Welcome to the Podiatry Arena forums

You are currently viewing our podiatry forum as a guest which gives you limited access to view all podiatry discussions and access our other features. By joining our free global community of Podiatrists and other interested foot health care professionals you will have access to post podiatry topics (answer and ask questions), communicate privately with other members, upload content, view attachments, receive a weekly email update of new discussions, access other special features. Registered users do not get displayed the advertisements in posted messages. Registration is fast, simple and absolutely free so please, join our global Podiatry community today!

  1. Have you considered the Clinical Biomechanics Boot Camp Online, for taking it to the next level? See here for more.
    Dismiss Notice
Dismiss Notice
Have you considered the Clinical Biomechanics Boot Camp Online, for taking it to the next level? See here for more.
Dismiss Notice
Have you liked us on Facebook to get our updates? Please do. Click here for our Facebook page.
Dismiss Notice
Do you get the weekly newsletter that Podiatry Arena sends out to update everybody? If not, click here to organise this.

Validity of comparing plantar pressures before and after callus reduction?

Discussion in 'Biomechanics, Sports and Foot orthoses' started by wdd, Sep 23, 2009.

  1. wdd

    wdd Well-Known Member


    Members do not see these Ads. Sign Up.
    Comparing plantar pressures before and after callus reduction.

    Research measuring differences in plantar pressure before and after callus reduction puzzles me a little.

    I assume that the aim of this type of research is to be able to say something meaningful about the forces applied to the living tissue (as changes in the underlying living tissue produces the changes in the dead stratum corneum), in two different situations. Firstly when the accumulation of the superficial dead layers of the epidermis is ‘normal’/ ‘minimal’ and secondly when the superficial dead layer is significantly thickened/ callused. In both cases the surface at which pressure measurements would more appropriately be taken is at the interface between the living epidermis and the dead epidermis. Ideally multiple measurements would be taken at various depths within the living tissue.

    The intervention of dead epidermis between the living tissue and the pressure measuring sensors is likely to modify the pressure on the underlying living epidermis and as the characteristics of the overlying callus change, eg thickness, hydration, rigidity, less and less reliability can be put upon pressure measurements taken on the outside of a ‘barrier’, ie callus, as an indicator of what is happening at a deeper level .

    I would consider research which measured pressure at the skin surface and then used it to make quantitative statements about the pressure on the periosteum of the underlying bone as flawed?

    Is it scientifically sound to measure the pressures at different interfaces, (ie callus/ground, normal stratum corneum/ground or reduced callus/ground) and then to make statements which seem to imply that a pressure difference at one interface can necessarily be compared to a pressure difference at another interface?

    Imagine a study to compare pressures on the head, when being hit over the head with a hammer, in two situations. Firstly wearing a crash helmet and secondly not wearing a crash helmet.

    Question. Where would you put the pressure sensors?

    Using the logic of callus pressure research it would seem that with the helmet on you would put the pressure sensor on the outside of the helmet and with the helmet off you would put it on the skin of the skull.

    To say anything meaningful about the pressure differences, the measurements must be taken at the same interface. Just because you can't take the pressure measurements at the same interface doesn't make it reasonable to compare pressures taken at different interfaces. You are not comparing like with like.

    Even if my comparison of callus to a crash helmet is not totally acceptable the basic principle holds, unless of course, it has been established that pressures are not significantly modified by their passage through callus?

    How flawed is this type of research? Is it flawed, significantly flawed or fundamentally flawed?


    Bill Donaldson
     
  2. wdd

    wdd Well-Known Member

    Hi Bill,

    You are looking a bit lonely sitting in the corner with no one talking to you so I thought I would put my tuppence worth in.

    I'm pretty sure I don't know anymore about this subject than you do but it seems to me that if this type of research is flawed then whatever the dimensions of the flaw it is undoubtedly augmented by the use of peak pressure? Is peak pressure, in isolation, related to callus production?

    Isn't peak pressure (force/area, just one of the components of the force at the skin sensor interface?

    OK maybe it's the biggest force but size isn't everything. I suppose it was used initially because that was all that the technology allowed researchers to measure?

    Now that the other components of force can be measured what kind of statements does or would this allow researchers to make about the force package/envelope (don't know what it's called)?

    Does it bring podiatrists any nearer to understanding how the application of force at the skin surface results in tissue change in the form of callus accumulation or for example blisters, corns, bursae, ulcers or gangrene?

