Contribution of Inertia on Venous Flow in the Lower Limb During Stationary Gait

Transcription

1 ORIGINAL RESEARCH Venous Flow in the Lower Limb 115 JOURNAL OF APPLIED BIOMECHANICS, 2004, 20, Human Kinetics Publishers, Inc. Contribution of Inertia on Venous Flow in the Lower Limb During Stationary Gait Jean-Thomas Aubert 1 and Christian Ribreau 2 1 Laboratoires Innothera, France; 2 Université Paris 12 Val de Marne Blood flows toward the heart through collapsible vessels, the veins. The equations of flow in collapsible tubes in motion show a strong dependence on body forces resulting from gravity and acceleration. This paper analyzes the contribution of body forces to venous blood flow during walking on level ground. It combines the biomechanics of gait and theory of collapsible tubes to point out that body forces due to gravity and limb acceleration cannot be overlooked when considering the determinants of venous blood flow during locomotion. The study involved the development of a kinematic model of the limb as a multi-pendulum arrangement in which the limb segments undergo angular displacements. Angular velocities and accelerations were determined and the body forces were calculated during various phases of the gait cycle. A vascular model of the leg s major venous system was also constructed, and the accelerations due to body and gravity forces were calculated in specific venous segments, using the data from the kinematic model. The results showed there were large, fast variations in the axial component (G x M x ) of the body forces in veins between the hip and the ankle. Acceleration peaks down to 2G were obtained at normal locomotion. At fast locomotion, a distal vein in the shank displayed values of (G x M x )/G equal to 3.2. Given the down-to-up orientation of the x-axis, the axial component M x was usually positive in the axial veins, and M x could shift from positive to negative during the gait cycle in the popliteal vein and the dorsal venous arch. Key Words: flow in collapsible tubes, kinematics and body forces The venous network consists of veins, which can be described as collapsible tubes undergoing deformations under varying transmural (internal minus external) pressures (see Comolet, 1984; Pedley, 1980; Ribreau, 1989; Shapiro, 1977). Gravity generates a hydrostatic gradient of pressure. Its effect on flow through collapsibles tube has been examined by Ribreau (1991) and by Thiriet, Naili, Langlet, and Ribreau (2001). In parallel, the external jugular veins of the giraffe have provided a useful physiological model because of the large range of pressure 1 Laboratoires Innothera, 7-9 Av. François-Vincent Raspail, BP Arcueil, France; 2 Laboratoire de Biomécanique et Biomatériaux Ostéo-Articulaires, Université Paris 12 Val de Marne, 61 Av. du Général De Gaulle, Créteil cedex, France. 115

2 116 Aubert and Ribreau gradients they show (Pedley, Brooks, & Seymour, 1996). In such a model, blood flow was proven to be necessarily time-dependant during postural changes (Brook & Pedley, 2002). However, postural changes were only modeled in terms of the time-dependant gravity variable. In other words, inertial acceleration was overlooked, even if in a 3-meter neck the skull of the giraffe underwent a fairly large radial acceleration up to 7.4 m s 2 for a 1-second head-raising movement. For human gait, in which the angular velocity peaks reach 3 rad s 1 at the hip (Winter 1990), the radial acceleration can reach 9 m s 2 at the ankle for a leg measuring 1 meter in length. In order to accurately address the question with this paper, the body forces are calculated in a 3-D lower limb (LL) during stationary gait, in the framework of the collapsible tube theory. According to this theory, each vein in the network is defined by a direction axis x of flow that can move in the Galilean system of reference. The cross-sectional area of the tube is described by the variable A(x). External stresses acting on the venous cross-section are defined as constant lateral pressures p e, but they are variable in time (Kamm & Shapiro, 1979). The intravascular pressure is denoted p i and U is the mean velocity through the venous crosssection. The density and the viscous forces per unit volume are written ρ and f v, respectively. The body forces resulting from movement are variable in time and space, whereas the x-axis component of gravity, denoted G x, is only variable in time. Then, if the x-axis component of the vectorial acceleration M (all vectors are shown in bold) is denoted M x, the well-known equation generally used for the flow is: (1) The issue at stake here is determining (G x M x ). In order to focus on this point, this paper is organized as follows. First a kinematic model for the lower limb will be described. The model will show how the accelerations vary quite specifically during one gait cycle. The most striking features observed are analyzed in the Results section; in particular, the body forces acting on different veins from the hip to the foot are highlighted. In the last section, the results are discussed in relation to the question, Do venous blood body forces affect the circulation during stationary gait? Finally, the Appendix will enable the reader to follow some of the phases in the calculations. Methods The gait cycle consists of two steps which equal one stride. It starts with one foot and includes the subsequent occurrence of contact with the ground by the same foot (during the stance phase). In the interim is the second stage, when that foot is no longer in contact with the ground (the swing phase) and the other foot takes over. The gait cycle is mainly defined in terms of three characteristic moments: (a) the moment of heel contact, when the heel of the foot initially contacts the ground (this marks the end of the swing phase and the start of the stance phase); (b) the toe-off moment when the toe leaves the ground (this marks the end of the stance phase and the start of the swing phase); and (c) the next moment of heel contact by the same foot.

