Estimating horizontal movement performance of patient beds and the impact on emergency evacuation time
Emergency evacuation of patients from a hospital can be challenging in the event of a fire. Most emergency evacuation studies are based on the assumption that pedestrians are ambulant and can egress by themselves. However, this is often not the case during emergency evacuations in healthcare facilities such as hospitals and nursing homes. To investigate emergency evacuations in such healthcare facilities, we performed a series of controlled experiments to study the dynamics of patient beds in horizontal movement. We considered a patient bed because it is one of the commonly used devices to tra
Прямых ссылок нет — публикация используется как справочный материал.
Объём оцифровки: 75,963 знаков. Ссылки внутри текста ведут на карточки источников и публикаций библиотеки.
Estimating horizontal movement performance of patient beds and the impact on emergency evacuation time
Jaeyoung Kwaka,∗ , Michael H. Leesb , Wentong Caic , Ahmad Reza Pourghaderid,f,a and Marcus
E.H. Onge,f a Complexity Institute, Nanyang Technological University, 61 Nanyang Drive, Singapore 637335, Singapore b Informatics Institute, University of Amsterdam, Science Park 904, Amsterdam 1098XH, The Netherlands c School of Computer Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore d Health Systems Research Center (HSRC), Singapore Health Services, 31 Third Hospital Avenue, Singapore 168753, Singapore arXiv:2006.02169v3 [physics.soc-ph] 23 Dec 2020
e Department of Emergency Medicine, Singapore General Hospital, Outram Road, Singapore 169608, Singapore f Health Services and Systems Research (HSSR), Duke-NUS Medical School, 8 College Road, Singapore 169857, Singapore
ARTICLE INFO ABSTRACT
Keywords: Emergency evacuation of patients from a hospital can be challenging in the event of a fire. Most emer-
Emergency evacuation gency evacuation studies are based on the assumption that pedestrians are ambulant and can egress
Patient bed by themselves. However, this is often not the case during emergency evacuations in healthcare facil-
Turning movement ities such as hospitals and nursing homes. To investigate emergency evacuations in such healthcare
Fatigue effect facilities, we performed a series of controlled experiments to study the dynamics of patient beds in
Movement duration horizontal movement. We considered a patient bed because it is one of the commonly used devices to transport patients within healthcare facilities. Through a series of controlled experiments, we ex- amined the change of velocity in corner turning movements and speed reductions in multiple trips between both ends of a straight corridor. Based on the experimental results, we then developed a mathematical model of total evacuation time prediction for a patient bed horizontally moving in a healthcare facility. Factoring uncertainty in the horizontal movement, we produced the probability distribution of movement duration and estimated the probability that an evacuation can be safely per- formed within certain amount of time. In addition, we predicted that the evacuation time would be longer than the prediction results from an existing model which assumes constant movement speed.
Our results from the model demonstrated good agreement with our experimental results.
1. Introduction pedestrian vertical movement through stairs such as down- ward moving speed and flow rate in a stairwell of a high-rise
Pedestrian emergency evacuation has been one of the building [4] and the influence of stair slope on such pedes- central topics in the field of fire safety engineering. In case of trian flow characteristics [5]. The pedestrian flow charac- life threatening incidents such as fire and hazardous chem- teristics have been studied for harsh moving conditions in- ical spills, a well-prepared emergency evacuation plan can cluding crawling in a room [6] and in a corridor [7], and the efficiently move occupants to the assembly points with min- existence of earthquake-induced falling debris [8].
imum amount of time and ensure the safety of evacuees. In most existing emergency evacuation studies, it is as-
Based on the degree of required aid during egress, occupants sumed that pedestrians are able to walk and egress by them- can be categorized into ambulant and non-ambulant occu- selves. However, this is often not the case during emergency pants. Ambulant occupants can egress to the place of safety evacuations especially in healthcare facilities such as hospi- without help from other occupants. On the other hand, non- tals and nursing homes. Healthcare facilities accommodate ambulant occupants need help when they move, especially considerable numbers of non-ambulant patients who have in case of an emergency. The non-ambulant occupants can limited mobility and need the help of medical staff with evac- be further categorized into subcategories such as bedridden uation devices during the evacuation process. To prepare occupants and wheelchair users depending on the device that emergency evacuation plans considering the non-ambulant they are using.
pedestrians, it is necessary to understand the performance of
Emergency evacuation of ambulant pedestrians has been evacuation devices such as their movement speed and move- investigated in experimental studies to understand pedestrian ment dynamics.
movement during emergency evacuations. Numerous exper- Previous experimental studies of non-ambulant pedes- iments have been conducted in horizontal evacuation scenar- trian evacuations analyzed video footage of the experiments ios in which pedestrians egress within the same floor. For in- to measure travel time between different reference points. By stance, researchers have studied evacuations through bottle- doing that, average movement speed was measured for dif- necks to understand the relationship between the bottleneck ferent sections of an evacuation route such as a straight cor- width and pedestrian flow, which is critical to the total evac- ridor and a stairwell. For example, Rubadiri et al. [9] per- uation time in room evacuations [1, 2, 3]. A considerable formed a horizontal evacuation exercise of manual and elec- number of experiments have been performed to characterize tric wheelchair users. They measured the evacuation per-
ORCID (s): formance index as a ratio of the wheelchair user movement
J. Kwak et al.: Preprint submitted to Elsevier Page 1 of 14
Patient Bed Horizontal Movement Performance Estimation
speed without assistance to the movement speed of ambulant in multiple trips between both ends of a straight corridor. pedestrians. Based on the measured evacuation performance Based on the experimental results, we then developed a move- index, they predicted evacuation time of the wheelchair users ment duration prediction model and then applied the model and then compared it with the actual measurement of the for a patient bed horizontally moving in a healthcare facil- wheelchair users. Strating [10] collected movement speed ity. Incorporating uncertainty in the horizontal movement, data of bedridden occupants in Dutch healthcare facilities. we predicted probability that an evacuation can be safely per- He reported an evacuation speed range from 0.54 to 1.34 m/s formed within certain amount of time. The experiment setup in Dutch hospitals and from 0.25 to 1.30 m/s in Dutch nurs- and data collection methods are described in Section 2. As ing homes. Hunt et al. [11] measured horizontal and verti- shown in Section 3, we analyze the dynamics of patient bed cal movement speed of evacuation devices including stretch- movement, develop a movement duration prediction model, ers, evacuation chairs, carry chairs, and rescue sheets. They and then apply the model for a patient bed horizontally mov- observed that the evacuation chair is the fastest among the ing in a healthcare facility. We summarizes the results with tested evacuation devices with an average speed of 1.5 m/s concluding remarks in Section 4. in the horizontal evacuation and of 0.83 m/s in the verti- cal evacuation. Based on the measured movement speed of those evacuation devices, they also estimated the total evac- 2. Experiment setup uation time of non-ambulant patients in a high-rise hospi- To collect patient bed movement data, we carried out a tal building. While those studies focused on measuring the series of controlled experiments in September 2019 at the average movement speed of non-ambulant pedestrian evac- Singapore General Hospital, Singapore. We recruited 8 in- uation devices, change of velocity during the movement has dividuals who were acting as handlers in the experiment: 4 not yet been studied in detail. males and 4 females aged between 25 to 35 without move- In pedestrian flow dynamics, it has been reported that ment impairments. In order to move a patient bed, the han- pedestrians continuously moving with heavy load tend to dlers were grouped into pairs of same gender handlers: male- have reduced movement speed due to fatigue [12]. Further- male and female-female handlers. Additionally, one male more, several studies investigated speed profiles of pedes- (27 years old, 189 cm, 87 kg) was lying on the patient bed trians moving near obstacles in front of an exit [13], cor- during its movement in order to substitute for a real non- ners [14], and merging areas [15]. Such structural elements ambulant patient. In total, 9 individuals (8 handlers and 1 of pedestrian facilities often act as a constraint on the ef- person acting as a bedridden patient) were volunteered for ficiency of pedestrian flow seemingly because pedestrians the experiment. Before starting the experiment, the handlers need to change their moving direction and speed. had an orientation session and conducted a few warm-up tri- Similar to the case of pedestrian flow dynamics, the move- als under the guidance of a nurse who works at the Singapore ment of an evacuation device becomes slower when the de- General Hospital Emergency Department. Through the ori- vice is moving in corners and areas of merging flow and entation session, the handlers became proficient enough in interaction with other devices coming from different direc- maneuvering the patient bed. In the experiment, we asked tions. In addition, there are often far more bedridden pa- all the handlers to move the patient bed as fast as possible tients than medical staff in healthcare facilities. It is ap- while making sure their safety. parent that, in emergency evacuations, every medical staff Figure 1 shows the sketches of the experiment setup: a needs to make multiple trips between the location of bedrid- 90° angled corner and a straight corridor. In the 90° angled den patients and the place of safety. One can expect that corner setup, each handler pair was maneuvering a patient the medical staff experience some level of exhaustion when bed in the intersection of 2 m Ч 2 m and two straight cor- they evacuate all the bedridden patients and it is likely that ridors of 2 m wide 5 m long. Handlers made left-turning their movement becomes slower as they travel longer dis- movements by moving from the top to the right branches, tance. Such a slow-moving evacuation device affects the and from the right to the top branches for right-turning move- flow of subsequent evacuation devices and pedestrians, es- ments. The handler pair repeated right and left turning move- pecially with limited passing opportunities in a narrow corri- ment at least 10 times. In the straight corridor experiment dor. Consequently, understanding these non-linear dynam- setup, each handler pair made 21 round trips with a patient ics of evacuation devices enables us to better estimate the bed in a straight corridor of 2 m wide 21 m long. By do- evacuation performance of the devices with taking into ac- ing that, the handler pair moved the patient bed a distance of count their speed profile and change of moving directions. 882 m in effect. In both experiment setups, there were not Among various evacuation devices, we considered the obstacles or doors on the experimental route. According to movement of a patient bed because it is one of the most com- the usual practice [16], a pair of handlers was moving the monly used devices to transport patients from one place to patient bed together in both experiments. In each pair, one the another in healthcare facilities. In this study, we per- handler was acting as a leading handler who was guiding the formed a series of controlled experiments to investigate the movement direction of the bed, while the other handler was dynamics of patient beds in horizontal movement. Through pushing the bed following the leading handler. Once the han- the controlled experiments, we examined the change of ve- dlers reached the end of the experiment area, they changed locity in corner turning movements and speed reductions their position in order to reverse their movement direction.