    Sorry I haven't got access to research papers or professional journals so I might be 'way behind the game.

    Keep up the good work Bill, at least you're keeping me amused.

    Bill Donaldson
     
  3. wdd

    wdd Well-Known Member

    Hi Bill,

    I can't believe it! All that time without someone putting you in your place or showing you the error of your ways. It hasn't been my impression that the users of Podiatry forum were backwards at coming forwards.

    I don't mind too much if I am completely wrong as the point I am trying to make seems too obvious. That is to say I am sure that it is something those who have done research in this area have considered and dismissed for a good reason. However as, I think I said elsewhere, if I am having problems getting my head round it then a lot of other podiatrists are too.

    I am sure that one of the people who has done research in this area will have read the original letter and will hopefully read this letter.

    Am I missing something? Please would someone whose in the know clarify the situation or someone whose not in the know scratch their head in public.

    The question neds to be answered as aspects of diabetic management are determined by research based upon or citing this type of research.



    Best wishes,

    Bill
     
  4. efuller

    efuller MVP

    Hi 2009 Bill and 2011 Bill,

    You need to think about Newton's 3rd law and do a free body diagram of the skin layers to answer your question. The pressure sensor measures the interface between the skin and the sensel of the force platform. (Most force platforms place individual sensors next to each other and measure the force applied to that particular location and then divide by the area of the sensor to give you the pressure value.) So, at the sensor, the force applied to the foot is equal and opposite to the force applied by the foot to the sensor. The deeper layer is the same. The difference in the force applied to the area above the callus is the weight of the callus. So, it is very close to the same force applied at each layer.

    Eric
     
  5. wdd

    wdd Well-Known Member

    Dear Eric,

    Many thanks for your reply. I hope you don't mind if I carry on thinking out loud, so to speak.

    My, bigger picture is to try and understand how that 'lump' of callus can modify the forces (not just those at right angles to the skin but possibly including them) to reduce trauma to the underlying soft tissue.

    At this stage I still want to keep my gaze on the forces at right angles to the skin or sensor.

    In light of your response I am now asking myself what characteristics would the callused area require to have if it was to be able to reduce traumatic forces (at right angles to the skin) reaching the underlying vital tissue? Of course, the peak force is just one of the characteristics of the force, but for the moment I would like to stick with it.


    At this stage I am making a number of assumptions:

    being able to produce callus gives a survival advantage in evolutionary terms;

    callus in some way protects the underlying vulnerable vital tissues, either by directly modifying the forces reaching the vital tissues or by helping to maintain
    homeostasis within the underlying living tissue;

    if the vitality of the soft tissue, directly underlying the callused area, could be mapped the areas of highest tissue vitality would be towards the periphery of the callus and the area of lowest, ie the most vulnerable area, would be towards the centre;

    the peak force at right angles to the skin is a major causative factor in callus production;

    under most circumstances a 'slight' reduction in peak force is all that is required to significantly increase the vitality of the most vulnerable area of underlying tissue.

    I am also assuming that it is likely that evolution, in all of its complexity, has given callus characteristics that go way beyond those of a 'foreign body'.

    Back to the characteristics that callus would require to be able to reduce the peak pressure on the most vulnerable area of vital tissue underlying the callus.

    If the callus either structurally and/or functionally formed a protective dome over the vulnerable tissue might it be able to reduce the peak pressure reaching the most vulnerable area at the centre of the dome?

    What other characteristics would callus require to have to be able to reduce the peak pressure to the most vulnerable area of underlying tissue?

    I hope that makes some kind of sense and that your reply doesn't just refer me back to Newton's 3rd law.


    Thanks again for your reply,

    Bill.
     
  6. Bill, there are a number of assumptions in your last post which probably need to be addressed, but lets start with this question: do you think callus will have a higher, lower or equal Young's modulus compared to "normal" epidermis?
     
  7. wdd

    wdd Well-Known Member

    Hi Simon,

    We are off to a flying start. First question and I don't know the answer to it.

    I would imagine that callus is anisotropic. If it is, would it be important to say in which direction Young's modulus is being measured?

    In tension I would imagine that callus has a higher initial Young's modulus.
    In compression I would imagine callus has a higher initial Young's modulus.
    I would imagine that callus also has a higher bending stiffness across the board.

