Talk:Pennate muscle/Archive 1

Archive 1

PCSA formula not consistent with PCSA definition

${\text{PCSA}}={\frac {{\text{muscle mass}}\cdot cos\Phi }{\rho \cdot {\text{fiber length}}}},$

This formula for PCSA is clearly wrong. Cos(Φ) should be deleted, otherwise (for instance in fog. 1A) it would coincide with the formula to compute ACSA! It is VERY easy to show that the formula is wrong, because the correct formula was built starting from this geometrical truth, that we learn in primary school:

Volume (V) = area (A) * length (L)

Also,

Volume (V) = Mass (M) / density (RHO)

Isn't this obvious? Well, then

A_fiber = V_fiber / L_fiber

That would be enough to prove my point intuituvely. However, let's apply it to the whole muscle. For the whole muscle, the cross section area PCSA is defined as the sum of all the orthogonal cross-setion areas of all the fibers, i.e.:

PCSA = SUM(A_fiber)

or

PCSA = SUM(V_fiber / L_fiber)

Figure 1 Pennate muscle fiber arrangements

If we assume that all fibers have same length L (see muscle A in figure 1, and imagine it were schematically drawn as a slanted parallelogram), this equation becomes

PCSA = SUM(V_fiber) / L_fiber

or

PCSA = V_muscle / L_fiber

and since V = M / RHO, then

PCSA = M_muscle / (RHO_muscle * L_fiber)

QED. This is the correct formula. Compare it with the wrong formula given above. Where's the cosine??? However, I have found many references which include the wrong formula. I only found the correct formula in a slide from a university course in Physiology, and can't find the textbook from were that slide was taken.

Let me be crystal clear. Since

${\text{ACSA}}={\text{PCSA}}\cdot cos\Phi .$

the wrong formula computes ACSA (the blue line in figure 1), rather than PCSA (the green line). In other word, if that formula were the correct definition of PCSA, it would not be necessary to define it, it would be enough to measure direclty ACSA (blue line), which is much easier!

− Paolo.dL (talk) 15:27, 3 September 2010 (UTC)

Fixed it - the problem is that it's muscle length, not fiber length.

Mokele (talk) 00:44, 4 September 2010 (UTC)

No, the problem is that you edited TWO formulas, and instead of fixing them, you broke them. Clearly, you do not completely understand the formulas. Let me explain. Notice that your version of the formula assumes that

L_fiber = L_muscle / cos(Φ),

obvioulsy equivalent to

L_muscle = L_fiber * cos(Φ),

and is obtained by substitution from this formula which I proved to be correct (see above):

PCSA = M_muscle / (RHO_muscle * L_fiber)

Now, please explain how you suppose to justify the assumption that

L_muscle = L_fiber * cos(Φ).

That's unthinkable. It means that the muscle is shorter than the fiber, unless you can prove that cos(Φ) is larger than 1 (which is impossible by definition of the cosine).

You also wrote

${\text{ACSA}}={{\text{PCSA}} \over cos\Phi }$

which is totally the opposite of the formula that I inserted in the article and in my comment above. Your formula implies

${\text{PCSA}}={\text{ACSA}}\cdot cos\Phi .$

Now, if you knew basic trigonometry, you would understand that this means that PCSA is smaller than ACSA (cos(Φ) is always less than 1 in a pennate muscle). Do I need to explain that this is totally the opposite of the truth?

I know that you are doing this in good faith, and I know you added some important references in the section about "anatomical gearing" (in which, by the way, now it became impossible to understand what is the meaning of the word "gearing", because you totally deleted the previous text, which could have been a good starting point, or an example in the form "what would happen IF", even knowing that IF never comes totally true).

However, I can't understand what drives you to correct something that you do not completely understand. This is always a bad idea. At least, before editing a section of an article which you do not totally understand, you should propose your suggestions in the talk page, especially when I showed (see my previous comment) that the literature proposes wrong versions of these formulas.

Moreover, you probably did not read my edit summaries, because I explained that the formula "Vol / length is useful, as muscle mass cannot be measured in living humans, whereas volume can be computed with MRI or CT scans". Also, in my prove above, I showed that the PCSA formula can be understood better if you start from volume (and eventually you obtain mass by substituting V = Mass/density). However, you deleted the relevant part of the formula, without even providing a rationale in your edit summary!