J. Kwak et al.: Preprint submitted to Elsevier Page 2 of 14
Patient Bed Horizontal Movement Performance Estimation
(a)
(b)
Figure 1: Schematic representation of experiment setup: (a) a 90° angled corner and (b) a straight corridor. In (a), Areas 1 and 3 indicate 3 m long 2 m wide straight corridor sections and Area 2 shows 2 m-by-2 m intersection. Blue rectangles depict patient beds and yellow arrows show their moving direction. Red circles and red arrows indicate the locations of cameras and their pointing direction, respectively.
(a) (b) (c) (d)
Figure 2: Schematic representation of handler positions with the direction of travel: (a) left-turning movement from the top to the right branches in the 90° angled corner setup, (b) right-turning movement from the right to the top branches in the 90° angled corner setup, (c) rightward movement in the straight corridor setup, and (d) leftward movement in the straight corridor setup.
Blue circles depict handlers and yellow arrows show their moving direction.
The formation of handlers in each experiment setup is illus- trajectory extraction software developed from Lund Univer- trated in Fig. 2. sity, Sweden [17]. The software has been applied in cyclist Both experiments were recorded using cameras with a traffic safety analysis [18] and pedestrian flow analysis [19]. frame rate of 30 frames per second. The cameras were mounted The conversion from video coordinates to the real world co- on tripods which were set on the top of tables. In the 90° an- ordinates was performed by T-calibration, a calibration soft- gled corner setup, we set two cameras near the corner to ware accompanied by T-analyst. In T-calibration software, closely observe the turning movement of the patient bed. In we placed calibration points and its real-world coordinates the straight corridor setup, two cameras were used to record and then performed TSAI-calibration algorithm [20]. After the bed movement from each end of the corridor. From video the calibration process, we manually annotated pedestrian footage of the 90° angled corner setup, we extracted patient positions for every 10 frames on average in the video footage bed movement trajectories using T-analyst, a semi-automatic with T-analyst software. Next, T-analyst software then ex-
J. Kwak et al.: Preprint submitted to Elsevier Page 3 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 1
Basic movement characteristics of handler groups in the corner area of 𝑥 ≤ 5 and 𝑦 ≤ 5.
Handler Movement Orientation (rad) No. of Distance Duration Average Entering Gender groups direction starting target trajectories (m) (s) speed (m/s) speed (m/s) LT_1 Female Left turn 1.57 0 12 7.59 ± 0.16 10.59 ± 0.57 0.72 ± 0.03 0.80 ± 0.06 LT_2 Female Left turn 1.57 0 13 7.45 ± 0.11 9.05 ± 0.67 0.83 ± 0.06 0.87 ± 0.09 LT_3 Male Left turn 1.57 0 10 7.58 ± 0.12 10.17 ± 0.73 0.75 ± 0.05 0.72 ± 0.15 LT_4 Male Left turn 1.57 0 15 7.59 ± 0.10 7.34 ± 1.04 1.05 ± 0.11 0.95 ± 0.12 RT_1 Female Right turn 3.14 1.57 12 7.67 ± 0.06 9.13 ± 0.54 0.84 ± 0.05 1.15 ± 0.07 RT_2 Female Right turn 3.14 1.57 13 7.59 ± 0.10 7.24 ± 0.42 1.05 ± 0.06 1.32 ± 0.11 RT_3 Male Right turn 3.14 1.57 10 7.98 ± 0.12 9.33 ± 0.61 0.86 ± 0.04 1.05 ± 0.08 RT_4 Male Right turn 3.14 1.57 15 7.82 ± 0.13 6.47 ± 0.56 1.22 ± 0.08 1.51 ± 0.24 Average 7.65 ± 0.19 8.51 ± 1.57 0.93 ± 0.18 1.07 ± 0.29
Table 2
Basic movement characteristics of handler groups in straight corridor experiment.
Handler
Gender speed (m/s) groups min. max. average
Group_1 Female 0.94 1.20 1.03
y (m)
Group_2 Female 1.34 1.64 1.47
Group_3 Male 0.98 1.13 1.04
Group_4 Male 1.31 1.70 1.53
tracted pedestrian trajectories. Details of pedestrian trajec- tory extraction process can be found from T-analyst manual available from its webpage [17]. Table 1 shows the basic movement characteristics of han- 0 1 2 3 4 5 dler groups in the turning movement experiment. We col- x (m) lected 10 to 15 trajectories from each handler group and in Figure 3: 𝑁 = 100 individual trajectories collected from the total 𝑁 = 100 trajectories were collected. The distance trav- corner turning experiment setup eled and the movement duration were measured for each tra- jectory in the area of 𝑥 ≤ 5 and 𝑦 ≤ 5 and then averaged for each handler group. The average speed was computed by di- ogous to the method presented by Chen et al. [21]. The av- viding the distance traveled by the movement duration. The erage speed was measured by dividing the distance traveled entering speed was measured when the patient bed is enter- (882 m) by the total travel time that the handlers moving the ing the area of 𝑥 ≤ 5 and 𝑦 ≤ 5. Handler groups RT_2 and patient bed. Table 2 summarizes the basic movement charac- RT_4 showed shorter movement duration than other han- teristics of handler group in the straight corridor experiment. dler groups seemingly because they entered the corner with Handler groups Group_2 and Group_4 showed higher speed higher entering speed and experienced smaller speed drop in but there is not significant difference between male and fe- the course of their movement. male. The differences between the groups in the same gen- In the straight corridor setup, based on the study of Luo et der (Group_1 vs. Group_2, and Group_3 vs. Group_4 in al. [12], we assumed that the fatigue experienced by the han- Table 2) seemingly attribute to the degree of risk aversion dlers is affected by the distance that they transported the pa- set by different groups. Although we asked all the handlers tient bed. The handlers were making multiple trips between to move as fast as possible, some of them might want to more one end to the other end of the corridor. We measured the focus on moving safely without getting injured, so they were time that the handlers took moving between the two ends not necessarily in a rush. of the corridor using the video footage. In doing that, we manually marked the transition period in which the handlers stopped and changed their position. After identifying the 3. Results and analysis transition period, we obtained the travel time in each one- 3.1. Turning movement way trip and then calculated the average speed for each trip. Figure 3 shows individual trajectories collected from the The initial speed was measured from the first one-way trip corner turning experiment setup. To understand the change in which the handlers moved first 21 meters in the straight of velocity in the course of turning movement, we firstly ob- corridor setup. Our approach of measuring the speed is anal- tained the average path of all the handler groups moving in
J. Kwak et al.: Preprint submitted to Elsevier Page 4 of 14
Patient Bed Horizontal Movement Performance Estimation
both movement directions. Similar to Hicheur et al. [22], we (a)
angular displacement (rad) resampled the collected trajectories into 𝑁𝑓 = 60 equal in- 1.8 tervals and then averaged the resampled trajectories for each rescaled time point 𝑡̂ ∈ [0, 1]: 1.5
1.2
1 ∑ ̂
𝑁 𝑥𝑎𝑣𝑔 (𝑡̂) = 𝑥 (𝑡),
𝑁 𝑖=1 𝑖 0.9
(1)
0.6
1 ∑ ̂
𝑁 𝑦𝑎𝑣𝑔 (𝑡̂) = 𝑦 (𝑡).