    Bill
     
  8. Cool. You were talking about normal forces before, so assumed that. Next question why does it hurt if you walk into a lamp post but not when you walk into a foam mattress?
     
  9. David Smith

    David Smith Well-Known Member

    Bill

    Sorry not to answer earlier, if I had noticed the post I would have replied, anyway
    I think your assumption about the crash helmet scenario would hold some water. It would seem reasonable that a protective mechanism would be protective. This protective mechanism, one might reasonably assume would indeed attenuate peak pressures by increasing the area and by changing the Young's modulus of the exterior surface i.e. callus is definentely harder than normal epidermis and dermis.
    Secondly the nature of the deflection of the ray would be changed i.e. to achieve the situation of all the rays/met heads (if we assume we are considering the met heads/MPJs) being level with the ground the one with callus build up would have a greater deflection and therefore stiffness than the met head on the non ray.

    This might have the effect of increasing local or discreet plantar forces and pressures. The point is, as you point out, we do not know without much more in depth investigation.

    As Eric mentioned the total discreet force would have to be the same in each scenario on either side of the callus interface but the discreet force distribution i. e. pressure may be changed.

    Regards Dave
     
  10. wdd

    wdd Well-Known Member

    Hi Simon,

    It hurts because the mass, of which ever part of me comes into contact with the lamp post, is decelerated very rapidly over a very short distance, which generates a high force and probably causes tissue damage. In the case of the foam matress the deceleration takes place over a greater period of time and distance so that the generated force is less and does not cause tissue damage. Also it is likely that the surface area of contact will be much greater with the foam matress.

    Bill
     
  11. wdd

    wdd Well-Known Member

    Hi Dave,

    Thanks for your reply.

    I think that you have highlighted my essential question. Does the pressure on one side of the callus have to the same as the pressure on the other side.

    I am not suggesting the following is true but imagine two rete pegs underlying the callus, are modified, ie become stiffer, as a direct or indirect of the pressure and callus, forming the supports of a 'bridge' of callus and that the area of vulnerable tissue corresponded to the tissue under the 'bridge' would the pressure in that area be equal to the pressure on the external surface of the callus?

    Is it possible to create an area underlying callus where the most vulnerable area of tissue is protected?

    Bill
     
  12. efuller

    efuller MVP

    Hi Bill

    The reason that I talked about Newton's third law was that you were asking about force/pressure deep in the callus. Now, I see that your question is getting a little more toward "Why callus?".

    Evolutionarily, Callus is also preventative of blisters. And blisters are more of a concern in terms of life threatening infections than protection of the underlying structures from pressure/ force.

    So, to prevent blisters you get callus build up. This can be come a problem in areas of high pressure on the bottom of the foot. Increased thickness on the bottom of the foot can become a mass lesion and then cause further pressure in that location. So, on the evolutionary plus side of callus formation you get fewer infections from blisters. On the downside you get increased pressure in areas of high pressure. The body has other mechanisms to adapt to high pressures. For example, you can see a long 2nd metatarsal that has increased cortical thickness as described by Morton.

    Cheers,

    Eric
     
  13. wdd

    wdd Well-Known Member

    Hi Eric,

    Sorry for the comment about referring me back to Newton's 3rd law. That was my insecurity talking. I thought that I was missing your point and that you would just repeat what you had said the first time round and would just keep doing so until I got the significance of what you were saying.

    I notice that your posting directly above this one and my posting diretly above that one are only three minutes apart. Possibly you didn't get a chance to read it before you replied? I think that my last posting makes it clearer although I wouldn't be suprised if it just muddies the waters more.

    At this stage I am only interested in callus and if there is any reasonable way in which the callus, possibly in combination with other tissue elements, could reduce the normal force transmitted to the most vulnerable areas below the callus.

    I always realised that talking could create miscommunication but at least when you are talking with someone you get all the non-verbal cues plus immediate feedback that lets you know if you have hit the mark but with the written word the potential for misunderstanding is multiplied.

    Bill
     
  14. javierdelgado

    javierdelgado Active Member

    Hello bill;
    action- reaction low says that the force you make to something is the same as you receive, so the pressure you make on a sensor is the same as the foot recieve, I think.
    It DOES NOT MATTER IN MY OPPINION WHERE YOU PUT THE SENSOR.
     
  15. wdd

    wdd Well-Known Member

    Dear Javier,

    Thanks for your reply.