3 Venous Flow in the Lower Limb 117 Figure 1 Kinematic data for one stride, normal speed: V = 1.5 m s 1. Stride period = 940 ms; Thin solid line = hip variable; Thick solid line = knee variable; Dotted line = ankle variable. Kinematic data obtained in the form of time-dependant joint angles, velocities, and accelerations (see below for a definition of the angles) have been studied by many researchers including Winter (1990), Bouisset and Maton (1995), and Allard and Blanchi (1996). The moment of heel contact, as determined using two force plates, was used by these authors to define the beginning and end of the gait cycle. Nevertheless, kinematic recordings of the experimental methods described in the Appendix were performed by us in order to get exhaustive biomechanics parameters (Figure 1). To calculate the acceleration of any point on the lower limb, the walking limb was taken to resemble a triple pendulum in the sagittal plane (Figure 2). The system of reference of the pelvis was approximated as being Galilean, although some vertical oscillations of the hip can be observed during stationary gait (Inman, Ralston, & Todd, 1981). In the model, the pelvis, thigh, shank, and foot are denoted S 0, S 1, S 2, and S 3, respectively. A pin joint links two consecutive segments. All the unit vectors k i associated with each pin joint, such that k i = k 0 (where k 0 = i 0 j 0 ), are perpendicular to the same sagittal plane (oriented with the two perpendicular unit vectors i 0, j 0 ), which for each segment is a symmetrical plane. The counterclockwise angles of this plane are positive. O i is the center of the joint at the distal end of S i when describing the multisegment system from the pelvis toward the foot. Each segment is defined by the vector O i 1 O i = s i = l i i i, where l i is the length of the solid i.

4 118 Aubert and Ribreau Figure 2 Multisegment model of lower limb. Each joint is defined by the vector O 0 O i = p i. The degree of freedom of segment S i to the previous one, S i 1, is θ i. This algebraic angle, which is less than π, is measured from the vector s i 1 to s i. The rotation vector of solid S i with reference to solid S n is denoted Ω i/n. The projection of p i in the frame n (O n ; i n, j n, k n ) is denoted p (n) i and the absolute acceleration of.. joint O i. is denoted p i, as opposed to the angular acceleration, which is denoted θ i (while θ i stands for the angular velocity). The LL venous network differs from one individual to another, but the scheme in Figure 3a is generally taken to be a fairly accurate picture. Actually the network consists of two subnetworks: a superficial network (saphenous, etc.) and a deep one (femoral, etc.). Many perforating and communicating veins connect the two networks together. The valves turn flows toward the heart, from superficial veins toward deep ones in all stages of the LL except the podal stage, when the valves turn flows from the deep veins toward superficial ones. From the body of angle data collected, accelerations of any point in a venous network can be computed in the Galilean system of reference. A standard 3-D venous network has been reconstructed in the framework of The Visible Human

5 Venous Flow in the Lower Limb 119 Figure 3 Lower limb venous network. (a) 3-D model. (b) Venous network and node N ijk ; Subscript i = segment number (1: Thigh; 2: Shank; 3: Foot); Subscript j = vein number; Subscript k = venous node number. Circled segments are the spots of the computations presented. Project ; it can be adapted to any subject morphology by similarity. Each vein is connected to each other by nodes N ijk (Figure 3b). Nodes also stand for the connectivity between two segments of the same vein. Subscript i is the number of the limb segment to which the venous segment belongs: 1 for the thigh, 2 for the shank, and 3 for the foot. Subscript j denotes the number of the vein in the venous network, and Subscript k is the number of the node. The acceleration of one node is computed from the acceleration of the articular centers and then expressed in its local venous frame. From the kinematics of O i with respect to S 0, we can obtain the whole fields of velocity and acceleration. Since Node N ijk is a point on the solid S i, its position is defined by O i N ijk = n ijk in the frame of S i. In the same way, one can define its position by O 0 N ijk = q ijk in the frame of S 0. The components a ijk, b ijk, c ijk of n ijk are expressed with cylindrical coordinates x ijk, r ijk, and ijk in the local frame of S i :

