2767 The Journal of Experimental Biology 211, 2767-2772 Publishe by The Company of Biologists 2008 oi:101242/jeb020073 The effects of gravity on human walking: a new test of the ynamic similarity hypothesis using a preictive moel Davi A Raichlen Department of Anthropology, University of Arizona, 1009 E South Campus Drive, Tucson, AZ 85721, USA e-mail: raichlen@emailarizonaeu Accepte 26 June 2008 SUMMARY The ynamic similarity hypothesis (DSH) suggests that ifferences in animal locomotor biomechanics are ue mostly to ifferences in size Accoring to the DSH, when the ratios of inertial to gravitational forces are equal between two animals that iffer in size [eg at equal Froue numbers, where Froue = velocity 2 /(gravity hip height)], their movements can be mae similar by multiplying all time urations by one constant, all forces by a secon constant an all linear istances by a thir constant The DSH has been generally supporte by numerous comparative stuies showing that as inertial forces iffer (ie ifferences in the centripetal force acting on the animal ue to variation in hip heights), animals walk with ynamic similarity However, humans walking in simulate reuce gravity o not walk with ynamically similar kinematics The simulate gravity experiments i not completely account for the effects of gravity on all boy segments, an the importance of gravity in the DSH requires further examination This stuy uses a kinematic moel to preict the effects of gravity on human locomotion, taking into account both the effects of gravitational forces on the upper boy an on the limbs Results show that ynamic similarity is maintaine in altere gravitational environments Thus, the DSH oes account for ifferences in the inertial forces governing locomotion (eg ifferences in hip height) as well as ifferences in the gravitational forces governing locomotion Key wors: strie length, Froue number, force-riven harmonic oscillator, inertial properties INTRODUCTION The ynamic similarity hypothesis (DSH) (Alexaner, 1976; Alexaner an Jayes, 1983) is one of the most general an useful principles in animal locomotion, allowing researchers to compare movement patterns in taxa that iffer greatly in size The DSH is the ynamic analog of geometric similarity an suggests that the biomechanics of geometrically similar animals can be mae ientical by multiplying all time urations by one constant, all forces by a secon constant an all linear istances by a thir constant Support for the DSH in empirical comparisons of gait inicates that locomotor ifferences are largely explaine by ifferences in size (Alexaner an Jayes, 1983) Thus, the DSH allows researchers to explore unerlying rules that govern animal locomotion (eg Alexaner an Jayes, 1983) an also provies a basis for unerstaning the evolutionary importance of eviations from ynamic similarity (eg Raichlen, 2006) Alexaner note that ynamic similarity is only possible, an therefore only testable, when two animals have equal ratios of the inertial an gravitational forces governing locomotion (Alexaner, 1976; see also Alexaner an Jayes, 1983) For terrestrial locomotion, the inertial force is generally assume to be the centripetal force that acts on an animal as it vaults over its stance leg, which acts as an inverte penulum (see Donelan an Kram, 1997) Therefore, the ratios of inertial to gravitational forces in two animals are equal when they walk at the same Froue number [velocity 2 /(gravity hip height)] Although many researchers rely on the DSH in stuies of comparative biomechanics, recent work has questione its valiity (see Donelan an Kram, 1997; Donelan an Kram, 2000) The purpose of the present stuy is to test the DSH using a preictive kinematic moel to assess its utility in unerstaning human locomotion Most tests of ynamic similarity examine the impacts of changes in inertial forces on ynamic similarity (eg through ifferences in limb length in comparative stuies) (see Alexaner an Jayes, 1983; Alexaner an Maloiy, 1984; Gatesy an Biewener, 1991; Bullimore an Burn, 2006) The DSH has been broaly supporte by stuies showing that animals that iffer in size generally use equal relative strie lengths (strie length ivie by hip height) when walking at the same Froue numbers an they transition from a walk to a run at equal Froue numbers (Alexaner an Jayes, 1983) Donelan an Kram, noting the importance of gravity in the DSH, suggeste that ynamic similarity shoul account for locomotor ifferences not only ue to size but also ue to changes in gravitational environments (Donelan an Kram, 1997) Thus, as gravitational forces change, iniviuals walking at the same Froue numbers shoul continue using equal relative strie lengths an shoul