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,

. 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,

, or major in-plane growth

^{‖} along
Langer’s lines combined with minor in-plane growth

^{} orthogonal to Langer’s lines,

. 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 .

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].