6 120 Aubert and Ribreau (2) The absolute acceleration of an articular center O i is given by the following equation (see explicit expression in Appendix): (3) The absolute acceleration of N ijk, a point belonging to S i, is given by the following equation: Node N ijk is chosen to be the origin of the local venous frame. The unit vectors of this local frame are u ijk, v ijk, and w ijk, respectively. Vector u ijk stands for the flow direction, and so the plane (N ijk ; v ijk, w ijk ) is perpendicular to the flow direction. The orientation of the vein segment with respect to the frame of S i is defined by two angles. The first angle ϕ ijk oriented by v ijk is defined by the two vectors u ijk and i i in the plane (N ijk ; i i, j i ). The second angle ijk oriented by i i is given by j i and v ijk in the plane (N ijk ; j i, k i ). Moreover, ϕ ijk [0; π], whereas ijk [0; 2π]. Since the flow usually goes from the bottom to the top of the articular segment, Angle ϕ ijk can be said to be greater than π/2. As shown by Equation 1, only the first component of the acceleration q ijk is required for the one-dimensional theory. It is defined on the flow direction by the dot product M x = q ijk u ijk, as well as by: (4) (5) The gravity component G x in the flow direction is also required by Equation 1. Three typical equations can be derived from the model in the thigh, shank, and foot, respectively. These expressions depend only on ϕ ijk, Ψ ijk, and the joint angles for a given posture. (6) Results The acceleration involved in Equation 1 has been calculated for any vein in the limb network. Three veins have been selected in the thigh (S 1 ), three in the shank (S 2 ), and two in the foot (S 3 ). Among these veins, we chose two perforating veins in the thigh and the shank and one superficial vein in the shank (Figure 3b). The coordinates and inclinations of the selected veins are listed in Table 1.

7 Venous Flow in the Lower Limb 121 Table 1 Coordinates and Inclinations of Selected Nodes and Veins Node coordinates Vein inclinations Node in S i (cm) in S i (rad) Vein number a ijk b ijk c ijk ijk ϕ ijk Popliteal vein N Perforating vein 3 N Common femoral vein N Internal saphenous vein N Perforating vein 1 N Peroneal vein N Venous arch N External marginal vein N Figure 4 Normalized acceleration of three veins of the thigh in one stride. Normal speed: V = 1.5 m s 1 ; stride period = 940 ms. Perforating vein 3: computed at Node N 112 on Segment [N 112 ;N 144 ]. Popliteal vein: computed at Node N 141 on Segment [N 141 ;N 142 ]. Common femoral vein: computed at Node N 146 on Segment [N 146 ;N 147 ]. Solid line: (G x M x )/G; Dotted line: G x /G.

8 122 Aubert and Ribreau Figure 5 Normalized acceleration of two veins of the foot in one stride. Normal speed: V = 1.5 m s 1 ; stride period = 940 ms. External marginal vein: computed at Node N 361 on Segment [N 361 ;N 362 ]. Dorsal venous arch: computed at Node N 311 on Segment [N 311 ;N 331 ]. Solid line: (G x M x )/G; Dotted line: G x /G. To be consistent with Equation 1, Figures 4, 5, and 6 give the body forces G x M x (solid line) and G x (dotted line) in the thigh, shank, and foot, respectively. All the results were obtained at a normal speed and normalized by the absolute value of gravity G. Depending on the orientation of the veins, the quantity G x /G is negative in almost all the veins. In the upright position, the perforating veins in the thigh and shank have a quasi-zero G x value. Other veins in the same segments are practically vertical, and their orientation with respect to the upright position depends on ϕ. For instance, G x /G in the popliteal vein is equal to 0.9 against ϕ = rad. Throughout the gait cycle these quantities vary with the angle of the different joints. The maximum range of peak-to-peak variation is about 25% of G around G x /G = 1 or 0, respectively. The corresponding variations in the veins of the foot are much higher, since they can reach up to 100%, for example in the dorsal venous arch, the orientation of which varies between and rad with respect to the vertical. In comparison to G x, the body forces G x M x show greater variations in amplitude. Indeed, they result not only from the multisegment angular velocities and angular accelerations but also from the distance from the node to the joints. For instance, the peak-to-peak variations in the body forces can reach 100% of G in the thigh and the shank, and 350% in the foot. As with G x, the body forces G x M x were negative during the entire gait cycle, at least in the nodes and veins examined here. The curves G x /G and (G x M x )/G can intersect during the gait cycle. This means that M x is equal to zero at the crossing point, negative above G x /G, and positive below (see Figure 4).

