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Biomech Model Mechanobiol. Author manuscript; available in PMC 2013 July 1.

Published in final edited form as:

Published online 2011 November 4. doi: 10.1007/s10237-011-0357-4

PMCID: PMC3425448

NIHMSID: NIHMS339988

Alexander M. Zollner,^{1} Adrian Buganza Tepole,^{1} Arun K. Gosain,^{2} and Ellen Kuhl^{1,}^{3,}^{4}

The publisher's final edited version of this article is available at Biomech Model Mechanobiol

See other articles in PMC that cite the published article.

Tissue expansion is a common surgical procedure to grow extra skin through controlled mechanical over-stretch. It creates skin that matches the color, texture, and thickness of the surrounding tissue, while minimizing scars and risk of rejection. Despite intense research in tissue expansion and skin growth, there is a clear knowledge gap between heuristic observation and mechanistic understanding of the key phenomena that drive the growth process. Here, we show that a continuum mechanics approach, embedded in a custom-designed finite element model, informed by medical imaging, provides valuable insight into the biomechanics of skin growth. In particular, we model skin growth using the concept of an incompatible growth configuration. We characterize its evolution in time using a second-order growth tensor parameterized in terms of a scalar-valued internal variable, the in-plane area growth. When stretched beyond the physiological level, new skin is created, and the in-plane area growth increases. For the first time, we simulate tissue expansion on a patient-specific geometric model, and predict stress, strain, and area gain at three expanded locations in a pediatric skull: in the scalp, in the forehead, and in the cheek. Our results may help the surgeon to prevent tissue over-stretch and make informed decisions about expander geometry, size, placement, and inflation. We anticipate our study to open new avenues in reconstructive surgery, and enhance treatment for patients with birth defects, burn injuries, or breast tumor removal.

One percent of neonates is born with congenital melanocytic nevi, dark-colored surface lesions present at birth [10]. Congenital nevi may vary in size, shape, texture, color, hairiness, and location, but they have one thing in common: their high malignant potential [23]. Birthmarks larger than 10 cm in diameter are classified as giant congenital nevi and have a prevalence of one in 20,000 infants [45]. Because giant congenital nevi place the child at an increased risk to develop skin cancer, surgical excision remains the standard treatment option [20]. Cosmetic deformity, significant aesthetic disfigurement, and severe psychological distress are additional compelling reasons for nevus removal, especially in the craniofacial region [23].

To reconstruct the defect, preserve function, and maintain aesthetic appearance, tissue expansion has become a major treatment modality in the management of giant congenital nevi [36]. Tissue expansion was first proposed more than half a century ago to reconstruct a traumatic ear and has since then revolutionized reconstructive surgery [43]. Today it is widely used to repair birth defects [5], correct burn injuries [3], and reconstruct breasts after tumor removal [46]. Tissue expansion is the ideal strategy to grow skin that matches the color, texture, hair bearance, and thickness of the surrounding healthy skin, while minimizing scars and risk of rejection [49].

Figure 1, left, shows a one-year old boy who presented with a giant congenital nevus concerning 25 percent of his forehead, extending to the right temporal scalp and cheek [21]. To resurface the nevus region and stimulate in situ skin growth, three simultaneous forehead, cheek, and scalp expanders are used. They are implanted in subcutaneous pockets adjacent to the defect, where they are gradually filled with saline solution. The amount of filling is controlled by visual inspection of skin color and capillary refill [49]. Multiple serial inflations stretch the skin and stimulate tissue growth over a period of several weeks [63]. Once enough skin is created, the expanders are removed, the nevus is excised, and the newly grown skin flaps are advanced to close the defect zone. Figure 1, right, shows the boy at age three, after completed forehead, scalp, and cheek reconstruction.

Tissue expansion for pediatric forehead reconstruction. The patient, a
one-year old boy presented with a giant congenital nevus involving 25 percent of
his forehead, extending to the right temporal scalp and cheek. Simultaneous
forehead, cheek, and scalp **...**

Figure 2 shows a schematic sequence of the mechanical processes during tissue expansion. Initially, at biological equilibrium, the skin is in a natural state of resting tension [54]. When the expander is implanted and inflated, skin is loaded in tension. Stretch beyond a critical level triggers a series of signaling pathways eventually leading to the creation of new skin [57]. On the cellular level, mechanotransduction affects a network of several integrated cascades including growth factors, cytoskeletal rearrangement, and protein kinases [11]. On the tissue level, skin growth induces stress relaxation and restores the state of resting tension [54]. The cycle of expander inflation, stretch, growth, and relaxation is repeated multiple times, typically on a weekly basis [63]. As demonstrated by immunocytochemistry, the expanded tissue undergoes normal cell differentiation and maintains its characteristic phenotype [61]. Skin initially displays thickness changes upon expansion, however, these changes are fully reversible upon expander removal [59]. When the expander is removed, the skin retracts and reveals the irreversible nature of skin growth, associated with growth-induced residual stresses [18,41]. Figure 3 shows a commonly used tissue expander to grow skin in reconstructive surgery.

