iMechanica - biomechanics - Comments

Internal energy -> 1/resonance frequency ?

Biswajit Banerjee — Thu, 09 Feb 2012 18:38:15 -0500

For a simple linear spring-mass system:

Internal Energy: W = 1/2 k d^2 where d = displacement, k = spring stiffness.

Resonance frequency: omega = sqrt(k/m) where m = mass.

Then

omega^2 = 2/(md^2) W

So what you're seeing could be due to changing m and d (or some inelastic effect).

-- Biswajit

\omega_n = \sqrt{1/fm}, f

tigerylx — Thu, 09 Feb 2012 04:44:59 -0500

\omega_n = \sqrt{1/fm}, f is flexural stiffness. For multi-freedom system, flexural matrix is the inverse of stiffness matrix. For single freedom system. K = 1/f;

potential energy

E = 0.5 u^2/f

u is displacement.

Hi Ravi, Thanks for the

sinornis — Wed, 08 Feb 2012 20:43:11 -0500

Hi Ravi,

Thanks for the response. Flexural stiffness of a body is inversely proportional to internal strain energy, though, isn't it? So when internal strain energy increases, natural frequency decreases.

What equations describing natural frequency can this relationship be seen in?

Roger

Hello Roger,

ravi_gutti — Wed, 08 Feb 2012 00:09:39 -0500

Hello Roger,

Natural frequency is directly proportional to stiffness of the body. This stiffness inturn is directly proportional to the internal strain energy of the body.

This means when the internal strain energy increases the natural frequency also increases and vice vers.

Anybody correct me if i am wrong...

ravi

Initial stress field

Alkiviadis Tsamis — Mon, 30 Jan 2012 08:41:01 -0500

Dear Ellen,

Dear Dr. Sacks,

Thank you for stimulating discussion on the important aspect of reference state. I agree it wouldn’t really matter what reference state we choose to calculate the deformed state, as long as we know the stress field in the reference state. For example, if one uses the end-diastolic configuration as reference state (as I have also done ) to calculate strains and stresses in the deformed state, without accounting for the already existing stress-field in the end-diastolic state due a blood pressure of say 80mmHg, that would be a limitation of the study.

And say that we know the stress field in the chosen reference state and we are interested in finding the real stress-free or traction-free state of the geometry. How can we do that if the geometry buckles under zero pressure? For example, a saccular cerebral aneurysm that has a very thin wall can easily buckle if one attempts to deflate it.

Another important aspect is the use of stress-free vs traction-free state. How can one define the ‘‘Opening Angle’’ in the cardiac or aneurysmal wall? If the geometry is considered thin-walled, one can approximate the zero-stress state by the load-free state. But if the geometry is thick-walled, one should account for the zero-stress state.

Best regards,

Alkis

Re: What does a reference state mean to a growing biological sys

Arash_Yavari — Mon, 23 Jan 2012 15:50:56 -0500

Dear Michael:

Very good questions. I don't think I'm the right person to answer most of your questions but regarding a reference state this is what I think. The first question would be: do biological systems experience any elastic deformations? If yes, then depending on the system one can define an elastic energy density. This energy density depends on the "elastic part" of your strain measure (of course, there are several such measures). A useful reference state would be one that is stress-free. Such a state may not even be realizable in our Euclidean 3-space. However, if one can find a three-dimensional manifold in which the body is stress-free (again a space that we can only visualize when living in the "rigid" 3D Euclidean space) then our problem would look like a classical nonlinear elasticity problem: reference configuration is being mapped to the deformed configuration. Having a constantly loaded system implies that this stress-free reference state is evolving (a manifold with an evolving geometry very much like space-time in Einstein's general relativity). In particular, the deformation gradient is purely elastic and everything anelastic would be buried in the reference state (which is explicitly time dependent).

Regards,
Arash

Re: surface growth

Arash_Yavari — Mon, 23 Jan 2012 15:32:34 -0500

Hi Amit,

If you don't like to add new material points in surface growth then your "change of mass density" has to be singular for the whole mass to increase (or decrease) as you can have non-vanishing mass density change only on a measure-zero set. So, I think for surface growth one has to consider a time-dependent underlying set. What do you think?

Regards,
Arash

Re: Re: bulk growth and invariance

Arash_Yavari — Mon, 23 Jan 2012 15:28:03 -0500

Hi Amit,

I cannot object to a bulk growth in which material points are not conserved. After all, these are all models. What I can say is that assuming that material points are conserved would be reasonable for bulk growth and this is what most people have assumed so far.

My understanding is that you postulate objectivity of energy balance (with some other assumptions) and then see what you get. For classical nonlinear elasticity, objectivity of energy balance and governing equations are "if and only if".

