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1 serdar göktepe *, jonathan wong **, ellen kuhl ** * department of civil engineering, middle east technical university, ankara, turkey ** departments of mechanical engineering, bioengineering, and cardiothoracic surgery, stanford university http://www.mursi.org growing skin a computational model of skin growth in reconstructive surgery

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2 serdar göktepe *, jonathan wong **, ellen kuhl ** * department of civil engineering, middle east technical university, ankara, turkey ** departments of mechanical engineering, bioengineering, and cardiothoracic surgery, stanford university http://www.mursi.org growing skin a computational model of skin growth in reconstructive surgery adrian buganza tepole chris ploch jonathan wong arun gosain ellen kuhl growing skin modeling skin simulating skin stretching skin

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3 growing skin

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4 resurfacing of large congenital defects. the patient, a one-year old girl, presented with a giant congenital nevus. three forehead and scalp expanders were implanted simultaneously for in situ forehead flap growth. the follow-up photograph shows the girl at age three the initial defect was excised and resurfaced with expanded forehead and scalp flaps. skin expansion in forehead reconstruction

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5 growing skin resurfacing of large congenital defects. the patient, a one-year boy, presented with a giant congenital nevus. simultaneous forehead, cheek, and scalp expanders were implanted for in situ skin growth. this technique allows to resurface large anatomical areas with skin of similar color, quality, and texture. the follow-up photograph shows the boy at age three after forehead reconstruction. skin expansion in forehead reconstruction

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6 growing skin tissue expanders in reconstructive surgery typical applications are birth defect correction, scar revision in burn injuries, and breast reconstruction after tumor removal. devices are available in different shapes and sizes, circular, square, rectangular, and crescent-shaped. they consist of a silicone elastomer inflatable expander with a reinforced base for directional expansion, and a remote silicone elastomer injection dome. mentor worldwide llc.

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7 growing skin schematic sequence of tissue expansion at biological equilibrium, the skin is in a physiological state of resting tension. a tissue expander is implanted subcutaneously between the skin and the hypodermis. when the expander is inflated, mechanical stretch induces cell proliferation causing the skin to grow. growth restores the state of resting tension. expander deflation reveals residual stresses in the skin layer. reference configurationloaded configurationgrown configurationunloaded configuration

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8 modeling skin carl ritter von langer [1819-1887] langer‘s lines - orientation of collagen fibers

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9 modeling skin amino acids collagen fibrils form collagen fiber three chains form collagen triple helix about 1000 amino acids form collagen chain prolinhydroxyprolinglycin collagen fibers - hierarchical microstructure

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10 modeling skin wormlike chain model kuhn [1936], [1938], porod [1949], kratky, porod [1949], treolar [1958], flory [1969], bustamante, smith, marko, siggia [1994], marko, siggia [1995], rief [1997], holzapfel [2000], bischoff, arruda, grosh [2000], [2002], miehe, göktepe, lulei [2004], kuhl, garikipati, arruda, grosh [2005], böl, reese [2008]

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11 modeling skin transversely isotropic eight chain model. individual chains are modeled as wormlike chains. eight chains are assembled in a transversely isotropic unit cell. the energy of each unit cell consists of the bulk energy, the energy of the eight individual chains, and their repulsive contributions. eight wormlike chainsunit cell bulkeight-chain model nonlinear, anisotropic, locking

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12 modeling skin uniaxial tension test. transversely isotropic wormlike-chain based eight chain model. dots represent experiments of rabbit skin tested parallel and perpendicular to langer's lines, lanir & fung [1974]. lines represent the corresponding computational simulation. the model nicely captures the characteristic features of skin, including the strong non-linearity, the anisotropy, and the locking stretches. 1.00 1.25 1.50 1.75 2.00 stretch [-] nominal stress [N/cm 2 ] 2.0 1.6 1.2 0.8 0.4 0.0 Langer’s lines || Langer’s lines nonlinear, anisotropic, locking