Schematic sequence of tissue expander inflation. At biological
equilibrium, the skin is in a physiological state of resting tension, unloaded
and ungrown. When an expander is implanted and inflated, the skin is stretched,
loaded and ungrown. Mechanical **...**

Tissue expander to grow skin for defect correction in reconstructive
surgery. Typical applications are birth defects, burn injuries, and breast
reconstruction. Devices consist of a silicone elastomer inflatable expander with
a reinforced base for directional **...**

To predict stress, strain, and area gain during tissue expansion in pediatric forehead reconstruction, we adopt a novel mechanistic approach [8, 9], based on the continuum framework of finite growth [50]. Originally developed for isotropic volumetric growth [14, 37], finite growth theories are based on the multiplicative decomposition of the deformation gradient into an elastic and a growth part [15, 38], a concept adopted from finite plasticity [34]. Depending on the format of their growth tensor, continuum growth theories have been refined to characterize isotropic [19, 32], transversely isotropic [48,56], orthotropic [17], or generally anisotropic growth [40, 42], either compressible [40] or incompressible [51]. Recent trends focus on the computational modeling of finite growth [22], typically by introducing the growth tensor as an internal variable within a finite element framework [16, 25], a strategy that we also adopt here. A recent monograph that compares different approaches to growth and summarizes the essential findings, trends, and open questions in this progressively evolving new field [2]. Despite ongoing research in growing biological systems, the growth of thin biological membranes remains severely understudied. Only few attempts address the growth of thin biological plates [12] and membranes [40]. Motivated by a first study on axisymmetric skin growth [55], we have recently established a prototype model for growing membranes to predict skin expansion in a general three-dimensional setting [8]. This study capitalizes on recent developments in reconstructive surgery, continuum mechanics of growing tissues, and computational modeling, supplemented by medical image analysis. It documents our first attempts to model and simulate skin expansion in pediatric forehead reconstruction using a real patient-specific geometry.

We adopt the kinematics of finite deformations and introduce the
deformation map , which, at any given time *t* maps the
material placement ** X** of a physical particle in the
material configuration to its spatial placement

$$\mathit{F}={\nabla}_{\mathit{X}}\mathit{\phi}={\mathit{F}}^{\mathrm{e}}\xb7{\mathit{F}}^{\mathrm{g}}$$

(1)

into a reversible elastic part
*F*^{e} and an irreversible growth part
*F*^{g}. This multiplicative
decomposition, reminiscent of the decomposition of the elastoplastic deformation
gradient [34], was first used to describe
growth of biologial tissues in [50].
Similarly, we can then decompose the total Jacobian

$$J=\text{det}(\mathit{F})={J}^{\mathrm{e}}{J}^{\mathrm{g}}$$

(2)

into an elastic part *J*^{e} = det
(*F*^{e}) and a growth part
*J*^{g} = det
(*F*^{g}). We idealize skin as a thin
layer characterized through the unit normal
*n*_{0} in the undeformed reference
configuration. The length of the deformed skin normal
** n** = cof(

$$\vartheta =\Vert \text{cof}(\mathit{F})\xb7{\mathit{n}}_{0}\Vert ={\vartheta}^{\mathrm{e}}{\vartheta}^{\mathrm{g}}$$

(3)

which we can again decompose into an elastic area stretch
^{e} =
‖cof(*F*^{e}) ·
*n*_{g}/‖*n*_{g}‖
‖ and a growth area stretch ^{g} =
‖cof(*F*^{g}) ·
*n*_{0}‖ [8]. Here,
*n*_{g} =
cof(*F*^{g}) ·
*n*_{0} =
*J*^{g}
*F*^{g − t} ·
*n*_{0} denotes the grown skin
normal, and cof(○) = det(○) (○)^{−t}
denotes the cofactor of the second order tensor (○). As characteristic
deformation measures, we introduce the right Cauchy Green tensor
** C** in the undeformed reference
configuration

$$\mathit{C}={\mathit{F}}^{\mathrm{t}}\xb7\mathit{F}={\mathit{F}}^{\text{gt}}\xb7{\mathit{F}}^{\text{et}}\xb7{\mathit{F}}^{\mathrm{e}}\xb7{\mathit{F}}^{\mathrm{g}}$$

(4)

and its elastic counterpart
*C*^{e} =
*F*^{et} ·
*F*^{e} =
*F*^{g − t} ·
** C** ·

$${\mathit{F}}^{\mathrm{e}-1}\xb7\mathit{l}\xb7{\mathit{F}}^{\mathrm{e}}={\mathit{L}}^{\mathrm{e}}+{\mathit{L}}^{\mathrm{g}}$$

(5)

which obeys the additive decomposition into the elastic velocity
gradient *L*^{e} =
*F*^{e−1} ·
^{e} and the growth
velocity gradient *L*^{g} =
^{g} ·
*F*^{g−1}. Here,
{○̇} =
_{t}{○}|_{X}
denotes the material time derivative of any field {○}
(** X**,

We characterize growing tissue using the framework of open system
thermodynamics in which the material density ρ_{0} is allowed to
change as a consequence of growth [26,
28]. The balance of mass for open
systems balances its rate of change _{0} with a
possible in- or outflux of mass ** R** and mass source

$${\dot{\rho}}_{0}=\text{Div}(\mathit{R})+{\mathcal{R}}_{0}$$

(6)

Similarly, the balance of linear momentum balances the density-weighted
rate of change of the velocity ρ_{0} υ =
ρ_{0} , with the momentum flux
** P** =

$${\rho}_{0}\dot{\mathit{\upsilon}}=\text{Div}(\mathit{F}\xb7\mathit{S})+{\rho}_{0}\mathit{b}$$

(7)

here stated in its mass-specific form [27]. ** P** and

$${\rho}_{0}\mathcal{D}=\mathit{S}:\frac{1}{2}\dot{\mathit{C}}-{\rho}_{0}\dot{\psi}-{\rho}_{0}\mathcal{S}\ge 0$$

(8)

typically contains an extra entropy source ρ_{0}
to account for the growing nature of living biological systems [26, 41]. Equations (7)
and (8) represent the
mass-specific versions of the balance of momentum and of the dissipation
inequality which are particularly useful in the context of growth since they
contains no explicit dependencies on the changes in mass [27].