And you even provided a reference which does not provide your version of the formula, but shows (pag. 31) the initial version of the formula, the one which I already proved to be wrong (see my comment above), referring to "fiber length", rather than "muscle length". So, your edit is not even based on the reverence you provided. To be crystal clear, both your version of the formula and the version provided by Lieber (2002) are wrong.

In sum, instead of fixing the TWO formulas,

1. you broke them both
2. you provided a reference which I already had proved to be not reliable
3. the reference you provided does not even support your edit

− Paolo.dL (talk) 09:51, 4 September 2010 (UTC)

Mokele, you should read the talk page, before editing. I proved that your formulas are absurd, replaced them with the correct ones, and you inserted them again. Do you realize that, with both your formulas, PCSA is smaller than ACSA? Do I need to explain how absurd this is? Do not undo my edits again, unless you can prove I am wrong, in the talk page. But first, read what I wrote with great attention and great respect, please. Do not assume you are right. You are a scientist, I am sure you can do it. − Paolo.dL (talk) 13:55, 4 September 2010 (UTC)

Mokele, you introduced, for the third time, a wrong formula. Stop this absurd edit war. I am sure there is a formula in which the pennation angle is considered, but my formula is correct. I might be able to find the formula you are seeking (a function of the pennation angle), but I will leave to you the task. And either provide a reliable reference, or prove it right, before inserting it in the article. Do not try math-fiction again.

Before doing that, I suggest you to fix the section on "anatomical gearing" where, after your edit (which introduced interesting concepts and references, but also deleted everything else), it became impossible to understand what "anatomical gearing" is! See my comment above for details.

− Paolo.dL (talk) 14:09, 4 September 2010 (UTC)

Wow, I made a slight error in the confusing wiki-code while editing, how shocking. It's almost like I have better things to do than learn petty markup details, especially while sleep-deprived. Clearly that justifies your arrogant little rant. Anyhow, I've fixed it again to something that's actually useful to measure, assuming I figured out how this strange code works. Measuring fiber-length is always tricky, because they'll frequently curve, while muscle length is as simple as a caliper measurement.

Also, I've done some looking around, and the formula *is* pretty inconsistent between sources, depending on what they're measuring.

Oh, and by the way, cram the arrogance. We're both trying to improve the article, and coming at it from different perspectives of human vs animals and what's measureable (you have the funding for MRIs, but we have fewer assumptions because we can kill our test subjects and cut them up). Try not assuming bad faith, especially since I'm not the one who wrote the article in the first place. Mokele (talk) 14:14, 4 September 2010 (UTC)

I told you that I am assuming good faith. But even if you wanted to fix, you inserted THREE TIMES wrong formulas, despite my patient explanations in this talk page. My patience is not infinite. If you are sleep deprived, do not edit, and do not keep fighting when you are proven to be wrong. I am sure you did not read with attention my comments, and of course you did not know that you were wrong, but this is another mistake of yours: before editing, read the discussion page. − Paolo.dL (talk) 14:24, 4 September 2010 (UTC)

Ok, look, here's the deal - we both accept that PCSA requires muscle volume to calculate, either from MRIs or, more realistically for most folks, muscle mass and muscle density. The issue is only whether to use fiber length or muscle length. In a non-pennate muscle, the two are the same (ignoring some subtleties), Cos phi = 1, and PCSA=ACSA. In a pennate muscle, the fibers are *shorter* than the whole muscle, in a manner related to their angle (look at the unipennate muslce - *muscle* length is from origin to insertion). In muscles with very small pennation angles (frog tibialis anterior), muscle length and fiber length are almost equal. In muscles with large pennation angles (turkey lateral gastrocnemius), fibers can be less than half the muscle length. A reasonable good approximation is that muscle length * cos(phi) = fiber length - the more pennate, the shorter the fibers. Substitution shows that we then get PCSA = muscle mass / (density * muscle length * cos(phi)) AND PCSA = muscle mass / (density * fiber length). Of course, the whole thing is an approximation - fiber orientation and length can vary within a single muscle, and overall orientation can vary in vivo with loading conditions.

