How to determine the parameters of the combined muscle after combining several similar muscles into a single muscle?

[Matthew Lee]

Dear Moco development team and Moco users,

Thank you all for your contributions to the biomechanics community! I have some questions about how to determine the parameters of the combined muscles in the adjusted model, I would appreciate it if anyone could give me some advice.

In studies focusing on the optimal control problem of lower limb movements (such as walking, jumping), it is a common method to combine multiple muscles with similar functions in the lower limbs into a single muscle in order to improve computational efficiency. It is common in the literature to simplify the muscles in the full body model or gait2392 model by combining muscles with similar functions in the lower limbs. After simplification, each leg usually has only 8 or 9 muscles. For example, the long head of the biceps femoris, semitendinosus, and semimembranosus can be combined into a single muscle called the hamstring. The three compartments of the gluteus maximus can be combined into a single muscle.

Assume that muscle A in the adjusted osim model is obtained by combining muscles B, C, and D in the original osim model. Since muscles B, C, and D have different geometry path and parameters,

1. How to determine the the muscle path of the muscle A?

2. How to determine muscle parameters of muscle A, such as
(1)optimal fiber length, (2)tendon slack length, (3)pennation angle, (4)optimal fiber velocity, (5)maximum isometric force, and parameters in the mechanical characteristic curves of (6)muscle fiber and (7)tendon?

I would be very grateful if someone could give me some advice.

Best Wishes,
Matthew

[Pasha van Bijlert]

Hi Matthew,

I guess I can weigh in here, since all my models have muscles that represent functional groups (I'm currently running simulations of the emu, an Australian running bird). I'd be interested to hear how other people have dealt with this problem.

I think how people tackle this issue typically depends on their research question / ultimate goal for the model. Presumably, if you're OK with lumping muscles together, you're not trying to get subject-specific results, and are thus looking to simulate general phenomena.

I sum the masses of each muscle together, and parametrize the new muscle as F_max = m * sigma / (rho * LO). Here, m is the total muscle mass, sigma is the specific tension (0.2 - 0.3 MPa), rho is the density of muscle (1060 kg/m^3) and LO is the optimal fibre length.

The issue is of course that we have a free parameter (LO). Studies have shown that many muscles are tuned to operate over about 0.5-1.5 (or 0.75-1.3) LO during normal use (e.g. Burkholder & Lieber). That means that if you have a line of action for the muscle in question, and you move the model over a physiological joint range, the change in length of the entire muscle in question will allow you to calculate a physiologically reasonable fibre length. In other words: you are ensuring that the model's optimal force range corresponds to the joint range that you are interested in.

Tendon length will then depend on the total muscle length, and what joint angle you expect the muscle to be exactly at L0. I ignore pennation angle, because I do not think it is still meaningful if you're lumping muscles together with completely different pennation angles. Alternatively, it might be reasonable to pick one existing muscle's parameters (so pennation angle, tendon length, fibre length, etc.), and only scale up F_max based on the total mass of the muscles you're combining.

Either way, this is a fairly rough way of dealing with this problem, and the absolute performance of the model could be reduced (since there is less functional specialization - you'd be lumping muscles with very short & very long tendons together if they have the same action). You can still get realistic simulations though (I think both Bill Sellers from UoM & Karl Bates from UoL have successfully used an approach like this - and my emu models match empirical data reasonably well).

Thomas Geijtenbeek published an alternative approach where I think he optimized some of the muscle parameters (origin, insertion, pathpoints, etc.) concurrently when optimizing gait activation patterns. However, he was simulating (hypothetical) bipeds, so that might be a less fruitful approach for you.

If you have more data available (which in humans, you typically do, like moment-joint angle curves), you could make more precise choices when simplifying the musculature. I think that's how Knoek van Soest parametrized the human models that he built for simulations of jumping.

Hopefully this helps you in some way!
Cheers,
Pasha

[Carlos Gonçalves]

Hello Matthew and Pasha,

I've been looking for references for scaling muscle max isometric forces and other parameters. And the reading suggestion from Nick (thanks Nick) from another post is excellent:

Handsfield, G.G., Meyer, C.H., Hart, J.M., Abel, M.F., Blemker, S.S., 2014. Relationships of 35 lower limb muscles to height and body mass quantified using MRI. J. Biomech. https://doi.org/10.1016/j.jbiomech.2013.12.002

The supplementary material has an average volume and scaling coefficient for Iliopsoas, Gluteals, Quadriceps, Hamstrings, and Triceps Surae. Hope it helps.

From the detailed explanation of Pasha, I can assume that the muscle volume should be in m3 to find the F_max, correct? In the article, everything is mentioned in cm3.

Best regards.

[Pasha van Bijlert]

From the detailed explanation of Pasha, I can assume that the muscle volume should be in m3 to find the F_max, correct? In the article, everything is mentioned in cm3.

I don't think it matters as long as you keep all your dimensions consistent. E.g., in my example:

F (in N) = m (in kg) * sigma (in Pa, so N m^-2) / rho (in kg m^3) / LO (in m).

It's easy to work out that the dimensions are consistent on both sides (as long as you plug in specific tension in Pa, not MPa).

Cheers,
Pasha

[Matthew Lee]

Hello Pasha and Carlos,

Thank you very much Pasha for your detailed reply! Also thanks to Carlos for the supplementary material! Sorry for the late reply (I've been busy with other matters recently and just had time to check the forum). I think your response contains rich knowledge and information, and I will take the time to carefully read the references you have provided. If I encounter any problems, I may come back and continue to listen to your suggestions. At the same time, anyone's opinions and help on this topic are also very welcome.

Best Wishes,
Matthew