Biological Growth and Remodeling: A Uniaxial Example with Possible Application to Tendons and Ligaments

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1 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 Biologial Growth and Remodeling: A Uniaxial Example with Possible Appliation to Tendons and Ligaments I. J. Rao 1, J.D. Humphrey 2, K.R. Rajagopal 3 keywords: mixture theory, kinetis of turnover, evolving natural onfigurations, hyperelastiity, mass prodution Abstrat: Reent disoveries in moleular and ell biology reveal that many ell types sense and respond via altered gene expression to hanges in their mehanial environment. Suh mehanotransdution mehanisms are responsible for many hanges in struture and funtion, inluding the growth and remodeling proess. To understand better, and ultimately to use e.g., in tissue engineering, biologial growth and remodeling, there is a need for mathematial models that have preditive and not just desriptive apability. In ontrast to prior models based on reation-diffusion equations or the onept of volumetri growth, we examine here a newly proposed onstrained mixture model for growth and remodeling. Speifially, we use this new model to present illustrative omputations in a representative, transversely-isotropi soft tissue subjeted to homogeneous deformations under uniaxial loading. Consequenes of various assumptions for the kinetis of mass prodution and removal are disussed, as are open problems in this important area of biomehanis. 1 Introdution D Ary Thompson 1917 said it well: Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obediene to the laws of physis that their partiles have been moved, moulded, and onformed. Whereas it has long been thought that biolog- 1 Dept. of Mehanial Engineering New Jersey Institute of Tehnology, Newark, NJ Corresponding Author Biomedial Engineering 233 Zahry Engineering Center Texas A&M University, College Station, TX jhumphrey@tamu.edu 3 Mehanial Egineering Texas A&M University, College Station, TX ial growth and remodeling is governed in large part by physial ations, reent disoveries in moleular and ell biology demonstrate that suh regulation ours at the level of gene expression Taber, That is, many ell types sense and onvert mehanial stimuli into bioeletrial or biohemial signals that ontrol gene expression and thus biologial struture and funtion, a proess now referred to as mehanotransdution. Unfortunately, the fundamental question remains What does the ell atually sense? Until, and likely after, the mehanisms of mehanotransdution are eluidated, phenomenologial ontinuumbased models will ontinue to play an important role in the design of experiments as well as industrial and linial proedures. Over the years many different lasses of models have been suggested. For example, muh of the literature on wound healing stems from the pioneering work of Turing 1952 who suggested that morphogenesis i.e., the development of shape of an organ an be modeled via reation-diffusion equations e.g., Tranquillo and Murray, In this approah, it is assumed that one should model the diffusion of moleules suh as growth fators, proteases, and ytokines as well as the migration of ells, whih together play key roles in governing the prodution and removal of onstituents. In ontrast, Skalak 1981 introdued the onept of volumetri growth whereby any finite growth or hange of form may be regarded as the integrated result of differential growth, i.e. growth of the infinitesimal elements making up the animal or plant. In this approah, one presribes growth a priori via kinematis based on assumed onstitutive relations for the evolution growth of stress-free onfigurations. See Rodriguez et al for an illustrative appliation. Yet, as noted by Fung 1995, growth neessarily results from the prodution and removal of onstituents, thus there is a need for onstitutive relations that relate mass prodution and stress. Fung did not offer a speifi model or approah to aomplish this, however. Reently, Humphrey and Ra-

2 44 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 jagopal 22 proposed a onstrained mixture model that melds ideas from the ontinuum theory of mixtures, the onept of evolving natural onfigurations, and a homogenization for onstrained mixtures that allows one to relate mass prodution/removal to the stress in the ontinuum these ideas were presented in general, however, without illustrative examples. The purpose of this paper, therefore, is to illustrate the possible utility of this new onstrained mixture model. In partiular, we fous on a lass of quasi-stati uniaxial problems wherein the deformation is homogeneous and the material symmetry group transverse-isotropy is preserved throughout growth and remodeling. Allowing growth and remodeling G&R to our over time-dependent deformations reveals onsequenes of indued material nonuniformities 4 and evolving loal natural onfigurations. We suggest that this illustrative example may have important linial impliations for tendons and ligaments, whih are often subjeted to uniaxial loading and they exhibit a transverseisotropy. 2 Balane Relations As we onsider G&R within a framework of mehanis, we must onserve mass, linear momentum and energy. By restriting our attention to isothermal proesses, however, we shall fous on mass and linear momentum. Beause tissues onsist of a number of solid onstituents, we onsider the balane of mass for eah onstituent as well as that for the whole tissue. As in mixture theory, we allow for o-oupany of the onstituents at eah point, thus mass balane for eah onstituent an be written as ρ i t +div ρ i v i = m i α m i ω = m i, i = 1,2,...N, 1 where the index i represents the ith of the N mehanially important onstituents omprising the tissue, m i α is the rate at whih mass of the ith onstituent is produed per unit volume of the mixture, and m i ω is the rate at whih the mass is removed per unit volume, with the differene between the two being the net prodution rate. The mass 4 Classial notions of material nonuniformity and homogeneity have to be modified to aount for the fat that the body in question is not a fixed set of partiles see Rajagopal 23 for a disussion of these issues. density and veloity of eah onstituent are denoted by ρ i and v i, respetively. We assume that none of the onstituents, suh as elastin or ollagen, diffuse with respet to eah other. In other words, the onstituents are onstrained to move together, thus v i = v, the mixture veloity. For suh a ase, the balane of mass for eah onstituent redues to ρ i t +div ρ i v = m i α m i ω = m i, i = 1,2,...N. 2 Total mass balane for the tissue is obtained by summing the mass balanes for eah onstituent and an be written as N ρ t +divρv= m i = m, 3 i=1 where ρ, the mass density of the tissue, equals the sum of the mass densities of the onstituents, namely ρ = N i=1 ρ i. 4 The net rate at whih mass is produed per unit volume of the mixture, m, is the differene between the rate of mass prodution and the rate of mass removal, m = m α m ω. 5 Here a few omments are in order. First, the net mass prodution rate per unit volume does not have to be identially zero as in most lassial mixture theories. This does not mean that matter is being produed or destroyed, however. Rather, we simply do not onsider the flux of speies suh as ells, raw materials, nutrients et., that lead to the prodution and removal of ollagen in the tissue, whih is an open system. Hene, the tait assumption is that the speies fluxing in and out do not have a signifiant effet on the overall mehanial response. This restrition an be lifted, but we do not add this additional level of omplexity herein. Our aim is to formulate a model that aounts for inreases or dereases in the

