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Multigenerational interstitial growth of biological tissues

Gerard A. Ateshian andColumbia University, New York, NY, USA, [email protected]

Tim RickenUniversity of Duisburg-Essen, Essen, Germany

AbstractThis study formulates a theory for multigenerational interstitial growth of biological tissues wherebyeach generation has a distinct reference configuration determined at the time of its deposition. In thismodel, the solid matrix of a growing tissue consists of a multiplicity of intermingled porouspermeable bodies, each of which represents a generation, all of which are constrained to movetogether in the current configuration. Each generation’s reference configuration has a one-to-onemapping with the master reference configuration, which is typically that of the first generation. Thismapping is postulated based on a constitutive assumption with regard to that generations’ state ofstress at the time of its deposition. For example, the newly deposited generation may be assumed tobe in a stress-free state, even though the underlying tissue is in a loaded configuration. The masscontent of each generation may vary over time as a result of growth or degradation, thereby alteringthe material properties of the tissue. A finite element implementation of this framework is used toprovide several illustrative examples, including interstitial growth by cell division followed by matrixturnover.

KeywordsBiological growth; Mixture theory; Residual stress; Reference configuration; Cell division

1 IntroductionBiological tissue growth is a process driven by chemical reactions among various fluid andsolid constituents of a mixture. For example, a cell may use amino acids that are present in aculture environment to build a protein and release it into the extracellular space; this proteinsubsequently may bind to the extracellular matrix, increasing its mass. In this example, aminoacids are considered to be fluid constituents (solutes in solution), whereas the extracellularmatrix is considered to be the solid constituent. Thus, growth alludes to changes in the massof the solid matrix of the mixture as a result of mass exchanges with the fluid constituents. Forporous mixtures, this mass exchange may occur in the interstitial space of the solid matrix(Cowin and Hegedus 1976), through which the fluid constituents may flow. Therefore, suchprocesses are described as interstitial growth, or equivalently, volumetric growth.

Fundamentally, interstitial growth is a process by which mass is deposited or removed fromthe interstitial space of a mixture. This process can be described by the equation of balance ofmass, taking into account mass exchanges among various constituents. Therefore, a naturalframework for describing growth of biological tissues is the theory of mixtures, which may

© Springer-Verlag 2010Correspondence to: Gerard A. Ateshian.

NIH Public AccessAuthor ManuscriptBiomech Model Mechanobiol. Author manuscript; available in PMC 2011 December 1.

Published in final edited form as:Biomech Model Mechanobiol. 2010 December ; 9(6): 689–702. doi:10.1007/s10237-010-0205-y.

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account for any number of fluid and solid constituents in a continuum analysis. Mixture theoryencompasses the classical frameworks of solid and fluid mechanics, and its formulation maybe generalized to account for chemical reactions among the constituents (Truesdell and Toupin1960; Bowen 1968, 1969; Ateshian 2007).

Mathematical growth theories must be cognizant of the underlying mechanisms responsiblefor growth in the physical world. Growth of non-living systems is simpler to model than growthof living systems. Non-living systems experience large mass deposition mostly in the form ofappositional, or surface growth. For example, stalactites and stalagmites may grow very largeas mineral-laden water flows over them, leading to slow deposition of mass onto the underlyingsolid structure; similarly, volcanoes grow in size as molten lava solidifies in successive layersover each other. Interstitial growth of a non-living system is typically much more limited inscope, since it can proceed only until the interstitial space has been filled. For example, porousfilters experience growth in mass as their pores progressively clog with retentate; this processends when all the pores have been filled, yet the external dimensions of the filter may not havechanged significantly.

In contrast, interstitial growth in biological systems can produce very large increases in massand volume over time, as evidenced by the growth of an embryo into a full-size adult humanbeing. This process occurs primarily as a result of cell division. In a recent study (Ateshian etal. 2009), we demonstrated that biological growth by cell division can be modeled within theframework of mixture theory by describing the interstitial growth of intracellular solid matrixconstituents and membrane-impermeant solutes. Since the cell membrane is semi-permeable,an intracellular increase in these constituents by uptake of solutes from the extracellular bathingsolution drives water into the cell via osmotic mechanisms during its synthesis phase. Thiswater uptake doubles the cell volume, a process needed to produce two nearly identicaldaughter cells during mitosis. With repeated cell division, the volume of the tissue can increasewith no theoretical upper limit, unlike the case of interstitial growth in non-living systems,because the osmotic swelling accompanying growth guarantees that the interstitial space willcontinually expand and never be filled.

This concept of growth by cell division reinforces the need to derive mathematical models thatare consistent with the underlying physical processes. However, regardless whether interstitialgrowth occurs by cell division or more prosaic processes, another challenge of growth theoriescenters on the concept of the reference configuration of the newly deposited solid matrix. Inclassical solid mechanics of non-reactive systems, where the mass of the body remainsconstant, it is common to posit a stress-free reference configuration from which the analysisof deformation may proceed. In a growing body, however, no consensus has yet emerged onthe manner by which reference configurations should be identified, though many alternativeshave been proposed (Rodriguez et al. 1994; Klisch et al. 2001; Humphrey and Rajagopal2002; Volokh and Lev 2005; Guillou and Ogden 2006; Ateshian 2007).

The objective of this study is to formulate a theory for multigenerational interstitial growth ofbiological tissues (and non-living systems), whereby each generation has a distinct referenceconfiguration determined at the time of its deposition. In this model, the solid matrix of agrowing tissue consists of a multiplicity of intermingled bodies, each of which represents ageneration, all of which are constrained to move together in the current configuration. Thisproposed framework builds on the concept of constrained mixtures of solids originallyformulated by Humphrey and Rajagopal (2002). The main distinction with the approach ofthese authors is that they treat the growing tissue as a mixture of different materials each havingan evolving reference configuration, whereas the current approach considers the tissue as amixture of constrained bodies representing multiple generations of material deposition, eachwith its own invariant reference configuration.

