Comput Mech (2014) 53:1183–1196DOI 10.1007/s00466-013-0959-z

ORIGINAL PAPER

Computational modeling of hypertensive growth in the humancarotid artery

Pablo Sáez · Estefania Peña ·Miguel Angel Martínez · Ellen Kuhl

Received: 12 August 2013 / Accepted: 7 December 2013 / Published online: 19 December 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Arterial hypertension is a chronic medical condi-tion associated with an elevated blood pressure. Chronic arte-rial hypertension initiates a series of events, which are knownto collectively initiate arterial wall thickening. However,the correlation between macrostructural mechanical load-ing, microstructural cellular changes, and macrostructuraladaptation remains unclear. Here, we present a microstruc-turally motivated computational model for chronic arterialhypertension through smooth muscle cell growth. To modelgrowth, we adopt a classical concept based on the multiplica-tive decomposition of the deformation gradient into an elasticpart and a growth part. Motivated by clinical observations,we assume that the driving force for growth is the stretchsensed by the smooth muscle cells. We embed our model intoa finite element framework, where growth is stored locallyas an internal variable. First, to demonstrate the features ofour model, we investigate the effects of hypertensive growthin a real human carotid artery. Our results agree nicely withexperimental data reported in the literature both qualitativelyand quantitatively.

Keywords Biomechanics ·Growth · Smooth muscle cells ·Hypertension · Finite element method

P. Sáez · E. Peña ·M. A. MartínezGroup of Applied Mechanics and Bioengineering, Aragón Instituteof Engineering Research, University of Zaragoza, Zaragoza, Spain

P. Sáez · E. Peña ·M. A. MartínezCentro de Investigación Biomédica en Red en Bioingeniería,Biomateriales y Nanomedicina (CIBER-BBN), Zaragoza, Spain

E. Kuhl (B)Departments of Mechanical Engineering, Bioengineering,and Cardiothoracic Surgery, Stanford University,Stanford, CA 94305, USAe-mail: ekuhl@stanford.edu

1 Motivation

Hypertension is characterized through a substantial elevationin blood pressure, which initiates the adaptation of vascu-lar tissue. The immediate change, on a short-term scale, isthe adaptation of smooth muscle cell contractile forces inthe connective tissue, which can reduce the lumen diame-ter to compensate for the extra stresses. In addition, we canobserve two long-term adaptation processes. The first long-term change is related to the secretion of collagen in theextracellular matrix, which gradually stiffens the arterial tis-sue. The second long-term change consists of the increase ofsmooth muscle cells both in volume and in number. Exper-imentally, this adaptation manifests itself in a thickening ofthe vessel wall. In this work, we focus on this latter phe-nomenon, the growth of smooth muscle cells in response tochronic hypertension.

Smooth muscle cells are responsible for vascular contrac-tion, also known as myogenic tone [6,62], which is directlyrelated to the intracellular calcium concentration and to theion channels of the cell [9]. Vascular contraction attempts tocompensate the over-stress in the vessel wall to maintain thesame vessel diameter [27,51].

Smooth muscle cells typically grow through hypertrophy,by increasing their volume, through hyperplasia, by increas-ing their number, or through a combination of both [52,54].This micro-structural change leads to the well-documentedthickening of the vessel wall to restore the homeostatic stressstate illustrated in Fig. 1. These changes are most pronouncedin small or resistance vessels [15,49]. A multitude of exper-imental findings is related to hypertension-induced thicken-ing of the arterial wall; many of them are related to drugperformance, to the genetic expression of substances, andto the plain case of thickening. Some of these studies lookat essential hypertension, while other assess hypertension

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normotensive hypertensive

Fig. 1 Hypertensive anisotropic arterial growth. In contrast to thenormtensive artery, left, the hypertensive artery, right, displays pro-gressive smooth muscle cell growth resulting in chronic arterial wallthickening

by means of hypoxic states or ligands of some particulararteries.

Wiener et al. [68] studied aortas of hypertensive rats andobserved a decrease in lumen diameter of 9 % and a thick-ness increase of 18 % from 182.5 µm in the normotensivestate to 215.5 µm in hypertension. Owens and Schwartz [54]reported an increase of 35 % in smooth muscle cell size dueto hyperplasia in the hypertensive rat aorta. The same authorsshowed an increase of 32.2 % in aortic mass and of 39.7 % insmooth muscle cell mass in Goldblatt hypertensive Sprague-Dawley rats [53]. Schofield et al. [61] studied the evolution ofsmall human arteries and observed that the internal diameterdecreased from 140 to 118 µm, while the thickness increasedby 46.39 %. Since smaller arteries display a more pronouncedcontractile response, such a pronounced decrease could becaused by contractile phenomena. In fact, the above studiesalso addressed changes in the miogenic response in responseto hypertension.