9 Venous Flow in the Lower Limb 123 Figure 6 Normalized acceleration of three veins of the shank in one stride. Normal speed: V = 1.5 m s 1, stride period = 940 ms. Perforating vein 1: computed at Node N 221 on Segment [N 221 ;N 231 ]. Peroneal vein: computed at Node N 241 on Segment [N 241 ;N 242 ]. Internal saphenous vein: computed at Node N 211 on Segment [N 211 ;N 212 ]. Solid line: (G x M x )/G; Dotted line: G x /G. Among the results given here, one can see that M x /G is close to zero in perforating veins, the axes of which are perpendicular to the acceleration. Moreover, M x /G is always positive in the proximal segment of the common femoral vein, in the distal segment of the internal saphenous vein, the peroneal vein, and the external marginal vein. A large negative range of M x /G values was observed in the case of the popliteal vein (Figure 4) and the dorsal venous arch (Figure 5). In Figure 4, the acceleration M x /G of the popliteal vein is equal to zero at 15 and 75% of the cycle, first during stance and then during swing. In that case, according to the pendulum... movement of the thigh, the M x component is proportional to the sum of θ 2 1 and θ 1, the first term being always positive and the second one positive or negative, in line with the direction of the movement. The peak in (G x M x )/G in the swing phase corresponds to the maximum flexion of the hip. Similar zero accelerations were observed in the internal saphenous vein and the peroneal vein during stance and swing (Figure 6). These points correspond to maximum knee flexion. Furthermore, when the angular velocity of the knee becomes maximum, (G x M x )/G reaches its minimum in the two veins. A similar pattern occurs in Figure 5, which gives the data obtained on the external marginal vein of the foot: zero acceleration M x /G occurred at maximum knee flexion and one minimum (G x M x )/G value occurred at maximum knee angular velocity. However, for the dorsal venous arch, there is a characteristic pattern of alternation in the midswing phase. At 75% of the cycle, the negative M x is

10 124 Aubert and Ribreau Table 2 Minimum (Maximum) Body Peak Forces (G x M x )/G in Vein Thigh Shank Foot Proximal common Distal internal saphenous External marginal femoral vein 1 ( 0.76) vein 1.68 ( 0.7) vein 2.08 ( 0.7) Popliteal vein 1.26 ( 0.68) Peroneal vein 1.74 ( 0.68) Dorsal venous Perf (0.07) Perf (0.05) arch 3.67 (0.31) close to G x and knee flexion is maximum. After 80% of the cycle, the M x sign reverses to positive. As expected, the amplitude of accelerations increased with distance from the veins to the hip, but the maximum values of the body force peaks were barely affected (Table 2). Indeed, the negative component of M x, which was reflected in the angular acceleration, was never large enough to cause any major variations. On the other hand, the minimum values depended greatly on this distance (Table 2). In a proximal vein such as the common femoral vein, the body force peak did not exceed 1G, whereas in a distal vein such as internal saphenous vein, 1.68G was observed. As a consequence, a long vein such as the internal saphenous, which starts from the ankle and goes up to the hip, undergoes great variations in the body forces it receives along its length. In the foot, more than 2G are reached in the dorsal venous arch or the external marginal vein. Discussion Three characteristic properties of body forces have been brought to light. First, the sign of the body forces was negative or close to zero in the nodes and veins examined here. Second, the normalized body force peaks (G x M x )/G measured at distal points from the joint were greater than 1. Third, the acceleration field M x could shift from positive to negative, especially in strategic veins of the venous return such as the popliteal vein and the dorsal venous arch. It is significant that these body forces seem to be responsible for unsteady blood flow through the vessels of the lower limb during stationary gait. Ribreau (1991) and Thiriet et al. (2001) have reported that the crosssectional area of the inclined collapsible tube, when filled by an upstream steady flow, was much more variable than in a horizontal one because of the negative G x. In light of this, the positive acceleration M x (or negative body force) should roughly reinforce the longitudinal changes in the cross-sectional area. For instance, in the borderline case where G x = 0 and M x > 0, the horizontal collapsible tube would have the same response as an inclined collapsible tube filled with an upstream flow. Since the flow through veins of the LL is mainly driven by negative (G x M x ) body forces, it is reasonable to assume that this scheme is applicable. Note that in Brook and Pedley s study (2002), for the jugular vein in the upright position, the blood flows downward from the heart and consequently the body forces are either positive or negative depending on the inclination of the neck and its angular velocity. In the giraffe s neck, the viscous forces counterbalance the positive body forces. Although at any given time the two components G x and M x seem to be equiva-