Ateshian and Ricken

Biomech Model Mechanobiol. Author manuscript; available in PMC 2011 December 1.

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The specific aim is to determine the form of constitutive relations for the solid constituents ofa multigenerational tissue based on constraints from the Clausius-Duhem inequality, to clarifywhich equations of linear momentum are sufficient to describe the motion of the constrainedsolids and unconstrained fluids in a biological tissue mixture and to provide illustrations thataddress fundamental problems in growth theory.

2 Mixture of fluids and constrained solidsThe detailed derivation of a mixture framework where the tissue consists of multiple fluid andsolid constituents, and where the solids are constrained to move together in the currentconfiguration, even though they may have distinct reference configurations, is given in theAppendix. The derivation appeals to the balance axioms of mass, momentum and energy, andproposes constitutive formulations consistent with the axiom of entropy inequality. Thoughthe detailed derivation is relatively involved, the final governing equations for a mixture offluids and constrained solids are remarkably simple. Since all the solid constituents areconstrained to move together, there is neither the need nor the possibility to solve for theconservation of linear momentum for each solid constituent. The balance of linear momentumfor the entire mixture may be used instead; when neglecting inertia and external body forces,this produces the familiar expression

(1)

where T is the mixture stress. When neglecting dissipative stresses (such as viscous stressesin the solids and fluids), and the contribution of diffusion velocities, and under the assumptionthat the mixture satisfies electroneutrality (see Appendix),

(2)

where σ denotes each of the solid constituents, and Fσ is its deformation gradient relating thecurrent configuration x = χσ (Xσ, t) to the reference configuration Xσ for that body, Fσ =∂χσ /∂Xσ. Since there are multiple solids constrained to move together, we may denote one ofthe solid constituents with σ = s and use it as a master reference configuration. Then, Js = detFs is the relative volume of the mixture when evaluated with respect to that referenceconfiguration, and W is the Helmholtz free energy density for the mixture (denoted by inthe Appendix), representing the free energy per unit volume of the mixture in the referenceconfiguration Xs. p represents the interstitial fluid pressure and arises from the assumption thatall fluid and solid constituents are intrinsically incompressible; however, since the mixture isporous and permeable, it may gain or lose volume as fluid enters or leaves a material regiondefined on the solid matrix (thus Js ≠ 1 under general conditions).

Since there can be multiple unconstrained fluid constituents in the mixture (such as multiplesolutes in a solvent), each fluid must satisfy its own equation of conservation of linearmomentum; when neglecting inertia, external body forces and dissipative stresses (such asviscous stresses), and under isothermal conditions, this equation reduces to

(3)

where ρι is the apparent density of fluid constituent ι, μ̃ι is its mechano-electrochemicalpotential, and represents the dissipative part of the momentum supply to constituent ι from

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all other constituents in the mixture; most commonly, models the frictional interactionsamong the constituents.

The state variables for W may be given by

(4)

where θ is the absolute temperature, is the apparent density of solid constituent σ relativeto the mixture volume in the reference configuration Xσ, and is the apparent density of fluidconstituent ι relative to the mixture volume in the reference configuration Xs. Since all thesolids are constrained to move together in the current configuration, it follows that

(5)

for all σ, even though the reference configurations Xσ are distinct. Thus, the referenceconfigurations of the various solid constituents may be related to the master referenceconfiguration via

(6)

where Fσs = ∂Xσ /∂Xs is a time-invariant transformation.

We may now define the effective stress tensor for each solid constituent as

(7)

such that it has the standard form for a hyperelastic solid. Importantly, it should be understoodthat is evaluated using that constituent’s Fσ (or any other related strain measure, such asthe right or left Cauchy-Green tensors, Cσ = (Fσ)T · Fσ and Bσ = Fσ · (Fσ)T). Therefore, themixture stress may be rewritten as

(8)

where

(9)

is a time-invariant quantity, and is the effective stress for the mixture.

Example 1Consider a 1-D analysis where the constitutive relation for is that of a linear spring,

, where kσ is the spring modulus and λσ is the stretch ratio for solid (spring)σ; note that Jσ = λσ in this case. Also assume that the fluid pressure has subsided, p = 0. For ahomogeneous deformation, we may write λσ = x/Xσ, where Xσ is the reference position for

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material points in solid σ. Consider that a traction f is applied on the mixture of two solids (σ= 1, 2); then according to (Eq. 8), if we pick Xs to coincide with X(1),

(a)

where

(b)

It is important to keep in mind that material points X(1) in solid (1) and X(2) in solid (2) areconstrained to move together, so that the ratio X(2)/X(1) for each constrained pair is constant,and their current position is given by x. The result for k(e) illustrates the fact that intermingledsolids act as springs in parallel, since their moduli sum up. In this example, because thedeformations are homogeneous, it is possible to represent the solid as a spring with modulusk(e) and equivalent reference configuration X(e).

Growth of the solid constituents is described by changes in as a result of chemical reactionsthat add mass to, or remove it from, constituent σ. According to the axiom of balance of mass(see Bowen 1969; Guillou and Ogden 2006; Ateshian 2007; Ateshian et al. 2009 andAppendix), when expressing in a material frame, , it satisfies

(10)

where is the mass supply to constituent σ from chemical reactions. This relation is easilyintegrated to produce

(11)

Note that is a material function that also depends on state variables such as those describedin (Eq. A.34), and this dependence must be described by experimentally validated constitutiverelations. Clearly, in the absence of growth, and the apparent density remainsinvariant, in which case it would no longer be needed as a state variable for W in (Eq. 4). Moregenerally, this mixture formulation distinguishes the effects of deformation from growth byletting W depend on Fσ and . Though changes in may influence deformation (e.g., viaosmotic alterations in the pressure p), and Fσ are independent state variables.

