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Journal Club Theme of January 2012: Mechanics of Growth

lncool's picture

"I can't understand how people are still working on growth. That stuff's all done." This was the beginning of the first lunch conversation at a recent Banff workshop on Mathematical Foundations on Mechanical Biology... somewhat frustrating for someone who is excited about growth. Fortunately, most of the presentations and discussions still focused on growth. Although "that stuff's all done".

Would anybody claim that plasticity was all done when Richard von Mises published his milestone work in 1913? Or was it all done when Geoffrey Ingram Taylor contributed his famous monograph on crystal plasticity in 1938? Or was it all done when Ekkehard Kröner introduced the concept of dislocations to explain the mechanistic origin of plastic slip in 1958? Or was it only all done when Juan Simo made it computationally manageable in 1985?

In a way, growth is like plasticity. It has its von Mises in Julius Wolff and D'Arcy Thompson, its Kröners in Steven Cowin and Dennis Carter, and its Simos in Rik Huiskes, Anne Hoger, Larry Taber, and Jay Humphrey. But... does that mean that "that stuff's all done"?

In the century of quantitative biology, mechanics has a lot to offer when it comes to exploring living systems. Continuum mechanics is a powerful tool, if not the only one, to bridge what a biologist sees in a dish and what a medical doctor diagnoses in a patient. As mechanics community we are very familiar with the tools to bring these two worlds together and characterize living systems across the scales, from the molecular to the subcellular, cellular, tissue, and organ levels. We are also familiar with the tools to characterize living systems across the fields, from mechanical, to biological, chemical, and sometimes even electrical.

We have to accept though that living systems are somewhat more complex than metal plasticity. Because living systems undergo a continuous turnover, they have the fascinating ability to adapt to their mechanical environment. From a continuum mechanics point of view, living systems are open systems interacting with their surroundings through exchanging mass. This allows them to grow in density and volume [5]. Excellent overviews classify these adaptation phenomena [15] and summarize recent trends [2,11].

Key to most soft tissue growth models is the multiplicative decomposition of the deformation gradient into an elastic and a growth part [14]. The specific format of the growth part, typically a second order tensor, depends on the particular type of tissue. It can be plain isotropic in the form of volume growth, e.g., for growing tumors [1], transversely isotropic in the form of area growth, e.g., for growing skin [18], transversely isotropic in the form of fiber growth, e.g., for growing muscle [8], or generally anisotropic [12]. While some models focus on the plain kinematic characterization of growth [3] and illustrate its geometric interpretation [16], others combine the growth equation with mechanical equilibrium and introduce either stress [14] or strain [4] as the driving force for growth. Alternative approaches adopt density changes, initially introduced to model hard tissue growth, to avoid the introduction of internal variables when characterizing growing soft tissues [17].

Interest in modeling growth is currently shifting from hard to soft matter [10], from infinitesimal to finite deformation [6], from phenomenological to mechanistic [15], from single scale to multiscale [7], from single field to multifield [9], from generic to subject-specific [13], and from reproductive to predictive modeling [2]. As such, growth of living biological tissues undoubtably remains one of the most challenging phenomena in continuum mechanics, and we hope that this Journal Club initiates interest amongst many of you to actively contribute to this rapidly evolving field.

If you don't think "that stuff's all done", please contribute additional references, comments, discussions, or figures. 


[1] Ambrosi D, Mollica F. On the mechanics of a growing tumor. Int J Eng Sci. 2002;40:1297-1316.

[2] Ambrosi D, Ateshian GA, Arruda EM, Cowin SC, Dumais J, Goriely A, Holzapfel GA, Humphrey JD, Kemkemer R, Kuhl E, Olberding JE, Taber LA, Garikipati K. Perspectives on biological growth and remodeling. J Mech Phys Solids. 2011;59:863-883.

[3] BenAmar M, Goriely A. Growth and instability in elastic tissues. J Mech Phys Solids. 2005;53:2284-2319.

[4] Buganza Tepole A, Ploch CJ, Wong J, Gosain AK, Kuhl E. Growin skin: A computational model for skin expansion in reconstructive surgery. J Mech Phys Solids. 2011;59:2177-2190.

[5] Epstein M, Maugin GA. Thermomechanics of volumetric growth in uniform bodies.
Int J Plast. 2000;16:951-978.

[6] Garikipati K. The kinematics of biological growth. Appl Mech Rev.2009;62:030801.1-030801.7.

[7] Göktepe S, Abilez OJ, Parker KK, Kuhl E. A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis. J Theor Bio. 2010;265:433-442.

[8] Göktepe S, Abilez OJ, Kuhl E. A generic approach towards finite growth with examples of athlete's heart, cardiac dilation, and cardiac wall thickening. J Mech Phys Solids. 2010;58:1661-1680.

[9] Humphrey JD, Rajagopal KR. A constrained mixture model for growth and remodeling of soft tissues. Math Mod Meth Appl Sci. 2002;12:407-430.