Studies on the carotid artery are also numerous. Bouto-uyrie et al. [5] reported a rise of the internal diameter inhuman carotids from 5.25 to 5.66 mm and an increase inthickness of 27.27 % from 0.487 to 0.572 mm. However, thesame work showed a slight decrease in the lumen diame-ter from 2.33 to 2.32 mm in the radial artery accompaniedby a 21.98 % increase in thickness. Fridez et al. [16] stud-ied the influence of hypertension in carotid arteries of rats.They showed that the internal radius increased in hyperten-sive rats but it increased at a slower rate than in the controlgroup. While the overall thickness increased by 58.4 % from142 to 225 µm, this represented just a 30 % increase withrespect to control group. The same group also observed thatthe thickness of the outer lamellar units grew more rapidlythan that of the inner units. Eberth et al. [10] studied morpho-logical changes in rat carotid arteries for a period of 56 weeksand found an increase in the internal radius from 531 to592 µm, accompanied by an increase in thickness from 23.1to 85.7 µm.

These results demonstrate how vascular adaptation candiffer, not only between different arteries and species, but alsowithin the same artery of the same species. Interspecimenvariation, different levels of smooth muscle cell activation,genetic disorders, age, life style, and many more factors cancause this variability. To point out a last example, Feihl etal. [12] collected data from different authors who all studiedsmall subcutaneous human arteries. While there was a strongquantitative variability in the degree of the adaptive response,the qualitative trends were all similar to the trends describedabove.

The computational study of growth has gained increas-ing attention in the theoretical and computational mechanicscommunity, [1,33,48,65]. Mechanical modeling of growthhas been addressed in different ways. Typically, growing bio-logical tissues are considered as open systems [40]. Their dif-ferent configurational settings and their numerical treatmentwithin the finite element method are discussed in Kuhl andSteinmann [39]. Typically, we can distinguish two fundamen-tal forms of growth: volume growth [28] and density growth[38]. The first allows for changes in volume while keepingthe density constant, whereas the second maintains a constantvolume while the density is allowed to change [41,46,47].The works of Skalak et al. [63] and Rodriguez et al. [57]pioneered the underlying kinematic theory of volumetricgrowth. Within the past decade, various authors and groupshave adopted and refined this kinematic concept [18,26,35].

An alternative approach towards growth is based on theconstrained mixture theory [67], where several constituentsof a tissue are allowed to growth independently [31,32].A similar approach [22,23,37] was recently extended toreactive mixtures [2]. First attempts are now underway tocorrelate cardiovascular growth and remodeling to dynamicalterations in pressure and flow [8] and to fluid structure inter-action phenomena [13,34]. We would also like to point outthe early work of Fung and Liu [17], which showed thatgrowth induces a change in the natural configuration associ-ated with the notion of residual stress [56].

The objective of this work is to introduce a computa-tional model for smooth muscle cell growth and embed itinto the anisotropic microstructure of vascular tissue. Wefocus on smooth muscle cell growth in response to chronichypertension and explore its impact on arterial wall thicken-ing. First, we briefly reiterate the general mechanical back-ground of soft tissue and growth mechanics. Next, we intro-duce smooth muscle cell growth as a stress- or strain-drivenprocess, activated by an elevated pressure or stretch. Weupdate the history of growth using an incremental, interativeNewton–Raphson scheme. To calibrate our model by meansof experimental data, we first simulate hypertensive smoothmuscle cell growth in an idealized circular arterial cross sec-tion. To demonstrate the regional variation of growth, we thensimulate growth in a human carotid artery reconstructed from

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clinical images. We close by discussing the main results andthe possible impact of our work.

2 Baseline elasticity of cardiovascular tissue

Two important aspects enter the mechanical characteriza-tion of cardiovascular tissue, the kinematically non-linearbehavior and constitutively non-linear behavior of quasi-incompressible materials. We will briefly summarize bothin the following subsections.

2.1 Kinematics of quasi-incompressiblity

First, we reiterate the basic equations of the kinematicallynon-linear behavior. Our most important kinematic quantityis the deformation gradient F, the tangent of the nonlineardeformation map ϕ,

F = ∇Xϕ, (1)

where the symbol ∇X denotes the spatial gradient withrespect to the referential coordinates X. We can then intro-duce the Jacobian J = det(F) and the right Cauchy Greendeformation tensor,

C = Ft · F . (2)

To account for the characteristic quasi-incompressibilebehavior of soft biological tissues, we adopt a volumetric–isochoric decomposition of the deformation gradient [14,20],

F = J1/3F . (3)

The overbar is associated with the prefix isochoric anddenotes the volume-preserving part. Accordingly, F denotesthe isochoric deformation gradient with det(F) = 1, and

C = Ft · F (4)

denotes the associated isochoric right Cauchy Green defor-mation tensor. It is convenient to introduce the first and fourthinvariants,

I1 = C : I and I4 = C : N (5)

where N = n ⊗ n denotes the structural tensor defined interms of the characteristic microstructural direction n.