We can have both formulas, but I think we *need* to have one with a cos in it to demonstrate the simple fact that, when I do in vitro muscle preps on frog muscles of roughly the same mass, I have to switch to the bigger, stronger motor if I moved from semimembranosus (non-pennate in frogs), to plantaris (highly pennate). The whole "punchline" of pennation is that it makes muscles stronger for a given volume, and we should have at least one formula showing how pennation angle increases PCSA.

Mokele (talk) 14:37, 4 September 2010 (UTC)

Please explain how it is "more realistic" to weigh a muscle than estimating its volume. To weigh a muscle you need to dissect it away from the body. It is only possible when you work with amimals or cadavers. In other fields, I would say it is impossible, rather than realistic.

Your assumption:

muscle length * cos(phi) = fiber length

is clearly wrong, that's why I deleted your latest formula. Muscle length is not a function of fiber length and cos(phi). Easy to understand the reason by comparing two muscle models, with same fiber length, fiber thickness, and pennation angle. In sum, two parallelograms with same height:

1. a parallelogram containing N fibers, and
2. another containing N/2 fibers (and therefore half as long)

Same fiber length, different muscle length... you draw the conclusion, please. I have to go. See you on Monday.

− Paolo.dL (talk) 15:07, 4 September 2010 (UTC)

Ok, I figured out what the problem is. I was preparing to point out the problem in your example (namely that two muscles of different PCSA but the same fiber length will also have different pennation angles, just due to the way muscles are built (embryologically as well as with respect to physical forces), and I realized the problem:

It's a model, not a formula. And the rule in science is All models are wrong, but some are more useful than others.

Think about it - Every single input to this equation WILL be wrong. Muscle mass / volume / density will be wrong due to both marcoscopic factors (capilaries, intramuscular fat, thick aponeuroses) and microscopic ones (ratio of sarcomeres to stores glycogen, mitochondria, and sacroplasmic reticulum). Muscle fiber length will also be wrong - they're almost never straight, and *always* vary in length throughout the muscle belly, not to mention that most muscles are long enough that fibers will bother begin and end within the muscle belly in a roughly staggered pattern. And pennation angle will vary within the muscle in a complex, 3-D way, as well as changing due to external force levels and aponeurosis strain. And that's just for a simple, unipennate muscle - imagine what a deltoid look like.

This brings us to the most important part, more important that any mathematical consideration - When applied to real muscles undergoing real maximum contractions in real conditions, it yields the correct answer to well within the range of experimental error. In the end, that's all that matters. It's an approximation, not an iron-clad definition of geometrical truth. Think of it like Newton's laws - we know they're wrong at very small scales, very big scales, and very big sizes (ie. QM & relativity), but in the day-to-day world, they give answers that are so close to correct as makes no difference.

It's a bit like cost-of-transport. In reality, we know that it's parabolic due to a series of ingenious experiments by Taylor & colleagues, but in the real world, between gait transitions and voluntary speed selection (and some interesting recent work on the role of the natural frequency of oscillation of the viscera), it's basically constant, and treating it as such gives us correct answers about organismal biology, feeding ecology, etc.

So, long story short, don't sweat it, it works when applied to real muscles. Mokele (talk) 23:46, 4 September 2010 (UTC)

Yes, all models approximate the real phenomenon. They are not perfect. And you are right about all those sources of error. So, approximation is due to:

1. imperfect model (but making it simple is good, because too complex models are impractical)
2. inaccurate input

But we cannot invent a model which is not plausible. The formula currently used in the article would give an almost perfect result if we were able to get a perfect measure of muscle mass or volume, density, and average fiber length. It can be compared to Newton's second law. The other formulas proposed by you or in the literature would give a result that is totally against evidence. As I already showed,

1. one of the formulas would give a PCSA that is smaller than ACSA,
2. the other would yield an identical PCSA for a 20 cm and a 40 cm long bipennate muscle (with same average fiber length and same average pennation angle).

These results are not plausible. It is impossible to publish these formulas. They are against everything we say in the text.

