biomechanics

Power to the planes?

I’ve just noticed that my blog is still displaying its Christmas card which feels a bit poor as we move into February. Thought I’d reflect on an issue that came up just before Christmas when we were looking at some joint power data from a cohort of amputees. My colleague first presented the data in “components” in the different planes. I commented that I regarded this as wrong and asked her to plot out the total joint power. She came back to me a little later to say that she couldn’t see how to compute this directly within Visual3D.  This caused us to look at their wicki which appeared to confirm what she had found with an explanation that this is how joint power was presented in Move3D, the software out of which Visual3D evolved, and that this “has become common in the biomechanics community”. My initial reaction was that “common in the biomechanics community” does not necessarily correlate with “correct”. To do Visual3D justice, the wicki also points out that you can calculate total power from its “components” but not vice versa which at least makes sense, even if you question how appropriate it is to refer to these as “components” . (Rather bizarrely feedback from C-motion after publication of this post makes it clear that a JOINT POWER SCALAR function has been available in the software for quite a long time as well as the original calculation of “components” and it was the wicki that was/is misleading! The original wording of this paragraph has been modified in acknowledgement of this).

Knee power

The confusion is quite widespread. When Vicon first produced Polygon it only allowed a graphing of total power and then one day I noticed an option to plot the different “components”. I dropped them an e-mail pointing out the mistake and was told that it wasn’t a mistake but a feature that a number of their customers had requested. It was clear that consumer demand was a more important driver of product development than the rigour of the biomechanics!

So what is the issue? As defined in physics, power is what we call a scalar, it cannot be related to any particular direction or plane. Think of it as a bit like your age, another scalar, it doesn’t really make any sense to talk about having age in a particular direction or plane does it. Well, to the classically trained physicist (me!) then talking about sagittal plane power doesn’t make any more sense than talking about sagittal plane age!

Or is it that simple? The quantities in physics that are related to direction are called vectors (position, speed, acceleration are common examples in biomechanics). Vectors are generally represented as a set of three number which are the components in a particular direction. Thus speed (v) is written (vx, vy, vz) with vx representing the component of speed in the x-direction. Joint power is the product of two such vectors, moment (mx, my, mz) and angular velocity (ωx, ωy, ωz) and under the laws of vector multiplication this gives the equation:

P = m.ω = mx ωx+my ωy.+mz ωy

and, although the physicist doesn’t think it has any significance, it is clear that the total power does appear to be made of three separate terms that involve quantities measured along different directions. It is these three terms that have come to be known as the “components” of power. (Notice that throughout this article I’ve put putted inverted commas around “component” when I’ve used it differently to the conventional definition in physics).

So if it is very clear what the terms mean, does it matter if we just choose to use it even if the physicists don’t think we should? I think the answer to this is “yes, it does matter” (I would though, I’m trained as a physicist). To me the whole point of biomechanics is that it allows us to understand the way the body works using rules and relationships that have been developed in the context of wider physics and engineering and which we know are true in all practical circumstances. If we start using terms which are not part of that understanding, no matter how convenient, then we lose that guarantee that they relate to each other in any particular way. It may seem sensible when you set out, but sooner or later it will lead you into trouble.

Power, in this context for example, is the amount of energy generated in a given time. The “components” of power (e.g. mx ωx ) can be negative as well as positive so if, for example, the x “component” is positive and the y and z “components” are negative, then the amount of energy generated in a given time in the x plane (if this is how it is regarded) is greater than the total energy generated in all the planes. This just doesn’t make sense. Are we saying that power is being generated at a joint in one plane at the same time as it is being absorbed in the other planes?! I hope even the non-physicists who read this can appreciate the problem.

The problem with calculating and using “components” of joint powers is that we don’t know under what other circumstances they lead us to nonsensical conclusions. Stick to the rules of physics and we know our conclusions will always be valid (as long as we’ve applied them properly of course!)

One defence of “sagittal plane joint power” which I have a little sympathy with is that, because the components of both angular velocity and moment tend to be considerably greater in the sagittal plane than others, the “sagittal plane joint power” is generally quite a good approximation to the total joint power. Given that in the modern world all these numbers just pop out of the computer anyway though its not at all clear how this is useful. If you want to know the total joint power why not calculate the total joint power? You also need to be careful that if you justify “sagittal plane power” as a good approximation to total joint power, then all you can really say about the transverse and coronal “plane powers” is that they represent the error in this approximation. Attributing physical significance to poorly defined error terms in a calculation is always going to end in tears.

In passing it may be worth commenting that kinetic energy can also be defined as a product of two vectors,

KE = ½mv.v = vx vx+vy vy.+vz vy

but I’ve never heard anyone talking of kinetic energy having components in different directions!