𝑁 𝑖=1 𝑖 0.3
Here, 𝑁 = 100 is the number of collected trajectories from 0 0.2 0.4 0.6 0.8 1 handler groups shown in Table 1. Likewise, we calculated rescaled time trajectory deviation 𝜎(𝑡̂) to quantify the difference between (b) the mean trajectory and an individual trajectory 𝑖: 0.3
0.2
√
xsd
√
√1 ∑ 𝑁
( )2 0.1 𝜎𝑥 (𝑡̂) = √ 𝑥𝑖 (𝑡̂) − 𝑥𝑎𝑣𝑔 (𝑡̂) , (2)
𝑁 𝑖=1
0 0.2 0.4 0.6 0.8 1
√ rescaled time
√
√1 ∑ 𝑁 (c)
( )2 𝜎𝑦 (𝑡̂) = √ 𝑦𝑖 (𝑡̂) − 𝑦𝑎𝑣𝑔 (𝑡̂) , (3) 0.3
𝑁 𝑖=1
0.2 ysd
√ 0.1 𝜎(𝑡̂) = 𝜎𝑥2 (𝑡̂) + 𝜎𝑦2 (𝑡̂). (4) 0 0.2 0.4 0.6 0.8 1 By making use of the rescaled time points, we identified rescaled time start and end of turning movement. We measured the angu- (d) traj. deviation
lar displacement 𝜃 and its average 𝜃𝑎𝑣𝑔 : 0.3
( ) 0.2 𝑒⃗0 𝑒⃗𝑖 𝜃(𝑡̂𝑗 ) = arccos , 0.1
‖𝑒⃗0 ‖ ‖𝑒⃗𝑖 ‖
‖ ‖‖ ‖
(5)
1 ∑ ̂
𝑁
0 0.2 0.4 0.6 0.8 1 𝜃𝑎𝑣𝑔 (𝑡̂𝑗 ) = 𝜃(𝑡 ),
𝑁 𝑗=1 𝑗 rescaled time
Figure 4: Angular displacement of the patient bed: (a) in- where 𝑒⃗0 is the initial moving direction which is set as (0, dividual angular displacement (blue circles) and the average
-1) for left-turn movement and (-1, 0) for right-turn move- value (red solid line). Black dashed lines indicate start (left ment. The moving direction at rescaled time 𝑡̂𝑗 is given as dashed line) and end (right dashed line) of turning movement.
𝑒⃗𝑖 = (Δ𝑥∕Δ𝑙, Δ𝑦∕Δ𝑙). Here, Δ𝑥 and Δ𝑦 denote the differ- (b)-(d) the corresponding graphs of 𝜎𝑥 (𝑡̂), 𝜎𝑦 (𝑡̂), and 𝜎(𝑡̂). It ence between the values of 𝑥𝑖 and 𝑦𝑖 at the current rescaled can be observed that the start and end of turning movement occur when either 𝜎𝑥 (𝑡̂) or 𝜎𝑦 (𝑡̂) is at peak value.
time points 𝑡̂𝑗 and the previous rescaled time point 𝑡̂𝑗−1 . The displacement between the positions at rescaled time 𝑡̂𝑗 and
√ 𝑡̂𝑗−1 is given as Δ𝑙 = Δ𝑥2 + Δ𝑦2 .
to work of Dias et al. [23], it was observed that the turning
Figure 4(a) presents angular displacement of the patient movement started before and ended after Area 2 (𝑥 ≤ 2 and bed against rescaled time points. One can observe that the 𝑦 ≤ 2). Although there exist similarities between the patient angular displacement curve is nearly straight between 𝑡̂ = bed and individual pedestrians in terms of the turning move-
0.358 and 𝑡̂ = 0.658, indicating that the patient bed is turning ment start and end locations, one can notice difference in the with a constant angular speed 0.24 rad/𝑡̂. When the patient mean shy-away distance to the corner boundaries. Here, the bed starts and ends the turning movement, either 𝜎𝑥 (𝑡̂) or mean shy-away distance was measured as the gap between 𝜎𝑦 (𝑡̂) is at peak value. Figures 4(b) and 4(c) indicate that the the mean trajectory and the corner boundaries. In this study, start and end of turning movement can be identified based on the mean shy-away distance of a patient bed turning around the components of trajectory deviation, i.e., 𝜎𝑥 (𝑡̂) and 𝜎𝑦 (𝑡̂).
a 90° angled corner was measured as around 1.1 m, which is
The identified locations of turning movement start and larger than that one measured for individuals in the work of end are presented with the mean trajectory in Fig. 5. Similar
J. Kwak et al.: Preprint submitted to Elsevier Page 5 of 14
Patient Bed Horizontal Movement Performance Estimation
(a)
1.6
speed (m/s)
1.2
0.8 y (m)
0.4 0 3 6 9 12 1 time (s) (b) 0 1.5 0 1 2 3 4 5
normalized speed x (m)
Figure 5: Average path (red solid line) with the locations of turning movement start and end (blue cross symbols).
0.5
Dias et al. [24]. This is seemingly because the patient bed
(0.8 m wide) requires larger lateral clearance than an indi- vidual (0.4 to 0.5 m wide) during the turning movement
0 0.2 0.4 0.6 0.8 1
Figure 6(a) shows examples of individual speed curves before resampling. Due to various initial speed and maneu- rescaled time
(c) vering duration, it is difficult to understand patterns in the 1.2 speed curves. We applied the idea of the rescaled time points normalized speed
to such speed curves. We resampled the instantaneous speed 𝑣𝑖 of individual handler groups into 𝑁𝑓 equal intervals and 0.8 then averaged the resampled speed for each rescaled time point 𝑡̂ ∈ [0, 1]: 0.6
0.4
1 ∑ ̂
𝑁 𝑣𝑎𝑣𝑔 (𝑡̂) = 𝑣 (𝑡). (6) 0.2
𝑁 𝑖=1 𝑖
0 0.2 0.4 0.6 0.8 1
In order to compare different speed curves, we normalized the speed curves with entering speed 𝑣0 . Figure 6(b) presents rescaled time the result of normalization performed against rescaled time Figure 6: Speed profile of handler groups as a function of rescaled time in the area of 𝑥 ≤ 5 and 𝑦 ≤ 5: (a) examples 𝑡̂ for curves in Fig. 6(a). We performed normalization for of individual speed curves. (b) the result of normalization per-
𝑁 = 100 individual speed curves and then averaged their formed against rescaled time 𝑡̂ for curves in (a). (c) the average normalized speed in rescaled time space. Figure 6(c) illus- of normalized speed curves (blue cross symbols) with its trend trates the average of normalized speed curves and its trend line (red solid line). The trend line begins to decrease and then line. The trend line is generated by the normalized speed increase when the patient bed is turning around a corner.
profile 𝑣(
̂ 𝑡̂).
According to minimum jerk principle (Hogan [25] and
Pham et al. [26]), people tend to minimize jerk (i.e., a third- The coefficients 𝑎0 , 𝑎1 , 𝑎2 , 𝑎3 , and 𝑎4 can be determined order derivative of position) while turning around a corner: from the average of normalized speed curves in Fig. 6(c).