    For the sake of argument imagine that a little metal table has been inserted under the skin with the table legs pointing down towards the unerlying bone.

    Would the peak pressure measured at the skin surface be the same as the pressure under the table?

    I think the pressure under the table would be significantly less than the pressure on top of the table?

    If that is correct is it pathophysiologically possible that callus, probably plus some other tissue component/s could form the equivalent of a table under conditions resulting in callus production?

    Best wishes,

    Bill
     
  16. Its my understanding that friction and shear forces are the problem with breakdown and callous formation rather than pressure.

    So how does stiffness of a material effect the friction between layers of tissue. I would assume that having the layer of callous would the point of friction to an area which has less stiffness ie under the callous.

    Direct pressure I can see a protection element but add friction and shear forces I guess callous would add to the problem. But I may be barking up the wrong tree
     
  17. Here something that discusses pressure and shear - Plantar shear: Casting light on ‘dark matter’

    Taken from paper -
     
  18. efuller

    efuller MVP

    Hi Bill,

    If there is a certain amount of force from the ground on the callus, that force has to be supported somewhere deep to the callus. The force above the callus has to equal the force below the callus, or the callus would accelerate relative to the floor (Newton's 2nd law).

    Regarding the callus as table analogy. It depends on what the bottom of the table legs are resting on. For example you could have two people sitting at the table and the table is resting on their thighs and the table legs don't reach the floor. Any object on the table will increase force on the thighs. For your analogy to work you have to find the anatomical equivalent of both the table legs and the floor.

    Eric
     
  19. javierdelgado

    javierdelgado Active Member

    hello again, bill,
    your right, the pressure made by the point is higher, but what I mean is hat the pressure made on the callus is more or less the same thay skin receives.
    Cheers
     
  20. David Smith

    David Smith Well-Known Member

    [​IMG]

    [​IMG]

    Scenario 1/ Gives an example how pressure possibly changes when force stays the same - NB just a change of 1mm radius of contact area = a drop of 25% pressure peak

    Scenario 2/ Gives a different example but also considers the thickness of soft tissue between GRF applied at the skin/callus/helomata and bone and how it compresses (in concept only)

    The main point of consideration is how can we know?

    Dave
     
    Last edited: Apr 8, 2011
  21. javierdelgado

    javierdelgado Active Member

    Dear all;
    p=f/s
    The deepest part of a callus is usually the smallest, so obviously, the pressure is higher.
    Cheers.
     
  22. David Smith

    David Smith Well-Known Member

    That maybe so, and is a reasonable assumption since a corn often hurts to stand on, but how does that validate the assumption that force measured in two different conditions, i.e. condition one (no callus or corn) is synonymous with force measured in condition two (with corn or callus) can apply to the pressure applied to one surface of interest i.e. the normal epidermis, this is the question of the OP. I would say that your statement says it is not.
     
  23. This is a very interesting thread.....I love the theoretical stuff....let me throw my 2 cents in.....

    First of all, Bill asked about the pressure reduction seen with pressure mats/insoles after callous trimming and whether that correlates to internal pressures. I agree with Eric's statement that if higher plantar pressure is present at the callous/ground interface then you can likely also assume that there is higher pressure on the plantar soft tissue and osseous structures that are directly dorsal to the callous with higher plantar pressure. As Eric noted, the high plantar pressure wouldn't exist if there wasn't some plantar structure directly dorsal to the area of high plantar pressure that was not resisting dorsal movement or dorsal deformation by ground reaction force.

    Therefore, I believe we can then safely assume that, for example, if we saw that there was a high pressure area at the plantar 2nd metatarsal head on our pressure mat, and we saw an intractable plantar keratosis also at the 2nd metatarsal head that there would also be an increase in compression pressure acting on the plantar structures of the 2nd metatarsal head including the long and short flexor tendons, plantar plate and plantar 2nd metatarsal head cartilage and subchondral bone.

    A second question to be asked, is what is the biomechanical function of the plantar callous where we may have an increase in thickness of skin which may have a higher elastic modulus and an increased bending stiffness relative to other areas of the skin of the plantar foot? To answer that question appropriately, we must be very precise in how we describe the callous/skin tissue, by describing as follows, since all these factors are important parameters which affect the plantar skin/soft tissue/osseous pressures:

    1. Total area of callous in specific area on the plantar foot.
    2. Thickness of callous at each specific area of plantar foot.
    3. Elastic modulus of callous at each specific area of plantar foot.
    4. Areas of distribution of callous on plantar foot.

