NUMERICAL STUDY OF CENTRAL NERVOUS SYSTEM, MUSCULO-SKELETAL SYSTEM, AND THEIR COUPLING TOWARD MOTOR DYSFUNCTION IN PARKINSON S DISEASE

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Blucher Mechanical Engineering Proceeing May 14, vol. 1, num. 1 www.proceeing.blucher.com.

1 Blucher Mechanical Engineering Proceeing May 14, vol. 1, num. 1 NUMERIAL STUDY OF ENTRAL NERVOUS SYSTEM, MUSULO-SKELETAL SYSTEM, AND THEIR OUPLING TOWARD MOTOR DYSFUNTION IN PARKINSON S DISEASE K. Shimizu 1, N. Yamamura, S. Takagi 1 1 Department o Mechanical Engineering, the Univerity o Tokyo (khimizu@el.t.utokyo.ac.jp) omputational Science Reearch Program, RIKEN Abtract. In thi reearch activity, we are now eveloping the numerical moel or the motor yunction, or example Parkinon ieae, on a high perormance computer. The inal target o the activity i to evelop the integration moel coniting o neural network (brain, central nervou ytem, an o on) an three-imenional muculo-keletal ytem, an to reprouce primary ymptom o Parkinon ieae in orer to clariy the caue o the motor yunction an obtain the knowlege about eective treatment. The evelope central nervou ytem inclue major motoneuron relate to a motion (e.g. α-motoneuron) an ecribe a imultaneou orinary ierential equation, in which communication o excitatory an/or inhibitory ignal with each other are expree mathematically The muculokeletal moel i bae on a nonlinear inite element ramework or incompreible hyperelatic material, an can treat both paive behavior o biological ot tiue an active behavior o mucle. The moel can ucceully reprouce the mucle-tenon behavior an generate orce uring an iometric contraction. Keywor: Parkinon ieae, central nervou ytem, muculo-keletal ytem 1. INTRODUTION Parkinon ieae i a egenerative iorer o central nervou ytem an caue motor yunction. The motor yunction i generally coniere to be caue by an abnormal motor comman rom the baal ganglia, which reult rom the eath o opamineproucing cell in a mibrain region. However, the caue o the epletion i not clariie yet. Thereore, treatment remain palliative an it evelopment cannot be toppe. In thi reearch, we are now eveloping a numerical moel or the motor yunction in Parkinon patient on a high perormance computer ytem. The inal target o thi tuy i to evelop the integrate moel o brain, central nervou ytem, an three-imenional muculo-keletal ytem, in orer to clariy the caue o the motor yunction in Parkinon patient an obtain the knowlege about eective treatment.

2 The inal objective o thi reearch, thereore, i to evelop a numerical moel coniting o a brain, neural network, an muculo-keletal ytem or a human motion an to reprouce primary ymptom o Parkinon ieae on a high perormance computer, in orer to clariy the caue o the motor yunction an obtain the knowlege about eective treatment. In the current tatu, central nervou moel an keletal mucle moel are evelope eparately, then thee moel are integrate.. NEURAL NETWORKS To reprouce a motion o a human in a numerical imulation, evelopment o the moel or brain, neural network, an muculo-keletal ytem an integration o their moel are require. Numerical reult by uturii [1] are given a a ucceul work about numerical imulation o Parkinon ieae. In hi moel, activitie o primary motoneuron an interneuron relate to an arm motion are expree a imultaneou orinary ierential equation. Hi reult how opamine epletion in a brain caue abnormal mucle behavior an neuron activitie which are imilar to rigiity in Parkinon patient. Meanwhile, many reearcher are now eveloping numerical moel or the brain (coniting o globu palliu externa, globu palliu interna, ubthalamic nucleu, an o on) o Parkinon patient an invetigating the inluence o opamine epletion generating an abnormal motor comman. In uch reearch activitie, membrane potential o neuron are expree a orinary ierential equation an the voltage epen on the behavior o ion channel. A mentione above, our inal target i to evelop the integration moel, an the central nervou ytem play a role in tranmitting ignal rom brain to mucle. Accoringly, an analogou trategy to a brain moel i aopte or the neural network or a motion in view o the connection o brain an mucle behavior. To realize thi concept, we ollow the ucce o the numerical moel evelope by ii et al. [], which imulation ytem coniting o the pinal cor circuitry an aociate mucle can ucceully reprouce orce generation an motonueron recruitment. Mathematical treatment or neuron orming neural network or motor comman communication are bae on a compartment moel an expree a orinary ierential equation or membrane potential. A an example, the expreion or an α-motoneuron are given below: I ion V t V t = I yn, = g yn g l ( V El ) gc ( V V ) Iion l ( V El ) gc ( V V = I, g ) Na 3 m h V 4 ( E ) + g n ( V E ) + g q ( V E ) Na K K K, (1), () K, (3) = π r l m, (4) = π r l m. (5)