transition from a walk to a run at the same Froue numbers In an innovative test of the effects of gravity on the DSH in humans, Donelan an Kram use a treamill that alters the effects of gravity on locomotion by introucing an ajustable upwar force on the boy through a harness system attache to the torso (Donelan an Kram, 1997; Donelan an Kram 2000; see also Kram et al, 1997) Results from these earlier stuies showe that humans eviate from ynamic similarity as gravity was reuce As gravity ecrease, humans walke with relatively shorter stries (Donelan an Kram, 1997), an the walk run transition occure at higher Froue numbers in very low gravitational fiels (Kram et al, 1997) Thus, the DSH may not be a governing principle of animal
2768 D A Raichlen locomotion an shoul be use with caution (Donelan an Kram, 2000) Gravity an swing phase One critique of these stuies, fully acknowlege by the authors, is that their novel experimental esign i not alter the gravitational environment for the limbs uring swing phase (Donelan an Kram, 1997; Donelan an Kram, 2000) Gravity shoul have an important effect on swing phase, an possibly on overall strie kinematics, because limbs act somewhat like suspene penula (see Hilebran, 1985) Therefore, the uration of swing phase is relate to limb mass istribution an gravity, an the natural perio of the limb (T) is: T = 2 π (1) g where g is gravitational acceleration (981ms 2 on earth) an is the length of the limb penulum (m): = I (2) ml In Eqn 2, I is the limb s mass moment of inertia about the hip joint (kgm 2 ), m is the limb s mass (kg) an L is the istance of the limb s center of mass from the hip joint (m) If all else is equal, a relatively long swing perio (ue to either a relatively large or to reuce g) will lea to a relatively long strie perio (the sum of swing an stance urations) an a relatively low strie frequency (the reciprocal of strie uration) Since velocity (v) is equal to the prouct of strie length an strie frequency, low strie frequencies lea to long stries at a given spee Both comparative an experimental stuies support these connections between penular limb swing an strie lengths an strie frequencies (Inman et al, 1981; Martin, 1985; Holt et al, 1990; Skinner an Barrack, 1990; Steuel, 1990; Mattes et al, 2000; Raichlen, 2004; Raichlen, 2005; Raichlen, 2006) For example, when weights were affixe to the ankles of ogs an humans, leaing to a large an, therefore, a longer natural swing perio, strie lengths increase an strie frequencies ecrease (Inman et al, 1981; Martin, 1985; Holt et al, 1990; Skinner an Barrack, 1990; Steuel, 1990; Mattes et al, 2000) Comparative stuies of natural variation in limb mass istribution also support the links between limb swing an overall strie kinematics (Preuschoft an Gunther, 1994; Raichlen, 2004; Raichlen, 2005; Raichlen, 2006) Animals with large values of ue to heavy muscles in the hans an feet (such as primates with grasping extremities) use relatively longer stries an lower strie frequencies than animals with more proximally concentrate limb mass (Raichlen, 2004; Raichlen, 2005; Raichlen, 2006) It is important to note that gravity can still play a role in etermining limb swing even if the limbs o not swing as completely passive penula Holt an colleagues introuce a moel that preicts kinematics at preferre walking spees by assuming that the limb acts like a force-riven harmonic oscillator (FDHO) uring swing phase, accounting for not only gravitational forces but also for some muscle action uring swing (Holt et al, 1990) This moel consiers the limb to be a mostly passive penulum but oes inclue a constant to account for the amping effects of muscles an tenons an provies a riving force The FDHO successfully preicts strie frequencies an strie lengths at preferre walking spees uner a variety of conitions incluing forwars an backwars walking (Holt et al, 1990; Schot an Decker, 1998) an walking with ankle weights (Holt et al, 1990) Thus, experimental stuies, comparative biomechanics an biomechanical moels support the hypothesis that the limbs swing as suspene penula assiste by some egree of muscular action an uner the influence of gravity The present stuy examines the effects of gravity on ynamic similarity using a very simple kinematic moel that links limb mass istribution an swing kinematics to overall locomotor kinematics as a function of spee an gravitational forces The moel presente here expans on the FDHO to preict strie lengths over a range of spees an explicitly preicts the walk run transition spee The moel will therefore examine the parameters that eviate from ynamic similarity in previous reuce gravity experiments (Donelan an Kram, 1997; Kram et al, 1997) I