[10] Jin L, Cai S, Suo Z. Creases in soft tissues generated by growth. EPL. 2011;95:64002,p1-p6.

[11] Menzel A, Kuhl E. Frontiers in growth and remodeling. Mech Res Comm, 2012;42,1-14.

[12] Menzel A. Modelling of anisotropic growth in biological tissues - A new approach and computational aspects. Biomech Model Mechanobiol. 2005;3:147-171.

[13] Pang H, Shiwalkar AP, Madormo CM, Taylor RE, Andriacchi TP, Kuhl E. Computational modeling of bone density profiles in response to gait: A subject-specific approach. Biomech Model Mechanobiol. 2012;11:379-390.

[14] Rodriguez EK, Hoger A, McCulloch AD. Stress-dependent finite growth in soft elastic tissues. J Biomech. 1994;27:455-467.

[15] Taber LA. Biomechanics of growth, remodeling and morphogenesis. Appl Mech Rev. 1995;48:487-545.

[16] Yavari A. A geometric theory of growth mechanics. J Nonlin Sci. 2010;20:781-830.

[17] Volokh KY. Stress in growing soft tissues. Acta Biomat. 2006;2:493-504.

[18] Zöllner AM, Buganza Tepole A, Kuhl E. On the biomechanics and mechanobiology of growing skin. J Theor Bio. 2012;297:166-175.

Lecture notes

ME337 "Mechanics of Growth", Winter 2012, Stanford University.


arash_yavari's picture

Hi Ellen:

This is a fascinating topic. Perhaps you could comment on how you computationally deal with the SO(3) ambiguity of the F=FeFg decomposition?

I think the other important issue is evolution of Fg. I have seen some simple 1D models. How do you deal with this computationally in 3D? I assume evolution of Fg explicitly depends on stress tensor? How?


lncool's picture

Hi Arash,

Great questions!

By the way, everybody, read Arash's excellent overview on the Geometric Theory of Growth Mechanics, it's a really cool introduction into the field!

I don't think there is one single unique evolution law for Fg, nor is there one single unique format of Fg. To identify useful candidates, we have found it useful to closely collaborate with clinical researchers.

In the cardiovascular system, for example, it's usually hypertension that drives growth. So, you are right, it's stress-driven. But depending on the type of disease, growth can also be initiated by overstretch. We have been extremely lucky to collaborate with cardiothoracic surgeons, Dr. Craig Miller and scientists in his lab, who perform controlled in vivo experiments, in which they induce growth and study it over a period of several weeks. 

In skin, for example, growth can be induced artificially by overstretch through tissue expansion. So, it's rather strain-driven. We have found an excellent reconstructive surgeon, Dr. Arun Gosain, who has helped us to create useful models for growing skin.

Unfortunately, and I guess that's what you are referring to, useful data on growth are rare and display tremendous variations. 

Can anybody recommend good literature on how to define growth laws?



Alkiviadis Tsamis's picture

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 


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 


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 


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.  



Serdar Goktepe's picture


Hi All,

Let me briefly address Arash's question regarding the non-uniqueness of the rotation tensor.

Evidently, the non-uniqueness of the rotational part of the deformation gradient is not specific to the kinematics of growth, but arises also for the finite plasticity of non-crystalline materials such as amorphous glassy polymers where two fundamental kinematic approaches are remarkable. While a group of researchers sticks with the Kroener-Lee split of the deformation gradient into F^e and F^p, another group of workers (e.g. Green, Naghdi, Miehe,..) alternatively proposes to model the plastic deformation through the notion of evolving plastic metric. The former group, on the other hand, often makes additional assumptions regarding the plastic spin tensor or the elastic part of the deformation gradient to ensure the uniqueness of the rotational part.

While the above mentioned approaches can be directly adopted to the theory of growth; since different classes of biological tissue possess well-defined micro-structure, in most cases one can relate F^g to the underlying architecture of the tissue and the growth phenomena associated with it. To be specific, if the problem is the high blood pressure-induced hypertrophic growth of the heart, direction of the incompatible growth can be chosen perpendicular to the myofibers by means of structural tensors and its amplitude may be assumed to be driven by over pressure. Compatibility of the final deformed configuration is then fulfilled by the elastic part F^e.




davide's picture


Hi Arash,

             these are two keypoints.

The constitutive equation for Fg must satisfy frame invariance, in the sense that F must transform correctly when the decomposition is applied.Very sketchy, if F=FeFg and F*=QF, of course it must be  Fe*Fg*=RFeFg. Lubarda and Hoger (2002) discuss the most general form of acceptable decomposition, in practice everything works if Fg is invariant with respect to rotations. 

 The remodelling ability of biological tissues  should be encoded in the evolution equation for Fg, that is of course problem-dependent, i.e. depends on the living system at hand. The earliest growth law I know is due to Taber, probably the pioneer in this respect, who introduces the idea of homeostatic stress. DiCarlo and Quiligotti address the same problem in a precise thermodynamical framework and they find that the Eshelby stress should be the correct measure of the the stress in the growth law. 