3 Here, V i a membrane potential or oma, V i a membrane potential or enrite (two compartment moel i employe or α-motoneuron), m, h, n, an q are gate variable or correponing ion channel. The three kin o ion channel (oium, at potaium, an low potaium) are coniere in the moel. The ize principle can be reprouce by mean o changing the ize (raiu : r, length : l, an o on) o each neuron. An example o the oma iameter i hown in Fig. 1. In thi igure, 8 S-type motoneuron, 5 FR-type motoneuron, an 5 FF-type motoneuron are illutrate, an they have linear lope in iameter. 6 Soma iamter [μm] 5 4 S-type FR-type FF-type ID o α-motoneuron Figure 1. Soma iameter or three type o α-motoneuron. Behavior o motoneuron obtaine by numerical tet or the above moel are illutrate in Fig., in which irect current o 5, 1, an 5 na are injecte. It i hown rom thi igure that mall irect current can caue iring o only mall α-motoneuron. When the injecte current become larger, larger α-motoneuron can be activate. In thi tet, all o the α- motoneuron are activate or the injecte current o 5 na. Thi reult how the ize principle i ucceully reprouce by thi numerical moeling or α-motoneuron. Figure 3 how generate orce when the irect current i injecte. The generate orce become large graually when the injecte current become large. The maximum orce i realize at the injecte current o 5 na. The increae in the injecte current can t change the generate orce when the current i larger than 5 na. Thi correpon to the behavior o motoneuron hown in Fig. 3, which inicate ucceul reprouction o ize principle o motoneuron.

4 1 I inj = 5 na I inj = 1 na I inj = 5 na Potential [mv] 5 1 Potential [mv] 5 1 Potential [mv] Time [m] Time [m] Time [m] Figure. Time train o membrane potential o α-motoneuron or three kin o injecte current. The column mean injecte irect current: 5, 1, an 5 na rom let to right. The row how the membrane potential o each α-motoneuron: the mallet S-type neuron, the larget FR-type neuron, an the larget FF-type neuron rom the top to the bottom. Generate orce [kg] I inj = 1 na I inj = 5 na I inj = 1 na I inj = 1 na Time [m] Figure 3. Generate orce againt injecte irect current.

5 3. MUSULO-SKELETAL SYSTEM The numerical moeling or muculo-keletal ytem i bae on a nonlinear inite element metho uner the aumption o incompreibility o biological ot tiue uch a mucle an tenon. Here, a mixe type iplacement-preure inite element ormulation, a total Lagrangian ormulation, an a ully implicit time integration proceure are aopte. One o the characteritic o the biological tiue i it ability to evelop internal orce, thereore, the tre tate in a mucle mut be expree a the reult o a uperpoition o paive an active part, i.e. [4] σ = σ + σ total pa act (6) total σ pa σ act σ where,, an are the total, the paive, an the active auchy tre normal component in the mucle irection. For the biological ot tiue, the train energy enity unction can be written a [5] W = W + W vol + Q. (7) The term W i the contribution o the iochoric train energy an i a unction o the aopte hyperelatic contitutive moel. The econ term W vol repreent the train energy aociate with the volumetric change, an the thir term Q, which come rom the introuction o incompreibility o the biological ot tiue, i the unction o both iplacement an eparately interpolate preure. The implemente moel or the paive part o the mucle i aume to be the aniotropic material one, i.e: [4] mucle matrix W = W + W. (8) matrix The irt term W correpon to the connective tiue o mucle an i eine a a Mooney-Rivlin material: W matrix = c ij i+ j= 1 i j ( I 3) ( I 3), 1 where c ij are material contant, I1 an I are reuce invariant right auchy-green train tenor, repectively. The econ term W repreent the mucle an connect the pa auchy tre in the irection ( ) to the tretch (λ) by the relation σ (9) λ W pa = σ σ λ = pa λ λ pa, pa pa where σ i a material contant an i the normalize paive orce given by (1) pa γ 1γ exp γ 1[ exp{ γ ( λ 1) } 1] (.4γ )( λ 1.4) + γ exp(.4γ ) 1 [ 1] or λ 1 or 1< λ 1.4 or λ > 1.4 (11)