3 Multigenerational interstitial growthThe basic framework of multigenerational interstitial growth advocated in this study is thatnew solid mass deposited within the interstitial space of tissue T, in a reference configurationthat differs from previously deposited mass, belongs to a new body Bσ, which is intermingledwith the bodies from earlier time points, such that all are constrained to move together.Consistent with the above presentation of mixtures of constrained solids, the body Bσ isrepresented by solid σ, and its reference configuration is Xσ. Since solid mass may be deposited

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over a continuous time spectrum, there may be an infinite number of bodies Bσ representingthe tissue. In the treatment adopted here, we consider that mass being deposited with a commonreference configuration over the time interval tσ ≤ t < tσ+1 belongs to the same generation.Thus, we refer to material deposited during this time interval as the σ–th generation, also knownas the body Bσ. The span of time intervals for each generation is entirely guided by the specificgrowth problem of interest, and there is no requirement or expectation that these intervals beof uniform length.

While all the bodies Bσ share the same domain as the tissue T, their mass content anddistribution, given by , will generally be different. By definition, the growth of bodyBσ implies that increases from zero over the time interval tσ ≤ t < tσ+1; however, someregions of T may nevertheless have . Indeed, the mass content, and thus the materialproperties, of Bσ may be inhomogeneous and evolve over time.

It follows that a growing tissue that has entered its γ-th generation consists of a total of γ bodies,B1, B2, …, Bγ. We may refer to the collection of these intermingled bodies as the tissue in itsγ-th generation, Tγ. In the continuum representation of the tissue, all these bodies co-existwithin an elemental material region of Tγ, though the mass content of each generation mayvary over time and even reduce to zero. Indeed, it is also important to recognize that eachgeneration’s Bσ may lose mass over time due to degradative processes; thus, the deposition ofmass sharing a common reference configuration defines a new generation or body in the tissue,but degradation could possibly lead to the total loss (solubilization) of the mass content of aparticular generation, equivalent to at some time t > tσ.

Each body Bσ has its own stress-free reference configuration Xσ, which has a one-to-onemapping with the master generation Xs, see (Eq. 6); normally, the most appropriate choice forXs is the first generation (σ = 1). Importantly, this mapping exists at every location Xs, evenif the mass content of Bσ at that point happens to be zero. The existence of this mapping isguaranteed by the fact that the reference configuration Xσ is posited by constitutiveassumptions, typically guided by ambient conditions prevailing during the time interval tσ ≤t < tσ+1. Therefore, the mapping does not depend on the current value of .

For example, we may assume that mass deposited in the interstitial space over the time intervaltime tσ ≤ t < tσ+1 is in a stress-free state, even though the underlying pre-existing tissue Tσ−1is in a loaded state. Thus, we may set Xσ = x (Xs, tσ), or equivalently, Fσs = Fs (Xs, tσ). In otherwords, the reference configuration of mass deposited in the σ-th generation coincides with thecurrent configuration of the tissue at the start of the generation. This is the simplest non-trivialexample of a constitutive assumption for determining the reference configuration of Bσ, thoughit may well be generally valid for most of the biological tissue growth problems. Otherassumptions may be similarly adopted, such that the newly deposited mass is not in a stress-free state at tσ, though such constitutive assumptions might require an inverse solution todetermine the reference configuration from the assumed state of stress at the time of massdeposition.

Most importantly, the reference configuration Xσ is invariant. Thus, even though Bσ may begaining mass over the time interval tσ ≤ t < tσ+1, the reference configuration Xσ does not evolvebecause, we have defined this time interval to represent mass deposition under the commonreference configuration Xσ. For example, if we assume constitutively that Xσ = x (Xs, tσ), it isimplied that tissue deformation remains unchanged (or nearly so) over the time interval tσ ≤t < tσ+1. Therefore, this assumption also guides the length of this time interval based on theprevailing loading conditions.

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The material properties of Bσ depend on its mass content , as well as the mass content of allthe other solid bodies (and fluid constituents) in the tissue, as is evident from (Eqs. 4 and 7);the precise form of this dependence needs to be provided by constitutive relations. Thisdependency is expected on physical grounds as well, since the bodies Bσ are porous and thematerial properties represent those of the porous structure, not the intrinsic properties of theskeleton material. For example, a porous solid matrix consisting of type II collagen will havedifferent properties depending on whether its porosity is 20 or 90%, even though collagenfibrils in the two cases have identical intrinsic properties. This is rooted in the fact that stresstensors in mixture theory are apparent stresses. Indeed, just as the apparent densities representmass per unit volume of the mixture, the apparent stresses have components that representforce per unit area of the mixture (not per unit area of a particular constituent in the mixture).This makes it possible to sum up the stresses of the various constituents in a consistent manner.

The mass density represents mass per unit volume of Bσ in its reference configuration. Whentracking growth and resorption of all solid constituents, there may be circumstances where itis more convenient to use the master reference configuration Xs to represent the mass contentof each body,

(12)

This relation shows that are related by the time-invariant factor Jσs.

Finally, it should be noted that body Bσ may itself consist of more than one solid species; forexample, in articular cartilage growth, chondrocytes may synthesize type II collagen andproteoglycans simultaneously, both of which contribute to its solid matrix, and it may beconstitutively assumed that these constituents are being deposited in a common referenceconfiguration over the time interval tσ ≤ t < tσ+1. Thus, solid σ may itself be a mixture ofconstrained solids, all sharing the same reference configuration. Conversely, if a constitutiveassumption is made that solid constituents deposited over the same time interval tσ ≤ t < tσ+1have different reference configurations, they may be considered to belong to different, though,overlapping generations, for the purpose of mathematical treatment. The generalization of thenotation adopted in the above equations for these two cases is straightforward.

