Complete Description of the Thelen2003Muscle Model

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1 Compete Description o the he23usce ode Chnd John One o the stndrd musce modes used in OpenSim is the he23usce ctutor Unortuntey, to my knowedge, no other pper or document, incuding the he, 23 pper describing this musce mode, contins the exct equtions used to impement the orce-gth, orce-veocity, nd tendon orce-strin retionships in the OpenSim source code hereore, it is described beow usce ode In simution studies, musce mode is n gorithm trnsorming musce ctivtion into musce orce Here we describe the musce mode used in our simution, which is sight modiiction o the Hi-type musce mode presented in he, 23 At given time t in simution, the musce mode s inputs re ctivtion t () ( re number between nd 1 incusive) nd iber gth ( t) he musce mode s outputs re the musce-tendon ctutor orce ( t) nd, or the next time step t+ Δ t, the timederivtive o ctivtion t ( +Δt) nd the time-derivtive o iber gth, ie, iber veocity ( t+δt ) Here we wi describe how the musce-tendon ctutor orce, timederivtive o ctivtion, nd iber veocity re ccuted

2 Activtion dynmics represents how n excittion ut ( ), unit-ess vue between nd 1 is trnsormed into n ctivtion t ( ), so unit-ess vue between nd 1, o musce When musce is neury excited, its ctivtion grduy increses, whie i musce s excittion decreses, its ctivtion so decreses, beit t sower rte thn the increse he time-derivtive o ctivtion is ccuted s: ( ) ( ) ut () t () / ct, ut () t () t τ ( +Δ t) = ut () t () / τ dect, ut () < t (), where τ ct = 1 ms nd τ dect = 4 ms Our musce mode consists o two prts: the musce ibers nd the tendon he musce ibers my be snted retive to the tendon t n nge ced the penntion nge he musce ibers consist o pssive eement tht produces pssive orce (ie, orce due to the ibers inherent stiness) nd contrctie eement tht produces ctive orce (ie, orce due to ctivtion o the musce) Our musce mode contins sever constnts Some constnts my vry between musces in our muscuoskeet mode o the body whie other constnts re the sme or musces he constnts tht my vry between musces re:, the mximum isometric orce o the musce, the optim iber gth o the musce s, the tendon sck gth, the tendon gth beow which the tendon (nd thereore the whoe musce) produces zero orce

3 α, the penntion nge o the musce ibers t the optim iber gth he constnts tht re the sme cross musces re: ε = 6, pssive musce strin due to mximum isometric orce k toe = 3, n exponenti shpe ctor ε = 33, tendon strin due to mximum isometric orce k = 1712 / ε, iner shpe ctor in ε = 69ε, tendon strin bove which tendon orce is iner with respect to toe tendon strin = , normized tendon orce bove which tendon orce is iner with toe respect to tendon strin PE k = 4, n exponenti shpe ctor or the pssive orce-gth retionship γ = 5, shpe ctor or the Gussin ctive orce-gth retionship A = 3, shpe ctor or the orce-veocity retionship = 18, mximum normized musce orce chievbe when the iber is gthening A br over vribes representing gths indictes normiztion with respect to the optim iber gth, eg, = / is the normized iber gth A br over vribes representing orces indictes normiztion with respect to the mximum isometric orce, eg, = / is the normized musce-tendon ctutor orce

4 he computtion o musce-tendon ctutor orce proceeds s oows rom the conigurtion o the muscuoskeet mode o the body t time t, we obtin the gth o the whoe musce-tendon unit, () t = () t cos α() t ( t) he tendon gth is then ccuted: rom this, normized quntity ced the tendon strin is ccuted: ε = ( s )/ s hen the tendon orce is ccuted s () t ( ε ) ( ) =, where k ε ε +, ε > ε in toe toe toe ktoeε / εtoe e 1 ( ε ) = 1( 1 + ε ) + toe, < ε ε k toe toe e 1, ε represents the normized tendon orce-strin retionship, so known s tendon compince or tendon stiness he extr term 1( 1+ ε ) exists to prevent the tendon rom going competey sck during simution he tendon ttches the musce ibers to the bones in the muscuoskeet mode, so the orce in the tendon is the orce generted by the musce mode s whoe, so ( t) = ( t) he computtion o iber veocity proceeds s oows he width o ech musce, kept constnt s w= sinα, nd the current iber gth ( t) re used to ccute the penntion nge

