Multi-Fidelity Gaussian Process Surrogate Modeling of Pediatric Tissue Expansion

Abstract

Growth of skin in response to stretch is the basis for tissue expansion (TE), a procedure to gain new skin area for reconstruction of large defects. Unfortunately, complications and suboptimal outcomes persist because TE is planned and executed based on physician's experience and trial and error instead of predictive quantitative tools. Recently, we calibrated computational models of TE to a porcine animal model of tissue expansion, showing that skin growth is proportional to stretch with a characteristic time constant. Here, we use our calibrated model to predict skin growth in cases of pediatric reconstruction. Available from the clinical setting are the expander shapes and inflation protocols. We create low fidelity semi-analytical models and finite element models for each of the clinical cases. To account for uncertainty in the response expected from translating the models from the animal experiments to the pediatric population, we create multifidelity Gaussian process surrogates to propagate uncertainty in the mechanical properties and the biological response. Predictions with uncertainty for the clinical setting are essential to bridge our knowledge from the large animal experiments to guide and improve the treatment of pediatric patients. Future calibration of the model with patient-specific data—such as estimation of mechanical properties and area growth in the operating room—will change the standard for planning and execution of TE protocols.

1 Introduction

Tissue expansion (TE) was first reported in 1957 and since then it has become a primary option to resurface large defects, from breast reconstruction after mastectomy, to burn wounds, to large congenital defects [1,2] (see Fig. 1 for an example of pediatric use of TE). The main principle behind this technique is that skin grows in response to applied stretch [3,4]. To achieve sustained skin growth, a balloon-like device, the tissue expander, is inserted subcutaneously and filled gradually with saline, with weekly or bi-weekly inflation steps of a few milliliters each. At the end of the inflation, 2 or 3 months after the expanders were first inserted, they are removed and new skin can be harvested for reconstruction of an adjacent defect [5]. Even though TE allows for reconstruction of large defects which would otherwise be impossible, a major limitation of TE stems from our current inability to predict the amount and shape of newly grown skin. In turn, this uncertainty results in protocols guided by trial and error and by individual surgeon's experience [6,7]. Common complications or suboptimal outcomes include lack of enough skin growth, closure of flaps under excessive tension leading to compromised healing, and need for multiple cycles of TE and reconstructive surgery [8]. Upon expander placement and inflation, the skin deforms into a three-dimensional (3D) dome with regional variations in the amount of stretch. The apex is stretched the most, and the deformation is less toward the periphery of the expanded area [9]. Growth then depends on this regional variation in the deformation. The process also depends on time. Thus, when an expander is taken out, the skin retracts, releasing some residual elastic deformation [10,11]. Only at that moment, the surgeon can actually assess the amount of newly grown skin by observing the relaxed skin dome that is left behind when the expander is removed. An example of this tiered process in a pediatric patient can be seen in Fig. 1. TE is thus a complex process involving spatio-temporal patterns of skin adaptation to stretch. This is why we are currently unable to accurately predict and thus plan for effective and safe tissue expansion.

Fig. 1.

(a) A pediatric patient with a giant nevus as indicated in the figure needed reconstruction of the scalp and face; (b)and (c) five tissue expanders were placed subcutaneously in the locations: scalp posterior, scalp anterior, forehead, anterior face, lower face. (d) and (e) End of the expansion process, the individual expanders were inflated to the volumes reported in the protocols Table 1.

Computational models of skin biomechanics and mechanobiology are key to improve TE. Coupled theories of large tissue deformation and tissue growth and remodeling have been developed over the past 30 years [12–14]. Within a continuum mechanics framework, the multiplicative decomposition of the deformation gradient into growth and elastic deformation, similar to plasticity, has been used to model growth of tissues, including skin [15,16]. Our model predicted the different patterns of deformation and growth from TE, which were then verified experimentally. The central hypothesis of the model was that growth rate was proportional to stretch, which was based on the existing literature on skin mechanobiology [3]. We confirmed this hypothesis with our experimental protocol on the swine [17]. Other work in the field of TE has also found similar results in a murine model [18]. We now have a calibrated computational model of TE, which shows that skin growth rate is proportional to the local amount of stretch with a characteristic time constant [19].

The pig is the closest animal model for TE because of the similarities in skin anatomy and mechanical properties between the porcine and human skin [20]. Additionally, the TE protocols in the porcine model mimic closely those used in the clinical setting [9]. However, the calibrated computational models based on the porcine animal experiments are not necessarily predictive of skin growth in humans. To deal with this uncertainty, we can use the models calibrated on the porcine data as a baseline and, assuming the mechanism of skin growth is analogous between the animal model and the human patients, we can introduce uncertainty in the parameters of the model to make predictions reflecting this uncertainty [19]. If additional data from the human cases were to become available, we would be able to recalibrate the models or at least reduce the uncertainties in our predictions.

Uncertainty propagation using computationally expensive computational models is not generally feasible [21]. Machine learning (ML) tools allow to replace the detailed models with inexpensive surrogates amenable for the uncertainty quantification tasks [22]. Our previous work in this area has established methods to replace finite element models of skin biomechanics and mechanobiology, including both mechanical properties and biological response uncertainties [23]. We follow a similar approach here. Two types of models are considered. A semi-analytical model is introduced, which is inexpensive but relatively inaccurate [24]. The semi-analytical model is referred to as the low fidelity model. The high fidelity model is based on finite element simulations, which are computationally more demanding but can capture the spatio-temporal variations of skin growth in great detail [25]. Both models are evaluated for a few combinations of the model parameters, and the input–output data pairs are then interpolated using multifidelity Gaussian process (GP) regression [26]. The GP is then used for prediction of skin deformation and growth under uncertainty. We apply the methodology to predict the deformation and growth of skin in clinical protocols of pediatric TE. We also show how, if observations of skin growth from human patients became available, the models could be recalibrated to reflect the new data.