Konstantin Volokh's picture

Hi guys,  

It is good to have a discussion on this topic. It is true that the multiplicative decomposition dominates the literature. It is also a misfortunate turn of events.  

The multiplicative decomposition has two drawbacks. First, it is physically meaningless because the abstract intermediate configurations are neither observable nor unique. Second, the multiplicative decomposition introduces internal variables that are truly superfluous. 

The whole framework of the soft tissue growth can be happily built upon the addition of the evolving mass density to the classical continuum mechanics formulation. No ambiguity, no internal variables. 


lncool's picture

Hi Kosta,

Great, thanks for your comment! I agree that especially for hard tissues, evolving mass density can capture most phenomena sufficiently well.

I still like the multiplicative decomposition though, especially for soft tissues. Are you aware of models for soft tissue growth that are characterized by density changes alone? It would be great to learn more about that. 

I agree, the most challenging aspect of growth is to find ways to calibrate and validate appropriate evolution laws and functional formats for those growth tensors. But I think it's super fun to talk to biologists and clinical researchers to try to identify correct formats together.

But you're right, those models will be most successful, if the internal variables have a true physical meaning. For example, for muscle, they can be related to the number of sarcomeres in a muscle cell. This number can be counted with relatively simple imaging techniques.

Any literature on density growth you would recommend?





Konstantin Volokh's picture


Though modesty forbids I give a link to my paper on soft tissue growth without internal variables: doi:10.1016/j.actbio.2006.04.002.

The basic idea is to use mass density analogously to the use of temperature in the finite thermoelasticity theories. In the latter case no multiplicative decomposition is required.

Of course, the formulation without internal variables might require some effort. That is fun, for my taste.

I agree with you that some quantitative description of the evolution of the tissue constituents is highly welcome. At least, these quantities should be measurable in principle if no experimental techniques are available yet.


lncool's picture

Hi Kosta,

Thanks for the link, very cool! Sorry for my ignorance, I had never seen soft tissue growth models based on density changes alone. Nice work! And, I agree, density change is a great feature that is not just a random internal variable with no physical meaning.

Would be cool to explore whether your ideas can be expanded to anisotropic growth as well.

The type of growth we model with Fg is usually anisotropic. In in vivo experiments, our collaborators in cardiac surgery have found that what we call "growth" is actually associated with a volume decrease of about 25%, although the heart itself grows in size. Since the number of heart muscle cells does not change in a life time, growth in the heart is rather associated with a rearrangement of microstructures than with a changes in density.

But you are right. Many types of tissue, like your tumors, might grow isotropically. In that case, your model seems perfect! I have come across a couple of people who have critizised the "mass flux", i.e., your equation (8), its mechanistic origin, and its parameters. Have you had questions of that kind, and what would you answer?




Konstantin Volokh's picture

Dear Ellen,

I think that anisotropy can be easily introduced by combining the mass densities of the constituents and the so-called structural tensors formed by tensor products of the unit vectors in the characteristic directions of anisotropy.

Mass flux is a must when a localized growth - surface growth - is considered.


psaez's picture

Hi Kosta!

 I think you are right about the introduction of anisotropy by means of structural tensors. I read a paper some time ago about it what I can't remember it right now...sorry! I don't think you have to use the direction of anisotropy, fibers for example, you could use any other direction, perpendicular or eigenvectors of stress for example, right?

About the "must" in mass flux when dealing with growth, I would say it can be neglected in some cases. For example, I will explain it with the case I am used to. When  endothelial cells feel changes in the wall shear stress they start to produce some substances, as MMP and TGF-beta, which go from the internal face of the vessel through the vessel wall. In this case I totally agree that mass flux is a must. When these and other substances move through the vessel wall they stimulate smooth muscle cells to segregate or degrade more extracellular matrix. In this case, tropocollagen molecules get out the cell and ensemble into the collagen fiber with almost any diffusion in the wall. So in this case I would say that the divergence of the mass flux can be neglected and the deposition or absorption of the new material could be considered just by the source term. Could you please give me your opinion about it?? Anyway, probably is an issue of scale, and if you move down enough, the flux mass would appear...


Konstantin Volokh's picture

Dear Pablo,

I think that the localized growth, e. g. surface growth, should be described by the solutions of the "boundary layer" type. To make such solutions possible it is necessary to increase the order of the differential equations. The equation of the mass flux serves the latter purpose.

For example, I considered the surface 'bone growth' in the following paper: doi: 10.3970/mcb.2004.001.147.



psaez's picture

Hi ellen!

Could you please give us a short intro of why some people critizise the mass flux?? Sometimes I was wondering If should introduce diffusion of some biological substances in the cardiovascular tissue...I think there are many examples in the literature where it is used.



Alkiviadis Tsamis's picture

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.  