3 Biologial Growth and Remodeling: A Uniaxial Example 441 primary strutural onstituents omprising the tissue and their effets on the gross mehanial response. It is lear from Eq. 2 that we need to onsider both the rate at whih the material is produed and the rate at whih it is removed. The manner in whih this is done is explained in the next setion. In general, both the size and mass density of the tissue an hange with G&R. Yet, it appears that the mass density of the tissue ρ 15 kg/m 3 does not vary signifiantly during or after G&R. This simplifiation has been widely used in previous works on the modeling of growth see Rodriguez et al., 1994; Taber, For suh a ase, Eq. 3 simplifies to divv= m ρ, 6 where ρ is the onstant mass density of the tissue. We, too, shall assume that the density of the tissue does not hange during the growth and remodeling proess. Following Humphrey and Rajagopal 22, we invoke a homogenization assumption for the onstrained mixture that allows us to use a rule-of-mixtures approah for the total Cauhy stress. Negleting the effets of inertia and body fores, the balane of linear momentum redues to a single equation for the mixture, divt =, 7 where T is the total or mixture Cauhy stress. Let us now onsider onstitutive equations that allow this general approah to be illustrated. 3 Constitutive Model for a Two Constituent Tissue For materials undergoing biologial G&R, there is a need to tie together relations for the stress tensor with those for the prodution and removal of onstituents. Toward this end, onsider a tissue of two onstituents in whih one does not turnover signifiantly during the G&R proess whereas the seond does turnover. Suh a ase may apply to, for example, growth and remodeling in tendons and ligaments wherein the primary onstituents are the elastin-dominated amorphous portion of the extraellular matrix, whih does not turnover, and the predominant onstituent, ollagen, whih does turnover note: the fibroblasts are assumed to regulate the matrix but not to ontribute to the mehanial integrity of the tissue unlike the myofibroblasts in a healing wound. Here it should be noted that the general methodology is not limited to two onstituents; it an be extended in a straightforward manner. Beause the elastin does not turnover, its natural onfiguration κ n at eah point and eah time remains unhanged, thus κ e n κ e o, the original natural onfiguration 5. Hene, the mass density for the elastin is given by detf κ e o = ρe o ρ e t, 8 where ρ e t is the density of elastin mass per mixture volume. Alternatively, referring to Figure 1, the mass density of the elastin at different times during the G&R proess an be omputed via the deformation gradient that relates stressed onfigurations at two different times, τ and t τ t, namely ρ e t=ρ e τdetf t τ, 9 where F t τ is the deformation gradient assoiated with a mapping from the urrent onfiguration bak to the onfiguration oupied by the tissue at time τ;itisgivenby F t τ := F κt κτ = F κ τf 1 κ t, 1 where κ is a suitable loal referene onfiguration unique for the elastin Figure 1. Note that detf t τ = detf τ t 1. Also, beause the density of the tissue does not hange appreiably during G&R, overall mass balane for the onstrained mixture an be written as divv=trḟ κ F 1 κ = m = m α m ω = m α m ω, ρ ρ ρ 11 where, m α denotes the rate at whih ollagen is produed per unit volume, m ω is the rate at whih ollagen is removed per unit volume, and F κ is the deformation gradient from a suitably hosen referene onfiguration κ. Thus, mass is gained or lost solely due to the prodution and removal of ollagen, i.e., m α = m α and m ω = m ω. Balane of mass for ollagen is given by ρ +ρ tr Ḟ κ F 1 κ = m α m ω, 12 5 A detailed disussion of the onept of natural onfiguration an be found in Rajagopal 1995, Rajagopal and Srinivasa 1995,1998 and Humphrey and Rajagopal 22. For our disussion it suffies to think of natural onfigurations as stress-free onfigurations with the response of the body being elasti from these onfigurations.

4 442 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 dv F dv F t L t F t dv t t t t 1 t F F F Figure 1 : Kinematis assoiated with growth where ρ e + ρ = ρ. The mass fration of elastin and ollagen at eah material point an be taken as α e = ρe ρ, α = ρ ρ, α +α e = It is possible to use Eq. 12 to alulate ρ and thus α at any instant by presribing the prodution and removal rates m α and m ω. It is not enough to know the total ollagen present in the tissue at any given time for ollagen produed at different times an exhibit different material symmetries or stiffness. We shall assume, however, that the material symmetry remains the same throughout a speifi ase of G&R, and so too for the stiffness relative to its updated evolving natural onfiguration. Rather than trak both m α and m ω, let us trak the prodution rate and the time of survival for the ollagen that is produed at a partiular instant. This is similar, in priniple, to traking the age distribution in population dynamis and is less involved than alulating the net population in terms of births and deaths. Let m α τ be the mass of ollagen produed per unit volume in the onfiguration oupied by the tissue at time, τ per unit time and G τ,t be the mass fration of ollagen that was produed at time τ that is surviving at urrent time t. Let dv t be a differential material volume element at time t that oupied the differential material volume element dv τ at time τ. These two differential volume elements are related through f. Figure 1, dv τ = detf t τdv t. 14 The mass of ollagen produed in the time interval between τ and τ + dτ, within the volume element dv τ,is given by m α τdv τ dτ. The amount of this mass of ollagen surviving at time t, denoted by dm is thus given by dm = m α τg τ,tdv τ dτ = m α τg τ,tdetf t τdτdv t. 15 Note, we have used Eq. 14 to obtain Eq The mass and density of ollagen in the urrent onfiguration are obtained by integrating Eq. 15 over time and volume V t, respetively, namely M t= ρ t= V t m α τg τ,tdetf t τdτdv t, 16 m α τ G τ,tdetf t τdτ. 17 Of ourse, the equation for the mass of ollagen an also