Regards,
Arash

Evolution of mass fractions as a function of wall stress

Alkiviadis Tsamis — Mon, 23 Jan 2012 08:28:09 -0500

Dear all,

Some experimental findings of hypertension-induced arterial remodeling suggest that the time variation in constituent mass fractions (or densities) is driven by wall stress. It was found that thickening in arteries, which serves to restore the circumferential wall stress to control after a sustained increase in pressure, was mainly due to enhanced collagen deposition in the media and adventitia. The smooth muscle content was also increased in the media. The change in mass fractions of the constituents appears to be driven by the deviation of circumferential stress from its normotensive value.

Hu, J. J., Fossum, T. W., Miller, M. W., Xu, H., Liu, J. C., and Humphrey, J. D., 2007, “Biomechanics of the Porcine Basilar Artery in Hypertension,” Ann. Biomed. Eng., 35(1), pp. 19–29.
Hu, J. J., Ambrus, A., Fossum, T. W., Miller, M. W., Humphrey, J. D., and Wilson, E., 2008, “Time Courses of Growth and Remodeling of Porcine Aortic Media During Hypertension: A Quantitative Immunohistochemical Examination,” J. Histochem. Cytochem., 56(4), pp. 359–370.
Xu, C., Zarins, C. K., Pannaraj, P. S., Bassiouny, H. S., and Glagov, S., 2000, “Hypercholesterolemia Superimposed by Experimental Hypertension Induces Differential Distribution of Collagen and Elastin,” Arterioscler., Thromb., Vasc. Biol., 20(12), pp. 2566–2572.
Walker-Caprioglio, H. M., Trotter, J. A., Little, S. A., and McGuffee, L. J., 1992, “Organization of Cells and Extracellular Matrix in Mesenteric Arteries of Spontaneously Hypertensive Rats,” Cell Tissue Res., 269(1), pp. 141–149.

Below you will see some recent studies which link the evolution of mass fraction or density of the arterial wall components (elastin, collagen, smooth muscle cells, water) to the local value of wall stress through the variation in geometrical dimensions, as compared to other models which describe the evolution of mass fractions by using functions of time which are not directly associated with wall stress.

Tsamis A, Stergiopulos N, Rachev A. A structure-based model of arterial remodeling in response to sustained hypertension. Journal of Biomechanical Engineering 131(10): 101004, 2009.
Rachev A, Gleason RL Jr., 2011, Theoretical study on the effects of pressure-induced remodeling on geometry and mechanical non-homogeneity of conduit arteries. Biomech Model Mechanobiol, 10(1), pp. 79-93.
Tsamis A, Rachev A, Stergiopulos N. A constituent-based model of age-related changes in conduit arteries. American Journal of Physiology - Heart and Circulatory Physiology 301(4): H1286-H1301, 2011.

Best,

Alkis

Literature on arterial growth

Alkiviadis Tsamis — Mon, 23 Jan 2012 07:42:42 -0500

Hi Ellen,

In terms of growth in arterial tissue, one can find the following 3 general categories of models in the literature.

Volumetric Growth Approach

Examples:

Taber, L. A., 1998, "A Model for Aortic Growth Based on Fluid Shear and Fiber Stresses," J. Biomech. Eng., 120(3), pp. 348-354.

Taber, L. A., and Humphrey, J. D., 2001, "Stress-modulated Growth, Residual Stress, and Vascular Heterogeneity," J. Biomech. Eng., 123(6), pp. 528-535.

Rachev A, Gleason RL Jr., 2011, Theoretical study on the effects of pressure-induced remodeling on geometry and mechanical non-homogeneity of conduit arteries. Biomech Model Mechanobiol, 10(1), pp. 79-93.

Global Growth Approach

Examples:

Rachev, A., Stergiopulos, N., and Meister, J. J., 1996, "Theoretical Study of Dynamics of Arterial Wall Remodeling in Response to Changes in Blood Pressure," J. Biomech., 29(5), pp. 635-642.

Rachev, A., Stergiopulos, N., and Meister, J. J., 1998, "A Model for Geometric and Mechanical Adaptation of Arteries to Sustained Hypertension," J. Biomech. Eng., 120(1), pp. 9-17.

Rachev, A., 2000. A model of arterial adaptation to alterations in blood flow. Journal of Elasticity 61, 83-111.

Tsamis A, Stergiopulos N. Arterial remodeling in response to hypertension using a constituent-based model. American Journal of Physiology - Heart and Circulatory Physiology 293(5): H3130-H3139, 2007.

Tsamis A, Stergiopulos N, Rachev A. A structure-based model of arterial remodeling in response to sustained hypertension. Journal of Biomechanical Engineering 131(10): 101004, 2009.

Tsamis A, Stergiopulos N. Arterial remodeling in response to increased blood flow using a constituent-based model. Journal of Biomechanics 42(4): 531-536, 2009.

Tsamis A, Rachev A, Stergiopulos N. A constituent-based model of age-related changes in conduit arteries. American Journal of Physiology - Heart and Circulatory Physiology 301(4): H1286-H1301, 2011.

Constrained Mixture Approach

Examples:

Brankov G, Rachev A, Stoychev S. 1975. A composite model of large blood vessels. Mech Biol Solids pp. 71–78.