use the moel presente here to test the hypothesis that humans o in fact walk with ynamic similarity in reuce gravity once the effects of altere gravity on limb swing are consiere Aitionally, since previous stuies have altere gravitational forces for the upper boy but not the limbs (Donelan an Kram, 1997; Kram et al, 1997; Donelan an Kram, 2000), the moel is use to examine the kinematic effects of altering gravitational acceleration on the boy alone in orer to compare moel preictions against previous treamill stuies MATERIALS AND METHODS Moel assumptions The moel evelope in the present stuy preicts strie length an the walk run transition spee base only on limb inertial properties an hip height There are three primary assumptions involve in the moel The first assumption is that swing phase follows the FDHO moel, where limb swing was moele as a suspene penulum, taking into account muscle an tenon amping of the oscillating limb (Holt et al, 1990) The secon assumption is that step length (the istance travele uring stance phase) remains constant over all walking spees an is equal to 95% of hip height This value is the maximum step length use uring walking in humans that still minimizes energy costs (see Srinivasan an Ruina, 2006) Finally, it is assume that at the walk run transition velocity, stance uration an swing uration are equal When walking, uty factor (stance uration/strie uration) is always greater than 050 (ie there is no aerial phase); when running, uty factor is less than 050, inicating an aerial phase Thus, the walk run transition shoul occur when uty factor is 050 (ie equal stance an swing urations) Moel evelopment Base on the assumption that the limbs swing as FDHOs, the perio of the limb is calculate following Turvey et al (Turvey et al, 1988; see also Holt et al, 1990) as: T = 2π m 2 (mg + kb 2 ) where k is the spring constant that represents the composite stiffness of limb muscles an tenons that ampen an rive limb oscillations (Nm 1 ) an b is the istance of the composite spring from the hip (m) Turvey an colleagues (Turvey et al, 1988; see also Holt et al, 1990) foun, experimentally, that kb 2, across mammals, is always some multiple of the gravitational forces (mg) acting on the limb, an thus the equation is reuce by assuming a constant ratio of kb 2 to mg It is important to note that the value of this ratio is somewhat arbitrary, may iffer among (3)
Gravity an ynamic similarity 2769 gaits an taxa an is foun by comparing moel preictions to experimental ata (see Turvey et al, 1988) The value that best fit human walking ata in the present stuy (see Results) was 35, an thus, Eqn 3 is reuce as follows: T = 2π m 2 (mg + kb 2 ) = 2π m 2 (45m g ) = 2π Because Eqn 4 represents one full oscillation, limb swing uration, (t sw ), is half this value: t sw =π Stance uration (t st ) is calculate by first assuming that step length (L st ; istance travele uring stance phase) oes not change with spee an is equal to: L st = t st v (6) Accoring to the secon assumption of the moel, human step lengths are equal to 95% of limb length (h) (see Srinivasan an Ruina, 2006) Thus, t st at any given velocity (v) is calculate from Eqn 6: t st = L st v 1 = 095hv 1 (7) All other spatio temporal kinematic variables are calculate from t st an t sw at a given velocity Strie uration (SD) is calculate as the sum of swing an stance urations Strie frequency (f) is the reciprocal of strie uration Finally, strie length (SL) is the prouct of SD an v: SD = t st + t sw, (8) f = 1/SD, (9) SL = vsd (10) By combining Eqns 8 10, SL at a given v is calculate base solely on h an limb : SL = v 095hv 1 +π Since the walk run transition occurs when swing an stance urations are equal, the walk run transition velocity (v wr ) is: v wr = L st t 1 sw = 095hπ Altering gravitational fiels If the ynamic similarity hypothesis is correct, when gravitational acceleration is altere in the moel, preicte relative strie lengths shoul be equal at the same Froue numbers In orer to preict the effects of altere gravity on kinematics an test for ynamic similarity, moel parameters are first converte into imensionless numbers Thus, a given velocity is converte into a Froue number (Fr) (imensionless velocity): Fr = v 2 gh Strie length is converte to a imensionless value (SL) as: 45g 45g 45g 1 (45g) (4) (5) (11) (12) (13) SL = SL h (14) Substituting Eqns 11 an 13 into Eqn 14, SL is calculate at any given velocity: a b π SL = 45g + 095h ghfr From Eqn 15, it is clear that gravity plays a major role in etermining SLs at a given spee in two ways (enote by the curve braces above): (a) by changing the calculation of swing uration, an (b) by changing calculation of velocity from Froue