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Sense of satisfaction

Modern academic research is largely a rather slow process taking small incremental steps. I’ve vented my frustration before about how dispiriting it can be to get lost in a fog of low-level research projects which often leave us more confused rather than enlightened. I thus feel I want to celebrate a rare occasion when I do feel a sense of completion of a substantial programme of research.

I was lucky enough to move to Belfast  shortly after Kerr Graham and Aidan Cosgrove  had completed their early work demonstrating the efficacy of Botulinum toxin injections first in hereditary spastic mice and then in children with cerebral palsy. Kerr had departed for Melbourne by the time I arrived but left Niall Eames, an orthopaedic surgeon, lined up to do some research to try and better understand the effect of the toxin. Given that the problem in CP is that the muscles are too short and that Botulinum toxin, by reducing the neural input to the muscle, allows them to elongate, we decided that we should do this by looking at the changes in muscle length. We thus started with some, by modern standards extremely crude, muscle length modelling of the gastrocnemius.

Niall graph

Response to botulinum toxin plotted against the pre-operative dynamic component (taken from Eames et al. 1999)

Having developed the model we applied it to a cohort of children with cerebral palsy having Botulinum toxin injections and were able to demonstrate that the action of the toxin was to reduce the “dynamic component” of reduced muscle length (see figure above). This makes a lot of sense as it is this component that is affected by the neural input to the muscle. The “fixed component” (contracture) is largely a consequence of changes to the composition and structure of the muscle and is unlikely to be affected by the toxin. The research also allowed us to understand that the variable response was largely due to children having a different dynamic component rather than of the toxin acting differently and led to reasonably simple prescription guidelines. Botulinum injections to the calf are most likely to be beneficial if the child has a large dynamic component (good range of passive dorsiflexion during physical examination but walking up on their toes). It further explained that the different response in children with diplegia  and hemiplegia was also attributable to them having different magnitudes of dynamic component.

Armed with this understanding I was then able to work with the pharmaceutical company Ipsen to set up a cliniucal trail to establish the most appropriate dose of the toxin. We couldn’t find enough children to study in the UK so had to extend the study to five centres in Poland. We divided children into one of four groups and injected them with either a placebo or one of three different doses. We used the same modelling technique which we had developed for the earlier study to analyse the results and came to  the conclusion that placebo didn’t work (very much) and that the middle dose was the most effective (see figure below). It was interesting that the biomechanical modelling came to clear logical conclusions whereas doctors’ subjective opinions were that the placebo was very nearly as effective as the drug and that they were so impressed by the “improvement” after placebo injection that they would have recommended repeating the process for two thirds of the children! (despite biomechanical evidence that the placebo had had no effect).

Baker graph

Reduction in dynamic component as a function of different doses of Botulinum toxin at 4, 8 and 16 weeks (Baker et al. 2002)

Having established the most appropriate dose on a single occasion the most obvious remaining question is, “How often should those injections be repeated?”. I’d moved to Melbourne to join Kerr by then and we applied to the Australian National Health and Research Council to fund a clinical trial to compare injections delivered either annually or every four months over a  two year period. Reflecting on the biomechanics we recognised that the long term goal of the injections had more to do with preventing the development of secondary fixed contractures than on the immediate effect on the dynamic component. We would have to measure relatively small changes over a two year time span and thus devised a method to standardise the measurement of passive dorsiflexion range as much as possible.

Which brings me to the stimulus for writing this post in that the results of that study have just been published . The first conclusion is that passive range of dorsiflexion was maintained over the two year period by both injection regimes. We had no true control, because by this stage it wasn’t considered ethical to inject placebo over such a long period, but these measurements were taken over an age range in a child’s life during which preserving dorsiflexion range would be extremely unlikely without injections. The second conclusion was that the more regular injections where only slightly more effective in preserving dorsiflexion range and therefore that there doesn’t appear to be any particular benefit in injecting more regularly than once a year.

Thus after nearly twenty years of research based on the application of thoughtful biomechanics to a clinical problem we finally have clear evidence of which children to inject, how much toxin to inject and how often to repeat this. As one leader of the western world was once heard to comment under less auspicious circumstances, “Mission accomplished!”

Footnote

Trials like this take so long to organise that we were not actually the first group to complete a study to establish the most appropriate injection frequency. This was actually published about 5 years ago. It was a very similar study (it had been sponsored by Ipsen as a follow-on our to earlier work and I’d had some involvement in its planning before I left for Australia) and arrived at a very similar result. Rather than feeling that there was competition here though it highlights the scientific importance of repeating studies to confirm results. With such an emphasis on innovation in modern clinical research the need to repeat and confirm earlier results, which is an important part of the scientific process, can very often be overlooked.