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  1. 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.
  2. 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.
  3. 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.



davide's picture


Hi Kostantin,

                      I like a lot the attitude of using the Occam's Razor in mechanics and I agree that internal variables should be inroduced with major parsimony. However my feeling is that the multiplicative decomposition does not introduce an internal variable in the sense of some field that cannot be directly measured. As  well described by  Rodriguez, Hoger and McCulloch, experimentally Fg is the gradient of deformation that takes from the unloaded grown, residually stressed (physically measurable, if like) configuration to the relaxed, possibly incompatible one. As a matter of fact, in some works (see Taber and Humphrey) Fg has been explicitely calculated from grown (physical) configurations cutted in a finite number of pieces.

The second point is the need of some notion of "directionality" to account for observed anisotropic growth in some biological systems. The prototype are arteries: they grow circumferentally for an increase in shear stress (blood flow rate), they grow radially  for an increase in pressure. Where does this directionality come from? Is it related to the mechanical properties of the material thanks some structural tensor, independently on the load? Or should it more likely depend on the principal directions of the stress? In both cases some anisotropy is needed, and a (growth) tensor seems to be the correct too  to address it.






Konstantin Volokh's picture

Dear Davide,

Thanks! In the case of homogeneous deformations the multiplicative decomposition might be observable. However, in the general case of the inhomogeneous deformation both Fg and Fe become internal variables. The piece-cutting business is quite vague because every new cut releases deformations from the previous cut rather than from the initial structure. I believe that the best cuts are those made by the Occam's razor Smile.

In my humble opinion there is no need in Fg to introduce anisotropy. The latter can be done, for example, by a direct account of the directions of evolving fibers etc.


likask's picture


Models based on evolution of mass density can be equivalently expressed by models based on multiplicative decomposition. Opposite case is not possible, pleas note that deformation gradient of growth is tensorial quantity whereas density is a scalar quantity.

In order to include anisotropy into model some additional quantities have to be aded to the model at each material point

In second case, utilizing Elshebian mechanics, a material momentum flux  work conjugate to growth's deformation  can be postulated.This is very attractive,  because material evolution can be controled not only by mass conservation equations but conservation of energy and material momentum flux.

I am new in this field, however I done some some small work following Ellen and others ideas,




psaez's picture

Hi all!

First of all, I really want to thanks Ellen for coming up with such an interesting, and not at all dead, topic. In fact, my thesis work is about growth and remodeling models in the cardiovascular system, so I can not say is dead something I am working on, right? :-)

Kosta, you have shown a very nice work using changes densities to deal with growth and without the DG decomposition. I would like to differentiate first about what could be understood, or at least I do, as volumetric and density growth. Density growth, or density changes, do not have to lead to induce stress in a material point. For example in blood vessels, a collagen fiber can become less dense, due to degradation of the tropocollagen molecules that made up collagen fibers, and do not induce stress but soften (or stiffen) the material. However, volumetric growth do lead to changes in the stress state. For example smooth muscle cells growth via hyperplasia or hypertrophy and they induce stresses in the body. Your model seems to deal with these both processes at once, which is pretty cool. I would also say that I like more models were density and volumetric growth formulations are totally split up, because you have more control over them, in a separate way.
You guys also have been discussing about the volumetric growth tensor, which by the way I dealing with right now. In terms of your point of view of internal variables, I think internal variable, in many cases, have a really clear and measurable value, e.g. in the evolution of Fg, as Serdar pointed out above. In many of the models with volumetric growth, e.g. the models from Ellen and coworkers, the internal variable defining the growth tensor Fg, is directly related to the % of volume expansion, whether isotropic or anisotropic. Ellen mentioned a nice example, where the increase in the muscle volume is due to the increase of sarcomeres which can be relatively easily measured. So, although in some, or many, cases internal variables have a superfluous and ambiguous meaning, in my opinion they do not in this case. Davide discuss it below better than I probably do..



Zhigang Suo's picture

Thank you, Ellen, for such an instructive post! I have not read much on models of growth, but you have made the subject extremely interesting. I’ll study the papers that you have suggested.

Several people have pointed out to me that swelling of elastomeric gels may serve as a model of growth. The model represents highly idealized situations, if they can be called growth at all. But the model does have a particularly clean structure. Specifically, it clearly describes how growth (swelling) depends on stress. The model comes up in polymer physics. But you may be able to relate to existing models of tissue growth.  Here I briefly list the main ideals of the model.

Elastomeric gels. Long, flexible polymers can be crosslinked by covalent bonds to form a three-dimensional network, an elastomer. Submerged in an environment containing solvent molecules, the network imbibes the solvent and swells, resulting in an elastomeric gel.