To close the set of equations, we introduce the constitutive equations
for the mass source _{0}, for the momentum flux
** S**, and for the growth tensor

$${\mathcal{R}}_{0}={\rho}_{0}\text{tr}({\mathit{L}}^{\mathrm{g}})$$

(9)

can be expressed as the density-weighted trace of the growth
velocity gradient tr (*L*^{g}) =
^{g} :
*F*^{g − t} [22]. We model skin as a hyperelastic
material characterized through the Helmholtz free energy ψ =
(** C**,

$${\rho}_{0}\mathcal{D}=[\mathit{S}-{\rho}_{0}\frac{\partial \psi}{\partial \mathit{C}}]:\frac{1}{2}\dot{\mathit{C}}+{\mathit{M}}^{\mathrm{e}}:{\mathit{L}}^{\mathrm{g}}-{\rho}_{0}\frac{\partial \psi}{\partial {\rho}_{0}}{\mathcal{R}}_{0}-{\rho}_{0}{\mathcal{S}}_{0}\ge 0$$

(10)

We observe that the Mandel stress of the intermediate configuration
*M*^{e} =
*C*^{e} ·
*S*^{e} is energetically conjugate
to the growth velocity gradient *L*^{g} =
^{g} ·
*F*^{g − 1}. From the
dissipation inequality (10), we
obtain the definition of the second Piola Kirchhoff stress
** S** as thermodynamically conjugate quantity
to the right Cauchy Green deformation tensor

$$\mathit{S}=2\phantom{\rule{thinmathspace}{0ex}}{\rho}_{0}\frac{\partial \psi}{\partial \mathit{C}}=2\frac{\partial \psi}{\partial {\mathit{C}}^{\mathrm{e}}}:\frac{\partial {\mathit{C}}^{\mathrm{e}}}{\partial \mathit{C}}={\mathit{F}}^{\mathrm{g}-1}\xb7{\mathit{S}}^{\mathrm{e}}\xb7{\mathit{F}}^{\mathrm{g}-\mathrm{t}}$$

(11)

According to this definition, the first derivative of the Helmholtz free
energy ψ with respect to the elastic right Cauchy Green tensor
*C*^{e} introduces the elastic
second Piola Kirchhoff stress *S*^{e}, while
the second derivative defines the elastic constitutive moduli
𝗟^{e}.

$${\mathit{S}}^{\mathrm{e}}=2\phantom{\rule{thinmathspace}{0ex}}{\rho}_{0}\frac{\partial \psi}{\partial {\mathit{C}}^{\mathrm{e}}}\text{and}{\U0001d5df}^{\mathrm{e}}=2\frac{\partial {\mathit{S}}^{\mathrm{e}}}{\partial {\mathit{C}}^{\mathrm{e}}}=4\phantom{\rule{thinmathspace}{0ex}}{\rho}_{0}\frac{{\partial}^{2}\psi}{\partial {\mathit{C}}^{\mathrm{e}}\otimes \partial {\mathit{C}}^{\mathrm{e}}}$$

(12)

To focus on the impact of growth, rather than adopting a sophisticated
anisotropic material model for skin [9,
30], we assume a classical
Neon-Hookean free energy ${\rho}_{0}\psi =\frac{1}{2}\lambda \phantom{\rule{thinmathspace}{0ex}}{\text{ln}}^{2}({J}^{\mathrm{e}})+\frac{1}{2}\mu \phantom{\rule{thinmathspace}{0ex}}[{\mathit{C}}^{\mathrm{e}}:\mathit{I}-3-2\text{ln}({J}^{\mathrm{e}})]$, introducing the elastic second Piola Kirchhoff stress
*S*^{e} = [λ
ln(*J*^{e}) −
μ]*C*^{e − 1} +
μ ** I**, and the elastic constitutive moduli ${\U0001d5df}^{\mathrm{e}}=\lambda {\mathbf{C}}^{\mathrm{e}-1}\otimes {\mathbf{C}}^{\mathrm{e}-1}+[\mu -\lambda \mathrm{ln}({J}^{\mathrm{e}})][{\mathbf{C}}^{\mathrm{e}}\overline{\otimes}{\mathbf{C}}^{\mathrm{e}}+{\mathbf{C}}^{\mathrm{e}}\underset{\xaf}{\otimes}{\mathbf{C}}^{\mathrm{e}}]$ . Motivated by clinical observations [49], we classify skin growth as a
strain-driven, transversely isotropic, irreversible process. It is characterized
through one single growth multiplier

$${\mathit{F}}^{\mathrm{g}}=\sqrt{{\vartheta}^{\mathrm{g}}}\phantom{\rule{thinmathspace}{0ex}}\mathit{I}+[1-\sqrt{{\vartheta}^{\mathrm{g}}}]\phantom{\rule{thinmathspace}{0ex}}{\mathit{n}}_{0}\otimes {\mathit{n}}_{0}$$

(13)

For this particular type of transversely isotropic growth, for which all
thickness changes are reversibly elastic [59], area growth is identical to volume growth, i.e.,
^{g} = det(*F*^{g}) =
*J*^{g}. Because of the simple rank-one update
structure in (13), we can invert
the growth tensor explicitly, ${\mathit{F}}^{\mathrm{g}-1}=1/\sqrt{{\vartheta}^{\mathrm{g}}}\mathit{I}+[1-1/\sqrt{{\vartheta}^{\mathrm{g}}}]\phantom{\rule{thinmathspace}{0ex}}{\mathit{n}}_{0}\otimes {\mathit{n}}_{0},$, using the Sherman-Morrison formula. This explicit
representation introduces the following simple expression for the growth
velocity gradient,

$${\mathit{L}}^{\mathrm{g}}=\sqrt{{\dot{\vartheta}}^{\mathrm{g}}}/\sqrt{{\vartheta}^{\mathrm{g}}}\mathit{I}+[1-\sqrt{{\dot{\vartheta}}^{\mathrm{g}}}/\sqrt{{\vartheta}^{\mathrm{g}}}]\phantom{\rule{thinmathspace}{0ex}}{\mathit{n}}_{0}\otimes {\mathit{n}}_{0}$$