    Lets look at one example first. In this foot, we have a callous that is of broad surface area covering all the weightbearing areas of the plantar foot that has not only increased thickness but also increased elastic modulus. This "thickened hide", probably present in the habitually barefoot populations, in fact, reduce the internal forces on the plantar osseous and soft tissue structures during walking or running on irregular surfaces while barefoot. Since the increased bending stiffness of this relatively non-compliant layer of "hide" would prevent surface irregularities from the ground (e.g. pebbles and stones) from causing a large stress-riser on the plantar soft tissue and osseous structures as would normally occur, the thicker soled, higher elastic modulus plantar foot skin would naturally cause reduced plantar soft tissue and osseous pressures.

    However, if another foot with thinner, more compliant skin (habitually shod population) had an isolated 1.0 square cm area of thickened skin plantar to the 2nd metatarsal head with higher elastic modulus, then this foot would have increased plantar pressures in the area of plantar skin that is thickened and has higher elastic modulus which would, in turn, transmit higher pressures into the soft tissue and osseous structures which are directly dorsal to this thickened, stiffer plantar skin lesion. Trimming this 1.0 square cm lesion so that the skin was thinner and the skin became more compliant would naturally reduce the plantar pressures at the skin/ground interface and in the soft tissue and osseous structures that were directly dorsal to the lesion.

    However, now going back to the foot that has the whole surface area of the plantar foot covered by a thick layer of stiff skin and then debriding down an equivalent area of plantar skin of 1.0 square cm area under the 2nd metatarsal head to make the plantar 2nd metatarsal head skin the thinnest of the plantar foot may, in this case, cause more potential for injury plantar to the 2nd metatarsal head if the individual is often exposed to uneven or irregular surfaces such as stones or pebbles on the ground.

    Therefore, we must consider the function of each individual's skin within their plantar skin contact-environment before we can arrive at the best solution for them to increase their functional capacity, decrease their pain and prevent further injuries from occurring within their feet within the future.

    This is an interesting and important subject and is one that definitely deserves a paper or two in one of the podiatry journals in order to describe the normal and abnormal biomechanical function of the plantar skin of the human foot.:drinks
     
  24. wdd

    wdd Well-Known Member

    Hi,

    I would like to stick with my table (but not as we know it) analogy a little longer and recognising that just considering the peak force at right angles to the skin is very limiting continue to consider just that aspect of that force.

    I am trying to establish whether (on a smooth flat surface) a complex anatomical structure, including the callus, could exist that shields part of the tissue underlying the callus from some of the pressure and would reduce the peak pressure in that area.

    Imagine that the table, I mentioned earlier, is a microscopic 'table' and that the length and or stiffness and or strength of the legs is directly related to characteristics of the callus, eg the smaller and thinner the callus the shorter and less stiff the legs and the larger and thicker the callus the longer/stiffer/stronger the legs.

    I have used the words longer, stiffer and stronger assuming these are the characteristics needed to minimize damage to parts of the underlying vulnerable tissue but the characteristics of the legs might be described in some other way (they might have more of the characteristics of the legs on a bouncy castle construction).

    Now imagine a great number of these tables joined together so that when you look at a cross section of them they are more like the arches on a viaduct or railway bridge.

    Of course thats only in two dimensions add the third dimension and in the model that's developing the callus becomes the table top and the table 'legs' are formed by the epidermal rete pegs and while the vulnerable tissue would be the epidemal basal cells the system might directly protect the blood supply.

    As I only want to establish a principal at the moment assume that I have hit the nail on the head so to speak. Would/could the above model create areas under the callus where the peak pressure was less than that measured at the external aspect of the callus?

    Bill
     
  25. Mike:

    Thanks for providing us with this nice paper by Keith Williams and coworkers. Keith is the head of the kinesiology department at UC Davis, my alma mater. Keith is a good friend of mine and I have helped him with a few of his research projects. Keith has been around a long time and he did quite a few papers with Peter Cavanagh in the early years of running biomechanics research. He is a very smart man.

     
  26. Bill:

    I am at a loss trying to understand your model here. Why don't you try doing a drawing of what you are trying to say here? I'm sure that many others would be able to follow you better with an illustration to demonstrate your "table concept".
     