Thermodynamics. In the simplest case, the state of an elastomeric gel is characterized by deformation of the network and the amount of solvent absorbed. The former is represented by the deformation gradient F, and the latter by the concentration of the solvent C. The free-energy density is a function of state, W(F,C). Once this function is given, one can obtain the equations of state that relate the stress and the chemical potential as the function of (F,C). Consequently, the function W(F,C) characterizes chemomechanical transduction: how stress affects swelling and how a change in the chemical potential generates stresses. The function W(F,C) itself may be determined by a combination of experiment and modeling. Perhaps the best-known example is the Flory-Rehner model.

Kinetics. The flux of solvent is proportional to the gradient of the chemical potential. This kinetic model is closely related to Darcy’s law.

Conservation of the number of solvent molecules. The change in the concentration is balanced by the divergence of the flux.

Balance of forces. Same as in continuum mechanics.

This model is discussed in detail in the following paper:

The model has been extended to treat polyelectrolyte gels, pH-sensitive gels, and temperature-sensitive gels:

I’ll be extremely grateful to hear critiques of this kind of model from you and other growthers.

Konstantin Volokh's picture

Hi Zhigang,

Of course, these type models are suitable for modeling growth. Just interpret C as a mass density of a constituent and you have a growth theory of the type I considered in the paper mentioned above. The only thing you should be careful with is the thermodynamic theology: growing systems are open systems.



lncool's picture

Hi All,

Zhigang, great comments, thanks!

There are a couple of awesome thermodynamics textbooks from the 50s and 60s, de Groot 1951, de Groot & Mazur 1962, Kestin 1966, who first discussed changes in mass using open system thermodynamics. Those initial models were indeed developed for chemomechanics rather than biomechanics. I personally think open systems are really exciting! Here is one of our first papers on open system thermodynamics illustrating the impact of density changes on all other balance laws.

Kuhl E, Steinmann P. Mass- and volume specific views on thermodynamics for open systems.
Proc Roy Soc. 2003;459:2547-2568.

Zhigang, it would super be fun to use swelling of elastomeric gels as a controlable and tunable model system for growth! I hope we can initiate some collaboration on this in the future! Great idea!


Amit Acharya's picture

Hi All:

Interesting things being discussed here!

I have a question for all who have thought about mechanics with a mass flux. When a mass flux is admitted, have you experienced any issues with invariance under superposed rigid body motions (SRBM) of (reduced) balance of energy?

I have played on-and-off with this idea of a mass flux in continuum mechanics and an 'interesting' constraint seems to come up without which I (and may be it's just me)  cannot seem to see how invariance under SRBM can hold. Perhaps you all have encountered a similar issue, but have good formulations to get around that.

Any thoughts/comments will be welcome.

- Amit

arash_yavari's picture

Hi Amit,

For bulk growth energy balance should be modified considering the mass source (in Ellen's reference [5]). If you start with that energy balance (or a geometric variant of it that I considered) invariance of energy balance gives you balance of mass (and the other conservation laws). I would be interested to know more about your "constraint".


Amit Acharya's picture

OK, Arash, here it is

Rough notes, of course. My considerations are in the context of modeling damage, but growth seems to be the other side of the coin! (I hope Ellen will excuse this intrusion of damage into her growth blog!)

Suppose one says that the practical objective is to
develop a dynamical model of damage. We want to link the dynamics to some sort of "basic principle" that should be valid even
under quasi-static balance of linear momentum. The physical idea of a
damaging solid seems to have some implication for balance of mass
since damage may be viewed as a local redistribution of mass in the "body",
whatever the latter means under such circumstances. We try to see what can be
done by trying to obtain damage dynamics as an appropriate statement of balance
of mass.

We want to do conventional continuum mechanics of particle motion; so topology of body has to remain unchanged in a motion to avoid headaches over fundamental kinematics. Therefore, we assume that mass is a field that may "move" relative to the (mathematical) particles comprising the body. The body consisting of these
particles moves according to Newton/Euler balance of linear and angular
momentum. So one can get into a situation where a subpart of the body has no

While I appreciate your interest, let me also warn you that you might be completely wasting your time on reading my ramblings and it could be physical garbage - cannot determine at this stage!

- Amit

Amit Acharya's picture


Energy balance for models for bulk growth (or void damage) can  be made invariant under SRBM by several different assumptions. In the notes I posted, it shows one way of doing it within a consistent mechanical model. My concern is about the physical implications of such assumptions, as is laid out in my remarks in the notes.

It seems to me that in all such cases what one is doing is a very fundamentally subtle thing and therefore my discomfort. At the end of the day, there are two choices - one is to say you are changing the set of particles that constitute the body during growth (or void damage) but the mass of any fixed set of particles remains fixed during the process. If this is the fundamental assumption, then the continuous kinematics of motion has to be abandoned, it seems to me - this is very, very BIG and I am almost certain technically very, very difficult. Note that this is different from the rocket burning fuel problem which can be done with the physical equations of conventional continuum mechanics several different ways - Eulerian, Lagrangian or on a domain (control volume) moving with an arbitrarily specified velocity.