2.2 Constitutive equations of quasi-incompressibility

Second, we summarize the basic equations of the consti-tutively non-linear behavior. For soft biological tissues itis common to adapt the framework of hyperlasticity, basedon the definition of a strain energy density function Ψ . Toaccount for quasi-incompressiblity, we additively decompose

this strain energy density function into volumetric and iso-choric parts,

Ψ = Ψvol(J)+ Ψiso(C) . (6)

The volumetric contribution Ψvol, is primarily related to thewater content in the tissue. The isochoric contribution Ψiso,is typically further split into an isotropic contribution relatedto the elastin content Ψela and an anisotropic contributionrelated to the collagen fibers Ψcol. From the evaluation of thedissipation inequality [29,45], we obtain the second Piola–Kirchhoff stress,

S = 2∂CΨ = Svol + Siso (7)

which consists of a volumetric part,

Svol = 2∂CΨvol = J ∂JΨvol C−1 , (8)

and an isotropic part,

Siso = 2∂CΨiso = 2∂CΨiso : ∂CC = J−2/3P : S . (9)

Here S = 2 ∂CΨiso is the fictitious second Piola–Kirchhoffstress and P = I − 1

3 C−1 ⊗ C denotes the fourth orderprojection tensor. Through the contravariant push forwardoperation, we obtain the Cauchy stress σ ,

σ = 1

JF · S · Ft , (10)

which is typically required for the finite element implemen-tation. The tensor of tangent moduli, a fourth order tensorthat relates incremental stresses and strains, is essential fora consistent finite element implementation. It represents thetotal derivative of the stress S with respect to the deformationtensor C,

C = 2dCS = Cvol + Ciso , (11)

and consists of a volumetric contribution,

Cvol = 2dCSvol = 2 J p C−1 ⊗ C−1 − 2 J p IC−1 , (12)

and an isochoric contribution,

Ciso = 2dCSiso = P : C : Pt + 2

3J−2/3 tr(Siso) P

−2

3[C−1 ⊗ S+ S⊗ C−1 ] . (13)

Here p denotes the hydrostatic pressure with p = p+ J ∂J p,and C = 4 J−4/3 ∂C⊗CΨiso are the fictitious elastic tangentmoduli [21]. In addition, we have introduced the follow-ing abbreviations for the fourth order tensors P = IC−1 −13 C−1 ⊗ C−1 and IC−1 = 1

2 [C−1⊗C−1 + C−1⊗C−1],where the non-standard fourth order tensor products takethe following interpretation, {•⊗◦}i jkl = {•}ik {◦} jl and{•⊗◦}i jkl = {•}il {◦} jk .

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Table 1 Elastic material parameters for media and adventitia of humancarotid artery (CCA) and of human internal carotid artery (ICA) [60,64]

μ (kPa) k1 (kPa) k2 (-)

CCA media 4.31 2.19 4.15

CCA adventitia 0.04 7.32 66.81

ICA media 11.01 2.14 20.72

ICA adventitia 0.04 15.97 51.01

Remark 1 (Elasticity of arterial tissue) The modeling of arte-rial tissue has been widely discussed in literature [30]. It iscommon to describe the isochoric response Ψiso by means ofan isotropic contribution for elastin, Ψela, and an anisotropiccontribution for collagen, Ψcol,

Ψiso(I 1, I 4) = Ψela(I 1)+ Ψcol(I 4) ,

where I 1 and I 4 are the first and fourth invariant of theisochoric part of the deformation according to Eq. (5). Theisotropic elastin part is typically parameterized in terms of asingle stiffness parameter μ, e.g., as

Ψela(I 1) = μ [ I 1 − 3 ] .The anisotropic collagen part has been discussed intensely inthe literature. O’Connell et al. [50] have studied the structuralorganization of the arterial layers, and found that the colla-gen fibers follow a helicoidal distribution from the outer tothe inner layer. Collagen fibers are bundled around smoothmuscle cells and around some secondary fibrils, which cross-link the main collagen fibers. In the adventitia, the outermostlayer of the artery, collagen displays a rather random orien-tation. In the media, the middle layer, collagen and smoothmuscle cells form organized sublayers with slightly varyingpreferred orientations. Garcia [19] reported an almost cir-cumferential orientation of the collagen fibers in the mediaof carotid arteries. Here, for simplicity, we adopt the follow-ing ansatz for the collagen part of the free energy [21],

Ψcol(I 1) = k1

2k2[ exp(k2[I 1 − 1]2)− 1] .

Based on experimental data [64], we identify the mechanicalparameters μ, k1, and k2 in different regions of the humancarotid artery. Table 1 summarizes the identified materialparameter values. Figure 2 illustrates the regional variationof the mechanical parameters along the carotid artery.

3 Growth of cardiovascular tissue

3.1 Kinematics of growth

Within the framework of finite growth, the key kinematicassumption is the multiplicative decomposition of the defor-mation gradient F into an elastic part Fe and a growth partFg [57],

µ [kPa] k1[kPa] k2 [-]

Fig. 2 Regional variation of material parameters μ, k1, and k2 alonghuman carotid artery

Fig. 3 Kinematics of finite growth. Multiplicative decomposition ofthe deformation gradient F into an elastic tensor Fe and a growth tensorFg

F = Fe · Fg . (14)

The underlying concept is adopted from the multiplicativedecomposition in finite plasticity [43] and illustrated in Fig.3. While we can think of the growth tensor Fg as a second-order variable to characterize arbitrary forms of isotropic oranisotropic growth, here we will parameterize the growthtensor exclusively in terms of a single scalar-valued variable,the growth multiplier ϑ [24]. This approach is conceptuallygeneric and can be easily adapted to model volume growth[28], area growth [7], and fiber growth [70]. We denote theJacobians of the elastic tensor and of the growth tensor asJe = det(Fe) and Jg = det(Fg), such that

J = Je Jg . (15)

We can then introduce the elastic right Cauchy Green tensor,

Ce = Fte · Fe = F−t

g · C · F−1g , (16)

and relate it to the total right Cauchy Green tensor C usingthe inverse growth tensor.