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13 illustration of covariant spatial metric g, deformation tensors C and C e, stress tensors S, P, P e and , and mappings F = F e · F g and F -t = F e-t · F g-t between tangent spaces TB and cotangent spaces TB* in the material configuration, the intermediate configuration, and the spatial configuration. modeling skin kinematics of finite growth

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14 modeling skin stretch-induced area growth with deformation gradient volume change area change with growth tensor lee [1969], rodriguez, hoger, mc culloch [1994], taber [1995], epstein, maugin [2000], lubarda, hoger [2002], ambrosi, mollica [2002], himpel, kuhl, menzel, steinmann [2005], goriely, ben amar [2005], menzel [2005], kuhl, maas, himpel, menzel [2007], garikipati [2009], goktepe, abilez, kuhl [2010]

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15 modeling skin himpel, kuhl, menzel, steinmann [2005], kuhl, maas, himpel, menzel [2007], goktepe, abilez, parker, kuhl [2010], goktepe, abilez, kuhl [2010], schmid, pauli, paulus, kuhl [2011], buganza tepole, ploch, wong, gosain, kuhl [2011], buganza tepole, gosain, kuhl [2011] growth tensor area growth weighting function growth criterion stretch-induced area growth

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16 modeling skin relaxation & creep temporal evolution of total area stretch, reversible elastic area stretch, and irreversible growth area stretch for displacement- and force-controlled skin expansion. displacement control induces relaxation, a decrease in elastic stretch, while the growth stretch increases at a constant total stretch. force control induces creep, a gradual increase in growth stretch and total stretch at constant elastic stretch. 5.0 4.0 3.0 2.0 1.0 normalized time [-] area stretch [-] u u u u F F F F constant total stretch constant elastic stretch 0.00 0.24 0.48 0.72 0.96 normalized time [-] area stretch [-] 0.00 0.24 0.48 0.72 0.96 5.0 4.0 3.0 2.0 1.0

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17 simulating skin local newton iteration algorithmic treatment check growth criterion calculate growth function calculate residual calculate tangent update growth multiplier check converngence calculate growth tensor calculate elastic tensor calculate elastic right cauchy green calculate piola kirchhoff stress calculate lagrangian moduli given and

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18 simulating skin skin is modeled as a 0.2cm thin 12 12cm 2 square sheet, discretized with 3 24 24=1728 trilinear brick elements, with 4 25 25=2500 nodes and 7500 degrees of freedom. the base surface area of all expanders is scaled to 148 elements corresponding to 37cm 2. this area, shown in red, is gradually pressurized from below while the bottom nodes of all remaining elements, shown in white, are fixed. expander inflation & deflation

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19 stretching skin fractional area gain & expander volume tissue expander inflation. expanders are inflated gradually between t=0.00 and t=0.08 by linearly increasing the pressure, which is then held constant from t=0.08 to t=1.00 to allow the skin to grow. under the same pressure, the circular expander displays the largest fractional area gain and expander volume, followed by the square, the rectangular, and the crescent-shaped expanders. 0.00 0.24 0.48 0.72 0.96 300 250 200 150 100 50 0 1.6 1.2 0.8 0.4 0.0 normalized time [-] fraction area gain [-]expander volume [cm 3 ] 0.00 0.24 0.48 0.72 0.96

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20 stretching skin tissue expander inflation. expanders are inflated gradually between t=0.00 and t=0.08 by linearly increasing the pressure, which is then held constant from t=0.08 to t=1.00 to allow the skin to grow. under the same pressure, the circular expander displays the largest area gain and expander volume, followed by the rectangular ||, the square, the crescent ||, the crescent , and the rectangular expanders. fractional area gain & expander volume 0.00 0.24 0.48 0.72 0.96 300 250 200 150 100 50 0 1.6 1.2 0.8 0.4 0.0 normalized time [-] fraction area gain [-]expander volume [cm 3 ] 0.00 0.24 0.48 0.72 0.96