On the other hand, one could say that the set of particles constituting the body remains fixed, but the mass content of fixed sets of particles can change. This is technically easier, the conventional definition of motion of a body can be preserved (which is what I seem to see in most of the literature without having done an exhaustive search), but then there are curious things that emerge in making balance of energy invariant. I suppose one needs to solve some carefully crafted, simple, but essential-in-content, problems to test the implications of these assumptions.

So, my concern is not so much about making balance of energy invariant and getting corresponding balance laws (or vice versa - postulating physically reasonable balances and then make the assumption that makes BoE invariant, as I did). It is whether what has been put in works out to be physically reasonable or not, as a theory. Right now, from all I have seen (including my own stuff on this issue), I cannot make up my mind......

- Amit

arash_yavari's picture

Hi Amit,

Bodies can grow in the bulk or on a surface(s). What is discussed here (as far as I understand it) is bulk growth.  This corresponds to your "second choice", i.e. material points are "conserved" and only mass density changes. In my formulation this means that the material manifold as a set is fixed but its geometry (metric) is dynamic. Now perhaps the more interesting problem is surface growth. Here you would have new material points added or removed from the body, i.e. the underlying set of the material manifold is time dependent. This would be more complicated to formulate I think (it's in my to do list). There are some existing theories mainly by Russians. If you're interested I can send you some papers.

I agree that in any case it would be helpful to look at some concrete examples. I still think energy balance and its invariance for bulk growth should be ok. But surface growth needs much more thinking.


Amit Acharya's picture


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

arash_yavari's picture

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".


Amit Acharya's picture


I agree with your observation on classical nonlinear elasticity (and what I did would recover that result in the absence of mass flux)- but here the whole issue is that nothing is classical once you fool around with balance of mass in its usual form.

Amit Acharya's picture


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

arash_yavari's picture

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?


psaez's picture

Hi Zhigang,

 I have taken a look to your first paper and seems really interesting and, in fact, I think It could be useful to my current work! I will read the papers you post, hopefully before this Club theme is done...





Zhigang Suo's picture

Dear Kosta, Ellen, and Pablo:  Thank you so much for coming back to me.  I hope to read some of the papers on growth.  As cited in our JMPS paper, the continuum theory of swelling of elastic solids is fully formulated in the Gibb (1878) paper.  His work looks unbelievably modern:  he gave a full large-deformation formulation by using the nominal stress (PK1) and deformation gradient.  He did not give any specific form of the free-energy function, however.

The remaining problem is to prescribe a free-energy function, which is material specific.  We and others have been using the Flory-Rehner theory.

In a recent paper, Shengqing Cai and I have discussed the interaction between chemistry and mechanics.  It seems that for elastomeric gels, one can say something remarkably general without talking about any special chemical interation or statistical mechanical model.  Here is the recent paper:

Shengqiang Cai, Zhigang Suo. Equations of state for ideal elastomeric gels.  EPL. In press. 

Idea like this may help us address a central difficulty:  how to desceribe chemomechanical interaction in a continuum model.

Cai Shengqiang's picture

Thank you, Ellen, for raising such an interesting topic.  I am inspired by your post and learn a lot from the papers you collected.  I am particularly interested in the mechanical instabilities in the tissues induced by inhomogeneous growth or growing under constraint. Several interesting examples I collected are simply described below. 

When the smooth muscle of airways shortens, the mucosa folds and obstructs the airways; the amount of obstruction increases in asthma due to the thickening of the airway walls[1,2]. Buckling can enable the invagination of embryos[3], and the primordial development of sunflowers[4]. Fingerprint patterns can result from the buckling of the layer of basal cells of the fetal epidermis[5]. The rippled edges of long leaves in terrestrial plants and a blooming lily have been demonstrated to be the result of wrinkling instability caused by in-plane differential growth[6,7]. 

[1] Wiggs B. R., Hrousis C. A., Drazen J. M. and Kamm R. D. 1997 J. Appl. Physiol. 83 1814 

[2] Hrousis C. A. et al 2002 J. Biomech. Eng. 124 334 

[3] Pauchard L. and Couder Y. 2004 Europhys. Lett. 66 667

[4] Dumais J. and Steele C. R. 2000 J. Plant Growth Regul. 19 7

[5] Kucken M. and Newell A. C. 2004 Europhys. Lett. 68 141 

[6] H-Y. Liang and L. Mahadevan, Proceedings of the National Academy of Sciences, 108, 5516-21, 2011

[7] H. Liang and L. Mahadevan, Proceedings of the National Academy of Sciences (USA), 106, 22049, 2009.

lncool's picture

Hi All,

Shengqiang, great comment! I am super excited by research on growth-induced instabilities. There are so many cool examples of things that grow! Thank you for mentioning some. Maybe we can collect some more?

I really like the work by Alain Goriely and Martine BenAmar. Here are two cool recent papers, one on the airway wall, like Shengqiang mentioned, and one on rhubarb:

[1] Moulton DE, Goriely A. Possible role of differential growth in airway wall remodeling in asthma. J Appl Physiology. 2011;110:1003-1012.