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Remark 2 (Growth of arterial tissue) Motivated by experi-mental observations of smooth muscle cell growth [52–54],we specify the growth tensor as

Fg = I+ [ϑ − 1 ]n⊗ n ,

where ϑ is the scalar-valued growth multiplier that character-izes the amount of growth and n is a characteristic structuraldirection [25]. The sensitivity of growth with respect to thegrowth multiplier then takes the following simple form,

∂ϑFg = n⊗ n ,

If we choose n to be alined with the radial direction, thisparticular format of the growth tensor characterizes smoothmuscle cell thickening in the radial direction while the cellslength remains constant. This type of growth is conceptuallysimilar to hypertrophic growth in the heart [55].

3.2 Constitutive equations of growth

Next, we embed the kinematic characterization of growthinto the hyperelastic baseline description introduced in Sect.2.2. To this end, we reparameterize the free energy functionΨ (Ce), which was initially parameterized in terms of theelastic deformation tensor Ce, in terms of the total deforma-tion tensor C and the growth tensor Fg, such that Ψ (C, Fg)

and

Ψ = ∂CΨ : C+ ∂FgΨ : Fg . (17)

Thermodynamic considerations motivate the introduction ofthe second Piola–Kirchhoff stress,

S = 2∂CΨ = 2∂CeΨ : ∂CCe = F−1g · Se · F−t

g . (18)

Here, Se = 2 ∂CeΨ is the classic elastic second Piola–Kirchhoff stress as introduced in Eq. (7). To derive theLagrangian tangent moduli, essential for a consistent finiteelement implementation, we evaluate the total derivative ofthe second Piola–Kirchhoff stress S with respect to the rightCauchy Green deformation tensor C,

C = 2dCS = Ce + Cg

= 2∂CS∣∣Fg+ 2∂CS

∣∣F

= 2∂CS∣∣Fg+ 2

[

∂Fg S : ∂ϑFg]⊗ ∂Cϑ

∣∣F. (19)

The first term of Eq. (19) represents the pull back of the elastictangent moduli to the undeformed reference configuration,

Ce = 2dCS∣∣Fg= [F−1

g ⊗F−1g ] : 2dCe Se : [F−t

g ⊗F−tg ] . (20)

Here, 2dCe Se are the classic elastic tangent moduli as intro-duced in Eq. (11). The second term of Eq. (19) is related tothe linearization of the growth model,

Cg = 2dCS∣∣F = 2

[

∂Fg S : ∂ϑFg]⊗ ∂Cϑ

∣∣F . (21)

The first term, ∂FgS, is conceptually generic,

∂Fg S = −[F−1g ⊗S+ S⊗F−1

g ]−[F−1

g ⊗F−1g ] :

1

2C : [F−t

g ⊗Ce+Ce⊗F−tg ]. (22)

The second term, ∂ϑFg, is specific to the format of thegrowth tensor Fg as discussed in Remark 2, i.e., in our case∂ϑFg = n⊗ n. The third term, ∂Cϑ , is specific to the evolu-tion of growth ϑ , which we will specify in Remarks 3 and 4for the stress- and strain-driven case. To obtain the Euleriantangent moduli c for the finite element implementation, wepush Lagrangian tangent moduli C of Eq. (19) forward to thedeformed configuration,

c = 1

J[F⊗F ] : C : [Ft⊗Ft ] . (23)

Last, we define the evolution of the growth multiplier [24],

ϑ = k(ϑ) φ(�) . (24)

Here Ξ represents driving force for growth and k is a limitingfunction that ensures that the tissue does not grow unbound-edly [44],

k = 1

τ

[ϑmax − ϑ

ϑmax − 1

]γ

(25)

with ∂ϑk = −γ k/[ϑmax − ϑ]. Growth evolves in timeaccording to three parameters, the maximum amount ofgrowth ϑmax, the adaptation speed τ , and the non-linearity ofthe growth process γ [28]. The function φ(Ξ) represents thegrowth criterion similar to the flow rule in the theory of plas-ticity. In the following, we discuss two different mechanismsto drive the evolution of growth, stress and strain.

3.3 Numerical implementation

For the numerical implementation, we integrate the evolutionof growth in time using an implicit Euler backward scheme,

ϑ = [ϑn+1 − ϑn]/Δt , (26)

where Δt denotes the current time increment. This allows usto introduce the discrete local residual,

R = ϑn+1 − ϑn − k(ϑ)φ(Ξ)Δt . (27)

To solve this non-linear equation, we expand the residualup to the first order term. We can then solve this equation,R + ∂ϑRΔϑ

.= 0, within an iterative Newton–Raphsonscheme. With the tangent of the residual,

K = ∂ϑR = 1− [ k ∂ϑφ + ∂ϑk φ ] Δt , (28)

we can incrementally update the growth multiplier ϑn+1 ←ϑn−R/K until the residual R has converged towards a smallenough user-defined tolerance. Table 2 summarizes the localalgorithmic treatment of the numerical procedure, which can

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Table 2 Algorithm for implicit Euler scheme of stress- or strain-drivengrowth

be easily embedded into any finite element setting at the con-stitutive level.