4 Examples4.1 Finite element implementation

A finite element implementation of multigenerational growth of constrained solids wasdeveloped based on a pre-existing custom code for growth of a single generation (Ateshian etal. 2009). The user specifies the number of generations and their starting time tσ in the inputfile; during the solution steps, the code automatically stores the current configuration x (Xs,tσ) at the start of each generation and assigns it to Xσ, so that Fσ may be subsequently evaluatedfor that generation. Any desired number of solid constituents may be assigned to eachgeneration, each solid having its own constitutive relation. Furthermore, material propertiesfor each solid may be prescribed to vary over time, to account for their alteration with growth.The code also accommodates osmotic alterations in the fluid pressure p as a result of celldivision, growth of charged extracellular solid matrix constituents and other forms of osmoticloading (Ateshian et al. 2009). This code was used for all the examples illustrated below.Relatively large loads are applied in these examples to produce visibly large deformations forease of interpretation. The constitutive relation adopted for the solid matrix is the isotropic,compressible, hyperelastic formulation of Holmes and Mow (1990), whose material propertiesare Lamé-like constants λs and μs and an exponential stiffening coefficient β. Compressibility

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is the expected behavior of porous solids, even if their skeleton is intrinsically incompressible,because pores need not preserve their volume under loading.

4.2 Cantilever beamConsider a cantilever beam T (10 × 1 × 1 mm) consisting of a single porous elastic solid B1 inits first generation (λs = 0, μs = 0.5 MPa, β = 0.1). Assume that there are no osmotic effects inthis problem, so that p = 0 at steady state. The beam is straight in its unloaded configuration(Fig. 1a). Upon loading with a uniformly distributed transverse load (−0.01 N resultant alongY), the beam deflects as expected (Fig. 1b), and the net effective stress in the beam is that ofB1, . Now let a second generation B2 grow into T while it remains subjected to thetransverse load. For convenience, B2 is given the same constitutive relation and materialproperties as B1. Since B2 is growing in a stress-free state, , there will be no alterationin the current configuration of T as a result of this growth (Fig. 1b); thus, .If the transverse load is now removed, the beam will recover only partially (Fig. 1c), becausethe effective stresses in B1 and B2 compete to produce a non-zero effective stress

in the unloaded beam, even though ∇ · Te = 0 according to the mixture balanceof momentum (Fig. 2). This example illustrates how growth can alter the unloaded state of atissue and produce residual stresses in it.

4.3 Thick-walled tubeThis example addresses a classical problem in the study of growth and residual stresses inarteries (Taber and Humphrey 2001). Consider a similar problem where a thick-walled tubeT (inner radius Ri = 0.5 mm, outer radius Ro = 1.0 mm), initially in its first generation (B1 withλs = 0, μs = 0.5 MPa, β = 0.5) under a stress-free state, is subjected to an internal pressure (1MPa). Consider three possible alternatives for growth: in case 1, growth of B2 (same materialproperties as B1) occurs only in the inner rim of the wall (Ri ≤ R ≤ (Ri + Ro) /2, Fig. 3); in case2, B2 grows only in the outer rim (Fig. 4); and in case 3, it grows throughout the wall thickness(Fig. 5). When growth ends, T is unloaded and returns to a residually stressed state, whichproduces a different distribution of Te in each case (Fig. 6). Now cut the tube radially at thebottom end, and in each case, the tube takes on a different, residually stressed, unloaded state:in case 1, the tube exhibits a positive opening angle (Fig. 3d), whereas in cases 2 (Fig. 4d) and3 (Fig. 5d), it produces a negative opening angle, though much more pronounced in the formercase. This example is illustrative of a potential mechanism by which residual stresses may beproduced in the arterial wall. As inferred from experimental observations by previousinvestigators (Vossoughi et al. 1993; Greenwald et al. 1997), a radial cut in the residuallystressed unloaded tube does not relieve the residual stresses (Fig. 6); in these examples, uponcutting, residual stresses come closest to a uniform zero value only in the case of homogeneousgrowth (Fig. 6c).

4.4 Multigenerational growth by cell division and matrix turnoverIn this last example, we consider multigenerational growth by cell division. As described inour recent study (Ateshian et al. 2009), in order for a cell to divide into two nearly identicaldaughter cells, it must double its mass content; this is achieved during the synthesis phase bydoubling the content of the intracellular solid matrix (denoted by ) and intracellularmembrane-impermeant solutes ( ), which produces a doubling of the water content viapassive osmotic uptake. When considering a multitude of cells, it is not necessary to constrainthe analysis to discrete jumps in cell number, and growth by cell division may be simplydescribed by the mass supplies as smooth functions of time. For simplicity, weassume that the extracellular matrix (ECM) is of neutral charge and that the water content ofthe ECM and intracellular space is the same; furthermore, consider that the external bathing

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solution of the tissue consists predominantly of NaCl at a concentration c* that remains isotonic;then, the osmotic pressure in the homogenized tissue (comprised of cells and ECM) is simplygiven by

where p* is the ambient pressure in the bath (typically taken as p* = 0), R is the universal gasconstant, χ is the volume fraction of cells in the tissue, and is the concentration of intracellularmembrane-impermeant solute in the current configuration, evaluated on a solution volumebasis. Based on the standard relation between concentration (on a solution volume basis) andapparent density (on a mixture volume basis), it can be shown that

where is the molecular weight of intracellular solutes, and is the true density of theintracellular solid matrix. The evolution of with growth is given by (Eq. 11).Furthermore, as cells divide, the volume fraction of cells in the tissue may evolve accordingto