(14)

which proves convenient to explicitly evaluate the mass source in equation (9) as ${\mathcal{R}}_{0}={\rho}_{0}[1+2\sqrt{{\dot{\vartheta}}^{\mathrm{g}}}/\sqrt{{\vartheta}^{\mathrm{g}}}]$. Motivated by physiological observations of stretch-induced skin expansion [21], we adopt the following evolution equation for the growth multiplier,

$${\dot{\vartheta}}^{\mathrm{g}}={k}^{\mathrm{g}}({\vartheta}^{\mathrm{g}}){\varphi}^{\mathrm{g}}({\vartheta}^{\mathrm{e}})$$

(15)

which follows a well-established functional form [37], but is now rephrased in a strain-driven format [17]. To control unbounded growth, we introduce the weighting function

$${k}^{\mathrm{g}}=\frac{1}{\tau}\phantom{\rule{thinmathspace}{0ex}}{\left[\frac{{\vartheta}^{\text{max}}-{\vartheta}^{\mathrm{g}}}{{\vartheta}^{\text{max}}-1}\right]}^{\gamma}$$

(16)

where 1/τ controls the adaptation speed, the exponent
γ calibrates the shape of the growth curve, and ^{max}
> 1 is the maximum area growth [22,37]. The growth criterion

$${\varphi}^{\mathrm{g}}=\langle {\vartheta}^{\mathrm{e}}-{\vartheta}^{\text{crit}}\rangle =\langle \vartheta /{\vartheta}^{\mathrm{g}}-{\vartheta}^{\text{crit}}\rangle $$

(17)

is driven by the elastic area stretch ^{e} =
/^{g}, such that growth is activated only if the
elastic area stretch exceeds a critical physiological stretch limit
^{crit}. Here, ○ denote the
Macaulay brackets.

Figure 4 displays the constitutive
response of the four-parameter growth model in equi-biaxial stretch. At a
prescribed piecewise constant total stretch , the growth stretch
^{g} increases gradually while the elastic stretch
^{e} decreases. This induces stress relaxation. Horizontal
dashed lines represent the elastic stretch limit beyond which skin growth is
activated ^{crit} and the maximum area growth
^{max}. Increased adaptation speeds 1/τ ↑
and decreased growth exponents γ ↓ both accelerate convergence
towards the biological equilibrium [22],
but do not affect the final equilibrium state [48, 51]. At all times, the
multiplicative decomposition of the deformation gradient
** F** =

We solve the coupled biological and mechanical equilibrium for skin
growth within an incremental iterative finite element setting [58]. To characterize the growth process at
each instant in time, we introduce the growth multiplier ^{g}
as an internal variable, and solve the biological equilibrium (15) locally at the integration
point level. For the temporal discretization, we partition the time interval of
interest into n^{stp} subintervals, $\mathcal{T}={\U0001d5e8}_{\mathrm{n}=1}^{{\mathrm{n}}_{\text{stp}}}[{t}_{\mathrm{n}},{t}_{\mathrm{n}+1}]$ and focus on the interval [*t*^{n},
*t*_{n+1}] for which Δ*t* =
*t*_{n+1} − *t*_{n}
> 0 denotes the current time increment. Our goal is to determine the
current growth multiplier ^{g} for a given deformation state
** F** at time

$${\U0001d5b1}^{\vartheta}={\vartheta}^{\mathrm{g}}-{\vartheta}_{\mathrm{n}}^{\mathrm{g}}-{k}^{\mathrm{g}}{\varphi}^{\mathrm{g}}\Delta t\doteq 0$$

(18)

We solve this nonlinear residual equation using a local Newton
iteration. Within each iteration step, we calculate the linearization of the
residual 𝖱^{} with respect to the growth multiplier
^{g},

$${\U0001d5aa}^{\vartheta}=\frac{\partial {\U0001d5b1}^{\vartheta}}{\partial {\vartheta}^{\mathrm{g}}}=1-[\frac{\partial {k}^{\mathrm{g}}}{\partial {\vartheta}^{\mathrm{g}}}{\varphi}^{\mathrm{g}}+{k}^{\mathrm{g}}\frac{\partial {\varphi}^{\vartheta}}{\partial {\vartheta}^{\mathrm{g}}}]\phantom{\rule{thinmathspace}{0ex}}\Delta t$$

(19)

with the derivatives of the weighting function
*k*^{g}/^{g} =
− γ *k* / [^{max} −
^{g}] and the growth criterion
ϕ^{g}/^{g} =
− / ^{g 2} introduced in equations (16) and (17). Within each iteration step,
we iteratively update the unknown growth multiplier ^{g}
← ^{g} − 𝖱^{} /
𝗞^{} until convergence is achieved, i.e., until the
local growth update Δ^{g} =
−𝖱^{} / 𝗞^{}
reaches a user-defined tolerance.