  27. efuller

    efuller MVP

    Bill, In your model you have to have table rest on something/ somewhere. It can't just float plantar to the bones. Kevin's idea of a "hide" type effect where the skin is thick could certainly spread ground reactive force over a wider area or more anatomical structures. This is in contrast to a keratoma where the forces would be concentrated on the above anatomical structure.


    Cheers,

    Eric
     
  28. wdd

    wdd Well-Known Member

    Hi Kevin,

    I can do a diagram without any problem. What I haven't mastered yet is attaching it so that it shows in my posting.

    Tell me how to do that and I'll put one online.

    Bill
     
  29. wdd

    wdd Well-Known Member

    OK

    I think I have managed to attach a diagram.

    here goes.

    Bill

    PS. The red dots represents the blood vessels and the areas where I think the peak pressure will be reduced.
     

    Attached Files:

  30. Bill:

    Now that you have sent us the diagram, please explain it to us. I have converted your illustration to a jpg image so it is easier for you to discuss.
     

    Attached Files:

  31. wdd

    wdd Well-Known Member

    Hi,

    I have spent a bit more time creating a diagram and writing a description of how certain areas directly underlying the callus do not transmit the pressure directly although of coutrse other area translit greater pressure to compensate.

    I am attaching the new and better (I hope) diagram and another attachment with the words that go with it.

    here goes.
     

    Attached Files:

  32. efuller

    efuller MVP

    Bill,

    Is there any evidence that your diagrams represent reality? I'm not up on the latest histology of callus and non callused skin.

    Eric
     
  33. wdd

    wdd Well-Known Member

    Hi Eric,

    My evidence is based on years of removing callus with a scalpel and noting how frequently the striations, under the callus, appeared further apart, thicker and deeper and how stratum corneum, to a greater of lesser degree, filled the valleys between the striae.

    Possibly my evidence is at the level of expert witness. However I am sure there are plenty of expert witnesses out their who will be able to confirm or refute my evidence.

    I am sure someone will be able to direct us to electron micrographs of normal and callused plantar skin?

    Bill
     
  34. wdd

    wdd Well-Known Member

    The attachments aren't quite what I want. One is of thick skin and the other is of thin skin.

    The thick skin is either from the palmar or plantar surface and shows the rete pegs and the stratum corneum following the same contour.

    Now I am looking for one of callused thick skin to see if the rete pegs are more exaggerated.

    Bill
     

    Attached Files:

  35. wdd

    wdd Well-Known Member

    Found a supportive reference.

    Lorimer D, French G, O'Donnel M, Burrow JG, Neale's Disorders of the Foot, 6th Ed. 2004, page 490, Fig.14.16. Churchill Livingstone.
     
  36. I suspect Kate Springett may have looked at comparative histology of callused and normal skin as part of her PhD studies- check her chapter in Assessment of the Lower Limb -Merrimen and Tollafield
     
  37. wdd

    wdd Well-Known Member

    Hi Simon,

    Thanks for the suggestion.

    I don't have access to any podiatry books but I did find a contact e-mail address for Kate online and have e-mailed her asking if she has any photos that demonstrate changes in the rete pegs in callused skin.

    Best wishes,

    Bill
     
  38. Bill:

    I wish I could say that I was impressed with your ideas, but I can't. I believe that the microscopic geometry of the rete pegs are insignificant mechanical factors when compared to the thickness and elastic modulus of calloused skin in producing the abnormal tissue stresses that create the pain from plantar callouses.

    However, please have a go at proving me wrong...I'm all ears.:drinks
     
  39. wdd

    wdd Well-Known Member

    Hi Kevin,

    I think that you may well be partly right but I don't think the two ideas are incompatible ie the geometry of the rete pegs may be 'slight' in comparison to the thickness of callused skin in producing the abnormal tissue stresses that create the pain from plantar callus without being 'insignificant'.

    Using the table analogy. For me the 'table' produced when the callus is thick doesn't have to be a perfect table it only has to be slightly better than the uncallused 'table' to be significant. 'Better' meaning able to protect, in a relative sense, a more vulnerable area, a little more effectively for a little longer.

    At this stage I think my idea is qualitatively sound but quantitatively it needs some support, ie yes theoretically it seems reasonable that a more protective arch is formed but the extent to which it shelters anything is likely to be very small possibly insignificant?

    I am going to spend it little more time thinking about the factors contributing to 'significance' in this case.

    Bill
     
Loading...

Share This Page