(13)

where ξ is the ratio of cell volume to ECM volume, related to χ via

(14)

and ξ̂ represents the rate at which this volume ratio increases. For cell division (or apoptosis),it is expected that the following growth rates be proportional to each other,

(15)

Let us now consider an example where the first generation B1 of a tissue T consists of a clumpof cells in a stress-free state at time t = t1; the shape of this clump is arbitrary and will bepreserved if we assume that growth rates are homogeneous throughout the clump (in the finiteelement analysis, any shape may be adopted, including the simplest case of a single brickelement). Assume that the ECM solids and fluids occupy 20% of the tissue volume (χ = 0.8,ξ = 4) and that the cells have negligible stiffness; for the ECM, let λs = 0, μs = 5 kPa and β =0.1. Let growth proceed by cell division such that and ξ increase by a factor of 100 attime t = t2; no ECM growth is assumed to occur over this time span. The tissue is still in itsfirst generation, because there is no need to define a new reference configuration during thisgrowth stage, owing to the fact that the solid matrix of cells has negligible stiffness. As seenin Fig. 7, the tissue volume has now increased by a factor of Js = 88.5, a value that dependssignificantly on the elastic properties of the ECM, since it is now in a swollen, stressed statebalanced by an increase in the interstitial osmotic pressure p to 124 kPa (Fig. 7). Indeed, for

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this case of homogeneous growth under traction-free conditions, T = 0 at all times, so that theeffective residual stress is given by Te = pI. Now consider that there is turnover of the ECM,which we model in two steps: first, the ECM of B1 progressively degrades (its elastic moduliare reduced to zero) over the time interval t2 ≤ t < t3; as a result of this degradation, the resistanceto the osmotic expansion progressively decreases, so that the osmotic pressure reduces to zero;and the tissue volume reaches 100 times its original volume at t3 (Fig. 7), consistent with the100-fold increase in the number of cells embodied in the equivalent increases in . Inthe second stage of this turnover, a new ECM is synthesized, forming a second generationB2 whose reference configuration is X(2) = x (X(1), t3). This process occurs over the time intervalt3 ≤ t < t4, during which there is no further change in tissue volume (Js = 100) or interstitialosmotic pressure (p = 0).

Because we idealized the turnover process to occur such that the second generation is depositedonly after the ECM of the first generation has completely resorbed, there is no residual stressin the solid matrix of the tissue at the end of this growth process. The deposition of ECM didnot produce any obligatory change in volume, as illustrated in the growth of B2 from t3 to t4.Instead, the tissue relative volume Js increased as a result of cell division from t1 to t2 and asa result of matrix degradation from t2 to t3.

Clearly, many other illustrative sequences of growth may be conjured, which may lead todifferent outcomes with regard to residual stresses. Evidently, the matrix turnover process neednot be assumed to occur over two consecutive steps of old matrix degradation followed by newmatrix deposition, as these processes may occur concurrently. Such concurrent processes maybe idealized in the above framework by modeling multiple generations over smaller timeincrements, and allowing new matrix deposition to occur prior to the complete degradation ofolder matrix generations. The availability of a computational framework facilitates suchanalyses considerably.

5 DiscussionThe main objective of this study was to demonstrate that interstitial growth may be modeledas the successive deposition of porous solids, each having its own reference configuration, thusdefining a growth generation. All the intermingled generations are constrained to movetogether in the current configuration (Humphrey and Rajagopal 2002). This approachrepresents a fundamental framework for interstitial growth, cognizant of the fact that growthrepresents a process of mass deposition and removal from the tissue solid matrix, as a resultof chemical reactions involving solutes in solution in the interstitial pore space.

In this approach, it is implicit that the resulting tissue can experience interstitial growth onlyuntil its interstitial space has filled; for biological tissues, however, the interstitial space neednever fill, because the growth of intracellular solid and solute constituents accompanying celldivision produces an osmotic driving force that continually swells the tissue, preventing fillingof that space (Ateshian et al. 2009). This mechanism was illustrated in an example where thenumber of cells increased 100-fold as a result of cell division (Fig. 7).

This study also demonstrates that the state of residual stress in a growing tissue may bepredicted in a straightforward manner when its growth history is known, as illustrated in thecase of the cantilever beam (Fig. 2) and the thick-walled cylinder (Fig. 6). As demonstrated inthe case of the unloaded intact and unloaded radially cut thick-walled cylinder, traction-freeconfigurations need not be stress-free because of persisting residual stresses induced duringgrowth. Therefore, unless its growth history is known, there is no possibility of deducing thestress-free configuration of a grown tissue under general conditions.

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The framework adopted in this study builds considerably on the concept of constrainedmixtures of solids advanced by Humphrey and Rajagopal (2002) and reprised by otherinvestigators (Wan et al. 2009). It differs from that original framework primarily by not usingthe concept of evolving natural (stress-free) configurations. These authors described their solidmixtures primarily on the basis of different material types (collagen, elastin and smooth musclecells); in their approach, each of these materials may have a different growth history, and thusan associated evolving natural configuration. In the approach advocated in the current study,a given material (e.g., collagen) may itself consist of multiple solid constituents σ, eachrepresenting a generation produced during a particular growth spurt, and each having its owninvariant stress-free reference configuration. When a tissue consists of multiple matrix types,each type may have multiple growth generations. An advantage of this approach is that eachgeneration can be treated using the conventional kinematics of continua, where the referenceconfiguration is stress-free and time-invariant.