11 Venous Flow in the Lower Limb 125 lent, one can observe that G x remained constant along the whole axis of the straight tube whereas M x was found to depend mainly on the distance to the point at which it is computed. In the forced unsteady flow induced by the gait cycle, one can expect at least volumetric variations with some variable distal flattening of the tube. The possible effect of compression stockings might be that they limit these volumetric variations (Gardon-Mollard & Ramelet, 1999). In the same way, one can expect unsteady flows to have consequences on cell physiology. This statement is partly based on a study by Haond, Ribreau, Boutherin-Falson, and Finet (1999) which showed that endothelial cells are very sensitive to the shear stress and its gradient in a collapsible tube. The results presented in this paper, between the hip and the ankle, show peak values of [(G x M x )/G] max and [G x /G] max in the [0; 2] and [0; 1] range, respectively. These results corroborate the study by Wu and Ladin (1996), who recorded peaks from 0 to less than 2G on shank accelerometers. This of course leads to the logical conclusion that even larger negative values can be reached in the case of the foot. In addition, the body force peaks were found to increase with the speed of locomotion. For instance, a distal vein in the shank (computed at the ankle) undergoes a minimum body force peak from 1.94 G at slow speed to 3.23 G at fast speed (see Appendix). Thus the LL veins probably undergo even greater body force peaks when the person is running or jumping. From all the above data, it follows that the term (G x M x )/G should not be omitted from Equation 1. The magnitude of the term (G x M x )/G and its fast variations is certainly one of the causes of venous unsteady flow in the lower limb. Unless the movement is quasi-static, any attempt to model the flow in moving tubes, collapsible or not, must include the body forces. When the vein is not parallel with the segment axis, the computations have shown that it is possible for the M x sign to change, for instance in the popliteal vein and the dorsal venous arch. It is noteworthy that these veins are at the output point of some important subnetworks. Any M x sign changes are likely to affect valve dynamics (Gottlob & May, 1986), especially the valve closure process. Such affirmation was evidenced in the study by Bitbol, Dantan, Perrot, and Oddou (1982). In addition to possible variations in the vein volume induced by body forces and/ or skeletal muscle forces, valve closure may be associated with a kinetic pump. This pump would differ from the calf venous pump (Van Cleef, Griton, Cloarec, Moppert, and Ribreau 1990) or the foot pump described by Lejars (1890). Except for the relationship of cause and effect between M x and valve closure dynamics, or between G x M x and unsteadiness, it can be expected that body forces actually act on a collapsible tube. But to have an effective action on such a tube, the body forces must act differently on both sides of the venous wall. Indeed, if the outside and inside mediums are equivalent liquids, the effects on the tube wall will be counterbalanced. This is the case when the veins are considered as enclosed within a more or less liquid medium, the density of which is not very different from that of the blood. Given this hypothesis, no effects of body forces are expected on the venous wall, just as occurs when one wears anti-g pants. Unfortunately, no biomechanical data are as yet available to resolve this question, except for some data on the cutaneous veins, where qualitative effects are easily observable. Obviously if one views any vein environment as a movable soft mass and/or