Table 3 shows the coefficient values. Following the work of
[( )2 ( )2 ] Dias et al. [24], we modeled the normalized speed profile 𝑑3𝑥 𝑑3𝑦
+ 𝑑 𝑡̂, (7) ̂ 𝑡̂) with the same values in the start and end of the curve:
𝑣(
∫0 𝑑 𝑡̂3 𝑑 𝑡̂3 ̂ = 1 and 𝑣(1) 𝑣(0) ̂ = 1.
where 𝑥 and 𝑦 represent position in 𝑥- and 𝑦-axis at a rescaled 3.2. Fatigue effect time point 𝑡̂. Based on the minimum jerk principle, the nor- As stated in the last paragraph of Section 2, we measured malized speed profile 𝑣(̂ 𝑡̂) can be represented by a fourth- the travel time in each one-way trip and then calculated the order polynomial equation of rescaled time point 𝑡̂: average speed for each trip. The handler group speed in the straight corridor setup is shown in Fig. 7. As can be seen
̂ 𝑡̂) = 𝑎0 + 𝑎1 𝑡̂ + 𝑎2 𝑡̂2 + 𝑎3 𝑡̂3 + 𝑎4 𝑡̂4 .
𝑣( (8)
J. Kwak et al.: Preprint submitted to Elsevier Page 6 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 3
Normalized speed profile coefficients 0.3
Coefficient Value 𝑎0
0.2 𝑎1 1.1651
fa 𝑎2 -10.0789 𝑎3 17.4769 0.1 𝑎4 -8.5632
2
0 150 300 450 600 750 900 s (m)
1.5 speed (m/s)
Figure 8: Relationship between the fatigue coefficient 𝑓𝑎 and
the distance traveled 𝑠. The measured fatigue coefficient 𝑓𝑎
1 from handler groups is denoted by blue circles (○). The corre-
sponding trend line is indicated by red solid line. Black dashed
line indicates the breakpoint location 𝑠Ч = 264.4 m at which
0.5 fatigue coefficient 𝑓𝑎 curve slope changes. Purple solid lines
with cross symbols (Ч) show the boundaries of 80% prediction
band with quantiles 0.10 and 0.90. The fatigue coefficient
0 150 300 450 600 750 900 curve of Luo et al. [12] is denoted by green solid line with tri-
s (m) angles (△). In their study, 𝑓𝑎 was measured for pedestrians
carrying heavy items (10∼20 kg) by hand.
Figure 7: The handler group movement speed in the straight
corridor setup. Different symbols represent the speed of differ-
ent handler groups: black cross (Ч) for Group_1, blue rectan-
gle (□) for Group_2, purple circle (○) for Group_3, and red We applied piecewise linear regression [27] to quantify
triangle (△) for Group_4. those two regimes in the fatigue coefficient curve:
{
𝛽1 𝑠 if 𝑠 ≤ 𝑠Ч
𝑓𝑎 (𝑠) = (10)
from Fig. 7, one can observe that the handler group move- 𝛽1 𝑠 + 𝛽2 (𝑠 − 𝑠𝑥 ) if 𝑠 > 𝑠Ч
ment speed decreases as the distance traveled 𝑠 increases,
especially for first 300 𝑚. where 𝑠Ч is the breakpoint and 𝛽1 and 𝛽2 are regression coef-
To further quantify the tendency of decreasing travel speed ficients. We used nls function in R package to fit the piece-
due to fatigue effect, we evaluated the fatigue coefficient 𝑓𝑎 wise linear regression model in Eq. (10). It was estimated
based on the speed reduction with respect to the initial move- that the fatigue coefficient 𝑓𝑎 curve slope changes at break-
ment speed 𝑣0 . According to the work of Luo et al. [12], the point 𝑠Ч = 264.4 m. The slope for the first and second
fatigue coefficient 𝑓𝑎 is given as: segments are estimated as 𝛽1 = 4.076 Ч 10−4 and 𝛽1 +
𝑣0 − 𝑣𝑖 𝛽2 = 5.84 Ч 10−5 , respectively. The slope coefficients 𝛽1
𝑓𝑎 = . (9) and 𝛽1 + 𝛽2 indicate that, the speed decreases by roughly
𝑣0
4.08% and 0.58% on average for each increase of 100 m in
Here, 𝑣𝑖 indicates the current speed which was calculated distance traveled for 𝑠 ≤ 𝑠Ч and 𝑠 > 𝑠Ч , respectively. Table 4
for each one-way trip from one end to the other end of the depicts the fatigue coefficients estimated by piecewise linear
straight corridor (see Fig. 1(b)). As shown in Eq. (9), the regression including the breakpoint and slopes. In addition,
fatigue coefficient examines speed reduction ratio, so the fa- we also applied quantreg function in R package to perform
tigue effect can be compared among different handler groups piecewise linear quantile regression based on Eq. (10) in or-
even if they have different speed. In addition, the concept der to generate 80% prediction band showing possible up-
of fatigue coefficient is useful when one wants to estimate per and lower boundaries of data points. Table 5 depicts
the speed at a certain traversed distance based on the initial the slope coefficients estimated by piecewise linear quantile
speed. Based on Eq. (9), we evaluated the fatigue coefficient regression for 80% prediction band with quantiles 0.10 and
𝑓𝑎 for each handler group indicated in Table 2. Note that 0.90.
the current speed 𝑣𝑖 and fatigue coefficient 𝑓𝑎 are given as We compared the value of fatigue coefficient 𝑓𝑎 obtained
functions of distance traveled 𝑠, and 𝑣𝑖 is the average speed in this study with that one reported in the study of Luo et
calculated from each one-way trip. Figure 8 shows the rela- al. [12]. In their study, the handlers were walking on a cir-
tionship between 𝑓𝑎 and 𝑠. One can observe that the fatigue cular track while carrying heavy items (10∼20 kg) by hand.
coefficient values tend to rapidly increase in the beginning For the same initial speed (between 1.25 m/s and 1.75 m/s),
and then slowly grow. That is compatible with the decreas- at distance of 325 m, the value of 𝑓𝑎 measured in this study
ing trend of handler group movement speed shown in Fig. 7. (𝑓𝑎 = 0.11) is lower than the value (𝑓𝑎 = 0.42) reported
J. Kwak et al.: Preprint submitted to Elsevier Page 7 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 4 speed profile 𝑣̂ at interval 𝑖. The movement duration in the The fatigue coefficients estimated by piecewise linear regres- corner area 𝑡𝑐 is given as: sion: the breakpoint and slopes. 𝑠𝑐 𝑁𝑡 Coefficient Value 𝑡𝑐 = ∑𝑁 𝑡 . (14) 𝑣0 𝑣̂ 𝑖=1 𝑖 𝑠Ч 264.4 ± 24.7 𝛽1 4.076 Ч 10−4 The average normalized speed profile 𝑣̂ 𝑎𝑣𝑔 is given as: 𝛽2 −3.492 Ч 10−4 𝛽1 + 𝛽2 5.840 Ч 10−5 𝑁𝑡 1 ∑ (15) 𝑣̂ 𝑎𝑣𝑔 = 𝑣̂ 𝑁𝑡 𝑖=1 𝑖 Table 5 The slope coefficients estimated by piecewise linear quantile and if we approximate Eq. (15) to continuous space by uti- regression for 80% prediction band with quantiles 0.10 and lizing Eq. (8), 𝑣̂ 𝑎𝑣𝑔 becomes 0.90. Quantiles Coefficient 𝑣̂ 𝑎𝑣𝑔 = 𝑣̂ 𝑖 𝑑 𝑡̂ 0.1 0.9 ∫0 𝛽1 2.520 Ч 10−4 5.476 Ч 10−4 |1 = 𝑎0 𝑡̂ + 𝑎1 𝑡̂2 + 𝑎2 𝑡̂3 + 𝑎3 𝑡̂4 + 𝑎4 𝑡̂5 || (16) 1 1 1 𝛽2 −1.976 Ч 10−4 −4.714 Ч 10−4 2 3 4 5 |0 𝛽1 + 𝛽2 5.447 Ч 10−5 7.619 Ч 10−5 ( ) 1 1 1 = 𝑎0 + 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 . 