To explore the interplay between growth and mechanics, we discretize the
deformation map as nodal degree of freedom, and solve the mechanical
equilibrium (7) globally at the
node point level. To solve the quasistatic mechanical equilibrium, Div
(** F** ·

$${\U0001d5e5}_{I}^{\phi}=\underset{\mathrm{e}=1}{\overset{{\mathrm{n}}_{\text{el}}}{\U0001d5d4}}{\displaystyle {\int}_{{\mathcal{B}}_{\mathrm{e}}}}\nabla {N}_{\phi}^{i}\xb7[\mathit{F}\xb7\mathit{S}]\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}{V}_{\mathrm{e}}\doteq \mathbf{0}$$

(20)

Here, the operator 𝗔 symbolizes the assembly of all element
residuals at the *j* = 1, ‥, n_{en} element nodes
to the global residual at the global node points *J* = 1,
‥, n_{el}. We evaluate the global discrete residual (20), once we have iteratively
determined the growth multiplier ^{g} for the given deformation
state ** F** and the given history ${\vartheta}_{\mathrm{n}}^{\mathrm{g}}$ as described in the previous section. Then we successively
determine the growth tensor

$${\U0001d5aa}_{\mathit{\text{IJ}}}^{\phi}=\frac{\partial {\U0001d5e5}_{I}^{\phi}}{\partial {\mathit{\phi}}_{J}}=\underset{\mathrm{e}=1}{\overset{{\mathrm{n}}_{\text{el}}}{\U0001d5d4}}{\displaystyle {\int}_{{\mathcal{B}}_{\mathrm{e}}}}\nabla {N}_{\phi}^{i}\xb7\mathit{S}\xb7\nabla {N}_{\phi}^{j}\phantom{\rule{thinmathspace}{0ex}}\mathit{I}+{[\nabla {N}_{\phi}^{i}\xb7\mathit{F}]}^{\text{sym}}\xb7\U0001d5df\xb7{[{\mathit{F}}^{\mathrm{t}}\xb7\nabla {N}_{\phi}^{j}]}^{\text{sym}}\mathrm{d}{V}_{\mathrm{e}}$$

(21)

The fourth order tensor 𝖱 denotes the Lagrangian constitutive
moduli which, we can determine directly from the linearization of the Piola
Kirchhoff stress ** S** with respect to the total right
Cauchy Green tensor

$$\U0001d5df=2\frac{\mathrm{d}\mathit{S}}{\mathrm{d}\mathit{C}}=2\frac{\partial \mathit{S}}{\partial \mathit{C}}{|}_{{\mathit{F}}^{\mathrm{g}}}+2\phantom{\rule{thinmathspace}{0ex}}[\frac{\partial \mathit{S}}{\partial {\mathit{F}}^{\mathrm{g}}}:\frac{\partial {\mathit{F}}^{\mathrm{g}}}{\partial {\vartheta}^{\mathrm{g}}}]\otimes \frac{\partial {\vartheta}^{\mathrm{g}}}{\partial \mathit{C}}{|}_{\mathit{F}}$$

(22)

The first term

$$2\frac{\partial \mathit{S}}{\partial \mathit{C}}=[{\mathit{F}}^{\mathrm{g}-1}\overline{\otimes}{\mathit{F}}^{\mathrm{g}-1}]:{\U0001d5df}^{\mathrm{e}}:[{\mathit{F}}^{\mathrm{g}-\mathrm{t}}\overline{\otimes}{\mathit{F}}^{\mathrm{g}-\mathrm{t}}]$$

(23)

represents the pull back of the elastic moduli
𝖱^{e} introduced in equation (12) onto the reference configuration. Here we have used
the abbreviations
{•○}_{ijkl} =
{•}_{ik}
{○}_{jl} and
{•⊗̲○}_{ijkl} =
{•}_{il}
{○}_{jk} for the non-standard fourth
order products. The second term

$$\frac{\partial \mathit{S}}{\partial {\mathit{F}}^{\mathrm{g}}}=-[{\mathit{F}}^{\mathrm{g}-1}\overline{\otimes}\mathit{S}+\mathit{S}\underset{\xaf}{\otimes}{\mathit{F}}^{\mathrm{g}-1}]-[{\mathit{F}}^{\mathrm{g}-1}\overline{\otimes}{\mathit{F}}^{\mathrm{g}-1}]:\frac{1}{2}{\U0001d5df}^{\mathrm{e}}:[{\mathit{F}}^{\mathrm{g}-\mathrm{t}}\underset{\xaf}{\otimes}{\mathit{C}}^{\mathrm{e}}+{\mathit{C}}^{\mathrm{e}}\overline{\otimes}{\mathit{F}}^{\mathrm{g}-\mathrm{t}}]$$

(24)

consists of two terms that resemble a geometric and a material stiffness in nonlinear continuum mechanics. The third term

$$\frac{\partial \mathit{F}}{\partial {\vartheta}^{\mathrm{g}}}=\frac{1}{2\sqrt{{\vartheta}^{\mathrm{g}}}}\phantom{\rule{thinmathspace}{0ex}}[\mathit{I}-{\mathit{n}}_{0}\otimes {\mathit{n}}_{0}]$$

(25)

and the fourth term

$$\frac{\partial {\vartheta}^{\mathrm{g}}}{\partial \mathit{C}}=\frac{1}{\tau}\frac{1}{{\vartheta}^{\mathrm{g}}}\phantom{\rule{thinmathspace}{0ex}}{\left[\frac{{\vartheta}^{\text{max}}-{\vartheta}^{\mathrm{g}}}{{\vartheta}^{\text{max}}-1}\right]}^{\gamma}\frac{1}{{\U0001d5aa}^{\mathrm{g}}}\Delta t\frac{1}{2}\vartheta {\mathit{C}}^{-1}\frac{1}{2}\frac{{J}^{2}}{\vartheta}\phantom{\rule{thinmathspace}{0ex}}[{\mathit{C}}^{-1}\xb7{\mathit{n}}_{0}]\otimes [{\mathit{C}}^{-1}\xb7{\mathit{n}}_{0}]$$

(26)

depend on the particular choice for the growth tensor
*F*^{g} in equation (13) and on the evolution
equation for the growth multiplier ^{g} in equation (15), respectively. For
each global Newton iteration step, we iteratively update the current deformation
state $\phi \leftarrow \phi -{\U0001d5de}_{\mathit{\text{IJ}}}^{\phi -1}\xb7{\U0001d5e5}_{I}^{\phi}$ until we achieve algorithmic convergence. Upon convergence, we
store the corresponding growth multipliers ^{g} at the
integration point level. Table 1
summarizes the algorithmic treatment of skin growth at the integration point
level.

