Thought Experiment #2: Effect of STJ Axis Location on Met Head GRF

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In this second in a series of podiatric biomechanics thought experiments, I have created a model of the foot including the tibia and talus as one rigid body, the talo-tibial unit, which is attached to a rigid foot model by way of the subtalar joint axis, that, in this model is a hinge perpendicular to the frontal plane. The foot has only two weightbearing surfaces at the approximate locations of the 1st and 5th metatarsal heads. F1 is the ground reaction force (GRF) acting plantar to the medial weightbearing surface of the foot and F2 is the GRF acting plantar to the lateral weightbearing surface of the foot. It may be assumed that both the total GRF acting on the plantar foot and the compression force acting through the talo-tibial unit to the STJ axis is equal to 400 Newtons.

In the three examples shown, the STJ axis is positioned in a medial location (left), a central location (center) and a lateral location (right). The distance from F1 to F2 in each foot is 8 cm. In the foot with the medially deviated STJ axis, the STJ axis is 2 cm from F1 and 6 cm from F2. In the foot with the central STJ axis location, the STJ axis is 4 cm from F1 and 4 cm from F2. In the foot with the laterally deviated STJ axis, the STJ axis is 6 cm from F1 and 2 cm from F2.

Here are my questions:

1. Determine how the magnitudes of GRF change at the medial and lateral weightbearing surfaces of the foot (i.e. F1 and F2) as the axis moves from a medial to central and lateral location.

2. Why does this change in weightbearing pattern occur on the plantar foot in all three examples if the tibia is still at the same angle to the foot in all the examples?

3. If the weightbearing surfaces of the foot were movable dorsally, which surface in which foot would tend to have more dorsiflexion force on it?

4. Which foot would have more equal tendency for dorsiflexion of the weightbearing surfaces?

5. In the foot with the medially deviated STJ axis, how could we better design the leg-foot model to have reduced GRF plantar to the medial weightbearing surface by adding tensile load-bearing structures to it?

6. In the foot with the medially deviated STJ axis,how could we better design the leg-foot model to have reduced GRF plantar to the medial weightbearing surface by adding compression load-bearing structures to it?

7. In the foot with the laterally deviated STJ axis, how could we better design the leg-foot model to have reduced GRF plantar to the lateral weightbearing surface by adding tensile load bearing structures to it?

Interested in anyone's thoughts on this little experiment.

 

Dear Kevin,

I have been looking at these pictures with much confusion and I must admit that my problem solving skills and physics skills are a little rusty (I have only recently joined the world of Podiatry Arena).

Is the model stationary, ie. the forces applied to the axis are not moving the foot (inverting or everting) but the GRF is the only variable here? And by redesigning the foot with tensile/compression load bearing structures, would that be simulating muscle or ligament around the STJ? I apologise if I have misinterpreted the question

Regards

Donna

 

Thanks for taking the interest in commenting on my thought experiment, Donna. I thought these thought experiments would generate more discussion but it seems that either most people are either uninterested or are too timid to try and answer them. This problem is fairly straightforward.

The model I have created is stationary and is in both translational equilibrium and rotational equilibrium. The movement of the STJ axis from a medial location to a central location and then to a lateral locatoin will change the GRFs at the two weightbearing points on the plantar foot.

Translational equilibrium means that all vertical forces and all horizontal forces are equal and opposite to each other so that the object is not translating. In other words, there is no acceleration occuring since the net force (F) equals 0. Since Newton's Second Law of Motion states F=ma, then if F equals O, then acceleration (a) must also equal 0. Translation desribes linear movement, in other words, an object moving from point A to point B in a straight line.

Rotational equilibrium means that all rotational forces acting about an axis of rotation are counterbalanced, or are in equilibrium, so that there is no net rotational force (i.e. moment) acting across that axis of rotation. Again, Newton's Second Law of Motion can be used here since a moment is simply a rotational force so that if the moments acting in one direction versus the opposite direction (i.e. pronation vs supination) are equal, then it follows that the rotational acceleration (i.e. angular acceleration) will be 0 at the axis of rotation. So rotational equilibrium will only occur in this model when the supination moments are equal to the pronation moments.

Before I give the answers away, let me give you a hint. The center of pressure (CoP) under the plantar foot in all three examples will be directly under the axis of rotation of the subtalar joint and the total ground reaction force (GRF) acting on the plantar foot will be equal to 400 Newtons (N). So that in the model on the left, the CoP will be 2 cm lateral to F1 and 6 cm medial to F2, in the center model the CoP will be 4 cm lateral to F1 and 4 cm medial to F2, and in the model on the right, the CoP will be 6 cm lateral to F1 and 2 cm medial to F2.

To solve the problem, you must find what magnitude of force both F1 and F2 must be, at their respective distances (i.e. moment arms) to the axis of rotation of the STJ, in order so that the pronation and supination moments acting from GRF are equal to each other (i.e. no net rotational force acting at the STJ axis). This would be much like the high school physics problem of trying to solve what would need to be the weights of two boys, one sitting at each end of a see-saw, in order to balance a see-saw in a horizontal position.

Go ahead ...... give it a try.