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21 2.25 37.00 41.19 1.11 108.42 0.002 0.33 2.26 37.00 44.40 1.20 122.06 0.002 0.34 2.35 37.00 50.63 1.37 186.77 0.002 0.41 2.36 37.00 58.74 1.59 257.45 0.002 0.42 crescent rectangular square circular expander pressure p/E [-] expander volume V [cm 3 ] absolute area gain A [cm 2 ] fractional area gain A/A 0 [-] initial area A 0 [cm 2 ] maximum growth [-] residual stress /E [-] stretching skin tissue expander inflation and deflation. maximum growth multiplier, absolute area gain, fractional area gain, and expander volume under constant pressure loading at time t=50 and maximum principal residual stresses upon unloading after a constant pressure growth until t=12 are are largest for the circular expander, followed by the square, the rectangular, and the crescent shape expanders. quantitative expander classification

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22 stretching skin area growth - isotropic skin model 1.00 1.35 1.70 2.05 2.40 tissue expander inflation. spatio-temporal evolution of area growth. under the same pressure applied to the same base surface area, the circular expander induces the largest amount of growth followed by the square, the rectangular, and the crescent-shaped expanders.

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23 stretching skin tissue expander inflation. spatio-temporal evolution of area growth. the circular expander induces the largest growth, followed by the square, the crescent-shaped, and the rectangular expanders. growth is smaller when the expanders are placed orthogonal to the strong direction, orthogonal to langer's lines. langer’s lines area growth - anisotropic skin model 1.00 1.35 1.70 2.05 2.40

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24 stretching skin langer’s lines tissue expander inflation. spatio-temporal evolution of area growth. the circular expander induces the largest growth, followed by the rectangular, the square, and the crescent-shaped expanders. growth is larger when the expanders are placed along material's strong direction, aligned with langer's lines. area growth - anisotropic || skin model 1.00 1.35 1.70 2.05 2.40

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25 stretching skin tissue expander deflation. spatio-temporal evolution of elastic area stretch. as the expander pressure is gradually removed, from left to right, the grown skin layer collapses. deviations from a flat surface after total unloading, right, demonstrate the irreversibility of the growth process. 0.90 0.95 1.00 1.05 1.10 elastic stretch - isotropic skin model

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26 stretching skin tissue expander deflation. spatio-temporal evolution of maximum principal stress. as the expander pressure is gradually removed, from left to right, the grown skin layer collapses. remaining stresses at in the unloaded state, right, are growth-induced residual stresses. 0.00 0.10 0.20 0.30 0.40 residual stress - isotropic skin model

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27 serdar göktepe *, jonathan wong **, ellen kuhl ** * department of civil engineering, middle east technical university, ankara, turkey ** departments of mechanical engineering, bioengineering, and cardiothoracic surgery, stanford university http://www.mursi.org growing skin a computational model of skin growth in reconstructive surgery

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28 serdar göktepe *, jonathan wong **, ellen kuhl ** * department of civil engineering, middle east technical university, ankara, turkey ** departments of mechanical engineering, bioengineering, and cardiothoracic surgery, stanford university http://www.mursi.org growing skin a computational model of skin growth in reconstructive surgery buganza tepole, ploch, wong, gosain, kuhl. growing skin: a computational model for skin growth in reconstructive surgery, 2011; submitted. buganza tepole, gosain, kuhl. stretching skin: the physiological limit and beyond, 2011; submitted. goktepe, abilez, kuhl. a generic approach towards finite growth, j mech phys solids. 2010;58:1661-1680. schmid, pauli, paulus, kuhl, itskov. how to utilise the kinematic constraint of incompressibility for modelling adaptation of soft tissues. comp meth biomech biomed eng. 2011; doi:10.1080/1025 5842. 2010.548325. ambrosi, ateshian, arruda, cowin, dumais, goriely, holzapfel, humphrey, kemkemer, kuhl, olberding, taber, garikipati. perspectives on biological growth and remodeling. j mech phys solids. 2011;59:863-883. http://biomechanics.stanford.edu/publications

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