5 Biologial Growth and Remodeling: A Uniaxial Example 443 be written as M t= V τ m α τg τ,tdv τ dτ, 18 where V τ is the volume oupied by the tissue at time τ. The funtion G τ,t an differ for matter produed at different times and an also depend on loal onditions suh as stress, deformation, et., as needed to aount for hanges in removal rates during periods of G&R. The lower limit of the integral in the above equations, given by negative infinity, an be replaed by some earlier but finite time, t, as long as the ollagen formed prior to t is either known or it has been removed and hene does not ontribute to the mass and density of the tissue. In many ases G&R is initiated from a given basal state; for suh a ase, the mass and the age distribution of the onstituent present in the tissue prior to the ommenement of G&R would have to be known. For instane, Eq. 16 and Eq.17 an be rewritten as M t= + V t V t t = M t+ ρ t= + t m α τg τ,tdetf t τdτdv t m α τg τ,tdetf t τdτdv t V t m α τg τ,tdetf t τdτdv t, 19 m α τ G τ,tdetf t τdτ m α τ G τ,tdetf t τdτ = ρ t+ m α τ G τ,tdetf t τdτ, 2 where G&R is initiated at time t =. M t is the ontribution to the mass of ollagen at urrent time t, due to ollagen originally present in the tissue prior to initiation of G&R, and similarly for ρ t. The total stress in the onstrained mixture is modeled using the notion of multiple natural onfigurations. Following Humphrey and Rajagopal 22, we assume that the tissue is mehanially inompressible and therefore, with two onstituents, the total Cauhy stress tensor takes the form T = pi +T E, T E = T e +T 21 where p is the Lagrange multiplier that arises due to the onstraint of inompressibility and T e and T are the onstituent stresses due to the elastin and ollagen, respetively. The form for stress given in Eq. 21 redues to the usual rule of mixtures approximation under ertain onditions as is shown later in this setion f. Eq. 29. The motivation to split the stress tensor arises from the observation that eah onstituent an have separate material properties and symmetries, and most importantly natural onfigurations that an evolve differently. Splitting the stress tensor in this manner affords us a diret way of onneting the behavior of the whole tissue to the amount and properties of the onstituents. Assuming that the natural onfiguration assoiated with elastin does not hange, and that its funtional form for stress remains unhanged during the G&R proess, the stress in the elastin is given by T e = f e κ e Fκ e t, 22 where κ e is the natural onfiguration assoiated with elastin, fκ e e is the response funtion assoiated with the stress and F κ e t is the deformation gradient from the natural onfiguration, κ e, to the urrent onfiguration assoiated with the tissue, κt, i.e., F κ e t := F κ e κt. 23 The onstitutive equation for ollagen should aount for hanges in stress due to the prodution of ollagen in onfigurations different from those during prior prodution. Moreover, the model should allow the funtional form of the stress tensor for the newly produed ollagen to aount for hanges in material properties and material symmetries during G&R. Both of these features are inorporated into the model by letting the natural onfigurations and the funtional form of the stress for the

6 444 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 newly produed ollagen eah to depend on loal onditions suh as stress, strain, et. Under these onditions the stress in the ollagen may be given by T = f κ nτ Fκ nτ t m α τg τ,tdetf t τ dτ, 24 where κ n τ is the natural onfiguration assoiated with the ollagen that was produed at time τ and fκ is the nτ orresponding response funtion for the stress. F κ n τ t is the deformation gradient from the natural onfiguration, κ n τ, to the urrent onfiguration oupied by the tissue, κt, i.e., F κ n τ t := F κ n τ κt. 25 Hene, to determine the stress, we need to presribe the natural onfigurations for the nasent ollagen. If, for example, the ollagen was produed in a stress-free state, then F κ n t t=i. A similar approah has been used in a multi-network theory for polymers Rajagopal and Wineman, 1994 and rystallization in polymers Rao and Rajagopal, 2; 21, wherein it is assumed that the material was onverted in a stress-free state. In general, however, F κ n t t an depend on the stress, mass prodution rates, mass densities, strains, et., i.e., F κ n t t=gt,m α,ρ From Eqs. 21, 22 and 24, the stress in the twoonstituent tissue redues to T = pi +fκ e e Fκ e t 27 + f κ nτ Fκ nτ t m α τg τ,tdetf t τdτ. The funtional form for fκ e e and f κ an be fixed by nτ hoosing a speifi form of the stored energy funtion. Assuming hyperelasti responses, Eq. 27 an alternatively be written as T = pi +2ρ e ψ e F κ e F T κ C e 28 κ e ψ + 2F κ n τ C κ n τ F T κ nτ m α τg τ,tdetf t τdτ, where C is the right Cauhy-Green tensor C = F T F and ψ i are Helmholtz potentials. Note: if the natural onfiguration and the form of the Helmholtz funtion for ollagen produed at different times remains unhanged, for instane during tissue maintenane, using Eq. 17, the above equation an be simplified to: T = pi +α e ψ e 2ρ F κ e F T ψ κ C e +α 2ρ κ e F κ n F T κ C κ n n = pi +α e ˆT e +α ˆT, 29 wherein κ n τ has been replaed with κ n to indiate no dependene of the natural onfiguration of ollagen on the time at whih it was produed. This shows that for the limiting ase of ollagen produed in the same state at all times, a simple rule-of-mixtures approximation is reovered, where ˆT e and ˆT an be interpreted as the stresses present in pure elastin and pure ollagen, respetively. Of ourse, for a single onstituent tissue, one of the mass frations vanishes while the other goes to unity, thus reovering the standard result from inompressible finite elastiity. For illustrative purposes, let us now onsider some speifi forms for the elastin and ollagen. The behavior of amorphous elastin, suh as that in a ligament or tendon, an be modeled as a neo-hookean material Dorrington and MCrum, Thus, letting ψ e = µe 2ρ trcκ e 3, 3 we obtain the following form for the stress due to elastin, T e = ρ e µe ρ B κ e = α e µ e B κ e, 31 where B is the left Cauhy-Green tensor B = FF T and µ e is the shear modulus having units of stress assoiated with elastin. For type I ollagen in a tendon or ligament, we assume that its response an be desribed by an exponential relation embodying transverse isotropy with respet to the urrent natural onfiguration. A possible form for the Helmholtz potential is thus, ψ = µ { [ 1 exp δ1 trcκ 2ρ δ n τ 3 ] 1 } µ { [ 2 exp δ 2 Nκ 4ρ δ n τ C κ n τn κ n τ 1 ] } 2 1, 2