2 3 4 by Luo et al. [12]. This is seemingly because maneuvering a Accordingly, Eq. (14) should read patient bed is physically less demanding than carrying heavy 𝑠𝑐 items by hand while walking. Although the data points ob- 𝑡𝑐 = . (17) tained from this study are scattered in Fig. 8, it still clearly 𝑣0 𝑣̂ 𝑎𝑣𝑔 illustrates the tendency of increasing fatigue coefficient 𝑓𝑎 Here, distance traveled in the corner area 𝑠𝑐 , entering speed as the distance 𝑠 increases. 𝑣0 , and average normalized speed profile 𝑣̂ 𝑎𝑣𝑔 are input pa- 3.3. Prediction of movement duration rameters. Based on Eq. (16), we can obtain 𝑣̂ 𝑎𝑣𝑔 = 0.874. Based on the average normalized speed curves shown in In order to reflect uncertainty in the movement duration Fig. 6(c), we can predict the movement duration of a patient of a bedridden patient (see Eq. (17)), we developed a prob- bed in the corner area (i.e., 𝑥 ≤ 5 and 𝑦 ≤ 5). We describe abilistic model based on the input parameters 𝑠𝑐 and 𝑣0 for the distance traveled in the corner area 𝑠𝑐 with the following movement duration prediction. Previous studies [28, 29] noted equation: that a probabilistic approach is beneficial to estimate evacu- ation time based on an empirical predictive model utilizing 𝑁𝑡 ∑ experimental data, especially when there exists notable vari- 𝑠𝑐 = 𝑣𝑖 Δ𝑡𝑖 ability in evacuation parameters such as premovement time 𝑖=1 (11) and movement time. In the presented experimental results, 𝑁𝑡 ∑ one can notice that the difference in basic movement charac- = Δ𝑡𝑖 𝑣𝑖 , teristics among handlers is considerable, thus our prediction 𝑖=1 model was developed in line with the probabilistic approach. where 𝑁𝑡 is the number of intervals and 𝑣𝑖 is the speed at Analogous to previous studies [9, 10, 11], the evacuation in interval 𝑖. The interval length Δ𝑡𝑖 can be obtained by rear- this study refers to the movement of a bedridden patient mov- ranging Eq. (11): ing with a group of handlers to the place of safety. In the development of probabilistic prediction model, we assumed that 𝑠𝑐 and 𝑣0 follow Gaussian distribution. Ta- 𝑠𝑐 Δ𝑡𝑖 = ∑𝑁𝑡 . ble 6 shows the probabilistic model input parameters. We set (12) 𝑣 𝑖=1 𝑖 the mean and standard deviation values of the input parame- ters same as in the experimental results. The maximum and The movement duration in the corner area 𝑡𝑐 is a summation minimum values were determined by adding and subtracting of Δ𝑡𝑖 , twice of the standard deviation to and from the mean value, 𝑁𝑡 respectively. Based on Eq. (17) with the probabilistic model ∑ 𝑡𝑐 = Δ𝑡𝑖 input parameters in Table 6, we performed 1000 simulations. 𝑖=1 (13) The frequency histogram of the movement duration is shown = 𝑁𝑡 Δ𝑡𝑖 . in Fig. 9, which has a mean of 8.85 s and a standard devia- tion of 2.79 s. Its range is [5.01, 18.62] s. The experimental The speed at interval 𝑖 can be obtained from 𝑣̂ 𝑖 by multiply- results of movement duration [5.67, 11.80] s all fall within ing entering speed 𝑣0 , i.e., 𝑣𝑖 = 𝑣0 𝑣̂ 𝑖 . Here, 𝑣̂ 𝑖 is normalized the simulated distribution. This indicates that our prediction
J. Kwak et al.: Preprint submitted to Elsevier Page 8 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 6
Probabilistic model input parameters.
parameters mean sd min max 𝑠𝑐 7.65 0.19 7.27 8.03 𝑣0 1.07 0.29 0.49 1.65
y (m)
Frequency
60 -15
-15 0 15 30 45 60 x (m)
30 Figure 10: Schematic representation of bedridden patient evacuation route in current Singapore General Hospital Emer-
0 gency Department (SGH ED). The evacuation route starts
4 6 8 10 12 14 16 from the critical care unit (red square) which is placed at (0, 0)
Movement duration (s) to the place of safety (green triangle) at (45, 55). The evac-
Figure 9: Frequency histogram of movement duration in a uation route includes three straight corridor sections and two corner area estimated from numerical simulation. Red shaded corner areas. The evacuation route is 2 m wide, same as the area indicates the range of movement duration measured from experiment condition.
the experiment.
Table 7
Bedridden patient evacuation route segments model of movement duration can successfully replicate the experimental data of movement duration. Segment type from to Length (m)
In order to reflect the effect of fatigue on travel time, 1 Straight corridor (0, 0) (0, 11) we considered the fatigue coefficient 𝑓𝑎 and the relationship 2 Corner (0, 11) (4, 15) 7.65 ± 0.19 with the distance traveled 𝑠 in the estimation of movement 3 Straight corridor (4, 15) (41, 15) duration. Like Eq. (13), the movement duration in a straight 4 Corner (41, 15) (45, 19) 7.65 ± 0.19 corridor 𝑡𝑠 is given as 5 Straight corridor (45, 19) (45, 55)
𝑁𝑠 ∑ 𝑡𝑠 = Δ𝑡𝑗 , (18) 3.4. Case study: Singapore General Hospital 𝑖=1 Emergency Department where Δ𝑡𝑗 is the travel time in segment 𝑗 and 𝑁𝑠 is the num- In this case study, we demonstrate how the presented ber of segments. Note that Δ𝑡𝑗 is not constant here, meaning approach can be applied to predict the movement duration that the travel speed is different for each segment. The nor- of a handler group transporting multiple bedridden patients. malized speed 𝑣̂ is obtained after we rearranged Eq. (9), We selected the current Singapore General Hospital Emer- gency Department (SGH ED) for the case study. In Fig. 10, 𝑣 we present a schematic representation of the bedridden pa- 𝑣̂ = 𝑣0 = 1 − 𝑓𝑎 , (19) tient evacuation route in the current SGH ED. The evacua- again, 𝑣0 is entering speed. The average travel speed in seg- tion route starts from the critical care unit (CCU) at (0, 0) ment 𝑗 is given as to the place of safety at (45, 55). The evacuation route is 2 m wide (the same width as our experiment condition), and Δ𝑠𝑗 𝑣𝑗−1 + 𝑣𝑗 includes three straight corridor sections and two corner ar- = , (20) eas, as shown in Table 7. Note that we modeled CCU and Δ𝑡𝑗 the place of safety as single points in order to focus on the where Δ𝑠𝑗 is the distance traveled in segment 𝑗 and 𝑣𝑗−1 is evacuation of bedridden patients between those areas. The the speed at the beginning point of segment 𝑗 and 𝑣𝑗 for the actual geometry of the areas is much more complicated, thus end point of segment 𝑗. After combining Eqs. (19) and (20), the movement duration within the areas is not considered for we obtained the travel time of segment 𝑗 as simplicity. As in Section 3.3, the evacuation time in this case ( ) study was estimated for a patient bed moving with two han- Δ𝑠𝑗 2 dlers. Another important point to note is that evacuation of Δ𝑡𝑗 = . (21) 𝑣0 𝑣̂ 𝑗−1 + 𝑣̂ 𝑗
J. Kwak et al.: Preprint submitted to Elsevier Page 9 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 8
Probabilistic model input parameters.