numbers To preict the effects of reuce gravitational forces on locomotion, the gravitational acceleration constant, g, is change to some fraction of earth s gravitational acceleration; if gravitational forces influence both the limbs an the boy, g is change in both places in the equation (ie a an b in Eqn 15) Altering gravitational forces in the velocity calculations alone (b in Eqn 15) will moel the limbs swinging in earth s gravity, while the rest of the boy experiences a ifferent gravitational fiel Testing the moel: sample To test the moel, strie lengths were preicte for a sample of humans an were measure uring treamill walking A sample of 11 iniviuals (five males, six females; see Table1) volunteere to participate in this project All subjects gave informe consent an all proceures were approve by the University of Arizona Human Subjects Committee Each subject performe a series of treamill walking trials at three spees (10, 15 an 20ms 1 ) Pressuresensitive footswitches were attache to the unersie of their feet at the heel an hallux (Delsys, Inc, Boston, MA, USA) to etermine the time of touch-own an toe-off Strie uration was calculate as the time elapse between two successive touch-owns of the same foot Using treamill velocity, strie lengths were calculate as the prouct of velocity an strie uration Limb inertial properties were calculate from limb length an boy mass after Winter (Winter, 1990) Hypothesis testing The effects of reuce gravity on walking were preicte by changing the gravitational acceleration constant in the moel Two cases were moele: (1) changing the gravitational acceleration constant for both the limbs an the boy (ie a an b in Eqn 15) an (2) changing the gravitational acceleration constant for the boy only (ie b in Eqn 15) The effects of gravity were moele in subjects over a range of Froue numbers (Fr=01, 02, 03, 04) at four ifferent gravitational accelerations (% of earth s g=100, 75, 50, 25) Moel preictions were compare to previous stuies in which gravity was altere for the entire boy incluing the legs (Newman, 1996) an for the upper boy only (Donelan an Kram, 1997; Kram et al, 1997) RESULTS Moel valiation The moel was valiate by comparing preicte strie lengths with strie lengths measure uring treamill walking Preicte strie lengths o not iffer significantly from observe strie lengths (ttest, P=008) An orinary least-squares regression line relating h * ghfr (15)
2770 D A Raichlen Table 1 Sample escription Subject Sex Mass (kg) Hip height (m) *(m) T (s) Subject 1 F 6591 087 061 074 Subject 2 M 7455 095 067 077 Subject 3 F 5818 084 059 072 Subject 4 F 6136 091 063 075 Subject 5 F 6364 083 058 072 Subject 6 F 6360 094 066 077 Subject 7 M 6362 092 065 076 Subject 8 M 7272 093 065 077 Subject 9 F 4500 084 059 073 Subject 10 M 6682 085 059 073 Subject 11 M 6272 092 065 076 * is calculate as istance in meters from the hip joint following equation 2 T is the swing perio preicte by the force-riven harmonic oscillator (FDHO) in secons Note that this value is for one full oscillation, an swing phase uration is half this value preicte to observe strie length oes not iffer significantly from the line of ientity (y=x; see Fig1) Aitionally, if this regression line is force through the origin (ie y-intercept=0; at zero velocity, both preicte an actual strie length must be zero), the slope an 95% confience intervals (CI) overlap with the line of ientity [slope (95% CI)=104 (007)] Testing ynamic similarity Dynamic similarity is maintaine across all Froue numbers an gravitational environments when the gravitational acceleration constant is change for both the limbs an the boy (Fig 2) However, the moel preicts lower imensionless strie lengths in reuce gravity when gravitational forces are altere for the boy alone (Fig 2) This pattern is consistent with the results of previous stuies for both walking an running where gravitational forces were altere for the boy alone (see Donelan an Kram, 1997; Donelan an Kram, 2001) Donelan an Kram (Donelan an Kram, 2001) compare relative strie lengths from treamill experiments with those from a stuy where locomotion was examine at a constant velocity onboar an airplane flying a parabolic flight path (Newman, 1996) These flights generate true reuce gravity at the apex of each parabola for short perios of time (Newman, 1996; Donelan an Kram, 2001) The Preicte strie lengths (m) 195 175 155 135 115 095 075 075 095 115 135 155 175 195 Actual strie lengths (m) Fig 1 Comparison of preicte an observe strie lengths in a sample of humans (N=11) The regression line (soli line) relating preicte an observe strie lengths oes not iffer significantly from the line of ientity [slope (95%CI)=087(016); intercept (95% CI)=023(023)] Broken line is the line of ientity (y=x) moel matches the pattern an magnitue of changes in strie lengths in reuce gravity on parabolic flights (Fig 3) Aitionally, moel results are similar to those from treamill experiments when gravitational forces are reuce for the boy only (Fig 3) Finally, the moel preicts that the human walk run transition shoul occur at the same Froue number (Fr=058) regarless of gravitational environment (Fig 4) These preicte values are slightly higher than the mean walk run transition Froue number foun in most experimental stuies (Fr~050) but are within the range of variation in these stuies [range=037 066 (Gatesy an Biewener, 1991; Hreljac, 1995; Dierich an Warren, 1995; Kram et al, 1997; Rubenson et al, 2004)] In treamill experiments (where gravity was reuce for the upper boy only), Kram an colleagues (Kram et al, 1997) showe that, as gravity was reuce to very low levels, the walk run transition occurre at higher Froue numbers (see Fig 4) When gravitational acceleration is altere in the moel for the upper boy only, a similar pattern emerges In this case, the moel preicts that humans will transition to a run at higher Froue numbers as the gravitational acceleration constant is reuce Relative strie length (m) 18 17 16 15 14 13 12 11 1g 075g 05g 025g 075g boy only 05g boy only 025g boy only 1 005 01 015 02 025 03 035 04 045 Froue number Fig 2 Effects of gravity on strie length Circles are moel preictions when gravitational acceleration is altere for both the limbs an the boy Squares are moel preictions when gravitational acceleration is altere for the boy only Smaller symbols enote reuce gravity Gravitational forces are presente as a fraction of earthʼs gravity (981 m s 2 ) Note that ata for a single subject are presente here for clarity Results for all subjectsʼ moels are ientical
Gravity an ynamic similarity 2771 Relative strie length (m) 28 26 24 22 20 18 16 14 12 10 0 02 04 06 08 10 12 Gravity (g) Fig 3 Comparison of moel preictions with parabolic flight ata Strie lengths calculate at a constant velocity (2 m s 1 ) for gravity acting on the boy an limbs (gray circles) an for gravity acting on the boy only (gray squares) Moel values are means ±1 s for all subjects Data from treamill experiments [open squares (Donelan an Kram, 2001)] an parabolic flights [open circles (Newman, 1996)] are presente for comparison Gravitational forces are presente as a fraction of earthʼs gravity (981 m s 2 ) Froue walk run transition 25 20 15 10 05 0 0 02 04 06 08 10 12 Gravity (g) Fig 4 The Froue number at the preicte walk run transition spee as a function of gravity Walk run transition Froue numbers preicte for two conitions: gravity altere for the limbs an the boy (gray circles), an gravity altere for the boy only (gray squares) Moel values are means for all subjects Data from treamill experiments [open circles (Kram et al, 1997)] presente for comparison Gravitational forces are presente as a fraction of earthʼs gravity (981 m s 2 ) DISCUSSION The results of this stuy suggest that the kinematics of human locomotion are strongly influence by the ynamics of the limbs swinging as suspene penula A very simple moel incorporating only anthropometric ata preicts strie lengths relatively well an preicts walk run transition spees that match experimental ata This moel was generate solely to explore the effects of gravity on strie length an the walk run transition; it is not intene to be an exhaustive epiction of human walking However, with a few simplifying assumptions, the moel oes effectively link swing ynamics to whole strie kinematics such that gravity can be altere inepenently on the limbs an the boy Notably, moel preictions matche observe strie lengths an walk run transition spees from previous altere-gravity experiments, inicating that the DSH remains vali in altere gravitational environments Moel Simplifying assumptions were mae to allow for a clear examination of the effects of reuce gravity on human strie lengths For example, step length is assume to be constant over all walking spees, although experimental ata show that human step lengths o change slightly with walking spee (eg Kuo, 2001) Aitionally, while the FDHO moel takes iniviual variation in limb mass istribution into account, it oes not account for possible variation in muscle an tenon stiffness For example, Obusek et al suggeste that there is some iniviual variation in the stiffness of the muscle tenon units that will alter the uration of swing perio (Obusek et al, 1995) Despite these assumptions, comparisons of the moel with experimental ata support its use in investigations of the effects of gravity on locomotion