A fundamental concept of this multigenerational growth framework is that the referenceconfiguration of each generation is formulated with a constitutive assumption, guided byexperimental observations of the manner by which new material gets deposited within theinterstitial space of preceding generations. One such constitutive assumption, Xσ = x (Xs, tσ),was adopted for illustrative purposes in Sect. 3 and the examples of Sect. 4. Any number ofalternative assumptions may be adopted, such as the simple choice Xσ = Xs, implying that thenewly deposited material is under the same state of strain as the underlying material (thus, ina non-zero state of stress). This case will not produce residual stresses from multigenerationalgrowth, and some physical mechanism would be needed to explain how newly depositedmaterial can learn and reproduce the substrate’s state of strain. More elaborate assumptionsmay be formulated, such as the possibility that newly deposited molecules alter the surfaceenergy of the substrate (at a microscopic scale) such that the deposited mass is not in a stress-free state once it binds to the substrate. Other mechanisms, such as electromagnetic forces,may similarly alter the state of stress of molecules being deposited unto the precedinggenerations. Such alternatives may be explored in future studies.

This framework differs more fundamentally from those of Skalak et al. (1982) and Rodriguezet al. (1994), who proposed that growth may be defined kinematically. In particular, Rodriguezet al. (1994) describe a growth model where the solid density remains constant, in contrast tothe pore-filling interstitial growth mechanism described by Cowin and Hegedus (1976) andadopted here. In the current framework, growth is represented by temporal changes in ,which is exclusively described by the equation of balance of mass, (Eq. 10), using the conceptof the pullback of the solid density (Bowen 1969; Guillou and Ogden 2006; Ateshian 2007),(Eq. A.22). In this context, growth is not tied to deformation via an obligatory relation; thus,Fσ and are both needed as state variables in a growth framework, (Eq. 4). Indeed, thedependence of the Helmholtz free energy on the solid density is also needed to produce arelation consistent with the classical relation for the thermodynamic admissibility of chemicalreactions, (Eq. A.41), which involves the chemical potential of all solid and fluid constituentsof a mixture.

In the phenomenological growth framework proposed by Guillou and Ogden (2006), thedeformation gradient F of the solid may be decomposed into F = Fe · Fr, where Fr is thetransformation from the stress-free reference configuration (ℬ0) to the residually stressed,traction-free configuration following growth (ℬr), and Fe represents the deformation relativeto ℬr as a result of loading in the current configuration (ℬt). In this context, ℬr (which is not astress-free configuration) may evolve due to growth, and indeed, the example of the thick-walled cylinder given above illustrates how ℬr differs from ℬ0 (compare Fig. 3a, c, and notethe residual stresses in the uncut tube in Fig. 6a). Therefore, the approach presented in the

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https://www.researchgate.net/publication/6595566_On_the_theory_of_reactive_mixtures_for_modeling_biological_growth?el=1_x_8&enrichId=rgreq-2bd47f90-4417-4246-a461-e008fe271dbd&enrichSource=Y292ZXJQYWdlOzQyMjU2MTY5O0FTOjEwMTE5NDY3NjU3MjE2N0AxNDAxMTM4MDk5NDk1

https://www.researchgate.net/publication/40806036_A_3-D_constrained_mixture_model_for_mechanically_mediated_vascular_growth_and_remodeling?el=1_x_8&enrichId=rgreq-2bd47f90-4417-4246-a461-e008fe271dbd&enrichSource=Y292ZXJQYWdlOzQyMjU2MTY5O0FTOjEwMTE5NDY3NjU3MjE2N0AxNDAxMTM4MDk5NDk1

current study provides a method by which ℬr may be obtained for the purpose of using theframework of Guillou and Ogden, given knowledge of the growth process.

From our perspective, this study extends our recent presentation of interstitial growth by celldivision, regulation of intracellular solute content and deposition of charged ECM constituents(Ateshian et al. 2009), where it had been assumed that all solid constituents belonged to thesame growth generation. These combined studies provide a comprehensive modelingframework for interstitial growth of biological tissues. They recognize that growth is a resultof chemical reactions; therefore, constitutive relations are needed to relate chemical reactionsto the state variables in an analysis. The formulation of such constitutive relations ties thegrowth process to chemistry and biology; as usual, such formulations require experimentalinvestigations.

AcknowledgmentsThis study was supported with funds from the National Institute of Arthritis and Musculoskeletal and Skin Diseases(AR46532), and the National Institute of General Medical Sciences (GM083925) of the U.S. National Institutes ofHealth, as well as the Scientific Exchange Program of the University of Duisberg-Essen.

A AppendixWe begin by describing biological tissues as unconstrained mixtures of multiple solids andfluids. The concept of constrained intermingled bodies resulting from multigenerational growthwill be addressed below. In mixture theory (Truesdell and Toupin 1960; Bowen 1976), multipleconstituents α can co-exist within an elemental region, including solid and fluid constituents.In the current configuration, the spatial position of an elemental mixture region is denoted byx. Since different mixture constituents may have different motions, each constituent α whichis currently at x originated from possibly distinct reference locations Xα. The motion of eachconstituent is thus described by its own function

(A.16)

A.1 Balance of massThe content of each constituent may be given by the apparent density ρα, which represents themass of constituent α in the elemental mixture volume. The equation of balance of mass foreach constituent is given by

(A.17)

where vα = ∂χα/∂t is the constituent’s velocity, and ρ ̂α is the mass density supply to constituentα from chemical reactions. The material derivative following constituent α is defined as

(A.18)

for any function f. The balance of mass for the entire mixture is given by

(A.19)

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where

(A.20)

and requires that

(A.21)

implying that any net mass added to some of the constituents must exactly match the mass lostby the others. ρ represents the mixture density, and v is the mixture velocity.