7 Biologial Growth and Remodeling: A Uniaxial Example 445 where µ 1 and µ 2 are material parameters having units of stress, δ 1 and δ 2 are non-dimensional material parameters, and the unit vetor N κ n τ represents the preferred i.e., ollagen fiber diretion, the material being transversely isotropi with respet to this diretion. This diretion an hange, in general, depending on the onditions under whih ollagen fibers are produed. That is, it is possible for the tissue to have a symmetry different from transverse isotropy if the ollagen fibers produed at different instants have different preferred diretions N κ n τ. For the form of the potential hosen, the stress in the ollagen redues to t T = 1 ρ µ 2 ] m αg τ,tdetf t τ {µ 1 [δ exp 1 trc κ n τ 3 B κ n τ + N κ n τ C κ n τn κ n τ 1 exp ] 2 [δ 2 N κ n τ C κ n τn κ n τ 1 } F κ n τn κ n τ N κ n τf T κ dτ. 33 nτ To proeed, we need to presribe a prodution rate for the ollagen, m α, and the fration of ollagen surviving, i.e., presribe G τ,t. The mass prodution rate m α is presribed through a onstitutive equation that may depend on the stress, mass densities, et. f. Eq. 26. For instane, one ould presribe an equation of the form: m α = f T,ρ, Fration surviving, G,t 1. t 1 L t 2 Time t Ht 1-Ht 2 t 2 G t 1, t Figure 2 : Fration of ollagen surviving with time The speifi form will be ditated by experimental data, one they are available. In the next setion we hoose illustrative forms and onsider a problem involving homogeneous uniaxial extensions. For G τ,t, we seek to speify a form that inorporates the main features assoiated with G&R without being overly ompliated. Here we look at a ase in whih ollagen is being produed and removed ontinuously, i.e., we do not onsider the situation in whih ollagen is produed and removed intermittently even though the urrent methodology ould be so generalized. In this work, we assume that all ollagen produed at a speifi material point at a given time is removed at a later time; this is tantamount to assuming that ollagen produed under idential onditions and experiening the same environment will have idential life spans. This need not be the ase, in general, as ollagen that is removed at a speifi material point i.e., in a representative volume element at a given time an have a distribution of ages. This is illustrated in Figure 2, wherein we show a possible funtion G τ,t, with τ = t 1. The ollagen produed at a given time t 1 in the figure will in general be removed gradually, in ontrast with our assumption that the ollagen produed at a given time t 1 is all removed at the same later time t 2. If the variation in the lifespan is narrow ompared to the average lifespan, this assumption is reasonable. For suh a ase, the funtion G τ,t an be represented using the Heavyside funtion, H, and has the form G τ,t=h τ H τ +L, 35 where L represents the mean lifespan of the ollagen formed at time τ. As a onsequene of these assumptions the ollagen being removed at urrent time t was formed at time t L t, where L t is the age of the ollagen being removed. We also assume that the ollagen that is produed earlier is removed earlier, therefore, the ollagen formed between t L t and t is present in the tissue whereas the ollagen formed before t L t has already been removed. Thus, the time, τ, at whih a speifi fration of ollagen was produed is related to the time, t, at whih it is removed through τ = t L t. 36 Note, at any time, t, we only need to presribe the age of ollagen being removed at that time, i.e., L t and not the spetrum of lifespans assoiated with eah fration of ollagen omprising the tissue that was produed between t L t and t. Realling that in Figure 1 F t τ is