 

A bit of a work out

While still reflecting on the way we use terminology so misleadingly within gait analysis it might be worth thinking a little about the concept of external work. It’s a concept that is even older than I am. Although previous workers (notably Fenn and Elftman) had used similar concepts it was Giovanni Cavagna who popularised it with his classic paper from 1963.  (Cavagna et al. 1963). The article starts with the sentence, “The work performed in walking can be considered as being made of two components, the internal work and the external work”. My response to this is that you can consider it like that if you want but you are likely to confuse people if you do!

Graphs from Cavagna's 1963 paper showing how horizontal components of speed and displacement are calculated from acceleration data. Note that his data was taken from an accelerometer worn on the body whereas it is more common these days for similar techniques to be used based on forde plate measurements.

Graphs from Cavagna’s 1963 paper showing how horizontal components of speed and displacement are calculated from acceleration data. Note that his data was taken from an accelerometer worn on the body whereas it is more common these days for similar techniques to be used based on force plate measurements.

Let’s be clear that there is no external work in walking. All the work required for walking is generated internally by the muscles. The result of muscles (and ligaments) exerting forces on the skeleton is that the foot exerts a force against the floor and generates the ground reaction (following Newton’s third law) but the ground reaction itself doesn’t do any work. It can’t. In order for a force to do work the point of application needs to move and the ground doesn’t move (well, not very often).

Whether its name is correct of not, the concept is important because it allows an estimate of the energy cost of walking on the basis of force plate measurements alone (cuts out all that nasty kinematics). The theory behind the calculations is generally presented as  straightforward but actually requires some quite subtle reasoning.

Although the ground reaction doesn’t do any work, it is a force applied externally to the body and will result in the centre of mass of the body being accelerated (Newton’s first law). If we measure the ground reaction we can thus calculate this acceleration and thus how the centre of mass is moving (its velocity and displacement).

Now if we wanted to move an equivalent mass through the same trajectory we could do so by applying an external force of the same magnitude and direction as the ground reaction directly to its centre of mass. If we did this then the point of application of this imaginary force would move and it would do work. Knowing the laws of physics it is reasonably easy to calculate what this work would be.

This can be taken as equivalent to the work that the muscles have to do to move the centre of mass, but it should be emphasized that the external force applied at the centre of mass is entirely imaginary, for the purposes of the calculation only. All the work is done internally by the muscles.

Of course this is one of those areas where people who understand the underlying concepts can cope with the fact that the name is wrong and get on with life … but I suspect that the terminology has the potential to be extremely misleading for those who don’t.

Additional note. It may also be worth being explicit that the muscles do other things as well as moving the centre of mass. They also move the segments with respect to the centre of mass and the work required to do this is not captured in the calculation outlined above. The calculation will thus always be an under estimate of the true mechanical cost of walking. It’s interesting that despite the extent to which these techniques have been used there have been very few studies of how much of an under-estimate, either for normal walking or for walking with pathology of different kinds.

Demonstrating the gravity of the situation

I’ve been modifying e-Verne recently to make him a little more friendly to use on tablets and phones, particularly those running under iOS or Android (follow this link for tips). This has been in preparation for a Tutorial session I’m presenting at the GCMAS meeting next week in Portland. While I was doing the maintenance something reminded me of a picture in Braune and Fischer’s, “The Human Gait” (which was originally released in chapter form between 1895 and 1904) of a device to calculate the position of the centre of gravity of the whole body once you know the positions of the centre of gravity of the individual body segments. I scanned through through the book and this is the image that I remembered from page 127. The ingenious scaffolding mechanism moves the black spot in the centre to illustrate where the centre of gravity is.

Gravity man

I thought it might be interesting to add this functionality into Verne and below you can see how it looks. Use the “Mass centres” button to toggle the centre of mass positions on and off. The individual segment masses are depicted in black. The red and blue symbols are the centre of masses for the different limbs (femur, tibia and foot segments combined) and the green one that of the overall body. The area of the symbols are proportional to the mass of the different segments with the masses and centre of mass positions for the individual segments based on the data in David Winter’s book. Drag on the different segments to move them around (there are more instructions on the Verne page of this blog-site.

Whilst checking to see if I could save myself the walk to the scanner by surfing to  find if the image from The Human Gait was already on-line I was fascinated to come across a similar picture.

It’s from the catalogue of the German scientific instrument maker Eduard Zimmermann published in 1904. It’s labelled as a “Schwerpunktmechanismus nach Fischer”. They obviously sold well because they are still listed in the 1928 catalogue where a fuller description is available. It doesn’t say how big this was but it weighed 4.6 kg so much have been a fair size.