Remark 3 (Stress-driven growth) Stress is the most popu-lar driving force for growth, in particular in the biomedicaland clinical communities, and it is widely used for growthwithin the cardiovascular system [24]. Accordingly, we canintroduce the evolution of the growth multiplier,

ϑ = k(ϑ)φ(Me) ,

using a stress-driven growth criterion [24,36],

φ(Me) = pe − pcrit with pe = tr(Me) .

Here, Me = Ce · Se is the elastic Mandel stress, its trace,pe = tr(Me), is a measure of the pressure in the arterial wall,and pcrit denotes the physiological pressure above whichgrowth occurs. The derivative of the growth criterion withrespect to the growth multiplier ϑ required for the consistentlinearization in Eq. (28) then reads

∂ϑφ = −∂ϑCe : Se + Ce : ∂ϑSe ,

with ∂ϑCe = −F−tg · ∂ϑFt

g ·Ce−Ce · ∂ϑFg ·F−1g and ∂ϑSe =

12 Ce : ∂ϑCe. Last, to complete the tangent moduli in (19), weprovide the derivative of the growth multiplier with respectto the Cauchy–Green strain tensor,

∂Cϑ = k Δt

K [ 1

2Ce : Ce + Se ] .

Within a finite element setting, stress-driven growth requiresan additional internal iteration to update the growth multi-plier, since its driving force, the current stress, is typicallynot known a priori.

Remark 4 (Strain-driven anisotropic growth) Strain is a pop-ular driving force for growth, in particular in cell biology andmechanotransduction, since cells are equipped with variousmechanisms to sense and respond to strains [72]. Adopting

the concept of strain-driven growth, we can introduce theevolution of the growth multiplier,

ϑ = k(ϑ)φ(Ce) ,

using a strain-driven growth criterion [7,24],

φ(Ce) = λe − λcrit with λe = [n · Ce · n ]1/2 .

Hereλcrit denotes a physiological stretch above which growthoccurs and n is a characteristic microstructural direction. Ifwe assume that our smooth muscle cells and collagen fibersare collectively aligned along this direction n, and that theyundergo an affine deformation, they sense the same elasticstretches, λe = λsmc = λcol, which act as driving force forgrowth. The remaining term for the local tangent in Eq. (28)then simply reads

∂ϑφ = −λ/λ2g ,

and the global tangent term in Eq. (19) becomes

∂Cϑ = k �t

K[

1

2 λn⊗ n

]

,

where λ = [n · C · n ]1/2 is the total stretch along the mir-costructural direction n. Unlike stress-driven growth, strain-driven growth does not require an additional internal itera-tion to update the growth multiplier, since its driving force,the current strain, is typically known a priori. Strain-drivengrowth is therefore computationally more stable and robust.

4 Arterial growth in hypertension

In this section, we illustrate the characteristic features of theproposed growth model by means of two examples. First, tocompare stress and strain driven growth, we simulate hyper-tensive growth in a circular cross section of an artery. Sec-ond, to illustrate the regional variation of growth, we simulatehypertensive strain-driven growth in a human carotid artery.Table 1 summarizes the three elastic material parameters,the elastic stiffness μ, the collagen stiffness k1, and the non-linearity of the collagen contribution k2 for both the mediaand the adventitia. Table 3 summarizes the three growth para-meters, the maximum amount of growth ϑmax, the adaptationspeed τ , and the non-linearity of the growth process γ forboth examples. To initiate growth, we adopt the followingsimulation protocol:

Table 3 Growth material parameters for circular cross section of anartery in Sect. 4.1 and for human carotid artery in Sect. 4.2 [16]

ϑmax (-) τ (1/days) γ (-)

4.1 Circular cross section 2.0 1.0 2.0

4.2 Human carotid artery 1.6 10.0 3.0

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normotensive

hypertensive

E [-]max

S [kPa]max

S [kPa]max

E [-]max

max principal strain max principal stress

max principal strain max principal stress

Fig. 4 Normotensive and hypotensive conditions in a cylindrical pro-totype artery with maximum principal strains, left, and stresses, right

1. Gradually pressurize an artery up to the normotensivestate at a physiological pressure level of 13.3 kPa.

2. Calculate the resulting profiles of the critical physiolog-ical stretch λcrit and pressure pcrit and store their valueslocally in a pointwise fashion.

3. Gradually increase the pressure up to the hypertensivestate at a chronically elevated pressure level of 16.0 kPa.

4. Allow the tissue to grow chronically to compensate forthe extra deformation and stress.

Existing models typically treat the critical physiologicalstretch λcrit and pressure pcrit as constant homogeneous mate-rial parameters, which do not vary in space. Here we asso-ciate these critical values with the normotensive state. Assuch, these threshold values for growth are naturally associ-ated with the physiological pressure and inherently displaytransmural and axial longitudinal variations across the arte-rial wall.