8 446 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 the deformation gradient between the urrent onfiguration and the onfiguration oupied by the tissue at time i.e., F 1 τ t, Eqs. 16 and 17 an be re-written as ρ t= M t= t L m α τdetf t τdτ, 37 V t L m α τdetf t τdτdv τ. 38 The quantities m α and L have to be presribed; they, too, may depend on loal onditions suh as the stress, et. The rate at whih ollagen is removed, mω, is related to m α and L beause matter removed at a given time was produed at an earlier time. The relationship between these quantities is given by a mass balane for ollagen inside an arbitrary material volume for an arbitrary interval of time. We do this by equating the mass of ollagen removed in a material volume during a speified period to the period in whih it was produed. The mass of ollagen removed in an arbitrary material volume for a period t 2 t 1 is given by 2 t 1 V τ m ω dv τdτ, 39 where the material volume, V τ, over whih the integral is evaluated, is a funtion of time that an hange with G&R. The quantity in Eq. 39 was born at an earlier time and is equal to 2 t 1 V τ m αdv τ dτ, 4 where the time variables in the above two equations are related through Eq. 36, i.e., τ = τ L τ. 41 Equating Eqs. 39 and 4, we obtain 2 t 1 V τ 2 m ωdv τ dτ = t 1 V τ Differentiating Eq. 41, we obtain m αdv τ dτ. 42 dτ dτ = 1 dl dτ. 43 Note further that we assume that L is a pieewise ontinuously differentiable funtion of time. Also, the material volume elements are related through Eq.14, namely dv τ = detf τ τ dv τ. 44 Substituting Eq. 41, Eq. 43 and Eq. 44 into Eq. 42, we obtain. 2 t 1 V τ m ω m α detf τ τ L τ 1 dl dτ dv τ dτ =. 45 Sine both the material volume V τ and the time interval are arbitrary, the integrand must vanish. Hene, m ω t=m α detf t t L 1 dl t. 46 dt For solutions to be physially reasonable L annot take on arbitrary values. For example, the rate of removal, m ω is positive. This ondition results in the mathematial restrition, dl dt < In addition, we also require that L be positive; if L =, matter disappears the instant it is produed whereas L < implies that matter is removed before it is produed, whih is a physial impossibility. In general, the life span of ollagen an be presribed as a rate equation of the form f. Eq. 34:

9 Biologial Growth and Remodeling: A Uniaxial Example 447 where m α,eq and L eq are the equilibrium prodution rate dl and the equilibrium life-span assoiated with ollagen = g T,m dt α,ρ,ρ e..., 48 in maturity prior to G&R that is, in a state of tissue maintenane. In addition, let the extant ollagen have the where the age of ollagen being removed an vary with same natural onfiguration beause we assume that the loal onditions. Alternatively, a onstitutive equation tissue is far enough into maturity that all tissue produed an be presribed for the removal rate, mω, and Eq. 46 during development has been removed. Of ourse, the an be used to derive a form for the dl dt. The speifi elastin and ollagen will not have the same natural onfigurations in general for the elastin was produed during hoie between these two methods will be ditated by the problem under onsideration and the experimental data development. Hene, let t = be a time during maturity available to formulate onstitutive equations for m ω or at whih G&R starts, whih we shall initiate by subjeting the tissue to the additional e.g., non-physiologial dl dt. For the examples onsidered in this paper, we speify onstitutive equations for dl dt by hoosing speifi uniaxial extension. See Figure 3. Speifially, we shall forms for the funtion g. With these simplifiations, the study the G&R response for two ases, namely, a onstant extension and a quasi-stati sinusoidally varying ex- stress tensor given by Eq. 28 redues to tension. It should be noted that we seek solutions given T = pi +α e µ e B κ e 49 presribed homogenous deformations within the ontext + 1 t m α detf t τ { µ 1 exp [ δ 1 trcκ ρ n τ 3 ] of a standard semi-inverse approah. B κ n τ + First, onsider the speimen in equilibrium prior to G&R. Under these onditions, the natural onfigurations assoiated with ollagen κ 2 n τ do not hange and detf t τ=1. t L τ µ 2 Nκ τ C n κ n τn κ n τ 1 exp [δ 2 Nκ n τ C κ n τn κ n τ 1 ] From Eq. 5, the stress in normaly i.e., the basal } F κ n τn κ n τ N κ n τf T state is given by κ dτ. nτ Let us now onsider a simple illustration of G&R during uniaxial extensions. 4 Uniaxial Extension To illustrate the model, onsider a uniaxial extension of a ylindrial speimen that is initially in a uniaxially stressed state with its lateral surfaes tration free. G&R is initiated by the appliation of an additional uniaxial streth. After applying this additional streth, the tissue is onstrained in the diretion of streth, onsequently, the prodution and removal of ollagen only hanges the radius of the speimen. Although G&R is initiated by ontrolling the streth, it ould also be initiated by hanging the axial load, i.e., by inreasing the stress. For suh a ase the prodution and removal of ollagen would hange both the axial length and radius of the tissue. We do not onsider this type of stress-ontrolled example here, instead we fous our attention on the strainontrolled ase. Consider the ylindrial speimen in a uniaxially stressed equilibrium state. By equilibrium, we imply that in addition to fore balane, m α = m ω = m α,eqand L = L eq, T basal = pi +α e µ e B κ e 5 +α { µ 1 exp [ δ 1 trcκ 3 ] B nτ κ n τ +, µ 2 Nκ nτ C κ n τn κ n τ 1 exp [δ 2 Nκ n τ C κ n τn κ n τ 1 ] 2 F κ n τn κ n τ N κ n τf T κ } for τ t <, nτ where, ρ e and ρ are the mass densities of elastin and ollagen in the tissue prior to G&R, and from Eq. 37, ρ = m α,eq L eq. 51 In general, the unit vetor N κ n τ an vary with time, but here we onsider a uniaxial extension wherein the ollagen fibers are oriented in the diretion of streth and the ollagen fibers that are subsequently laid down retain this original orientation. The vetor N κ n τ for this example problem is thus an unhanging unit vetor given by 1,,. We hoose the natural onfigurations of elastin and ollagen prior to G&R to be suh that they are related to the initial onfiguration of the tissue through uniaxial