Changes in ρα may occur as a result of changes in the mixture volume resulting from loadingor as a result of chemical reactions that alter the mass of constituent α in the elemental mixturevolume. It is possible to isolate the effect of growth by formulating the apparent density andmass supply of constituent α relative to the mixture volume in the reference configuration forthat constituent (Bowen 1969; Guillou and Ogden 2006; Ateshian 2007),

(A.22)

where Jα = det Fα and Fα = ∂x/∂Xα. Substituting this relation into (Eq. A.17) produces

(A.23)

Thus, the quantity will only change as a result of chemical reactions, allowing an explicitdistinction between the effects of growth and loading on the apparent density. This equationis very important in the theory of growth, as addressed in our prior studies (Ateshian 2007;Ateshian et al. 2009).

A.2 Balance of linear momentumThe equation of balance of linear momentum for each constituent is given by

(A.24)

where aα = Dαvα/Dt is the acceleration, Tα is the apparent Cauchy stress, bα is the externalbody force on constituent α per unit mass, and p ̂α is the momentum supply to constituent αfrom interactions with all other constituents. The balance of momentum for the mixture is givenby

(A.25)

where a = Dv/Dt and

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(A.26)

and the following constraint must be satisfied,

(A.27)

In this relation,

(A.28)

is known as the diffusion velocity of constituent α relative to the barycentric mixture velocity.In biological tissues, the term ραuα ⊗ uα is typically negligible compared to Tα, so that themixture stress is very nearly equivalent to its inner part,

(A.29)

In most problems, inertia terms given by ραaα in (Eq. A.24) and ρa in (Eq. A.25) are alsonegligible compared to other terms in those equations; therefore, differences between mixturetheory and Biot’s theory of consolidation (Biot 1962), which appear in the inertia terms, areinconsequential in most biological applications.

A.3 Constitutive restrictionsIn biological tissues, it is common to model solid and fluid constituents as intrinsicallyincompressible and to enforce electroneutrality at every point in the continuum. Intrinsicincompressibility implies that the true density of a constituent is invariant; the apparent andtrue density are related via

(A.30)

where φα if the volume fraction of that constituent. In a saturated mixture, the volume fractionssatisfy

(A.31)

The electroneutrality condition is given by

(A.32)

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where zα is the charge number and Mα is the molecular weight (always invariant) of constituentα. Constitutive restrictions may be imposed on mixtures by the second law of thermodynamics.The incompressibility and electroneutrality constraints may be introduced into the Clausius-Duhem inequality for mixtures using Lagrange multipliers.

As shown in previous studies, the multiplier for the incompressibility constraint represents apressure p (Bowen 1980), while that for electroneutrality is Fcψ (Huyghe and Janssen 1997),where ψ is the electric potential in the mixture and Fc is Faraday’s constant. In such mixtures,the Clausius-Duhem inequality for mixtures becomes

(A.33)

where ψα and ηα are, respectively, the specific Helmholtz free energy and entropy forconstituent α, Lα = ∇vα is the velocity gradient, qα is the heat flux vector for constituent α, andθ is the absolute temperature.

To proceed from this point, it is necessary to posit a set of state variables for ψα, ηα, Tα, p ̂α,qα and ρ̂α. While we may appeal to the principle of equipresence and provide a long list ofstate variables, it is more expedient to adopt the following minimum set needed for ourpurposes,

(A.34)

where σ denotes solid constituents, ι denotes fluid constituents, and β refers to all of theconstituents. In this choice of state variables, we are including deformation and growth of thesolid matrix via the respective inclusion of Fσ and , since changes as a result of growth,(Eq. A.23). Solid matrix remodeling is ignored here; it is also assumed that these functions donot depend on the history of these state variables but only on their current value.

Using these state variables, the chain rule of differentiation on Dαψα/Dt yields

(A.35)

Though ψα appears as a convenient variable in the formulation of the Clausius-Duheminequality, (Eq. A.33), constitutive relations are generally formulated in terms of the freeenergy density for the entire mixture. The free energy density for constituent α is given byΨα = ραψα and that for the mixture is ΨI = ∑α Ψα. Then it can be shown that

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(A.36)

Substituting these relations into (Eq. A.33) and grouping like terms yields

(A.37)

where we used the identities

(A.38)

and

(A.39)

Following standard arguments, the entropy inequality of (Eq. A.37) is satisfied for arbitraryprocesses if and only if

(A.40)

and

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(A.41)

where

(A.42)

represent the chemical, mechano-chemical, and mechano-electrochemical potential ofconstituent α, respectively. In these expressions, is the dissipative part of the stress (suchas viscous stresses), and is the dissipative part of the momentum supply (such as frictionalinteractions with other constituents). The reduced entropy inequality in (Eq. A.41) spells outthe terms that may dissipate free energy in a thermodynamic process. Making use of (Eqs. A.30)–(A.33), the inner part of the mixture stress becomes

(A.43)

It should also be noted that the set of state variables for ΨI is now reduced based on (Eq. A.40), such that

(A.44)

Substituting the general expression for p ̂α from (Eq. A.40) into the constraint of (Eq. A.27)produces a constraint on the choice of constitutive relations for ,

(A.45)

Of particular interest is the fact that for fluid constituents, the equation of balance of linearmomentum, (Eq. A.24), when combined with the expressions for Tι and p ̂ι in (Eq. A.40), nowbecomes

(A.46)

and it becomes apparent that the gradient in the mechano-electrochemical potential is animportant driving force in the momentum balance of fluids. While a similar substitution maybe performed for the solid constituents, the resulting expression does not simplify as nicely asfor fluids. Nevertheless, for a mixture of unconstrained solids, that equation would be neededto describe the momentum response of each solid constituent. Since we are not interested inthis more general case at this time, we proceed with the next step of constraining the solids tomove together.