Humphrey, J. D., Rajagopal, K. R., 2003. A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomechanics and Modeling in Mechanobiology 2, 109-126.

Gleason, R. L., Taber, L. A., Humphrey, J. D., 2004. A 2-D model of flow-induced alterations in the geometry, structure, and properties of carotid arteries. Journal of Biomechanical Engineering 126, 371-381.

Gleason, R. L., and Humphrey, J. D., 2004, "A Mixture Model of Arterial Growth and Remodeling in Hypertension: Altered Muscle Tone and Tissue Turnover," J. Vasc. Res., 41(4), pp. 352-363.

Gleason, R. L., and Humphrey, J. D., 2005, "A 2D Constrained Mixture Model for Arterial Adaptations to Large Changes in Flow, Pressure and Axial Stretch," Math. Med. Biol., 22(4), pp. 347-369.

Alford, P. W., Humphrey, J. D., and Taber, L. A., 2008, "Growth and Remodeling in a Thick-walled Artery Model: Effects of Spatial Variations in Wall Constituents," Biomech. Model. Mechanobiol., 7(4), pp. 245-262.

Best,

Alkis

Reference state and driving force

lncool — Thu, 19 Jan 2012 01:35:17 -0500

Michael, great comments! I think you hit it right on the head!

But, to be even more provocative... for growth and remodeling theories to be successful, does it even matter whether there is a single unique reference state? Does it matter whether this state is stress free or not? And does it matter whether growth is stress or strain driven?

For example, in your work on the mitral valve leaflet, I believe it is a huge step already to be able to say that leaflets can grow if they are stretched beyond their physiological limits. I think it's cool that continuum mechanics allows us to quantify this growth and to identify heterogeneous growth patterns, both in space and time. Maybe, one day, this may help us to identify what truly triggers growth on the cellular or even subcellular levels.

I really like the plain mathematical challenges that come with modeling growth, which have been discussed on this site so far. But I believe that growth theories should be able to do more than just reproduce what we see. In that sense, I see Jay as a true role model who is using these theories to generate new hypotheses with the bigger picture to learn more about the behavior of living matter.

To further stimulate this discussion, to what extent will standard continuum theories without growth and remodeling even be able to tell us something useful about a living system?

What does a reference state mean to a growing biological system?

Michael S. Sacks — Wed, 18 Jan 2012 23:04:00 -0500

Folks,

While we all clearly want put G&R on a firm mathematical/physical ground, we must recall we are working with complex biological systems.

So, in the interest of stimulating discussion on another point of view, consider the following:

1) What does a ref state mean to constantly loaded biological tissue that never sees (or probably never senses) an unloaded state?

2) Does the concept of strain really have any true biological meaning or relevance?

3) Does it matter if G&R is deformation or force driven?

4) Recall that Humphrey posited that growth takes place in a fully loaded state - never in the unloaded state as in Rodriguez and Hoger. How does this affect our fundemental approaches when using plasticity-derived decomposition theories, especially when considering the underlying biological mechanisms?

Recall also that cells, which drive the entire process of G&R, sense forces and displacements only. Linking this simple but elusive concept to tissue-continuum concepts is critical.

W. A. “Tex” Moncrief, Jr. Simulation-Based Engineering Science Chair
Professor of Biomedical Engineering
Institute for Computational Engineering and Sciences (ICES)
The University of Texas at Austin
201 East 24th Street, ACES 5.438
1 University Stati

Modeling aneurysm growth

lncool — Tue, 17 Jan 2012 20:07:20 -0500

Hi Baek,

Thank you this is great! What a nice overview! I have started to read the papers you have suggested and I really like the ones on swelling gels.

Thanks a lot!

Ellen

surface growth

Amit Acharya — Mon, 16 Jan 2012 23:21:57 -0500

Arash,

Even for surface growth if one did not change the set of particles constituting the body but dumped new mass near/on the boundary particles and the density was smaller than in bulk regions, it would be hard to tell if the body was/wasn't growing at the surface....

- Amit

Re: bulk growth and invariance

Amit Acharya — Mon, 16 Jan 2012 17:21:10 -0500

Arash,

Bulk growth does not have to correspond to "material points conserved" - this is the relatively easier game (which I would opt for anytime!).

I would tend to agree that energy balance should be invariant under SRBM (perhaps only because of my conventional prejudice) - but what it implies when allowing mass to grow or redistribute relative to the particles is what worries me. In what I did, the 'constraint' comes about. Now if you look at that statement carefully, while the possibility may exist that it allows the 'J part' of the stress tensor to be objective, it is definitely not obvious from the statement that should be the case - in fact, it is quite to the contrary. Were this to be true - a part of the stress not being objective, that gives me the heebie-jeebies - and I suspect this is not the case in my tentative formulation alone! This is what I was asking about in the first place - do people have formulations for this situation where balance of energy as well as the whole stress end up being objective. It would make me very happy if someone were to answer in the affirmative so I could learn from there.

- Amit