10 448 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 strethes see Figure 3, namely F e := F κ e t=diag Λ e 1, Λ e, 1 Λ e, for t < 52 F := F κ n τ t=diag Λ 1, Λ, 1 Λ,for τ t <. 53 The basal stress an be obtained by substituting Eqs. 52 and 53 into Eq. 5. If the lateral surfaes are tration free, we obtain T 11 basal = α e µ e Λ e 2 1 Λ e +α { [ µ 1 exp δ 1 Λ ] Λ 3 Λ 2 1 Λ [ 2 } +µ 2 exp δ 2 Λ 1] 2 Λ 2 1 Λ 2, 54 all other omponents of stress being zero. Equilibrium, div T =, is thus identially satisfied as are the lateral tration boundary onditions. Given that G&R is initiated by subjeting the tissue to an additional uniaxial extension at t =, let the deformation gradient from the onfiguration just prior to the imposition of streth to the onfiguration just after the streth be denoted by F 1 see Figure 3, where F 1 = diag Λ 1, 1 Λ1, 1 Λ1. 55 The deformation gradients from the natural onfigurations of elastin and ollagen to the onfiguration oupied by the tissue immediately after the extension are thus given by F κ e t = =diag Λ e 1 Λ 1, Λ e, 1 Λ 1 Λ e Λ 1, 56 F κ n τ t = =diag Λ 1 Λ 1, Λ, 1 Λ 1 Λ Λ 1,for τ <. 57 The stress in the tissue immediately after this extension is given by T 11 = α e µ e Λ e Λ α Λ e Λ 1 { [ µ 1 exp δ 1 Λ Λ Λ Λ 3 1 +µ 2 exp δ 2 [Λ Λ 1 2 1] 2 Λ Λ ] Λ Λ Λ Λ 1 } Λ Λ 1 2, for t =. 58 Of ourse, at this instant no new ollagen has been produed and onsequently the stress is determined only from onstituents present prior to the inrease in streth. The inrease in stress above the basal value will inrease the rate of prodution of ollagen and in addition redue the life span of the extant ollagen, resulting in an inrease in the rate of removal. Consequently, the tissue will grow and remodel. Beause the tissue is onstrained from growing in the diretion of streth, the radius of the ylindrial speimen will inrease if there is a net mass prodution. This is a highly speialized ase in that the problem ditates how the material will be laid down. Additional onstitutive assumptions will be needed in general, partiularly when the material symmetry evolves. Nonetheless, here we denote the deformation gradient from the onfiguration oupied by the tissue immediately after the appliation of the streth to any later grown onfiguration by F 2 Figure 3, where F 2 := F κ κt = diag Λ 2 t, bt, bt, for t, 59 F 2 = I, for t =. 6 The value of Λ 2 t is presribed. For a step hange in length, it is Λ 2 t=1, for t, 61 whereas for a step hange followed by sinusoidal variation in length, it is Λ 2 t=1 +Asin2π f z t, for t, 62 where f z is the frequeny of the osillations and A is the amplitude. Again, we emphasize that inertial effets are ignored. The funtion bt represents growth in the radial diretion and is obtained as a result of the alulations.

11 Biologial Growth and Remodeling: A Uniaxial Example 449 t 1 Maturity e ollagen F F e n n 1 elastin F 1 t 2 Initially Perturbed 2 ollagen F F 2 3 n 3 Remodeling F 2 4 t 3 Growth & 1 F t 4 ollagen F n 4 Growth & Remodeling 4 Figure 3 : Shemati of the kinematis assoiated with G&R Choosing the initial equilibrium onfiguration of the tissue to be the referene onfiguration κ, the deformation gradient F κ an be expressed in terms of F 1 and F 2 as F κ = F 2 F 1 = diag Λ 1 Λ 2 t, bt Λ1, bt Λ1, for t. 63 Mass balane, by substituting Eq. 63 into Eq. 11, requires ḃt bt = 1 m α m ω Λ 2 t ρ Λ 2 t The mass density of elastin after G&R begins is ρ e = ρ e detf κ = ρ e Λ 2 tbt Sine the mass density of the tissue is assumed to be a onstant, the density of ollagen is given by ρ = ρ ρ e. 66 Eq. 13 speifies the assoiated mass frations. For elastin, F κ e t after the initiation of G&R beomes F κ e t=f 2 F 1 F e = diag Λ e Λ 1 Λ 2 t, bt Λ e Λ 1, bt Λ e Λ 1, for t, 67 on whih the stress in the elastin depends. For a period after t = until t L t=, a part of the ollagen present was produed prior to the initiation of G&R whereas the remainder is produed after G&R begins. For the ollagen that was produed prior to the begining of G&R, the deformation gradient is F κ n τt=f 2 F 1 F = diag Λ Λ 1 Λ 2 t, bt bt Λ, Λ 1 Λ Λ 1, for τ < t. 68 Beause the newly produed ollagen is produed in a stressed state, we need to know the deformation gradient