 

 

 

 

 

Rockers or rollers?

Writing about the movement of the hind-foot the a couple of weeks ago and about projection angles last week has led me to reflecting a little on Jacquelin Perry’s rockers. As with many of the concepts that we have in gait analysis, the rockers can give us some really useful insight into how we walk but can also prove misleading if we don’t remain conscious of their limitations.

I don’t recognise the word “rocker” as meaning anything in particular in this context and had assumed it was an American word meaning pivot or fulcrum. I happened to mention this to a couple of American colleagues a couple of years ago, however, and found that they didn’t recognise the word either. It would appear that Perry simply made it up. Not that it matters much, the word seems to get the concepts across readily enough.

The rockers provide mechanisms for the tibia to move forward over the foot and hence for the passenger unit to be carried forward in stance. If we look at the angle the tibia makes to the vertical (above) then we can see that it starts off about 20° behind vertical at foot contact and progresses forwards reasonably steadily (with a bit of a wobble) to reach about 50° in front of vertical at foot off.

tibial progression

Perry explains this in terms of three rockers.  Early on the whole foot rotates about the heel. Later on the tibia rotates over the foot about the ankle and then finally the whole foot rotates about the forefoot (see below). Easy eh!

rockers

There is no doubt that all three mechanisms make important contributions to tibial progression. I’m not quite so convinced by Perry’s implication that these occur as a sequence of discrete mechanisms. To investigate this we need to look at the dorsiflexion graph which tells us when ankle rocker occurs and the foot projections graph that tells us when the heel and forefoot rockers are active (see graphs below, note that is impossible to distinguish the timing of the rockers from the ankle angle graph alone ).

Rocker graphs

 

Heel rocker starts off at foot contact and proceeds until the foot is flat at about 8% of the gait cycle (in red above). It should be noted that this is considerably longer than the period to maximum plantarflexion in early stance that it is sometimes related to. Ankle rocker is the period over which the dorsiflexion angle increases which we can see from the ankle angle graph is from about 5% of the gait cycle to about 45%. There is thus a short period of overlap when both the heel and ankle rockers are active.

Forefoot rocker starts with heel lift which Perry suggests occurs at mid-stance (30% gait cycle). The data depicted above suggests it might commence even earlier (20%?) and it continues until the end of stance. It is thus clear that there is a considerable period from about 20% of the gait cycle until 45% when both ankle and forefoot rockers and simultaneously active.

The conclusion is that whilst the rockers are undoubtedly the mechanisms which allow the tibia to progress they form an overlapping progression rather than a series of discrete events. Indeed for the majority of stance two rockers are active simultaneously.

Since Perry introduced the concepts there has been some slippage in how the terms have been applied which is best avoided. As far as I can see, Perry always talked about heel, ankle and forefoot rockers and never first, second and third rockers. I think this is good practice as quite a lot of our patients don’t have a first rocker (they make contact with the forefoot rather than the heel). It’s always seemed a little illogical to me for someone to have a second rocker if they’ve never had a first rocker!

The other common misconception is that the rockers are alternative labels for phases of the gait cycle. Again Perry never used them in this sense, for her they are mechanisms that allow the tibia to move forward over the foot not phases of the gait cycle. It is particularly erroneous to apply these terms to phases of pathological gait. Many kids with CP never make heel contact and it is thus completely inappropriate to refer to early stance as the phase of heel rocker.

This reinforces the fact that the rockers are mechanisms of normal gait and great care is required in applying the terms to walking with pathology. If a child with CP makes contact with the toe after which the foot comes flat later in stance then they must use a mechanism that might best be described as a reverse forefoot rocker during which the heel is being lowered to the ground rather than being raised. Similarly if they employ a vault to assist clearance of the swing limb then they will often have a reversed ankle rocker during which plantarflexion (rather than dorsiflexion) increases.

Referring back to the work I described in my blog the week before last strongly suggests to me that, in bare feet, the heel rocker is actually a heel roller with the movement being a rolling on the curved surface of the posterior-distal calcaneus rather than a pivot about a particular point on the heel. On the other hand if walking in a shoe with a reasonably stiff heel it is more likely that a rocker like mechanism does occur. The appropriateness of this terminology may thus depend on footwear as well as gait pathology.

PS. In the second edition of Gait Analysis, Perry and Burnfield describe a fourth toe rocker very late in stance.  This can certainly be seen on slow motion videos but I’m not aware of any detailed studies of its biomechanical significance. It looks to occur very late on and I suspect only after most of the load has been taken off the foot but it would be nice to see a more definitive analysis of this.