4.1 Growth in a cylindrical cross section of an artery

In the first example, we simulate hypertensive growth in acircular cross section of the arterial wall. In particular, weseek to answer the question, which driving force is moresuitable for hypertensive growth, stress or strain. We simulatean idealized circular arterial slice, discretized with separatemedial and adventitial layers. We systematically comparestress-driven growth, φ(Me) = p(Me) − pcrit according to

strain driven

Smax[kPa][-]2.00

1.90

1.95

growth multiplier max principal stress

S [kPa]max[-]

2.00

1.90

1.95

growth multiplier max principal stress

stress driven

Fig. 5 Stress driven, top, and strain driven, bottom, growth in a cylin-drical prototype artery with growth multiplier, left, and maximum prin-cipal stresses, right

Remark 3, with strain-driven growth, φ(Ce) = λe(Ce)−λcrit

according to Remark 4.Figures 4 and 5 illustrate a finite element simulation of

growth in a cylindrical cross section of the arterial wall, dis-played in the deformed current configuration. Figure 4 dis-plays the maximum principal strains and stresses under nor-motensive and hypertensive conditions. Figure 5 comparesthe growth multiplier and the maximum principal stressesfor strain- and stress-driven growth. The arterial wall thick-ness and the regional profiles of the growth multiplier differsubstantially for the strain- and stress-driven cases.

For the stress-driven case, as displayed in Fig. 5, bot-tom row, growth increases radially from inward to outward.Since the adventitial layer is much stiffer than the mediallayer, this results in higher stresses in the adventitia, mak-ing it grow much faster in the stress-driven case. Moreover,the higher growth rate in the adventitia initiates a markedperpendicular expansion towards the center thereby decreas-ing the circumferential dimension of the media. For thestrain-driven case, growth also increases radially, but sig-nificantly smoother than for the stress-driven case. For thefirst time, in contrast to previous studies [28,58], we havesimulated growth in an artery wall with distinct layers anddistinct stiffnesses. Our results show that it is critical toaccount for the different properties in different layers topredict experimental findings. As we have discuss in theintroduction, arterial thickening occurs mainly in the media,caused by hypertrophy and hyperplasia of the smooth mus-

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common carotid artery

external carotid artery

internal carotid artery

Fig. 6 Human carotid artery with three representative cross-sectionsin the common carotid artery, the internal carotid artery, and the externalcarotid artery, highlighted in dark

cle cells. For the stress-driven stimulus, growth seems tooccur mainly in the adventitia, which would contradict thesefindings.

Strain-driven growth, as displayed in Fig. 5, top row, dis-plays a more uniform growth distribution across the thick-ness. Since the circumferential stretch is more uniform thanthe stress distribution, growth occurs in a more distributedway, which seems to favorably agree with experimental find-ings.

The debate about the most adequate driving force fordifferent processes in cells, e.g., cell differentiation or cellmigration, remains vivid and still ongoing. Along these lines,the choice of a stress- or strain-driven approach to charac-terize hypertensive growth still remains unclear. From ourresults we conclude that growth in arterial tissue is highlysensitive to the choice of the underlying mechanical stimu-lus. In the particular case of growth in arteries it is knownthat growth occurs primarily in the media, since this is wherethe smooth muscle cells are situated. To better characterizethe constitutive response of the arterial wall, we will fromnow on assume that smooth muscle cell growth is allowedto occur only in the medial layer. Moreover, we assume that

the smooth muscle cell stretch is the mechanical variable thatdrives the growth process.

4.2 Growth in a human carotid artery

In the second example, we simulate hypertensive growth ina human carotid artery. In particular, we explore the regionaldistribution of growth across the bifurcated arterial segment.Figure 6 displays the geometry of the artery, which wediscretize with 166,408 tri-linear brick elements. The dis-cretization accounts for separate medial and adventitial lay-ers. Following Remark 4, we adopt a strain-driven approachtowards arterial growth based on the following growth crite-rion, φ(Ce) = λe−λcrit . This implies that the smooth musclecell stretch λe = λsmc = [n·Ce ·n ]1/2 is the driving force forgrowth. To illustrate the features of the proposed approach,we compare our simulation with experimental findings [16].We begin by studying the growth process in a representativeslice of the common carotid artery as indicated in Fig. 6.

Figure 7, top, shows the evolution of a representativesmooth muscle cell in the medial layer of the common carotidartery. The smooth muscle cell size increases in the radialdirection while the longitudinal direction remains constant.Figure 7, bottom, shows the evolution of growth in a circularsection of the deformed common carotid artery at differenttime steps. In response to hypertension, the maximum growthmultiplier increases to ϑ = 1.47, indicating an increase ofsmooth muscle cell thickness of almost 50 %.