12 45 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 from the stress-free state to the onfiguration oupied by the tissue, i.e., F κ n t t to know the stress in this newly produed material. Here, we assume that the new ollagen is laid down with F κ n t t equal to F, i.e., F κ n t t=f = diag Λ 1, Λ, 1 Λ, for t. 69 This is tantamount to assuming that ollagen produed during G&R is laid down at a streth equal to the streth in the ollagen prior to G&R. This assumption will allow us to illustrate the potential utility of the overall model; various hypothesis with regard to this issue of natural onfigurations for tissue produed in stressed states will need to be explored similarly. Moreover, the deformation gradient assoiated with ollagen produed at a previous time following the onset of G&R, i.e., F κ n τ t is given by F κ n τt=f t τ 1 F Λ =diag Λ 2 t bt, bt Λ, bτ Λ 2 τ, Λ bτ for τ < t, 7 In addition, the following relationship holds: F t τ=diag Λ2 τ Λ 2 t, bτ bt, bτ bt, for τ < t. 71 Utilizing Eq.5, Eq.67, Eq.68, Eq.7 and Eq.71 the stress during adaptation when ollagen produed prior to the onset of adaptation is still present in the tissue is given by T 11 = α e µ e Λ e Λ 1 Λ 2 t 2 bt2 Λ e Λ 1 [ Λ + µ 1 exp δ Λ 2 t bt Λ 2 τ Λ 3] bτ Λ Λ 2 t 2 1 bt 2 Λ 2 τ Λ m Λ 2 τbτ 2 α bτ Λ 2 tbt 2 dτ [ Λ + µ ] 2 exp δ Λ 2 t 2 2 [ 2 1 Λ Λ 2 t 2 1] Λ 2 τ Λ 2 τ Λ Λ 2 t 2 m Λ 2 τbτ 2 α Λ 2 τ Λ 2 tbt 2 dτ + t L t µ 1 exp δ 1 [Λ Λ ] 1 Λ 2 t 2 + 2bt2 Λ Λ 3 1 Λ Λ 1 Λ 2 t 2 bt2 Λ Λ 1 m α 1 Λ 2 tbt 2 dτ + t L t 2 µ 2 exp δ 2 [Λ Λ 1Λ 2 t 1] 2 Λ Λ 1Λ 2 t 2 1 Λ Λ 1Λ 2 t 2 m 1 α Λ 2 tbt 2 dτ, for t L t <. 72 When all the ollagen produed prior to the onset of adaptation has been removed the stress is given by T 11 = α e µ e Λ e Λ 1Λ 2 t 2 bt2 Λ e Λ 1 [ Λ + µ 1 exp δ Λ 2 t bt 2 Λ 2 τ Λ 3] t L bτ Λ Λ 2 t 2 1 bt 2 Λ 2 τ Λ m Λ 2 τbτ 2 α bτ Λ 2 tbt 2 dτ [ Λ + µ ] 2 exp δ Λ 2 t 2 2 [ Λ 2 1 Λ 2 t 2 1] Λ 2 τ Λ 2 τ t L Λ Λ 2 t 2 m Λ 2 τbτ 2 α Λ 2 τ Λ 2 tbt 2 dτ for t L t. 73 To illustrate this model, we simulate a G&R proess taking plae after a uniaxial extension using a mass prodution rate, m α,givenby m α = K m1 [ T 11 avg T 11 basal ]+m α,eq, 74 where K m1 is a positive onstant. T 11 avg is a measure of time-averaged stress, whih ould be the stress at the urrent instant or a value averaged over past times. Here we use T 11 avg = 1 L T T 11 dτ, 75 t L T where L T is the time over whih the value of stress is averaged. For the results presented here we assume that L T = L. Hene an inrease in stress above the basal value auses an inrease in the prodution of ollagen; when the average stress equals the basal value of stress, the ollagen prodution rate equals the basal value of the

13 Biologial Growth and Remodeling: A Uniaxial Example 451 prodution rate. For the lifespan of the ollagen, we onsider dl dt = K L1 [ T 11 avg T 11 basal ]+K L2 L eq L, 76 where K L1 and K L2 are positive onstants. Hene, an inrease in the stress auses the lifespan of ollagen to derease. Note: in the above two equations we have ignored the possible dependene of mα and L on other quantities suh as mass densities, et. Given the urrent lak of experimental data, it is emphasized that the above equations for mα and L were simply hosen to illustrate the model; preferred forms will have to ome from experimental data. In the alulations that follow, the stress is non-dimensionalized by an average modulus, µ µ e + µ 1 + µ 2, time with L eq, and the prodution rate with m α,eq. The non-dimensional parameters that result from this are K m1 µ / m α,eq, K L 1 µ, K L2 L eq and f zl eq. For the results presented in the next setion the value of K L2 L eq is set to unity. Numerial Results Λ 1.2 Λ Λ δ 1.5 δ 2.1 µ 1 1 µ e +µ 1 +µ 2 3 µ 2 1 µ e +µ 1 +µ 2 3 µ e 1 µ e +µ 1 +µ 2 3 Table 1 : Values of different parameters used for the alulations Figure 4 : Stress versus time for different values of K m 1 µ m α,eq the inrease in streth, after whih it drops beause the newly produed ollagen, born in a less stressed state, replaes the ollagen that is removed. The form of the equations hosen fores the stress to return to a value lose to its basal value. For larger values of K m1 µ / m α,eq an initial osillatory response is observed. The normalized radius of the tissue is plotted in Figure 5 for three different values of K m1 µ / m α,eq. Initially there is a drop In the following alulations, those parameters that remain unhanged are given in table 1. First, onsider the ase wherein the tissue is subjet to a step hange in length. Due to this inrease in streth, the stress inreases thus ausing more ollagen to be produed and an inrease in the rate at whih it is removed. This an result in either an inrease or derease in the radius, i.e., volume of the tissue. The non-dimensional stress is shown in Figure 4 for different values of K m1 µ / m α,eq, with K L1 µ =.3. The stress inreases immediately after Figure 5 : Radius versus time for different values of K m 1 µ m α,eq in the radius due to the applied streth, after whih the radius of the tissue inreases due to growth. The rate at whih ollagen is produed is shown in Figure 6; the inrease in stress following the inrease in streth auses an inrease in the value of mα. After this transient, it set-