− Paolo.dL (talk) 21:23, 5 September 2010 (UTC)

Neither of those objections hold water. The first we've already established was due to a mis-code on my part with the confusing markup language. The second is wrong - aside from being biologically irrelevant, it's incorrect - The formula (volume)/(muscle length X cos(phi)) would give correct results, since you're increasing PCSA and volume in the simulation suggested. Mokele (talk) 02:39, 6 September 2010 (UTC)

In the simulation suggested, you are doubling mass, but also doubling length, which is in the denominator of the fraction! So PCSA stays the same. I gave you a detailed proof, and your conclusion is not consistent with my proof. If you can't find some fault in my proof, you should accept my conclusion. And my conclusionj, given on 5 sep 2010 (see above), is: "It is impossible to publish these formulas. They are against everything we say in the text." Let me only add today (after reading your comments below): "or vice versa", meaning that we can publish formula 1 only if we change the text (see below), which is currently not consistent with it.

Paolo.dL (talk) 13:03, 6 September 2010 (UTC)

Original source about PCSA in the literature

Well, I found the original source, Sacks & Roy 1982. The original formula was correct - PCSA = (volume * cos (phi))/(fiber length), and I should have stuck with it rather than doubting myself. The reason is that PCSA isn't just a calculation of cross-sectional area, but also takes into account that the muscle will have reduced force due to acting at an angle. Apparently some folks in the literature are as confused as you were, and have used the formula w/o the cosine. Because most muscles don't have huge pennation angles anyway, the mistake is hidden within experimental noise. Anyhow, I'm restoring the original formula, since this has all been for naught. Mokele (talk) 09:47, 6 September 2010 (UTC)

Good job, but either the formula is wrong, or the text is wrong. My formula (see above) is consistent with the definition of PCSA given in text:

PCSA = "total area of the crossections perpendicular to the muscle fibers"

The original formula implies another definition:

PCSA = "total area of the crossections perpendicular to the muscle fibers, multiplied by cos(φ)"

This definition, as I have shown above, implies that PCSA = ACSA in muscles such as that in figure 1A.

By the way, the original formula would make sense to me if it were the formula to compute muscle force, rather than PCSA. I mean, if PCSA is computed with the formula I showed to be compatile with the definition given in text (see above), it would be by all means plausible that:

F = PCSA * cos(φ)

or

${\text{Muscle force}}={\frac {{\text{muscle mass}}\cdot cos\Phi }{\rho \cdot {\text{fiber length}}}}.$

Would you mind to send me by e-mail a copy of the original bibliographic source? Thank you.

Paolo.dL (talk) 12:28, 6 September 2010 (UTC)

I've fixed the text of the article too. I'll forward you the PDF once you reply to my email - I can't send files via the WP email thingy. And yes, the name is a bit wonky, but pretty par for the course - remember, this is the field where an active muscle stretching is still labeled a "contraction". Mokele (talk) 13:48, 6 September 2010 (UTC)

Figure 1B and 1C is wrong

There's another more important problem. Figure 1 is wrong. It does not show the entire PCSA (it does only for muscle A). This also means that the formula ACSA = PCSA * cos(φ) is correct only for figure 1A, and although it gives a PCSA larger than ACSA, it may underestimate PCSA. Namely, PCSA may be much larger, especially in long muscles such as the biceps femoris [NOTE: I meant RECTUS FEMORIS. 12:43, 6 Sep 10] (you can understand this if you use my parallelogram models). I am afraid we need to delete this formula (which I deduced from figure 1, but now I realize figure 1 is wrong).

Are you able to edit figure 1, indicating the width of ALL the fibers in 1B and 1C? (not only those close to the center of the muscle). If you are, would you mind to do it, please? In the meantime I'll look for another plausible version of the main formula, containing φ, and not fiber length. I think it will not be difficult to find it. Give me a week. Maybe I won't be able to work on it before next week end. − Paolo.dL (talk) 21:23, 5 September 2010 (UTC)

Um, the long head of the BF is pretty much parallel-fibered. The short head is more convergent, like the pectoralis, than pennate. It's got a funny tendinous bit at the end for insertion, but so does the biceps brachii, and that's your classic parallel-fibered muscle. Trust me on this, I've cut up a lot of corpses.

As far as the figure, it's fine. It demonstrates the key principle - muscle fibers can be oriented at an angle (in several ways), resulting in shorter fibers and more PCSA.