The moel preicts strie lengths very well in a sample of humans walking in normal gravity an preicte that changes in strie lengths match those from experimental stuies when the effects of gravitational forces are altere for the boy only It is possible that changes in step length with velocity coul impact preicte step lengths in reuce gravity However, the moel results agree with strie length ata from parabolic flight experiments of locomotion in reuce gravity These experiments must be consiere the gol stanar since alterations in gravity are real an felt by all boy parts The moel matches ata from these experiments better than treamill stuies Thus, the close corresponence between moel preictions an experimental results from parabolic flights further supports use of this moel to examine the effects of gravity on locomotion Gravity an the DSH This moel supports the preiction that humans will walk with ynamic similarity in ifferent gravitational environments As gravity is altere, imensionless strie lengths are ientical at equal Froue numbers Dynamic similarity shoul occur only when the ratio of inertial to gravitational forces governing locomotion are equal When gravity is reuce, equivalent Froue numbers are only possible at lower absolute velocities If gravity influences stance phase only, then we shoul expect reuce strie lengths in lower gravitational fiels because absolute velocity will be slower at the same Froue number However, reuce gravity also increases the perio of limb swing, which leas to longer strie urations an longer strie lengths The moel suggests that, as gravity is reuce, the increase in swing uration offsets the reuction in velocity at a given Froue number such that strie lengths remain constant The moel preictions also support previous analyses of locomotion on other planets The moel preicts that the walk run transition will occur at equal Froue numbers as gravity is altere Thus, as preicte by Minetti, the walk run transition velocity will ecrease as gravity is reuce (Minetti, 2001) This fining explains why Apollo astronauts reporte ifficulty walking on the lunar surface an instea preferre running an jumping (Minetti, 2001) Confirmation of these results can improve our unerstaning of how locomotion will be constraine in future manne missions to the moon or Mars For example, it may be possible to walk more easily on Mars than on the moon since larger gravitational forces on Mars woul allow humans to transition to a run at a higher velocity Conclusions A simple moel, base on few assumptions, was able to preict strie lengths in earth s gravity for a sample of iniviuals an
2772 D A Raichlen successfully preicte the effects of reuce gravity on human locomotion The DSH is well supporte by the moel, an its use remains a vali way to account for the effects of gravity on locomotion Movement in reuce gravity will clearly affect both swing an stance phase, an analyses of swing-phase or whole-strie kinematics may require either true-reuce gravity experiments (eg parabolic flights) or the use of preictive locomotor moels These types of kinematic ata may be essential for planning the next generation of space exploration Moels, properly valiate by parabolic flight experiments, may be the best way to gather necessary ata for how locomotion will change when walking on other planets b SL f FDHO Fr g h I k L L st m SD SL T t st t sw v LIST OF SYMBOLS AND ABBREVIATIONS istance of composite spring from hip penular length of the limb imensionless strie length strie frequency force-riven harmonic oscillator Froue number gravitational acceleration hip height limb mass moment of inertia Spring constant (composite stiffness of limb muscles an tenons) istance of the limb center of mass from the hip step length limb mass strie uration strie length natural penular perio of the limb stance uration swing uration velocity I woul like to thank Herman Pontzer an Daniel Lieberman for helpful iscussions an comments on this manuscript Aam Foster helpe with ata collection an processing The comments of two anonymous reviewers greatly improve this manuscript Support for this project was provie by the University of Arizona REFERENCES Alexaner, R McN (1976) Estimates of spees of inosaurs Science 261, 129-130 Alexaner, R McN an Jayes, A S (1983) A ynamic similarity hypothesis for the gaits of quarupeal mammals J Zool (Lon) 201, 135-152 Alexaner, R M an Maloiy, G M (1984) Strie lengths an strie frequencies of primates J Zool 202, 577-582 Bullimore, S R an Burn, J F (2006) Dynamically similar locomotion in horses J Exp Biol 209, 455-465 Dietrich, F J an Warren, W H Jr (1995) Why change gaits? 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