 

I'm guessing its either really simple, or really complicated, so i will have a crack at the simple:

(1) Moving from Left to Right:

a. F1 300N, F2 100N

b. F1 and F2 200N

c. F1 100N, F2 300N

2.

3. a. F2 b. Same c. F2

4. Centrally located Joint axis (b)

5. Change the direction of the force applied by the tibia? Move the force moving from medial to lateral or in this case more left to right looking at the diagram. This could be done by moving F2 dorsally (applying a varus wedge or removing a valgus wedge of bone...ouch!), which would tilt the foot offer , causing the tibia to apply its force as described above.


6. & 7. What the difference between tensile and compression forces again?

 

PF3:

Your answers are correct for the F1 and F2 magnitudes. Good job...it wasn't that difficult after all, now was it?

Here are the rest of my questions and my answers:

Because the frontal plane angle of the tibia does not determine the rotational forces acting across the STJ axis or the GRF acting on the plantar foot.

The medial weightbearing surface in the medially deviated STJ axis and the lateral weightbearing surface in the laterally deviated STJ axis would tend to have the most dorsiflexion force on it. Hence, the medial metatarsal rays will tend to be the metatarsal rays that have the most dorsiflexion moment on them in the medially deviated STJ axis foot and the lateral metatarsal rays will tend to be the metatarsal rays that have the most dorsiflexion moment on them in the laterally deviated STJ axis foot. Could forefoot varus and forefoot valgus then be caused by, respectively, a medially deviated STJ axis and a laterally deviated STJ axis due to plantar ligament creep (over time) in each respective column of metatarsal rays?

The foot with the centrally located STJ axis .

We might add a passive tensile load-bearing structure (like a deltoid ligament) from the medial tibia to the medial foot that placed a supination moment across the STJ axis when it was stretched. We could add a contractile tensile load-bearing element from the medial tibia to the medial foot (like a posterior tibial muscle) to create a variable magnitude of STJ supination moment in order to decrease the magnitude of F1 and increase the magnitude of F2.

We could add a wedge of compression-resistant material in the area between the talo-tibial unit and lateral-dorsal calcaneus (like a subtalar arthroeresis) to increase the STJ supination moment which would decrease the magnitude of F1 and increase the magnitude of F2.

We might add a passive tensile load-bearing structure (like a calcaneo-fibular ligament) from the lateral tibia to the lateral foot that placed a pronation moment across the STJ axis when it was stretched. We could add a contractile tensile load-bearing element from the lateral tibia to the lateral foot (like a peroneus brevis muscle) to create a variable magnitude of STJ pronation moment in order to decrease the magnitude of F2 and increase the magnitude of F1.

 

Dear Kevin,

I am now kicking myself for being so timid and not just answering the question to start with! I am now waiting for #3 to appear so I can have a go at that one!


Donna

 

Kevin,

Thanks for bothering to design these questions, or 'thought experiments'. They are really good!

It helps a lot with understanding the significance of STJ axis position. I have a much better understanding of it now.

Phil

 

Phil, thanks for the encouragement. Maybe I'll do some more.

You may be interested that great scientists such as Galileo Galilei, Ernst Mach and Albert Einstein used thought experiments in order to further their knowledge http://en.wikipedia.org/wiki/Thought_experiment

Mach first coined the phrase "Gedankenexperiment" (German for "thought experiment"). However, Einstein is probably best known for his use of thought experiments to help him make progress in his theories of relativity. They are useful tools that I have found to be helpful in the past by allowing me to better understand complex mechanical subjects.

 

Sorry for being a nit picker, but only the average axis foot is in rotational equilibrium as drawn. The medially deviated foot has a net pronation moment as drawn and no counter moment that would create equilibrium. So the foot is translational equilbrium, but not rotational equilibrium

 

Eric:

When you plug in the GRFs for the medial and lateral forefoot, at 300 N and 100 N respectively, the medially deviated foot will be in rotational equilibrium.

 

Sorry, it was a quick look at the pictures that fooled me. The arrows are the same length so I thought that you meant to have the force be equal.

Eric

 

"The medial weightbearing surface in the medially deviated STJ axis and the lateral weightbearing surface in the laterally deviated STJ axis would tend to have the most dorsiflexion force on it. Hence, the medial metatarsal rays will tend to be the metatarsal rays that have the most dorsiflexion moment on them in the medially deviated STJ axis foot and the lateral metatarsal rays will tend to be the metatarsal rays that have the most dorsiflexion moment on them in the laterally deviated STJ axis foot. Could forefoot varus and forefoot valgus then be caused by, respectively, a medially deviated STJ axis and a laterally deviated STJ axis due to plantar ligament creep (over time) in each respective column of metatarsal rays?"


Isn't that the whole theory of a forefoot supinatus?

 

That is not only a theory to explain what has been previously called "forefoot supinatus" in the medially deviated STJ axis foot but will also explain a "forefoot valgus" or "plantarflexed first ray" in the foot with a laterally deviated STJ axis. The $64,000 question then becomes: what came first, the abnormal STJ axis location or the forefoot to rearfoot "deformity"??

By the way, can you give any reference on forefoot supinatus that explains this "whole theory"?