Sure, maybe some 3D model rotating through space in an animation might show PCSA in a more technically correct way, but this figure demonstrates the key principle accurately and succinctly, without too much extraneous crap. Add too much, and figures become cluttered and unreadable. Mokele (talk) 02:48, 6 September 2010 (UTC)

Sorry, I meant rectus femoris, not biceps femoris. I did not refer to complex 3D information. I meant the figure (and the formula) should simply respect the definition given in the text (or vice versa):

PCSA = "total area of the crossections perpendicular to the muscle fibers"

In other words, figure, text and formula should be consistent, and currently they are not.

− Paolo.dL (talk) 12:43, 6 September 2010 (UTC)

As noted above, I've fixed the text - now it reads (more lengthily), PCSA = blue line * cos phi. Mokele (talk) 13:50, 6 September 2010 (UTC)

Fortunately, your edit was not the same as what you wrote here. Your formula: "PCSA = blue line * cos phi" is wrong. As I repeated millions of times, even if PCSA were a line (and it is not):

blue line = green line * cos(φ)

or

ACSA = PCSA * cos(φ) (but this is valid only for figure 1A)

Your "blue line * cos phi" is something smaller than the blue line! It cannot be equal to PCSA (represented by a line larger than the blue line). Also, you keep neglecting that the figure is still not consistent with text and formula! For instance, if the original formula is correct, in Fig.1A we should draw a blue line with the same length as the green line, and parallel to it, because according to that formula:

PCSA = ACSA (in fig. 1A).

Furthermore, this makes obviously incorrect even this sentence in the article: "This measure ... (PCSA)... increases with pennation angle." Are you ready to write that PCSA always stays the same as ACSA, whatever is the pennation angle?

− Paolo.dL (talk) 16:38, 6 September 2010 (UTC)

Wow, you found a typo, congrats! The text on the page itself is right. Furthermore, this is right for all figures - just look at them as a series of parallel unipennate muscles glued together, and you'll see. You can evaluate them part-by-part.

Furthermore, you're still confusing PCSA with actual fiber cross section, which it's not - it's the fiber cross-section corrected for loss of force due to off-axis force production. PCSA is not the green line, it's the green line cos (phi). Mokele (talk) 21:53, 6 September 2010 (UTC)

I fail to understand your point. You repeated a concept that I explained to you in my previous comments, accusing me for not understanding it! And you ignored that, according to the same concept, PCSA = ACSA (in fig. 1A), which means it does not change with pennation angle (in fig. 1A). Paolo.dL (talk) 14:45, 11 September 2010 (UTC)

Original source about PCSA in the literature

I found new sources. Here's the complete list (including the source kindly provided above by Mokele), sorted by year of publication:

• Alexander, R. McN. and Vernon, A. (1975). The dimension of knee and ankle muscles and the forces they exert, Journal of Human Movement Studies, 1:115–123.
• R.D. Sacks & R.R. Roy (1982). Architechture of The Hind Limb Muscles of Cats: Functional Significance. Journal of Morphology, 185-195.
• Narici MV, Landoni L, Minetti AE (1992). Assessment of human knee extensor muscles stress from in vivo physiological cross-sectional area and strength measurements. European Journal of Applied Physiology & Occupational Physiology. 65(5):438-444.
• Maganaris CN, Baltzopoulos V, Sargeant AJ (1998). In vivo measurements of the triceps surae complex architecture in man: implications for muscle function. J Physiol., 512:603-614.
• Maganaris, C.N., Baltzopoulos V. (2000). In vivo mechanics of maximum isometric muscle contraction in man: Implications for modelling-based estimates of muscle specific tension. In Herzog W. (Ed). Skeletal muscle mechanics: from mechanisms to function. Wiley & Sons Ltd, p.267-288.

So, Alexander is the original source. All these authors use the original formula, except for Sacks and Roy (which is not the original source).

I created a new article: Physiological cross sectional area, and explained everything there. Then I summarized the explanation in this article (where the main article is linked). If you read the new article with attention, you will understand that pictures 1B and 1C still need to be fixed, as they are currently compatible with none of the two definitions (while 1A is compatible with the original definition by Alexander).

Notice that there's a book in which Zatsiorsky uses the formula by Sacks and Roy (PCSA2), but gives it the original interpretation, which is incompatible with it. The same mistake was made by Wikipedia's editors before I corrected the article (see my first comment on this talk page).

− Paolo.dL (talk) 18:43, 11 September 2010 (UTC)