parameters mean sd min max 𝑠𝑐 7.65 0.19 7.27 8.03 𝑣0 1.07 0.29 0.49 1.65 𝑡𝑝 7.96 5.19 4.10 17.64 𝑡𝑞 6.25 2.60 2.54 10.25 𝑠Ч 264.4 24.7 215.0 313.8 𝛽1 4.076 Ч 10−4 7.780 Ч 10−5 2.520 Ч 10−4 5.476 Ч 10−4 𝛽2 −3.492 Ч 10−4 7.580 Ч 10−5 −4.714 Ч 10−4 −1.976 Ч 10−4 𝛽1 + 𝛽2 5.840 Ч 10−5 8.895 Ч 10−6 5.447 Ч 10−5 7.619 Ч 10−5
a bedridden patient with a handler group cannot be simply extended to the case of a population of bedridden patients with multiple handler groups.This is mainly because the in- teractions among the handler groups are likely to affect their
Frequency movement speed We estimated the movement duration of a handler group carrying a bedridden patient from CCU to the place of safety five times. It is reasonable to suppose that the handler group in this case study were expected to move the patient bed as fast as possible and their movement characteristics are the same as in the experimental study. Equations (17) and (21) were applied to estimate the movement duration in the cor- 600 900 1200 1500 1800 2100 2400 2700 ner areas and straight corridor sections, respectively. We as- Movement duration (s) sumed that the fatigue effect is in effect both in the outgoing and return trips to the CCU, thus we applied the same fatigue Figure 11: Numerical simulation results of movement duration for five round trips between CCU and the place of safety. The coefficient function in Eq. (10) for the both trips. probability of finishing the movement within the reference time In addition to the distance traveled in the corner area 𝑠𝑐 𝑇𝑟𝑒𝑓 = 999.09 s is 23.6%. However, 76.4% of prediction results and entering speed 𝑣0 shown in Table 6, we introduced addi- are longer than the reference time (indicated by the red shaded tional input parameters to the probabilistic model: prepara- area), indicating that the reference time is not likely to be tion time 𝑡𝑝 , positioning time 𝑡𝑞 , breakpoint 𝑠Ч , and fatigue achievable. Here, the reference time 𝑇𝑟𝑒𝑓 is computed with coefficient curve slopes 𝛽1 and 𝛽2 . The preparation time is the traditional deterministic evaluation method using the mean length of time required to prepare a patient bed for move- value of input parameters. ment from CCU and the positioning time is the time taken to place a patient bed at the place of safety after its arrival. We set the mean and standard deviation of 𝑡𝑝 and 𝑡𝑞 based on bility that an evacuation can be safely performed within cer- the observation reported by Strating (2013) [10], and based tain amount of time. Table 9 shows the probability and the on Tables 4 and 5 for 𝑠Ч , 𝛽1 , 𝛽2 , and 𝛽1 + 𝛽2 . corresponding available safe egress time (ASET) that a han- Based on our movement duration prediction model with dler group can safely evacuate five bedridden patients. Note the probabilistic model input parameters in Table 8, we per- that the handler group can transport one bedridden patient formed 10000 simulations. In each simulation, the handler to the place of safety for each round trip. For instance, the group makes five round trips between CCU and the place of probability of successful evacuation is estimated as 90% if safety. The frequency of histogram of the movement dura- ASET is 1773.08 s. In fire evacuation studies, ASET is the tion is shown in Fig. 11, which has a mean of 1272.5 s and duration measured from the start of a fire to the onset of haz- a standard deviation of 375.46 s. The estimated movement ardous conditions in which people cannot be evacuated due duration ranges from 718.1 s to 2699.7 s. One can observe to the fire. ASET is influenced by various factors including a large spread of the numerical simulation results presented fire detection devices, location of fire origin, growth of fire, in Fig. 11 due to the right-skewed distributions of 𝑡𝑐 and 𝑡𝑠 . and ventilation paths [30]. We refer the readers to Appendix A for further details. In order to quantify the effect of uncertainty in the in- By introducing a probabilistic model of movement dura- put parameters, we compared the movement duration pre- tion prediction, we can consider the effect of uncertain fac- diction results against the reference time which is based on tors in evacuation process, like entering speed, preparation the traditional deterministic evaluation method. As stated in time, positioning time, and fatigue effect coefficients. Based Zhang et al. [29], the value of reference time was computed on the results shown in Fig. 11, one can estimate the proba- based on the mean value of input parameters: evacuation
J. Kwak et al.: Preprint submitted to Elsevier Page 10 of 14
Patient Bed Horizontal Movement Performance Estimation
Table 9 moving patient beds. It is also noted that the handlers in The probability and the corresponding available safe egress the experiment were not necessarily in a rush as they would time (ASET) that a handler group can safely evacuate five be in the real emergency evacuation. Thus, there exists the bedridden patients. Note that the handler group can transport possibility that the movement speed of handlers in the real one bedridden patient to the place of safety for each round emergency evacuation can be higher than the speed reported trip. in this study. The presented experiment results can be gener- alized with larger number of handler groups having different corresponding Probability conditions like age and proficiency. ASET (s) There are noteworthy limitations in terms of the study 10% 892.89 scope. This study was performed for the case of a patient 20% 973.72 bed movement. The patient bed movement was not inter- 30% 1041.17 rupted by other patient beds and ambulant pedestrians, thus 40% 1109.65 their influence was not reflected in the presented results. In emergency evacuations, multiple groups of handlers are try- 50% 1181.31 ing to evacuate bedridden patients in a short time and they 60% 1266.27 are likely to move in the same evacuation route at the same 70% 1368.30 time. In addition, some healthcare facilities do not have a 80% 1506.48 dedicated evacuation route for the bedridden patients, so it 90% 1773.08 is expected that a considerable number of ambulant pedes- trians share the evacuation routes with the bedridden pa- tients. Consequently, the number of interactions among pa- route length (𝑙 = 99.3 m), preparation time (𝑡𝑝 = 7.96 s), tient beds and ambulant pedestrians is increasing, potentially positioning time (𝑡𝑞 = 6.25 s), and entering speed (𝑣0 = leading to longer evacuation time. Further experimental stud- 1.07 m/s). For a handler group making five round trips be- ies need to be carried out in order to consider the interac- tween CCU and the place of safety, the reference time 𝑇𝑟𝑒𝑓 tions among bedridden patients and ambulant pedestrians is computed as and their impact on the evacuation performance of patient beds. 𝑇𝑟𝑒𝑓 = 𝑀(𝑡𝑝 + 𝑡𝑞 + 𝑡𝑡 ), (22) Future studies can be also planned from the perspective of emergency evacuation simulations. Hunt et al. [31] pointed where 𝑀 is the number of round trips that the handler group out that the movement of evacuation devices has not been makes and 𝑡𝑡 = 2𝑙∕𝑣0 is the time required for the handler appropriately considered in existing evacuation simulation group to make a round trip between CCU and the place of models although the movement of such devices is critical for safety excluding the time for preparation and positioning. As vulnerable patients. We are currently developing an evac- can be seen from Fig. 11, the probability that the handlers uation simulation model that can incorporate our findings evacuate five bedridden patients within the reference time in patient bed movement dynamics to explicitly reflect their 𝑇𝑟𝑒𝑓 = 999.09 s is 23.6% while 76.4% of prediction results movement during the emergency evacuations in healthcare are longer than the reference time. It appears that the tra- facilities. The evacuation simulation models can check po- ditional deterministic evaluation method based on the mean tential conflicts with other evacuees and geometric elements value of input parameters underestimates the movement du- such as doors and corners, and predict the total evacuation ration. time for the scenarios in which patient beds are fleeing with other evacuees. 3.5. Limitations and future work The experiment data was collected from a small num- ber of handlers and they are all young adults who are not 4. Conclusion professionally trained. When we set up the experiment, we We performed a series of controlled experiments to study were mainly interested in observing difference in male and the dynamics of patient beds in horizontal movement. In our female groups, so we did not consider a group of mixed gen- experiments, we examined the change of velocity in corner ders. The weight of the bedridden patient was fixed and turning movements and speed reductions in multiple trips medical equipment was not connected to the patient. The between both ends of a straight corridor. In the corner turn- fatigue effect was modeled as a function of distance traveled ing movements, we observed that the start and end of turn- only, not considering other factors such as the temperature ing movement can be identified based on the trajectory de- and stress level perceived by the handlers. In addition, the viation. We also quantified common patterns in different movement speed at which the handlers can move might be speed curves by means of the normalized speed profile with affected by the mental and physical condition of the patients, rescaled time. In the straight corridor experiment, we dis- for instance, the anxiety of patients and use of intravenous covered experimental relationship between the fatigue co- fluid drips. The experiment results might be different for the efficient and distance traveled 𝑠 ≤ 882 m. Based on the mix of genders in the handler groups, weight of bedridden experiment results, we developed a movement duration pre- patient and medical equipment, and handlers’ proficiency in diction model and then applied the model for a patient bed
J. Kwak et al.: Preprint submitted to Elsevier Page 11 of 14
Patient Bed Horizontal Movement Performance Estimation
∑
horizontally moving in a healthcare facility. In order to re- 𝑡𝑠,𝑗 :
flect uncertainty in the horizontal movement, we introduced
a probability distribution to the horizontal movement param- 𝑇1 = 𝑡𝑝 + 𝑡𝑞 + 𝑡𝑐 + 𝑡𝑠
eters like entering speed, preparation and positioning time, 𝑁𝑐
∑ 𝑁𝑠
∑ (23)
and fatigue coefficients. According to our case study results, = 𝑡𝑝 + 𝑡𝑞 + 𝑡𝑐,𝑖 + 𝑡𝑠,𝑗 ,
one can estimate the probability that an evacuation can be 𝑖=1 𝑖=1
safely performed within certain amount of time. In addition,
where 𝑁𝑐 = 4 is the number of corner areas in a round trip
it is highly probable that the horizontal movement duration
and 𝑁𝑠 is the number of straight corridor segments in a round
would be longer than the prediction results from an exist-
trip. Travel time in corner area 𝑖 is denoted by 𝑡𝑐,𝑖 and 𝑡𝑠,𝑗
ing model which assumes constant movement speed. The
for straight corridor section 𝑗. Parameters 𝑡𝑝 and 𝑡𝑞 were
case study results demonstrated that our model has potential
assumed to follow normal distribution. The distribution of
in predicting emergency evacuation time of patient beds in
𝑡𝑐 can be explained by means of the ratio distribution and
healthcare facilities.
