Musc Struc: | Outline | Musc In-Out | CE-SE-PE Struc | CE Force-Vel-Len | Musc Reflex |
Hill Model: Contractile Element in Series and Parallel with Elastic Elements
Essential phenomena underlying Hill Model structure†
One key finding of Hill and colleagues systematic input-output modeling experiments was the observation that for a given sustained level of neural excitation n, a sudden change in force (or length) would result in nearly instantaneous change in length (or force).†† This suggests the relationship of a spring (Hill, 1950):
where k is often called the spring constant.†† Because this property is nearly instantaneous, i.e. doesnít depend on the past history of loading but only on this sudden change, we can view it as the behavior of a lightly damped spring.† Interestingly, under other situations (e.g., changes in n), muscle does appear to exhibit damping (e.g., inherent sensitivity of muscle force to muscle velocity).† This suggests that a spring-like element could be conceptualized as being connected to the node on the right of muscle black box, but not to the node for the neural drive n.† Letting the primary contractile tissue be called the contractile element (CE), we have the classic Hill model for muscle, shown in Fig 1, with lightly-damped spring-like elements both in series (SE) and in parallel (PE) with CE (Hill, 1938, 1950).†
In trying to understand how this model works, think of the CE as being a bit sluggish, unable to move instantaneously.† We also know that for spring-like elements in:
∑ series: forces are the same, and† extensions add,
∑ parallel: extensions are the same, and forces add.†
Using the analogy of Fig 1b, we see that with a sudden change in load (i.e., change to a new factive), xm moves right away but xce does not.† Thus, in this case Dl in is D(xm - xce.).† †For Fig 1a, this is not quite true because PE is also spring-like, and as we noted above for such springs in parallel, the forces add:† ftotal = factive + fpassive (where the former is that across SE and the latter† across PE).
Fig 1.† A.† Most common form of the Hill muscle model, with CE (representing the active contractile machinery) bridged by light-damped springs both in series (SE) and in parallel (PE).† B. Alternative form.† Note that the constitutive relations for SE and PE differ for the two cases (e.g., with passive stretch where the force across CE is zero, only PE is stretched in A, yet both in B).† However, because SE is usually much stiffer than PE over the primary operating range for most muscles (see Fig 4), it almost doesnít matter which form is used.
So why do most modelers use the form shown in Fig 1a?† First, for lengths below the muscle rest length, fpe is essentially zero, and thus SE can be directly obtained.† But letís also consider the case where muscle is not excited, i.e. n=0 and the force across CE is zero. In Fig 1a, since CE and SE are in series, this implies that the force across SE is also zero, i.e.† fce = fse = factive = 0.† Thus as the muscle is extended, for Fig 1a the measured force due to passive stretch, fpassive, is literally that across PE.† Any observed force is then simply that due to PE, which can be thought of as the passive force of the musculotendon unit that is due to connective tissue infrastructure such as the musculotendon sheath, the muscle fiber membranes, and the overall fluid environment within which muscle tissue lives.†† However, for Fig 1b, where SE and PE are in series, for extensions above rest length both will be stretched (although PE will likely stretch more since it is typically more compliant, i.e. less stiff).† Then for a given overall extension xm we have (in a good model) the same ftotal for both models, but in Fig 1b the PE extension equals xce rather than xm.† This is because of the confounding effect of SE still being present.† Another advantage of the form of Fig 1a is that there are many other passive spring-like elements, structurally in parallel, that also cross joints, and these can be mathematically lumped together.† Indeed, for the human system the experimental data that is available is usually already for lumped passive joint properties.