14 452 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 Figure 6 : m b versus time for different values of K m 1 µ m α,eq Figure 8 : Mass fration of ollagen versus time for different values of K m 1 µ m α,eq Figure 7 : L versus time for different values of K m 1 µ m α,eq Figure 9 : Stress versus time for different values of K L1 µ tles to a new steady value, higher than the initial value. Figure 7 shows the variation of the assumed life span of ollagen over time. The inrease in stress auses the life span to drop initially, but after the transient it reahes a new basal value. The mass fration of ollagen is shown as a funtion of time in Figure 8. As the tissue grows, the invariant elastin is spread over a larger volume while the density of the tissue is onstrained to remain onstant; this auses the mass fration of ollagen to inrease to a new basal value this is why m α settles to a higher basal value after the transient see Figure 6. These plots are repeated for different values of K L1 µ Figures 9 to 13 for a value of K m1 µ / m α,eq = 1.. For larger values of K L1 µ, the life span dereases rapidly after the initiation of G&R. For a sinusoidally varying streth, the stress is plotted for three different values of frequeny Figures 14 and 15. The following values were used: K m1 µ / m α,eq = 1., K L1 µ =.3 and A =.5. Note that the stress after adaptation osillates about the basal value. The radius of the tissue is plotted for these three ases in Figures 16 and 17. Conlusions To address the need for mathematial models apable of prediting mehanially indued G&R, Humphrey and Rajagopal 22 reently proposed a onstrained mixture model. In this paper, we applied this model to a two onstituent tissue, of whih one onstituent ontinu-

15 Biologial Growth and Remodeling: A Uniaxial Example 453 Figure 1 : Radius versus time for different values of K L1 µ Figure 12 : L versus time for different values of K L1 µ Figure 11 : m b versus time for different values of K L 1 µ Figure 13 : Mass fration of ollagen versus time for different values of K L1 µ ously turned over ollagen while the seond onstituent elastin remained unhanged. Collagen was assumed to exhibit a transversely isotropi behavior and elastin an isotropi behavior. The natural onfigurations of ollagen produed at different times during G&R evolve depending on the loal onditions. The amount and age of the ollagen present at eah material point in the tissue was presribed via mass prodution rates of ollagen per unit volume and the age of the ollagen being removed. Both of these quantities depend on loal onditions and an vary with stress, mass frations et.; illustrative equations were used to model these quantities. The behavior of the model was investigated for two uniaxial exten- sions: a step hange in the length and a step hange in length followed by a sinusoidal variation in length about the new value. These types of deformations are ommonly enountered in ligaments and tendons. Results showing the evolution of stress, radius, mass frations, and mass prodution rates over G&R time were plotted for different values of material onstants that arise in the model. These results illustrate the influene of the mass frations of the different onstituents and the kinetis assoiated with mass prodution and removal on the stress and shape of the tissue. It is hoped that the results presented in this paper will motivate further work on biologial growth and remodeling and in partiular aid in identifying the experiments that are needed to formulate

16 454 Copyright 23 Teh Siene Press CMES, vol.4, no.3&4, pp , 23 Figure 14 : Stress versus time for different values of f z L eq Figure 16 : Radius versus time for different values of f z L eq Figure 15 : Stress versus time for f z L eq = 1 Figure 17 : Radius versus time for f z L eq = 1 physiologially meaningful onstitutive equations. Aknowledgement: One of the authors I. J. Rao aknowledges support provided by the New Jersey Institute of Tehnology under Grant No This work was also supported, in part, by federal grants from the NIH HL-58856, HL and the NSF BES Referenes Humphrey J. D. 22: Cardiovasular Solid Mehanis: Cells, Tissues, and Organs. Springer-Verlag, NY. Humphrey J. D.; Rajagopal K. R. 22: A on- strained mixture model for growth and remodeling of soft tissues. Math. Model. Meth. Appl. Si. vol. 12, pp Rajagopal, K. R. 1995: Multiple onfigurations in ontinuum mehanis. In: Reports of the Institute for Computational and Applied Mehanis, University of Pittsburgh 6. Rajagopal, K. R. 23: The Art and Craft of Modeling, to be published. Rajagopal, K. R.; Srinivasa, A. R. 1995: On the inelasti behavior of solids: Part-I- Twinning. Int. J. Plast., vol. 11, pp

17 Biologial Growth and Remodeling: A Uniaxial Example 455 Rajagopal, K. R.; Srinivasa, A. R. 1998: Inelasti behavior of materials: Part-I- Theoretial underpinnings. Int. J. Plast., vol. 14, pp Rajagopal, K. R.; Wineman, A. S. 1992: A onstitutive equation for nonlinear solids whih undergo deformation indued mirostrutural hanges. Int. J. Plast., vol. 8, pp Rao, I. J.; Rajagopal, K. R. 2: Phenomenologial modeling of polymer rystallization using the notion of multiple natural onfigurations. Interfaes and Free Boundaries, vol. 2, pp Rao, I. J.; Rajagopal, K. R. 21: A study of strain indued rystallization in polymers. Int. J. Sol. Str., vol. 38, pp Rodriguez, E. K.; Hoger, A. and MCulloh, A. D. 1994: Stress-dependent finite growth in soft elasti tissues. J. Biomeh., vol. 27, pp Skalak R. 1981: Growth as a finite displaement field. In: Carlson DE, Shield RT, eds., Proeedings of the IU- TAM Symposium on Finite Elastiity. Martinus Nijhoff Publishers, The Netherlands, pp Taber, L. A. 1995: Biomehanis of growth, remodeling and morphogenisis. Appl. Meh. Rev., vol. 48, pp Thompson D Ary 1961: On Growth and Form. Cambridge University Press, Cambridge. Tranquillo R. T.; Murray J. D. 1992: Continuum model of fibroblast-driven wound ontration: Inflammation-mediation. J Theoret. Biol., vol. 158, pp Turing A. M. 1952: The hemial basis of morphogenesis. Phil. Trans. Roy. So. London, vol. 237, pp

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