To simulate skin growth on an anatomically exact geometry, we create a finite element mesh on the basis of three-dimensional computer tomography images shown in Figure 5. Figure 6 summarizes the sequence of steps to generate our patient-specific geometric model. First, we identify the skin region by a distinct grey scale value in the computer tomography scans and extract point cloud data of its boundary. Figure 6, left, mimics the discrete nature of the extracted point cloud, with high point densities in the scanning plane and low point densities between the distinct planes. To smoothen the data and decreases the overall number of points, we homogenize the point cloud using a median filter. Next, we create a triangular surface mesh from the smoothened point cloud by applying a ball-pivoting algorithm [7]. Ball-pivoting algorithms are particularly suited for surface reconstruction of large data sets. After placing an initial seed element, the ball-pivoting algorithm rotates a sphere over the edges of this element and sequentially creates new elements whenever the sphere touches three data points. However, since our data are based on plane-wise computer tomography scans, ball-pivoting algorithms typically fail to automatically create smooth surfaces. Unfortunately, other fully automated meshing strategies such as convex hull or shrink wrap algorithms are not suitable for non-convex geometries like the face, which possesses several non-convexities in the eye, nose, mouth, and ear regions [24]. Accordingly, we smoothen the triangular surface mesh semi-manually, as illustrated in Figure 6, middle.

Three-dimensional computer tomography scans from the skull of a one-year
old child. We create a patient-specific geometric model using discrete boundary
points extracted from sliced image sections across the skull.

Mesh generation from clinical images. From the computer tomography
scans, we extract discrete point cloud data (left), which we filter and mesh
using a ball-pivoting algorithm. This generates a triangular surface mesh, which
is further smoothened (middle) **...**

From the smoothened surface mesh, we finally create a one-element thick
volume mesh of the pediatric skull, discretized with 61,228 nodes, 183,684
degrees of freedom, and 30,889 tri-linear brick elements. As a first
approximation, we assume that all eight integration points within each element
posses the same skin plane normal *n*_{0},
corresponding to the normal from the initial surface mesh. We virtually implant
three tissue expanders as shown in Figure
6, right, motivated by the tissue expansion case illustrated in Figure 1. First, we implant an expander in
the scalp, discretized with 4,356 nodes, 13,068 degrees of freedom, and 2,088
tri-linear brick elements, covering an initial area of 50.4 cm^{2},
shown in red. Second, we implant an expander in the cheek, discretized with
2,542 nodes, 7,626 degrees of freedom, and 1,200 trilinear brick elements,
covering an initial area of 29.3 cm^{2}, shown in yellow. Third, we
implant and expander in the forehead, discretized with 3,782 nodes, 11,346
degrees of freedom, and 1,800 tri-linear brick elements, covering an initial
area of 48.5 cm^{2}, shown in blue. To simulate tissue expansion, we fix
all nodes and release only the expander degrees of freedom, which we then
pressurize from underneath.

We illustrate the impact of tissue expansion at three characteristic
locations of the skull, in the scalp, the forehead, and the cheek. For the elastic
model, we assume Lamé constants of λ = 0.7141 MPa and μ =
0.1785 MPa, which would correspond to a Poisson’s ratio of ν = 0.4
and a Young’s modulus of E = 0.5 MPa in the linear regime [1, 52].
For the growth model, we assume a critical threshold of ^{crit} =
1.1, a maximum area growth of ^{max} = 4.0, a growth exponent of
γ = 3.0, and an adaptation speed of 1/τ = 12. We gradually
pressurize the tissue expanders, 0.0 < *t* ≤ 0.125,
then hold the pressure constant to allow the tissue to grow, 0.125 <
*t* ≤ 0.75, and finally remove the pressure to visualize
the grown area, 0.75 < *t* ≤ 1.0.

Figures 7, ,8,8, and and99 illustrate the
tissue expansion process in the scalp. Figure
7 displays the temporal evolution of the normalized total area,
elastic area, and growth area upon subsequent expander inflation, constant
pressure, and expander removal. Once the elastic area stretch reaches the
critical threshold of ^{crit} = 1.1, slightly before the total
pressure is applied, at *t* = 0.125, the tissue starts to grow.
As the expander pressure is held constant, growth increases gradually causing
the total area to increase as well. Then, at *t* = 0.75, the
pressure is decreased to remove the expander. The elastic area retracts
gradually, while the grown area remains constant. The vertical dashed lines
correspond to the discrete time points, *t* = 0.225,
*t* = 0.300, *t* = 0.375 and
*t* = 0.750, displayed in Figure 8. Figure 8 illustrates
the spatio-temporal evolution of area growth ^{g}. Growth is
first initiated at the center of the expander, where the elastic stretch is
largest. As growth spreads throughout the entire expanded area, the initial area
of 50.4 cm^{2} increases gradually as the grown skin area increases from
70.07 cm^{2}, to 84.25 cm^{2}, to 95.73 cm^{2}, and
finally to 121.87 cm^{2}, from left to right. Figure 9 summarizes the final outcome of the expansion in
the scalp in terms of the remaining deformation upon expander removal. The
elastic area strain of 0.95 ≤ ^{e} ≤ 1.05
indicates an area change of ±5% giving rise to residual
stresses, left. The area growth of 1.0 ≤ ^{g} ≤
3.5 shows that skin has more than doubled its initial area, right. This is in
agreement with the final fractional area gain of 2.44, corresponding to an area
growth in the scalp of 122.8 cm^{2}.

