# Muscle force equation and force-velocity relationship

Hi,

I am working on determining the muscle force for various upper-extremity movements.
But as it turns out, there are multiple force elements available in Anybody. I studied about the inverse dynamics process and i am guessing fm is the output of the inverse dynamics operation.
Could you please let me know what is fout and also the mathematical equation behind that? According to the documentation, it says that fout is the output from dynamical analysis but that is a bit confusing as fm seems to be output of the inverse dynamics. So, could you please elaborate that with its mathematical equation? I am using the 3E hill type muscle model ?
My second question is that, will the software take the velocity of the motion into consideration when calculating the force?
My final question is: are there any difference between the way the force is being calculated in agonist and antagonistic muscle?

Hi James,

Please see this tutorial it list the different output from a three element hill muscle.
https://anyscript.org/tutorials/Muscle_modeling/lesson5.html

Fout:

“Fout” means that it is the “output force” from dynamic analysis, which for the muscle implies that it holds passive as well as active forces of the muscle-tendon unit.

Fm:

The force in the muscle’s contractile element. For this muscle type it is equal to the total force in the muscle-tendon unit because the muscle does not have any parallel components

So in other words Fm is the force in the contractile element it do not include the passive force.

Concerning the question on velocities yes this is accounted for same for accelerations.

Mathematically the antagonist and agonist muscles are treated the same way in the muscle recruitment, but for a hill type muscle the strength will be different because of different speed, so in that sense it will treat them different, but it is not the optimizer it will come from difference in strength.

Hope it helps

Best regards
Søren

Hi,

Thanks for the quick response.
So, i could just consider fm divided by strength as final activity.

1. I just need a clarification on the explanation you gave. The objective function in inverse dynamics can predict the active force by a muscle needed to be generated under a given scenario. What about passive force? How does Anybody calculate the passive force when muscle is stretched to its fullest or in any other scenario?
2. Concerning the answer about velocity, could you please let me know how velocity is taken into consideration ? There is no time or velocity component in the objective function. Could you let me know how velocity affects the force output ?

Hi James.

The passive force calculation is part of the three element muscle model and will be included into the optimization problem as a passive force.

The velocity and accelerations will create dynamics loads to the model, these loads needs to be balanced by the muscles like any of the other loads in the model .

Please see e.g. this paper for more details.

2006 Damsgaard M, Rasmussen J, Christensen ST, Surma E, de Zee M (2006), " Analysis of musculoskeletal systems in the AnyBody Modeling System ", Simulation Modelling Practice and Theory, vol. 14, pp. 1100-1111. [DOI, WWW]

Best regards
Søren

Hi Soren,

Thanks for the response.
I have just one more question and i think this was discussed before but i am not clear on this.
I am trying to find the MVC for each muscle. Is there a specfic way to obtain this value in Anybody?
I am have tried performing the MVC movements but sometimes, the force obtained from a regular movement is much higher than force from MVC which means my MVC values are wrong.
I found in a thread that MVC can be found by multiplying F0 and PCSA. But, i am not sure where to find these values.
Could you please let me know how to get the MVC values ? or is there a more accurate way of getting the MVC values?

Thank you.

Hi James,

To get the maximal strength for the muscle you should use the property F0 it is the maximum isometric strength of the muscle.

So except for a muscle working eccentric this is the maximal force the muscle can have. This values if found in different ways depending on which data has been available (it can be seen in the models). Typically it has been calculated using muscle volume, pennation angles etc.

Best regards
Søren

Hi Soren,

Thanks for the response.
I came across some literature that talks about the force generation in eccentric movement and more specifically in extensor muscles. I am specifically interested in replicating the force-velocity relationship of the muscle. If i had to calculate the normalized force that is in line with this force-velocity relationship, can i extract the force parameter from the model and then multiply or divide a normalized velocity component depending on the concentric or eccentric movement ?
That is, multiply the normalized velocity component for eccentric movement (because the force increases with increase in velocity during eccentric movement) while dividing a normalized velocity component for concentric movement (because the force decreases with increase in velocity during the concentric movement).
I understand this might complicate certain calculations such as determining an appropriate velocity component that can be multiplied or divided. But, if i had to calculate the normalized force without using the strength parameter, would this method theoretically yield the proper result?
Thank you.

Regards,
James

Hi James,

I am not sure I fully understand what you are trying to calculate, so sorry if I have misunderstood things.

The 3 element hill model does account for velocity and will consequently give different strength depending on speed. This change of strength will change how muscles are being recruited and thus change the loads in the model.

Are you seeking to extract some additional output from model only relating to the muscles or would you like to incorporate your own muscle model into the model?

There is a very old webcast available on muscle models in AnyBody please see

It explains some of the muscle model features.

Best regards
Søren

Hi Soron,

Thanks for the response and my apologies for over complicating the question.
I am trying to show the force-velocity relationship curve of the muscle.
That is, in concentric contraction, increasing the movement velocity decreases the muscle force whereas in eccentric contraction, increasing the movement velocity increases the muscle force.
I am using elbow flexion to showcase this. By performing the elbow flexion/extension at various speeds, i am looking at how biceps and triceps forces change as the velocity of movement changes.
I would like to use the Fm parameter (not the strength parameter) from the model. And i have the velocity of movement. Using these two values, i want to show the force-velocity relationship.
For this, would it be appropriate to extract the Fm parameter and divide the biceps force by a normalized velocity when concentric contraction occurs while multiplying the triceps force (Fm) by normalized velocity when eccentric contraction occurs?
The reasoning behind multiplying or dividing Fm with a velocity parameter is to make sure the force value abide by the force-velocity relationship i.e. force increase linearly with increasing velocity in eccentric contraction and force decrease linearly with increasing velocity in concentric contraction.
I am not sure if my explanation is still a bit confusing. The thing is i don't want to use the strength parameter and want to use only the raw muscle force parameter (Fm) to showcase the force-velocity relationship. I would want to do this force compensation based on velocity outside anybody in python and show the force-velocity relationship in python.

Regards,
James

Hi James,

The Fm value will depend on the muscle activation, so this depends on the dynamics/loads in the model, not just the muscle model. The Strength on the other hand is not dependent on the actual loads applied to the model just the the length and speed of the muscle.

When you run the model you will get strength as a function of both length and speed in combination so if you want to isolate their contribution you will need to make a parameter study where e.g. the model is in one position but speed varies, so just one frame at the time with diff speeds.

Best regards
Søren

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