classifier." Iteratioal Joural of Computatioal Itelligece i Bioiformatics ad Systems Biology 1(2): 177-192. 7. Cochra, J. K., ad Bharti, A. (2006). "A multi-stage stochastic methodology for whole hospital bed plaig uder peak loadig." It J Id Syst Eg 1(1/2): 8-36. 8. de Bruij, L. M., A. Hasma, et al. (1997). "Automatic SNOMED classificatio--a corpus-based method." Comput Methods Programs Biomed 54(1-2): 115-22. 9. Dobso, G., H.-H. Lee, et al. (2010). "A Model of ICU Bumpig." Oper. Res. 58(6): 1564-1576. 10. Durbi, C. G., Jr. ad R. F. Kopel (1993). "A casecotrol study of patiets readmitted to the itesive care uit." Crit Care Med 21(10): 1547-53. 11. Emergecy departmet overload: A growig crisis. (April 2002). The Lewi Group aalysis of AHA ED ad hospital capacity survey data. 12. Escobar, G. J., J. D. Greee, et al. (2008). "Riskadjustig hospital ipatiet mortality usig automated ipatiet, outpatiet, ad laboratory databases." Med Care 46(3): 232-9. 849
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