Figure 8 shows the evolution of the growth multiplier overa period of 100 days, both for the experimental measure-ments in [16] and for the computational simulation with ourmodel. Results displays a similar tendency throughout the

[-]

t = 0d t = 33d t = 67d t = 100d

evolution of arterial cross section

evolution of smooth muscle cell size

t = 0d t = 33d t = 67d t = 100d

1.0 1.5

Fig. 7 Growth of a representative smooth muscle cell in the mediallayer of the common carotid artery and its effect of the circular sectionat characteristic time steps. In response to hypertension, the growth

multiplier gradually increases to ϑ = 1.47, indicating an increase ofsmooth muscle cell thickness of almost 50 %

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Comput Mech (2014) 53:1183–1196 1191

0 20 40 60 80 1001.0

1.1

1.2

1.3

1.4

1.5

time [days]

experiment

simulation

grow

th [-

]temporal evolution of growth

Fig. 8 Temporal evolution of growth in human carotid artery. Circlesrepresent experimentally measured growth [16]; squares represent com-putationally simulated growth

0 20 40 60 80 100

0

10

15

time [days]

inner layermedia layer

max

prin

cipa

l str

ess

[kP

a]

outer layer

-5

5

temporal evolution of stress

Fig. 9 Temporal evolution of maximum principal stresses in inner,medial, and outer layers of human carotid artery

growth process. Our results slightly overestimate the amountof growth during the first few days, and slightly underesti-mate the amount of growth long time.

Figure 9 displays the evolution of the maximum princi-pal stresses in inner, medial, and outer layers of a humancarotid artery. Stresses in the medial layer decrease as a resultof smooth muscle cell growth. Smooth muscle cell growth,myogenic tone, and other physiological process have thecommon goal to reduce elevated wall stress caused by hyper-tension. Their ultimate goal is to bring the wall stress backto its physiological range, i.e., reduce its values towards thenormotensive situation. As the growth multiplier ϑ increases,the growth tensor Fg increases and causes the elastic tensorFe to decrease. This reduces the elastic stress Se. While themedia grows and its stresses decrease, stresses in the adven-titia increase markedly. The stress increase in the adventitiais caused by the radial expansion of the media.

section iv

section iii

section i

section ii

longitudinal sections

iiiiiiiv

Fig. 10 Growth in different longitudinal sections of the human carotidartery at t = 100 days of hypertension

transverse sections

i ii iii iv

section ivsection iiisection i section ii

Fig. 11 Growth in different transverse sections of the human carotidartery at t = 100 days of hypertension

Figures 10 and 11 display growth in four representativelongitudinal and transverse sections of the human carotidartery. The contour plots are displayed on the deformed cur-rent configuration after t = 100 days of hypertension. Simi-

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1192 Comput Mech (2014) 53:1183–1196

t = 5d t = 15d t = 40d t = 100d

evolution of growth in the common carotid artery

evolution of max principal stress in the media

evolution of max principal stress in the adventitia

Fig. 12 Spatio-temporal evolution of growth in a representative slice of the common carotid artery. Growth multiplier, top, maximum principalstresses in the adventitia, middle, and maximum principal stresses in the media, bottom at 5, 15, 40, and 100 days of hypertension

lar to the findings in Sect. 4.1, growth is heterogeneous acrossthe wall thickness, with smaller growth at the inner and largergrowth at the outer layer. Growth also displays variationsalong the direction of flow.

Last, we summarize the spatio-temporal evolution ofgrowth in three representative slices of the human carotidartery as highlighted in Fig. 6. Figures 12, 13, and 14 showthe evolution of the growth multiplier and of the maximumprincipal stresses in the adventitia and media of the common,external, and internal carotid artery. The snap shots displaythe growth and stress contours on the deformed current con-figuration at different points in time. Growth and stress dis-play similar trends in all three sections. This is in agreementwith the smooth variations in stretch between the normoten-sive and the hypertensive states. Stresses in the adventitiaincrease by 20 %, while stresses in the media decrease by25 % in the common carotid artery, by 13 % in the exter-nal carotid artery, and by 7 % in the internal carotid artery,respectively.

5 Discussion

Growth and remodeling of living systems has advanced to arapidly growing field of research within the past decade [1].Many recent studies focus on shedding light on different kine-matic formulations, alternative balance equations, appropri-ate evolution equations, and suitable mechanical stimuli [48].Here we have adapted the classical kinematic decomposi-tion of the deformation gradient into an elastic and a growthpart [43,57]. We have discussed microstructurally-motivatedevolution equations for growth [25], and systematically com-pared different mechanical stimuli for the growth process[42]. To discretize the governing equations in time and space,we have applied an implicit Euler backward finite differencescheme in time and a geometrically nonlinear finite elementscheme in space. To efficiently and robustly solve the setof governing equations, we have linearized the growth for-mulation consistently and embedded it into a local Newtoniteration at the integration point level. The algorithm con-

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t = 5d t = 15d t = 40d t = 100d

evolution of growth in the internal carotid artery

evolution of max principal stress in the media

evolution of max principal stress in the adventitia

Fig. 13 Spatio-temporal evolution of growth in a representative slice of the internal carotid artery. Growth multiplier, top, maximum principalstresses in the adventitia, middle, and maximum principal stresses in the media, bottom at 5, 15, 40, and 100 days of hypertension

verged quadratically within a few iterations both locally andglobally. The simulations of the circular arterial cross sec-tions finished within a few minutes; the simulations of thehuman carotid artery finished within a few hours.