Tissue expansion in the scalp. Temporal evolution of normalized total
area, elastic area, and growth area upon gradual expander inflation, 0.0
< *t* ≤ 0.125, constant pressure 0.125 <
*t* ≤ 0.75, and deflation 0.75 <
**...**

Tissue expansion in the scalp. Spatio-temporal evolution of area growth
displayed at *t* = 0.225, *t* = 0.300,
*t* = 0.375 and *t* = 0.750. The initial area
of 50.4 cm^{2} increases gradually as the grown skin area increases from
70.07 cm^{2}, to 84.25 cm^{2}, to 95.73 **...**

Figures 10, ,11,11, and and1212 summarize the tissue expansion process in the forehead. Figure 10 displays the temporal evolution of the normalized total area, elastic area, and growth area upon gradual expander inflation, constant pressure, and gradual expander removal.

Tissue expansion in the forehead. Temporal evolution of normalized total
area, elastic area, and growth area upon gradual expander inflation, 0.0
< *t* ≤ 0.125, constant pressure 0.125 <
*t* ≤ 0.75, and deflation 0.75 <
**...**

Tissue expansion in the forehead. Spatio-temporal evolution of area
growth displayed at *t* = 0.225, *t* = 0.300,
*t* = 0.375 and *t* = 0.750. The initial area
of 48.5 cm^{2} increases gradually as the grown skin area increases from
66.56 cm^{2}, to 76.54 cm^{2}, to 85.96 **...**

Tissue expansion in the forehead. Remaining deformation upon expander
removal. The elastic area strain of 0.95 ≤ ^{e}
≤ 1.05 indicates an area change of ±5% giving rise to
residual stresses (left). The area growth of 1.0 **...**

Similar to the expansion in the scalp, growth begins at stretches beyond
the critical threshold level, then increases gradually upon constant pressure,
and remains constant upon expander removal. Figure 11 illustrates the spatio-temporal evolution of area growth
^{g} at four characteristic time points indicated through
the vertical dashed lines in figure 10.
The growth process starts in the center of the forehead and spreads out
throughout the entire forehead area. As it does, the initial area of 48.5
cm^{2} increases gradually as the grown skin area increases from
66.56 cm^{2}, to 76.54 cm^{2}, to 85.96 cm^{2}, and
finally to 116.55 cm^{2}, from left to right. Figure 12 displays the remaining deformation upon expander
removal. The final fractional area gain during forehead expansion is 2.44,
corresponding to an area growth of 118.1 cm^{2}.

Figures 13, ,14,14, and and1515 document
the tissue expansion process in the cheek. Figure
13 summarizes the temporal evolution of the normalized total area,
elastic area, and growth area upon gradual expander inflation, constant
pressure, and gradual expander removal. Again, the growth process is initiated
once the stretches reach the critical threshold of ^{crit} =
1.1. Upon constant pressure, growth increases gradually. Upon pressure removal,
growth remains constant displaying the irreversible nature of the growth
process. Figure 14 illustrates the
spatio-temporal evolution of area growth ^{g} in the cheek.
Again, growth begins in center of cheek, where the elastic area stretch is
largest. As the growth process spreads out throughout the entire cheek area, the
initial area of 29.3 cm^{2} increases gradually as the grown skin area
increases from 42.74 cm^{2}, to 52.03 cm^{2}, to 59.39
cm^{2}, and finally to 76.86 cm^{2}, from left to right.
Figure 15 summarizes the outcome of
the expansion in the cheek with a final fractional area gain of 2.64,
corresponding to an area growth of 77.4 cm^{2}.

Tissue expansion in the cheek. Temporal evolution of normalized total
area, elastic area, and growth area upon gradual expander inflation, 0.0
< *t* ≤ 0.125, constant pressure 0.125 <
*t* ≤ 0.75, and deflation 0.75 <
**...**

Tissue expansion in the cheek. Spatio-temporal evolution of area growth
displayed at *t* = 0.225, *t* = 0.300,
*t* = 0.375 and *t* = 0.750. The initial area
of 29.3 cm^{2} increases gradually as the grown skin area increases from
42.74 cm^{2}, to 52.03 cm^{2}, to 59.39 **...**

Tissue expansion is one of the basic treatment modalities in modern reconstructive surgery. Inducing controlled tissue growth through well-defined overstretch, it creates skin that matches the color, texture, hair bearance, and thickness of the surrounding healthy skin, while minimizing scars and risk of rejection [21]. Despite its wide-spread use, the choice of the appropriate tissue expander is almost exclusively based on the surgeon’s personal preference, and the discrepancy between recommended shapes, sizes, and volumes remains enormous [36]. The current gold standard for expander selection is to predict tissue growth by calculating the difference between the inflated and non-inflated expander surface [13, 53]. From an engineering point of view, it is quite intuitive, that this purely kinematic approach severely overestimates the net gain in surface area [60]. With a discrepancy of up to a factor four, these models assume that the entire deformation can be attributed to irreversible growth, completely neglecting the elastic deformation, which is reversible upon expander removal [36]. In an attempt to account for this error, empirical correction factors of 6.00, 3.75, and 4.50 have been proposed for circular, rectangular, and crescent-shaped expanders [60]. This demonstrates the vital need to rationalize criteria for a standardized device selection.

Motivated by a first study on axisymmetric skin growth [55], we have recently established a prototype model for growing membranes to simulate tissue expansion in a general three-dimensional setting [8]. We have applied our model to quantitatively compare four commonly available tissue expander geometries, round, square, rectangular, and crescent [9], however, only on initially flat geometries. Here, for the first time, we demostrate the potential of the model during tissue expansion in pediatric forehead reconstruction using a real patient-specific model. To embed the solution into a nonlinear finite element environment, we discretize the governing equations for in-plane area growth in time and space. To solve the nonlinear set of equations, we apply an incremental iterative Newton-Raphson solution strategy based on the consistent algorithmic linearization. The resulting algorithm is remarkably efficient, stable, and robust. It is capable of predicting tissue expander inflation, tissue growth, and expander deflation at different locations of a human skull within the order of minutes on a standard laptop computer. Because of its geometric flexibility, our general algorithm could also be adapted to predict tissue expansion in the trunk [4] or in the upper and lower extremities [20].