where λ i the normalize tretch eine a λ = l / λ. The moel or tenon i aume to be iotropic an expree a Mooney-Rivlin moel. The three-imenional Hill-type moel [4], which i written a Eq.

6 where λ i the normalize tretch eine a λ = l / λ. The moel or tenon i aume to be iotropic an expree a Mooney-Rivlin moel. The three-imenional Hill-type moel [4], which i written a Eq. (1), i ue or the mucle active behavior. σ act = σ λ iom l v t. λ (1) iom In the above expreion, σ i the maximum iometric tre occurring at the optimum tretch λ. l, v, an t are the activation unction, length-epenence unction, an velocity-epenence unction, repectively. Numerical reult or iometric contraction o a human tricep urae mucle are hown in Fig. 4. In thi calculation, moele urae mucle i bae on meical image. It i een that each mucle caue a contraction in the irection. The Achille tenon i tretche by the aponeuroi exiting between gatrocnemiu an oleu, reulting in the orce generation. Figure 4. Simulation reult o the iometric contraction o a human tricep urae mucle. Four igure in the let-han ie illutrate the eormation an iplacement o the longituinal irection. The right-han ie igure how the noal orce vector at the en o the contraction. The eect o the maximum iometric tre on the generate orce at the inertion o Achille tenon at the en o the activation are alo examine. Figure 5 how the numerical reult o generate orce againt maximum iometric tre which i varie rom.5 to 1. MPa bae on the literature. The maximum iometric tre increae the generate orce. The larget orce, iom σ iom = 1. MPa, i about 9.7 time larger than the mallet one, σ =.5 MPa.

7 Figure 5. Generate orce againt the maximum iometric tre at the inertion o Achille tenon at the en o the activation. 4. SUMMARY In thi paper, numerical moel or the central nervou ytem an three-imenional muculo-keletal ininte element moel are ecribe. It i conirme that the threeimenional mechanim o the orce generation at Achille tenon i obtaine rom the preent muculo-keletal moel bae on a inite element approach. An the numerical moel or motoneuron reprouce recruitment uring orce generation. The inal target o our reearch activity i to integrate thee moel together an to reprouce a human motion, primary ymptom o Parkinon patient on a high perormance computer. In the uture, thereore, we are going to integrate thee moel, an urthermore brain moel. 3. REFERENES [1] Vaili uturii, Origin o a repetitive an co-contractive biphaic pattern o mucle activation in Parkinon ieae, Neural Network, 4, 59-61, 11. [] Steven J. Schi, Towar moel-bae control o Parkinon ieae, Phil. Tran. R. Soc. A, 368, 69-38, 1. [3] Rogerio R. L. ii, Anre F. Kohn, Simulation ytem o pinal conr motor nuclei an aociate berve an mucle, in a Web-bae architecture, J. omput. Neuroci., 5, 5-54, 8. [4] Johanon, T., Meier, P., Blickhan, R., A Finite-Element Moel or the Mechanical Analyi o Skeletal Mucle, J. Theor. Biol., 6, ,. [5] Suman, T., Bathe, K.-J., A FE Formulation or Nonlinear Incompreible Elatic an Inelatic Analyi, omputer & Structure, 6, , 1987.

ANALYSIS OF SECTION. Behaviour of Beam in Bending

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