We have shown that our model is capable of simulatinghypertensive growth in arterial tissue. In particular, we havefocused on growth-induced smooth muscle cell hypertrophy.We have assumed that when subjected to mechanical stim-uli, smooth muscle cells thicken in the radial direction, whiletheir length remains virtually unchanged [25]. After compar-ing the two most common mechanical stimuli for growth,stress and strain, we have decided to choose the local smoothmuscle cell stretch as the stimulus for smooth muscle cellthickening. The resulting mathematical model allows us toexplore how microstructural changes on the smooth musclecell level translate into a macrostructural thickening of thearterial wall. This is conceptually similar to studying howmicrostructural changes of heart muscle cells translate intoa macroscopic thickening of the ventricular wall in cardiac

hypertension [55]. While microstructural changes in cardiacmuscle cells [25] and skeletal muscle cells [70] have beenattributed to sarcomerogenesis, the creation and deposition ofnew sarcomere units within the cell, microstructural changesin smooth muscle cells are far more complex and less welldocumented.

Our simulations display an excellent qualitative and quan-titative agreement with experimental findings, both in termsof thickening and growth rates [16]. To demonstrate thepotential of the proposed approach, we have shown the finiteelement simulation of a real patient-specific carotid geome-try. Cerebral arteries are an object of intense investigation,since they are at high risk of uncontrolled growth, aneurysmformation [3,66], and rupture [69]. Our results indicate ahomogeneous growth throughout the media along the entirecarotid length. Only small portions of the carotid bifurcationdisplayed slightly elevated growth. While previous studieshave mainly modeled the growing arterial wall as a single-layer system [28,41], here, we have modeled the media

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t = 5d t = 15d t = 40d t = 100d

evolution of growth in the external carotid artery

evolution of max principal stress in the media

evolution of max principal stress in the adventitia

Fig. 14 Spatio-temporal evolution of growth in a representative slice of the external carotid artery. Growth multiplier, top, maximum principalstresses in the adventitia, middle, and maximum principal stresses in the media, bottom at 5, 15, 40, and 100 days of hypertension

and the adventitia as distinct layers with distinct mechani-cal properties [30]. This approach raises an interesting ques-tion related to the stress distribution. Stresses in the mediadecrease due to the radial growth of smooth muscle cells.This is a well-accepted mechanism in the adaptation of bio-logical tissue. Smooth muscle cells respond, both acutelyand chronically, to hypertension with the goal to bring thewall stress back to its physiological baseline state. Variousinterconnected mechanotransduction pathways in the cellsare responsible for sensing the underlying mechanical stim-uli [59,71]. In particular, smooth muscle cells are known tosense stretch in the extracellular matrix. Since smooth musclecells are mainly present in the media and not in the adven-titia, we have assumed that the medial layer grows, whilethe adventitial layer remains purely elastic. Accordingly, thestresses in the adventitia increase up to a 20–30 %, causedby combined effects of the elevated pressure and the growthof the media. These results reflect the general understandingthat the adventitia acts as a protective outer layer.

Despite these promising first results, we would liketo address a few fundamental limitations of the currentapproach. First, our current model is based on purely passivebaseline elasticity. Smooth muscle cells display an importantactive response, the myogenic tone, which allows the arte-rial wall to contract or expand acutely to maintain a base-line lumen [62]. Our arterial model would improve by theinclusion of this feature, although, up to date, only a fewcomputational models of myogenic tone are available in lit-erature [4,11]. The underlying stimuli for growth could pos-sibly include this basal tone. However, myogenic tone is aresponse to the overstretch of the smooth muscle cells, whichwe utilize as stimulus in our current model. We expect that theadditional inclusion of basal tone could quantitatively scaleour current growth response, but it would not qualitativelymodify the growth response overall. Second, our current evo-lution equation for growth pre-imposes both the growth leveland its rate. Its constitutive parameters could be calibratedexperimentally to describe specific arteries and other car-

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diovascular tissues. At this point though, they lack of a realphysiological interpretation. The challenge to tie the parame-ters to mechanotransduction pathways in arterial cells andtissue introduces an additional limitation. Experiments areusually performed in the normotensive state and in the finalhypertensive state, but not in longitudinal studies to explorethe transition between the two. Accordingly, most reportedexperiments do not allow us to calibrate our rate parameters.Moreover, experimental characterizations display huge vari-ations across different species, across different arteries fromthe same species, and even across different specimen fromthe same artery.

In summary, we have adopted a canonical framework forvolumetric growth to simulate hypertensive thickening ofthe human carotid artery. Our model makes a first attemptto link the key kinematic variable of growth, the macro-scopic growth tensor, to microstructural changes in smoothmuscle cell size. The characterization of growth in terms ofmicrostructural variables such as cell size and cell orientationsmoothly integrates the multiscale nature of the living sys-tem into the mathematical model. Hypertensive arterial wallthickening is a critical medical condition since it may lead todecreases blood flow and other related complications. Com-putational models like ours can help to understand the under-lying mechanochemical processes and provide a frameworkfor biological and clinical researchers to jointly enhance thepharmagolocial or surgical management of hypertension.

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