the distribution of 𝑡𝑠 can be approximated by the reciprocal
As a first step to study the dynamics of patient bed move-
normal distribution.
ment in horizontal space, we focused on the case of a patient
As presented in Eq. (17), travel time in corner area 𝑖 is
bed movement based on simple experiment setups. Due to
given as
the scope and setups of this study, the presented results are
not applicable to the scenarios in which the movement of the 𝑠𝑐
patient bed is interrupted by other patient beds and ambulant
𝑡𝑐,𝑖 =
𝑣0 𝑣̂ 𝑎𝑣𝑔
. (24)
pedestrians. In addition, the readers should bear in mind
that the experiment data was collected from a small num- Here, the distance traveled in the corner area 𝑠𝑐 and the en-
ber of handlers and the bedridden patient was transported tering speed 𝑣0 are assumed to follow the normal distribu-
without medical eqipment. To generalize the findings of this tion while the average normalized speed profile 𝑣̂ 𝑎𝑣𝑔 is set
study, the presented experiment should be replicated with as a fixed value based on the experimental study result. That
larger number of handlers and different conditions of bedrid- is, 𝑡𝑐,𝑖 follows a distribution of a ratio of two independent
den patients. Further experimental studies need to be car- normally distributed variables 𝑠𝑐 and 𝑣0 . According to Cur-
ried out with more bedridden patients and ambulant pedes- tiss [32] and Springer [33], the ratio distribution of 𝑡𝑐,𝑖 can
trians to quantify the impact of interactions with them on the be described by means of the joint distribution of 𝑠𝑐 and 𝑣0 :
evacuation time. Another interesting extension of the pre-
sented study can be planned from the perspective of emer-
gency evacuation simulations to predict the evacuation time 𝑓 (𝑠𝑐 , 𝑣0 ) = Ч
for various scenarios. 2𝜋𝜎𝑠,𝑐 𝜎𝑣,0
{ [( ) ( ) ]}
1 𝑠𝑐 − 𝜇𝑠,𝑐 2 𝑣0 − 𝜇𝑣,0 2
exp − + ,
Acknowledgements 2 𝜎𝑠,𝑐 𝜎𝑣,0
This research is supported by National Research Founda- (25)
tion (NRF) Singapore, GOVTECH under its Virtual Singa-
pore program Grant No. NRF2017VSG-AT3DCM001-031. where 𝜇𝑠,𝑐 = 7.65 and 𝜎𝑠,𝑐 = 0.19 are the mean and standard
The experiment was organized with the help of Ms. Yo- deviation of Gaussian distributed 𝑠𝑐 > 0, and 𝜇𝑣,0 = 1.07
geswary Pasupathi (Research nurse at the Singapore General and 𝜎𝑣,0 = 0.29 for Gaussian distributed 𝑣0 > 0. The prob-
Hospital Emergency Department) and members of Complex- ability function of 𝑡𝑐,𝑖 is given as
ity Institute, Nanyang Technological University, Singapore.
We thank Mr. Joshua for his help in processing video records. 𝑃 (𝑡𝑐,𝑖 )
+∞ ( )
= |𝑣 | 𝑓 𝑠 , 𝑣 𝑑𝑣
A. Distribution of movement duration ∫−∞ | 0 | 𝑐 0
+∞ ( )
In Sections 3.3 and 3.4, we presented the distribution of = |𝑣 | 𝑓 𝑣 𝑡 𝑣̂ , 𝑣 𝑑𝑣
movement duration in a corner area and in the case study, ∫−∞ | 0 | 0 𝑐,𝑖 𝑎𝑣𝑔 0 0
+∞
respectively (see Figs. 9 and 11). This appendix provides
= 𝑣0 Ч
evidence to suggest that the spread of movement duration 2𝜋𝜎𝑠,𝑐 𝜎𝑣,0 ∫0
can be large. { [( ) ( ) ]}
1 𝑣0 𝑡𝑐,𝑖 𝑣̂ 𝑎𝑣𝑔 − 𝜇𝑠,𝑐 2 𝑣0 − 𝜇𝑣,0 2
According to the setup of our numerical simulation study exp − + 𝑑𝑣0 .
presented in Section 3.4, the movement duration 𝑇1 for a 2 𝜎𝑠,𝑐 𝜎𝑣,0
round trip is given as a sum of preparation time 𝑡𝑝 , position- (26)
ing time 𝑡𝑞 , and total travel time in the corner areas 𝑡𝑐 =
∑
𝑡𝑐,𝑖 , and total travel time in the straight corridors 𝑡𝑠 = Based on Eq. (26), we theoretically predicted the frequency
histogram of movement duration in a corner area and com-
pared the result against one obtained from the numerical
J. Kwak et al.: Preprint submitted to Elsevier Page 12 of 14
Patient Bed Horizontal Movement Performance Estimation
150
120
Frequency
Frequency
90
60
30
0 0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 Movement duration (s) Movement duration (s) Figure 12: Frequency histogram of movement duration in a Figure 13: Frequency histogram of movement duration in the corner area estimated from numerical simulation (blue bars) straight corridor sections (total length 𝑠𝑗 = 168 m) estimated and theoretically predicted by the ratio distribution (red solid from numerical simulation (blue bars) and theoretically pre- line). We performed 1000 numerical simulations. dicted by the ratio distribution (red solid line). We performed 1000 numerical simulations.
simulation, see Fig. 12. As can be seen from Fig. 12, the Table 10 ratio distribution produced a right-skewed curve and the nu- Mode values of parameters merical simulation result is in good agreement with the the- oretical prediction. parameters mode note
As shown in Eq. (21), the travel time in the straight cor- 𝑡𝑝 7.96 Normal distribution ridor section 𝑗 is given as 𝑡𝑞 6.25 Normal distribution
( ) 𝑡𝑐 29.6 Ratio distribution
Δ𝑠𝑗 2 𝑡𝑠 138.7 Reciprocal normal distribution 𝑡𝑠,𝑗 = 𝑣0 𝑣̂ 𝑗−1 + 𝑣̂ 𝑗
(27) where 𝜇𝑣, 𝑗−1 = 1.07 and 𝜎𝑣,𝑗−1 = 0.29 are mean and stan-
2Δ𝑠𝑗
= . dard deviation of 𝑣𝑗−1 , respectively. Based on Eq. (29), we 𝑣𝑗−1 + 𝑣𝑗 theoretically predicted the frequency histogram of movement duration in the straight corridor and compared the result against
Here, Δ𝑠𝑗 is a fixed length of straight corridor section 𝑗, and one obtained from the numerical simulation, see Fig. 13. As the speed at the beginning and end points of the straight cor- can be seen from Fig. 13, the reciprocal normal distribution ridor section are denoted by 𝑣𝑗−1 and 𝑣𝑗 , respectively. It is generated a right-skewed curve that is consistent with the assumed that the value of 𝑣𝑗−1 follows the normal distribu- numerical simulation result.
tion and the value of 𝑣𝑗 is determined based on 𝑣𝑗−1 and the
We computed an estimate of the movement duration for fatigue effect. It can be inferred from Section 3.2 that the fa- a round trip by summing the mode values of 𝑡𝑝 , 𝑡𝑞 , 𝑡𝑐 , and 𝑡𝑠 tigue effect becomes considerable when the handler group shown in Table 10, i.e., 𝑇1,𝑒𝑠𝑡 = 7.96 + 6.25 + 29.6 + 138.7 = travels several hundred meters. In a round trip, the total
∑ 182.51 s. One can notice that the total travel time in the cor- length of straight corridor sections is 𝑠𝑗 = Δ𝑠𝑗 = 168 m, ner and straight corridor areas (i.e., 𝑡𝑐 and 𝑡𝑠 ) take a signif- thus it may be reasonable to suppose that the difference be- icant portion of movement duration 𝑇1,𝑒𝑠𝑡 . Due to the right- tween 𝑣𝑗−1 and 𝑣𝑗 is not significant. For simplicity, we ap- skewed distributions of 𝑡𝑐 and 𝑡𝑠 , the spread of the simulated proximate 𝑣𝑗 to 𝑣𝑗−1 and then simplify the equation of 𝑡𝑠,𝑗 as results presented in Figs. 9 and 11 is large. Given that the a function of the total length of straight corridor sections 𝑠𝑗 movement duration is inversely related to the entering speed, and the speed at the beginning point 𝑣𝑗−1 :
a small decrease in the entering speed might yield a large in- 𝑠𝑗 crease in the movement duration, extending the boundary of 𝑡𝑠,𝑗 ≈ , (28) the histogram graph rightward.