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A.4 Constrained mixture of solidsWe are now ready to consider a mixture where all the solid constituents are constrained tomove together, (Eq. 5), even though they may have distinct reference configurations Xσ. Theconstraint may be simply expressed as vσ = vs where vs represents the common velocity of allsolid constituents. For the purpose of incorporating this constraint into the entropy inequality,it may be rewritten as Lσ = Ls and introduced into (Eq. A.33) using a tensorial lagrangemultiplier Λσ, thus adding the term

(A.47)

to that expression. In this case, all of the constitutive relations of (Eq. A.40) remain the sameexcept for the solid stress, which now becomes

(A.48)

where

(A.49)

to satisfy the entropy inequality unconditionally for arbitrary changes in Ls; the resultingreduced entropy inequality keeps the same form as (Eq. A.41). Using these relations, themixture stress also reduces to the same form as (Eq. A.43). Note that Λσ remains anindeterminate quantity in a mixture of constrained solid constituents. Therefore, the equationof balance of momentum for each solid constituent is no longer useful; the balance ofmomentum for the mixture, (Eq. A.25), should be used instead, to solve for the deformationof the constrained mixture of solids.

The expression for TI in (Eq. A.43) may be simplified further if the mixture free energy densitywas expressed relative to the mixture volume in some reference configuration (Biot1972;Bowen 1980). Since all the solid constituents are constrained to move together, let usdefine the reference configuration Xs for the mixture such that it coincides with one of thereference configurations Xσ and then recognize that the various constrained solid constituentssatisfy (Eq. 6). We would like to define a Helmholtz free energy per unit volume of the mixturein the reference configuration Xs,

(A.50)

where Js = det Fs. Equation 6 establishes a fixed relation between the reference configurationsXs and Xσ for the solid constituents; we also need to define a similar relation between theconfiguration Xs and the fluid constituents. Thus, let

(A.51)

represent the mass of fluid constituent ι in the current configuration, per unit volume of themixture in the reference configuration Xs; and perform a change of state variables such that

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(A.52)

By evaluating the total differential of (Eq. A.50) and using (Eqs. A.44), (A.51) and (A.52), itcan be shown that

(A.53)

Substituting these relations into (Eq. A.43) yields the final desired result for the mixture stress,

(A.54)

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Biot MA. Theory of finite deformations of porous solids. Indiana U Math J 1972;21(7):597–620.Bowen RM. Thermochemistry of reacting materials. J Chem Phys 1968;49(4):1625–1637.Bowen RM. The thermochemistry of a reacting mixture of elastic materials with diffusion. Arch Ration

Mech An 1969;34(2):97–127.Bowen, RM. Theory of mixtures. In: Eringen, AE., editor. Continuum physics. Vol. vol 3. New York:

Academic Press; 1976. p. 1-127.Bowen RM. Incompressible porous media models by use of the theory of mixtures. Int J Eng Sci 1980;18

(9):1129–1148.Cowin SC, Hegedus DH. Bone remodeling-1. Theory of adaptive elasticity. J Elasticity 1976;6(3):313–

326.Greenwald SE, Moore JEJ, Rachev A, Kane TP, Meister JJ. Experimental investigation of the distribution

of residual strains in the artery wall. J Biomech Eng 1997;119(4):438–444. [PubMed: 9407283]Guillou, A.; Ogden, RW. Growth in soft biological tissue and residual stress development. In: Holzapfel,

GA.; Ogden, RW., editors. Mechanics of biological tissue. Berlin: Springer; 2006. p. 47-62.Holmes MH, Mow VC. The nonlinear characteristics of soft gels and hydrated connective tissues in

ultrafiltration. J Biomech 1990;23(11):1145–1156. [PubMed: 2277049]Humphrey JD, Rajagopal KR. A constrained mixture model for growth and remodeling of soft tissues.

Math Mod Meth Appl S 2002;12(3):407–430.Huyghe JM, Janssen JD. Quadriphasic mechanics of swelling incompressible porous media. Int J Eng

Sci 1997;35(8):793–802.

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Klisch S, Van Dyke T, Hoger A. A theory of volumetric growth for compressible elastic biologicalmaterials. Math Mech Solids (USA) 2001;6(6):551–575.

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Truesdell, C.; Toupin, R. The classical field theories. In: Flugge, S., editor. Handbuch der Physik. Vol.vol III/1. Berlin: Springer; 1960.

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Vossoughi, J.; Hedjazi, Z.; Borris, FS. Adv Bioeng. ASME; 1993. Intimal residual stress and strain inlarge arteries; p. 434-437.

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Fig. 1.Growth of a cantilever beam. a Stress-free reference configuration of T1 (first generation). bLoaded configuration of T1, also loaded configuration of T2, following growth of secondgeneration. c Unloaded (residually stressed) configuration of T2

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Fig. 2.Effective normal stress distribution Txx at fixed end of cantilever beam

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Fig. 3.Thick-walled tube, growth in inner rim only; due to symmetry, only the right side of the tubeis modeled. a T1 in stress-free configuration. b T1 (and T2) with internal pressurization. c T2in unloaded configuration. d T2 in unloaded configuration, after radial cut; the opening angleis positive

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Fig. 4.Thick-walled tube, growth in outer rim only. a T1 in stress-free configuration. b T1 (and T2)with internal pressurization. c T2 in unloaded configuration. d T2 in unloaded configuration,after radial cut; the opening angle is negative

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Fig. 5.Thick-walled tube, homogeneous growth. a T1 in stress-free configuration. b T1 (and T2) withinternal pressurization. c T2 in unloaded configuration. d T2 in unloaded configuration, afterradial cut; the opening angle is negative

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Fig. 6.Radial distribution of residual circumferential normal effective stress at apex (Θ = π/2), forthick-walled tube in its second generation (T2) unloaded configuration, before and after radialcut. a Inner rim growth; b outer rim growth; c homogeneous growth

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Fig. 7.Growth by cell division, followed by matrix turnover. Js indicates the relative change in volumeof T over time; p is the interstitial fluid pressure

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