Although the proposed model for skin growth represents a significant
advancement over the axisymmetric growth model previously proposed [55], we would like to point out that some
limitations remain. First, motivated by experimental observations, which report
normal cell differentiation upon tissue expansion [61], we have assumed that the material microstructure remains unaffected
by the growth process, ${\mathit{F}}^{g}=\sqrt{{\vartheta}^{g}}\mathit{I}+[1-\sqrt{{\vartheta}^{g}}]{\mathit{n}}_{0}\otimes {\mathit{n}}_{0}$. Here, for the sake of simplicity, we have modeled this
microstructure as isotropic and elastic. We have recently shown that it is
straightforward combine our growth model with in-plane anisotropy, introduced
through pronounced stiffness along Langer’s lines [9, 30]. It might also be
interesting to elaborate out-of-plane anisotropy and model the different skin layers
individually [39]. We have demonstrated how
to model the growth process itself as anisotropic as well [16]. This could imply growth ^{‖}
exclusively along specific microstructural directions such as Langer’s
lines, ${\mathit{F}}^{\mathrm{g}}=\mathit{I}+[{\vartheta}^{\Vert}-1]{\nu}_{0}^{\Vert}\otimes {\nu}_{0}^{\Vert}$, or major in-plane growth ^{‖} along
Langer’s lines combined with minor in-plane growth
^{} orthogonal to Langer’s lines, ${\mathit{F}}^{\mathrm{g}}={\vartheta}^{\Vert}{\mathit{\nu}}_{0}^{\Vert}\otimes {\mathit{\nu}}_{0}^{\Vert}+{\vartheta}^{\perp}{\mathit{\nu}}_{0}^{\perp}\otimes {\mathit{\nu}}_{0}^{\perp}+{\mathit{n}}_{0}\otimes {\mathit{n}}_{0}$. Similarly, we could even introduce a progressive reorientation of
the collagen network to allow for the material to align with the maximum principal
strains [33,42]. Ideally, the growth law would be tied to the underlying
mechanobiology [11]. Comparative tissue
histology of grown and ungrown tissue samples could help to identify the mechanisms
that trigger skin growth to validate or, if necessary, refine our evolution equation (13) for the growth
tensor.

Second, for the sake of simplicity, our finite element mesh consists of one single brick element with two integration points across the skin thickness. We have previously studied the sensitivity of growth with respect to thickness refinement using a higher resolution across the thickness [8, 9]. However, our results were rather insensitive to mesh refinement. This insensitivity might be explained by the fact that, upon expander inflation, the skin is almost in a pure membrane state. During deflation, however, we observe buckling associated with strain gradients across the skin thickness, which might play a critical role in the development of residual stresses. To explore these residual stresses further, we are currently refining our model utilizing a shell kinematics [47] with a higher resolution across the thickness direction. This will also allow us to simulate the individual skin layers [35, 54] and their interaction during the expansion process, which we believe to be a major source of residual stress in real tissue expansion cases [41, 62].

Third, for the sake of simplicity, we have modeled tissue expansion only implicitly through controlling the applied pressure. In real tissue expansion, the external control parameter is the expander volume [36]. This implies that our virtual tissue expansion displays creep under constant loading, while clinical tissue expansion might rather display relaxation under constant deformation [9], similar to our parameter study in Figure 4.

Fourth, here, we have assumed that the expander is connected tightly to the expanded tissue, neglecting effects of interface sliding and shear [55]. This seems to be a reasonable first assumption though, since most current expanders have well-designed textures to promote mild tissue in-growth, primarily to prevent expander migration [6]. To address these potential limitations, we are currently refining the elastic model, the growth model, and the boundary conditions, to render our future simulations more realistic.

Last, while our computational model seems well suited to provide qualitative guidelines and trends, at its present state, it is not recommended for quantitative statements. We will need to perform acute and chronic in vitro and in vivo experiments to truly calibrate the underlying material parameters, to potentially refine and fully validate our model, to eventually make it applicable for clinical practice. Nevertheless, we believe that using the equations on nonlinear continuum mechanics represents a significant advancement over the current gold standard to predict tissue growth exclusively in terms of kinematic quantities [53, 60].

We have presented a novel computational model to predict the chronic adaptation of thin biological membranes when stretched beyond their physiological limit. Here, to illustrate the features of this model, we have demonstrated its performance during tissue expansion in pediatric forehead reconstruction. We have quantified reversibly elastic and irreversibly grown area changes in response to skin expansion in the scalp, the forehead, and the cheek of a one-year-old child. In general, our generic computational model is applicable to arbitrary skin geometries, and has the potential to predict area gain in skin expansion during various common procedures in reconstructive surgery. A comprehensive understanding of the gradually evolving stress and strain fields in growing skin may help the surgeon to prevent tissue damage and optimize clinical process parameters such as expander geometry, expander size, expander placement, and inflation timing. Ultimately, through inverse modeling, computational tools like ours have the potential to rationalize these parameters to create skin flaps of desired size and shape. Overall, we believe that predictive computational modeling might open new avenues in reconstructive surgery and enhance treatment for patients with birth defects, burn injuries, or breast tumor removal.

This work was supported by the Claudio X. Gonzalez Fellowship CVU 358668 and the Stanford Graduate Fellowship to Adrián Buganza Tepole and by the National Science Foundation CAREER award CMMI-0952021 and the National Institutes of Health Grant U54 GM072970 to Ellen Kuhl.

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