12 126 Aubert and Ribreau solid structure, not just liquid, the LL acceleration will affect, in addition to blood flow, the venous wall response. References Allard, P., & Blanchi, J.P. (1996). Analyse du mouvement humain par la biomécanique [Biomechanical analysis of human movement]. Québec: Décarie. Bitbol, M., Dantan, P., Perrot, P., & Oddou, C. (1982). Collapsible tube model for the dynamics of the closure of the mitral valve. Journal of Fluid Mechanics, 114, Bouisset, S., & Maton, B. (1995). Muscles, posture et mouvement [Muscles, posture and movement]. Paris: Hermann. Brook, B.S., & Pedley, T.J. (2002). A model for time-dependant flow in (giraffe jugular) veins: Uniform tube properties. Journal of Biomechanics, 35, Comolet, R. (1984). Biomécanique circulatoire [Biomechanics of circulation]. Paris: Masson. Gardon-Mollard, C., & Ramelet, A.A. (1999). Compression therapy. Paris: Masson. Gottlob, R., & May, R. (1986). Venous valves. Wein New York: Springer-Verlag. Haond, C., Ribreau, C., Boutherin-Falson, & Finet, M. (1999). Laminar flow through a model of collapsed veins. Morphometric response of endothelial vascular cells to a longitudinal shear stress non uniform cross-wise. European Physical Journal Applied Physics, 8, Inman, V.T., Ralston, H.J., & Todd, F. (1981). Human walking. New York: Williams & Wilkins. Kamm, R.D., & Shapiro, A.H. (1979). Unsteady flow in a collapsible tube subjected to external pressure or body forces. Journal of Fluid Mechanics, 95, Lejars, F. (1890). Les veines de la plante du pied [The veins of the sole of the foot]. Archives de Physiologie, 5e série. Pedley, T.J. (1980). Fluid mechanics in large blood vessels. Cambridge: Cambridge University Press. Pedley, T.J., Brook, B.S., & Seymour, R.S. (1996). Blood pressure and flow rate in giraffe jugular vein. Philosophical Transactions of the Royal Society of London, Series B, 351, Ribreau, C. (1989). Hémodynamique veineuse, équations de base [Fundamental equations of venous flow]. Journal des Maladies Vasculaires, 14, Ribreau, C. (1991). Sur la loi d état, la loi de perte de charge, et la nature de l écoulement permanent en conduite collabable inclinée [On the tube law, the head loss and the nature of steady flow through tilted collapsible tubes]. Thesis, Université Paris 12 Val de Marne, Créteil. Shapiro, A.H. (1977). Steady flow in collapsible tubes. Journal of Biomechanical Engineering, 99, Thiriet, M., Naili, S., Langlet, A., & Ribreau, C. (2001). Flow in thin walled collapsible tubes. In C. Leondes (Ed.), Biofluid methods in vascular and pulmonary systems, Vol. 4. Series Biomechanical Systems. Techniques and Applications (chap. 10, pp. 1-43). Boca Raton, FL: CRC Press. Van Cleef, J.F., Griton, P., Cloarec, Moppert, M., & Ribreau, C. (1990). Modèle dynamique de la pompe musculaire du mollet [Dynamic model of the calf muscle]. Phlébologie, 43, Winter, D.A. (1990). Biomechanics and motor control of human movement. New York: Wiley. Wu, G., & Ladin, Z. (1996). The study of kinematic transients in locomotion using the integrated kinematic sensor. IEEE Transactions on Rehabilitation Engineering, 4,

13 Venous Flow in the Lower Limb 127 Appendix Data were recorded on one healthy 22-year-old woman (170 cm, 55 kg). Normal speed is what she achieves when walking as naturally or freely as possible (1.50 m s 1 for a stride period of 940 ms). Slow and fast gait velocities according to her personal comfort (1.16 m s 1 for a stride period of 1,220 ms and 1.96 m s 1 for a stride period of 860 ms). At each gait velocity, three trials were conducted in order to obtain average curves. Four passive markers were stuck on the right LL: (a) the greater trochanter (O 0 ), (b) medial condylis (O 1 ), (c) lateral malleolus (O 2 ), and (d) second metatarsal (O 3 ). The length of each segment was: O 0 O 1 = 44 cm (thigh), O 1 O 2 = 43 cm (shank), and O 2 O 3 = 16.5 cm (foot). The hip and knee curves did not require any smoothing prior to numerical differentiation, contrary to the ankle curves, which were noisy because of the impacts of the foot on the walkway (acquisition sampling rate: 50 Hz). Angular velocities and accelerations were calculated by numerical differentiation and then smoothed by performing correlations in parts (degree of polynomial: 6 or more). All polynomials were fitted so that the whole differentiated curve formed a continuum and also in order to obtain cyclical curves. Acceleration of O 1 (knee), O 2 (ankle), and O 3 in the frame of S 1, S 2, S 3 respectively: (7) (8) (9)

14 128 Aubert and Ribreau Acceleration of N 1jk, N 2jk, N 3jk on the thigh (S 1 ), the shank (S 2 ), and the foot (S 3 ), respectively: (10) (11) (12) Acknowledgments We thank Christophe Gillet and the Laboratoire d Automatique, de Mécanique et d Informatique Industrielles et Humaines, Université de Valenciennes et du Hainaut- Cambrésis, for their support with the experiment. This study was subsidized by Laboratoires Innothera, Arcueil, France.