[2] Vandiver R, Goriely A. Differential growth and residual stress in cylindrical elastic structures. Phil Trans R Soc A. 2009;367:3607-3630.

And here's another really fun one on plant growth with amazing images, check it out:

[3] Prusinkiewicz P, de Reuille PB. Constraints of space in plant development. J Exp Botany. 2010;61:2117-2129.

Does anybody know any other cool examples of growth?


Kejie Zhao's picture

Thanks Prof. Kuhl for bringing such an intriguing topic. Another example system of growth, beyond the soft matters, is the electrodes in lithium-ion batteries.  During charge and discharge cycles, lithium ions transport from one electrode to the other through diffusion. Such diffusion causes swelling or contraction of the electrodes which are typically ceramics or metals. The overall volumetric strain of one electrode during lithiation, depending on the host material, varies from a few percent to a few hundreds percent. The growth can dramatically alternate the physical properties of the host material, such as modulus, hardness, density, electric conductivity, among many others.     The concurrent mass transport and deformation (mostly anistropic) induces a field of stress. The stress causes plasticity, fracture, fatigue of the electrodes - the elements of mechanics, it also significantly affects the electrochemical lithiation process.   This field has been attracting more interest recently, not only because of the practical significance of designing a system with tolerance and compatibility of growth, also for the richness of the interaction of multiphysics.  Interested readers can find more discussions on this thread:

Matt Pharr's picture

As Zhigang has mentioned, one component of growth models is the kinetics.  In many systems, the kinetic processes are controlled by diffusion.  In such systems, theoretical models incorporate Fick's law (Darcy's law), which expresses that the rate of transport through a unit area is proportional to the gradient in chemical potential (measured normal to the area).  In these "Fickian" systems, the mass uptake is proportional to the square root of time. 

In contrast, some systems are seen to have mass uptake that is linear in time.  One example of this phenomenon occurs in certain glassy polymers.  If a cylindrical piece of glassy polymer has one end dipped into a bath of water, the water is seen to move linearly in time through the length of the tube.  As water penetrates the glassy polymer, a transformation to a rubbery state occurs.  Hence, the material behind the reaction front is in the rubbery state while that ahead of the front remains in the glassy state.  The region separating the two phases is usually found to be quite sharp. 

One common explanation for this process is as follows:  to absorb more solvent, the solvent must enter into the polymer, transport through the rubbery phase to the front, and "react" at the front.  The "reaction" at the front is a time dependent process that involves local rearrangement of molecules.  If this reaction time is small compared to the time for diffusion through the rubbery phase (with a representative time scale of L^2/D, where L is the length of the rubbery region and D is the diffusivity), then diffusion is the rate-limiting process, and the overall process is diffusion controlled (i.e. "Fickian").  However, if the time for reaction is long compared to the time for diffusion through the rubbery phase, then the reaction is the rate-limiting process, and the overall process is "reaction" controlled (i.e. "non-Fickian"). 

A typical molecular picture of the "reaction" is as follows.  Initially, polymer chains form a network with little space to accommodate solvent.  However, as some solvent penetrates into the polymer, it creates an osmotic pressure.  This pressure causes the network to swell at the reaction front, opening up space for further solvent penetration.  The rate of this opening-up process depends on the creep deformation of the polymer chains.  There is a representative time scale for creep deformation of the chains -- the relaxation time.  Thus, the overall uptake of mass scales with this relaxation time. 

Another example of "reaction-controlled" kinetics is found in lithium-ion batteries.  During the initial insertion of lithium into crystalline silicon, silicon and lithium react, forming an amorphous phase of lithiated silicon.  It has been seen that the boundary between crystalline silicon and the amorphous phase of lithiated silicon is atomically sharp.  Furthermore, this boundary is found to move linearly in time.  Thus, the movement of the reaction front is not limited by diffusion of lithium through the amorphous phase.  Instead, it is limited by the short-range processes at the front, such as breaking and forming atomic bonds.  For this system, we have presented a model of concurrent reaction and plasticity.  The growth-induced volumetric expansion is accommodated by plastic deformation.  For more details about this model, please see .

