Proposed Model

In order to study foot biomechanics, a multi-segment foot model is presented comprising of 13 muscles (10 extrinsic and 3 intrinsic) based in the one studied by Morlock and Nigg [1991]. Soft tissue behaviour under three loading places (heel pad, metatarsal head and hallux) is also modelled and results are presented in chapter 6. The musculoskeletal part of the model is depicted in figure 5-3, showing the path of the analysed muscles, where GAS is Gastrocnemius, EDL is Extensor Digitorum Longus, TA is Tibialis Anterior, EHL is Extensor Hallu- cis Longus, FHB is Flexor Hallucis Brevis, PL is Peroneus Brevis, TP is Tibialis Posterios, FDL is Flexor Digitorum Longus, FHL is Flexor Hallucis Longus and FDB+AH+ADM refers to the group of muscles Flexor Digitorum Brevis, Abduc- tor Hallucis and Abductor Digit Minimi respectively. The muscles were selected because they are the ones that contribute the most to motion activities. Each muscle-tendon complex is modelled using a the Hill muscle model introduced in the previous section. Given that the strongest movement the foot perfoms is in the sagittal plane, this study focuses on the forces and moments produced in that direction. The plantarflexors (red, black and green in figure 5-3) are the strongest muscles, working against gravity to keep balance and also providing propulsion when needed.

Chapter 5. Multi-segment foot model

5.2.1

Free body diagrams

A useful tool to perform this kind of study is the free-body analysis, in which we input all the forces and moments that affect the system. Note that the forces are represented by arrows that are only indicative of the line of action.

5.2.1.1 Ankle joint

In order to study the ankle joint, we have to analyse all the muscles crossing the joint of interest. A free-body diagram is presented in figure5-4(a). As can be seen, there are ten muscles (considering Gastrocnemius as lateral and medial) crossing the ankle joint (muscles are as defined in figure 5-3).

Figure 5-4 Foot free body diagram

Modified fromRobertson et al.[2013]

In static situation, the equations describing the model are the following: X

Fx = 0⇒Fax +FGRFx = 0 (5-27) X

Fy = 0 ⇒Fay+FGRFy−mfg = 0 (5-28)

For dynamic situations, the equations of motion that govern the model can then be derived as follow:

X

Fx =mfax ⇒Fax +FGRFx =mfax (5-29) X

Chapter 5. Multi-segment foot model

X

Ma=Ifα⇒Ma+ [ra−COM ×Fa] + [rCOM−GRF ×FGRF] =Ifαf (5-31)

whereFx is the force at x direction, Fy is the force at y direction, mf the mass of the foot, Fax is the force at the ankle in x direction, Fay is the force at the ankle in y direction, ax is foot acceleration in x direction, ay is foot acceleration in y direction, Ma is the moment at the ankle, If is the moment of inertia of the foot, αf the angular acceleration, ra−COM the distance from the ankle to the center of mass of the foot, rCOM−GRF is the distance to the point of application of ground reaction force to the center of mass of the foot, g is gravity. Rearraging in 5-31, yields:

Ma =Ifαf −[ra−COM ×Fa]−[rCOM−GRF ×FGRF] (5-32) The moment at the ankle is the outcome of all the muscle forces applied at the ankle joint: Ma = nf X i=1 Fi×ri − ne X i=1 Fi×ri (5-33)

This expression refers to the extension and flexion moments generated by muscles, calculated as the force generated by the muscle (Fi) times the moment arm (ri). For comprehension pursposes, the abbreviation of each muscle will be used as subscripts instead of numbers. Expansion of equation 5-33 is:

Ma=FGASM ×ra−GASM +FGASL×ra−GASL+FP L×ra−P L +FP B×ra−P B +FT P×ra−T P+FF HL×ra−F HL+FF DL×ra−F DL−FEDL×ra−EDL−FEHL×ra−EHL

−FT A×ra−T A (5-34) The moment at the ankle can be obtained from experiments, and therefore it is possible to, through fitting techniques, estimate the unknown parameters. Given that the force produced by the muscle must be equal to the one produced by the tendon, it is possible to first estimate tendon parameters and obtain an estimate of muscle force. This can be done by replacing the force transmitted by tendons by the tendon displacement times its stiffness (i.e. FT_{(t) = k}

t(LT(t)−LTo) from equation 5-2) in equation 5-34.

Chapter 5. Multi-segment foot model

5.2.1.2 Metatarsophalangeal joint

The metatarsophalangeal joint can be analysed in the same way as the ankle joint. However, the moment at this joint is not a model output from Vicon system, and therefore needs to be calculated. In this case, the mass and inertial moments of the phalanx are neglected [Miyazaki and Yamamoto,1993]. The free body diagram is displayed in figure 5-5. The governing equations are:

Mmtp =FGRF ×rmtp−GRF (5-35) Mmtp = nf X i=1 Fi×ri− ne X i=1 Fi ×ri (5-36)

Figure 5-5 Phalanx free body diagram

Modified fromRobertson et al.[2013]

Applying the same process explained in the previous section, it is possible to estimate tendon stiffness.