5 α, () t = or w/ () t = ( ) < < π /2, w/ ( t) 1 1 () t sin w/ () t, w/ () t 1 he ctive orce in the musce ibers is computed s ( ) ( ) () t = () t () t = () t () t ( t ) 2 t = e ( ()) ()1 / γ, where is Gussin unction representing the normized ctive orce-gth retionship or musces in our muscuoskeet mode he pssive orce in the musce ibers is PE PE computed s () t = ( () t ), where ( ()) PE t ( ( ε )) PE k 1 + 1, 1 + > + ε ε = PE k ( 1/ ) ε e PE, 1+ ε k e is unction representing the normized pssive orce-gth retionship or musces in our muscuoskeet mode his unction is ine or rge orces (the irst cse in the bove eqution) nd is otherwise exponenti (the second cse in the eqution) he tot eect o the normized pssive nd ctive orce-gth retionships yieds unction tht increses, eves o, decreses sighty, nd then increses rpidy A musces in our muscuoskeet mode o the body hve this sme normized orce-gth property It hs been shown tht the shpe o the tot orce-gth curve or dierent musces in the body re not identic (Greis et, 1992): whie some musces hve sight decrese ter the eve portion o the tot orce-gth curve, others hve signiicnt decrese,

6 nd some hve n increse insted o decrese hereore, our mode is n pproximtion o the tot orce-gth retionships o musces in re humns he orce in the contrctie eement is then ccuted s () t PE () t = () t cos α( t) he distinction between the orce in the contrctie eement nd the ctive orce ccuted bove is tht the ctive orce does not incude the eects o the orce-veocity retionship o the musce ibers We ccute this orce-veocity sce ctor s ( ( )) () t V t+δ t = () t V is n invertibe unction representing the normized orce-veocity retionship he normized iber veocity is ccuted s ( ) ( ) 1 () t t+δ t = V, () t where, or ny normized orce,

7 1 1+ 1, < A 1, < 1 1+ A 1 ( ) ( ) ( 1)( 1) V ( ) =, 1 < 95 A 1( 1) 2 ( 1 ) , A he exct impementtion o ( ) 1 V is sighty dierent to djust or possibe numeric issues s described beow he iber veocity (un-normized) is ( +Δ ) = ( +Δ ), t t Vmx t t where Vmx ( 5 5) Note tht = + is dierent rom, which we ccuted bove, is the iber veocity normized (ie, divided) by, whie V mx is the iber veocity normized (ie, divided) by : d d 1 d = dt = = = dt dt he origin normized orce-veocity retionship is

8 v + 1, v < A v + 1, 1 v < v 1 A 2 2+ v + 1 v, v < v A A ( ) = A 1( 1)( 95 1) V 1 1+ v A , 2( 1 ) 1( 1 ) 1 ( en 1)( 95 1) 1 1+ A v his unction ws inverted to obtin the expressions or ( ) 1 V bove, but in the impementtion, some ddition constnts ξ = 5 ( pssive dmping ctor or the 6 orce-veocity retionship) nd ε = 1 re incorported to prevent possibe numeric errors he compete impementtion o the inverted orce-veocity retionship is s oows

9 1 ( V ) (, ) ε +, < ε ε ξ + ξ + ξ + + A, < + + ξ = A, < ( ) A + ξ 1 95 v + ( v 1 v), 95 ε, where v 95 = A 1 + ξ nd v1 ( 95 ε ) + = ( 5 ε ) A 1 + ξ he curves representing the ctive orce-gth, pssive orce-gth, nd orce-veocity properties o ibers nd the esticity o tendon re shown in igure 1 beow

10 iber orce-length iber orce-veocity endon Esticity Normized orce tot 1 ctive pssive 1 Normized iber Length 1-1 shortening gthening 1 Normized iber Veocity 1 endon Strin ɛ igure 1 Curves representing the intrinsic properties o the he, 23 musce mode used in our study he curves shown here re normized, mening tht or speciic musce in our muscuoskeet mode, the ctu curves woud be obtined by scing the normized curves verticy by mximum isometric orce ( ) nd scing the curves in the three grphs horizonty by optim iber gth ( ( V mx ), nd tendon sck gth ( s ), mximum contrction veocity ), respectivey, nd then shiting the tendon esticity curve horizonty by the tendon sck gth ( s ) he ctive orce-gth curve nd orce-veocity curve woud hve smer vertic rnge when the musce is ess thn uy ctivted ( <1) Reerences he DG (23) Adjustment o musce mechnics mode prmeters to simute dynmic contrctions in oder duts J Biomech Eng 125: 7-77 Greis H, Soomonow, Brtt R, Best R, D'Ambrosi R (1992) he isometric gthorce modes o nine dierent skeet musces J Biomech 25:

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