𝑣𝑗−1
indicating that the distribution of 𝑡𝑠,𝑗 is can be described by a reciprocal normal distribution. According to Gurarie et al. [34], the probability distribution function of 𝑡𝑠,𝑗 is given as
[ ( )2 ] 𝑠𝑗 1 𝑠𝑗 − 𝜇𝑣, 𝑗−1 𝑡𝑠,𝑗
𝑃 (𝑡𝑠,𝑗 ) = √ exp − ,
2𝜋𝜎𝑣,𝑗−1 𝑡𝑠,𝑗 2 𝜎𝑣,𝑗−1 𝑡𝑠,𝑗
(29)
J. Kwak et al.: Preprint submitted to Elsevier Page 13 of 14
Patient Bed Horizontal Movement Performance Estimation
References The formation of trajectories during goal-oriented locomotion in hu- mans. I. A stereotyped behaviour, European Journal of Neuroscience
[1] T. Kretz, A. Grьnebohm, M. Schreckenberg, Experimental study of 26 (2007) 2376–2390.
pedestrian flow through a bottleneck, Journal of Statistical Mechan- [23] C. Dias, R. Lovreglio, Calibrating cellular automaton models for ics: Theory and Experiment 2006 (2006) P10014+. pedestrians walking through corners, Physics Letters A 382 (2018)
[2] S. Hoogendoorn, W. Daamen, Pedestrian behavior at bottlenecks, 1255–1261.
Transportation Science 39 (2005) 147–159. [24] C. Dias, M. Abdullah, M. Sarvi, R. Lovreglio, W. Alhajyaseen, Mod-
[3] A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht, eling and simulation of pedestrian movement planning around cor-
W. Klingsch, New insights into pedestrian flow through bottlenecks, ners, Sustainability 11 (2019) 5501.
Transportation Science 43 (2009) 395–406. [25] N. Hogan, An organizing principle for a class of voluntary move-
[4] F. Huo, W. Song, L. Chen, C. Liu, K. Liew, Experimental study on ments, Journal of Neuroscience 4 (1984) 2745–2754.
characteristics of pedestrian evacuation on stairs in a high-rise build- [26] Q. C. Pham, H. Hicheur, G. Arechavaleta, J. P. Laumond, A. Berthoz, ing, Safety Science 86 (2016) 165–173. The formation of trajectories during goal-oriented locomotion in hu-
[5] S. Burghardt, A. Seyfried, W. Klingsch, Performance of mans. II. A maximum smoothness model, European Journal of Neu- stairs–fundamental diagram and topographical measurements, Trans- roscience 26 (2007) 2391–2403.
portation Research Part C: Emerging Technologies 37 (2013) 268– [27] R. Hocking, Methods and applications of linear models: regression
278. and the analysis of variance, John Wiley & Sons, Hoboken, NJ, 3rd
[6] R. Nagai, M. Fukamachi, T. Nagatani, Evacuation of crawlers and edition, 2013.
walkers from corridor through an exit, Physica A: Statistical Me- [28] E. Ronchi, E. D. Kuligowski, R. D. Peacock, P. A. Reneke, A proba- chanics and its Applications 367 (2006) 449–460. bilistic approach for the analysis of evacuation movement data, Fire
[7] R. Kady, J. Davis, The effect of occupant characteristics on crawling Safety Journal 63 (2014) 69–78.
speed in evacuation, Fire Safety Journal 44 (2009) 451–457. [29] G. Zhang, D. Huang, G. Zhu, G. Yuan, Probabilistic model for safe
[8] X. Lu, Z. Yang, G. Cimellaro, Z. Xu, Pedestrian evacuation simula- evacuation under the effect of uncertain factors in fire, Safety Science tion under the scenario with earthquake-induced falling debris, Safety 93 (2017) 222–229.
Science 114 (2019) 61–71. [30] L. Y. Cooper, A concept for estimating available safe egress time in
[9] L. Rubadiri, D. Ndumu, J. Roberts, Predicting the evacuation ca- fires, Fire Safety Journal 5 (1983) 135–144.
pability of mobility-impaired occupants, Fire Technology 33 (1997) [31] A. Hunt, E. Galea, P. Lawrence, I. Frost, S. Gwynne, Simulating
32–53. movement devices used in hospital evacuation, Fire Technology
[10] N. Strating, Evacuation of bedridden building occupants, Master’s (2020) 1–32.
thesis, Eindhoven University of Technology, The Netherlands, 2013. [32] J. H. Curtiss, On the distribution of the quotient of two chance vari-
[11] A. Hunt, E. Galea, P. Lawrence, An analysis and numerical simulation ables, The Annals of Mathematical Statistics 12 (1941) 409–421.
of the performance of trained hospital staff using movement assist [33] M. D. Springer, The algebra of random variables, John Willey & Sons, devices to evacuate people with reduced mobility, Fire and Materials New York, NY, 1979.
39 (2015) 407–429. [34] E. Gurarie, J. J. Anderson, R. W. Zabel, Continuous models of
[12] L. Luo, Z. Fu, X. Zhou, K. Zhu, H. Yang, L. Yang, Fatigue effect on population-level heterogeneity inform analysis of animal dispersal phase transition of pedestrian movement: experiment and simulation and migration, Ecology 90 (2009) 2233–2242.
study, Journal of Statistical Mechanics: Theory and Experiment 2016
(2016) 103401.
[13] X. Shi, Z. Ye, N. Shiwakoti, D. Tang, J. Lin, Examining effect of ar- chitectural adjustment on pedestrian crowd flow at bottleneck, Phys- ica A: Statistical Mechanics and its Applications (2019) 350–364.
[14] C. Dias, O. Ejtemai, M. Sarvi, N. Shiwakoti, Pedestrian walking char- acteristics through angled corridors: An experimental study, Trans- portation Research Record 2421 (2014) 41–50.
[15] N. Shiwakoti, Y. Gong, X. Shi, Z. Ye, Examining influence of merging architectural features on pedestrian crowd movement, Safety Science
75 (2015) 15–22.
[16] Aneurin Bevan Health Board, Guidance for the transfer of patients between wards and departments within hospitals, Technical Report,
2012.
[17] Lund University Traffic Research Team, T-Analyst Software
(https://bitbucket.org/TrafficAndRoads/tanalyst/wiki/Manual), 2019
(last accessed September 15, 2020).
[18] A. Fyhri, H. Sundfшr, T. Bjшrnskau, A. Laureshyn, Safety in numbers for cyclists – conclusions from a multidisciplinary study of seasonal change in interplay and conflicts, Accident Analysis & Prevention
105 (2017) 124–133.
[19] S. Nielsen, R. Gade, T. Moeslund, H. Skov-Petersen, Taking the tem- perature of pedestrian movement in public spaces, in: W. Daamen,
D. Duives, S. Hoogendoorn (Eds.), The Conference on Pedestrian and
Evacuation Dynamics 2014 (PED2014), Delft, The Netherlands, pp.
660–668.
[20] R. Tsai, A versatile camera calibration technique for high-accuracy
3D machine vision metrology using off-the-shelf TV cameras and lenses, IEEE Journal on Robotics and Automation 3 (1987) 323–344.
[21] J. Chen, J. Wang, J. Wang, X. Liu, T. Li, P. Lin, An experimental study of individual ascent speed on long stair, Fire Technology 53
(2017) 283–300.
[22] H. Hicheur, Q. C. Pham, G. Arechavaleta, J. P. Laumond, A. Berthoz,
J. Kwak et al.: Preprint submitted to Elsevier Page 14 of 14