Dear all,

It is great that we have such interesting discussion going on here. The discussion about swollen polymer and growth model is also very interesting. In my opinion, however, the effort of using the idea of mechanics of swollen polymer for modeling biological growth is somehow already on-going. In fact, one type of model frequently used in vascular growth and remodeling (G&R) is a constrained mixture model, which has been developed based on the mixture theory by Humphrey and Rajagopal [1]. Among many contributions of Dr. Rajagopal in continuum mechanics, modeling of a swollen rubber and diffusion through it was one of his favorite topics in 90’s [2].  In 2002, Humphrey and Rajagopal introduced a new theoretical framework, called a constrained mixture model for modeling G&R of soft tissues. They presented the modeling framework that utilizes ideas from classical mixture and homogenization theories while avoiding the technical difficulties associated with mixture theory. Especially they claimed that the constrained mixture model is one that is based on a fundamental process by which growth and remodeling occur- the continual production and removal of constituents, which is conceptually different from the volumetric growth models.
Since the most focus in this discussion has been on multiplicative decomposition, I just wanted to add that there is another stream of modeling approach for soft tissue G&R. -Baek

[1] J.D. Humphrey and K.R. Rajagopal, A constrained mixture model for growth and remodeling of soft tissues, Mathematical Models and Methods in Applied Sciences, 2002, 3:407-430
[2] K.R. Rajagopal and L. Tao, Mechanics of Mixtures, World Scientific, 1995

lncool's picture

Dear Baek,

Absolutely! The Humphrey & Rajagopal 2002 paper is my favorite intro into mixture theories for growth. Mixture theories are a great way to interpret growth from a microstructural point of view.

But you're way too modest! I think you should mention some of your great contributions to the field fo aneurysm growth as well. Could you recommend two or three of your papers for us to read?


Hi Ellen,

As you know, I am one of many who have applied a constrained mixture approach into modeling of aneurysms. It should be interesting to see similarities and differences between them:
M. Kroon, G.A. Holzapfel, A model for saccular cerebral aneurysm growth by collagen fibre remodeling, Journal of Theoretical Biology, 247:775-787, 2007
P.N. Watton, N.A. Hill, Evolving mechanical properties of a model of abdominal aortic aneurysm, Biomechanics and Modeling in Mechanobiology, 8:25-42, 2009
S. Zeinali-Davarani, A. Sheidaei, S. Baek, A finite element model of stress-mediated vascular adaptation: application to abdominal aortic aneurysms, Computer Methods in Biomechanics and Biomedical Engineering, 9:803-817, 2011

During aneurysm initiation and progression, hemodynamics factors such as wall shear stress play important roles as key mediators of growth and remodeling. Hence, a new computational framework called Fluid-Solid-Growth (FSG) simulation has been developed to account for the effect of hemodynamics factors in modeling aneurysm growth:
C.A. Figueroa, S. Baek, C.A. Taylor, J.D. Humphrey, A computational framework for fluid-solid-growth modeling in cardiovascular simulation, Computer Methods in Applied Mechanics and Engineering, 198:3583-3602
P.N. Watton, N.B. Raberger, G.A. Holzapfel, Y. Ventikos, Coupling the hemodynamic environment to the evolution of cerebral aneurysms: computational framework and numerical examples, Journal of Biomechanical Engineering, 131: article no. 101003, 2009
A. Sheidaei, S.C. Hunley, S. Zeinali-Davarani, L.G. Raguin, S. Baek, Simulation of abdominal aortic aneurysm growth with updating hemodynamic loads using a realistic geometry, Medical Engineering & Physics, 33:80-88, 2011

Most papers above are based on a membrane approach but 3D constrained mixture models are also developed:
I. Karsaj, J. Soric, J.D. Humphrey, A 3-D framework for arterial growth and remodeling in response to altered hemodynamics, International Journal of Engineering Science, 48:1357-1372, 2010
I.M. Machyshyn, P.H.M. Bovendeerd, A.A.F. van de Ven, P.M.J. Rongen, F.N. van de Vosse, A model for arterial adaptation combining microstructural collagen remodeling and 3D tissue growth, Biomechanics and Modeling in Mechanobiology, 9:671-689, 2010
A. Valentin, J.D. Humphrey, G.A. Holzapfel, A multi-layered computational model of coupled elastin degradation, vasoactive dysfunction, and collagenous stiffening in aortic aging, Annals of Biomedical Engineering, 39:2027-2045, 2011

By the way, you may find interesting some of my work on modeling swollen gels:
S. Baek and A.R. Srinivasa, Modeling of the pH-sensitive behavior of an ionic gel in the presence of diffusion, International Journal of Non-Linear Mechanics, 39:201-218, 2004
S. Baek and T.J. Pence, Inhomogeneous deformation of elastomer gels in equilibrium under saturated and unsaturated conditions, Journal of the Mechanics and Physics of Solids, 59:561-582, 2011

Thank you. -Baek  

lncool's picture

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!


Justin Dirrenberger's picture

Thank you Ellen for this very interesting topic. I did not have a chance to go through all the references yet, but I came upon this article by Dunlop et al.:

J.W.C. Dunlop, F.D. Fischer, E. Gamsjäger and P. Fratz, A theoretical model for tissue growth in confined geometries , Journal of the Mechanics and Physics of Solids 58, 1073-1087 (2010)

This is quite far from my topics of research, but from what I understand this article could help the discussion since the model presented, relating stress distribution to growth kinetics, is confronted to experimental data.


Michael S. Sacks's picture


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

lncool's picture

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?

Alkiviadis Tsamis's picture

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 Smile) 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,


arash_yavari's picture

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).


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