2 Methods

2.1 Patient Protocols.

Two pediatric cases of TE were considered. The details of the expansion protocols that were performed clinically are indicated in Table 1. For one patient, five tissue expanders were placed, three of them rectangular 30 × 60 mm² expanders, another rectangular expander of dimensions 60 × 120 mm², and a crescent-shaped expander with base 50 × 130 mm². Figure 1(a) shows the patient just prior insertion of the tissue expanders. The expanders were placed subcutaneously, one in the lower face, another in the anterior side of the face, one on the forehead, one in the posterior scalp, and finally one in the anterior scalp, see Figs. 1(b) and 1(c). The final stage of the TE expansion protocol, approximately 3.5 months after expander insertion, is shown in Figs. 1(d) and 1(e). By the end of the inflation protocols, the expanders were filled to the desired volumes, which ranged from 33 ml for the smaller expanders, up to 243 ml for the expander in the anterior scalp. For the second patient, only one expander was considered, a rectangular expander places underneath the left clavicle, of dimensions 50 × 100 mm², filled to a final volume of 133 ml.

Table 1.

Clinical tissue expansion protocols considered in this study

Patient 1: Anterior face, rectangular expander 30 × 60 mm2
Time (days)	0	7	21	35	42	49	56	63	70	77	84	91	98	105
Volume (ml)	6	11	15	20	23	26	28	29	30	32	34	36	38	38
Patient 1: Lower face, rectangular expander 30 × 60 mm2
Time (days)	0	7	14	21	28	35	42	49	56	63	70	77	—	—
Volume (ml)	15	20	23	26	28	29	30	32	34	36	38	38	—	—
Patient 1: Forehead, rectangular expander 30 × 60 mm2
Time (days)	0	7	21	35	42	49	56	63	70	77	84	91	98	105
Volume (ml)	6	10	14	17	20	23	24	25	26	27	29	31	33	33
Patient 1: Posterior scalp, rectangular expander 60 × 120 mm2
Time (days)	0	7	21	35	42	49	56	63	70	77	84	91	98	105
Volume (ml)	20	47	74	89	106	123	141	156	166	181	196	211	221	221
Patient 1: Anterior scalp, crescent expander 50 × 130 mm2
Time (days)	0	7	21	35	42	49	56	63	70	77	84	91	98	105
Volume (ml)	20	47	74	97	124	142	157	172	188	203	218	233	243	243
Patient 2: Left clavicle, rectangular expander 50 × 100 mm2
Time (days)	0	7	14	21	28	35	42	49	56	63	70	91	—	—
Volume (ml)	8	18	38	58	78	94	109	123	126	128	133	133	—	—

2.2 Semi-Analytical Model.

Tissue expansion produces nonhomogeneous deformation in three dimensions with greater deformation at the apex and less in the periphery. Finite element methods are ideal for the coupled-mechanics and growth but more computationally expensive [25]. Instead, we developed a semi-analytical formulation in Ref. [24], illustrated in Fig. 2. Details are covered in Ref. [24], but the model is briefly explained here. The stretch of skin from tissue expander is λ_s. From the growth theory we introduce a split into growth and elastic contributions

$${\lambda }_s={\lambda }_s^e{\lambda }_s^g$$ (1)

Fig. 2.

Semi-analytical, low fidelity model of tissue expansion. (a) The expansion process is solved as a plane strain problem in terms of simple analytical assumptions. The control variable is the volume of the expander V_b. Given a value of growth ${\lambda }_s^g$, linear momentum balance result is the total stretch of the skin λ_s. Additionally, an equation for update of growth is given ${\dot{\lambda }}_s^g$. (b) Results for patient 1, lower face expander, with the protocol in Table 1.

For growth to occur, an ordinary differential equation analogous to our recent work is introduced [19]

$${\dot{\lambda }}_s^g=k({\lambda }_s^e-{\lambda }_s^{\text{crit}})$$ (2)

where k is a rate parameter and ${\lambda }_s^{\text{crit}}$ is a parameter that indicates that growth only takes place past a certain physiological range.

The stretch of the expander is λ_b. Both skin and expander stretches, λ_s and λ_b, can be obtained from geometry

$$\begin{array}{cc}{\lambda }_b\hfill & =\frac{1}{1+\text{cos}\left({\theta }_s\right)}\left(R_c{\theta }_s+R_b(\pi -{\theta }_s)+c_b\right)\hfill \\ {\lambda }_s\hfill & =\frac{1}{a_s}\left(\frac{a_s-c}{\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_s}+R_c{\theta }_s\right)\hspace{0.17em}\hfill \end{array}$$ (3)

where the main variables are θ_s and c. The parameters a_s, a_b, and d are related to the placement of the expander and dimensions of both expander and skin pockets. The variables R_c, R_b, and c_b can be expressed as a function of θ_s, c, and d through

$$\begin{array}{cc}{R}_c\hfill & =\frac{c}{\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_s}\hfill \\ R_b\hfill & =\frac{d+(a_s-c)\text{tan}\hspace{0.17em}{\theta }_s}{1+\text{cos}\hspace{0.17em}{\theta }_s}\hfill \\ c_b\hfill & =c-R_b\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_s\hfill \end{array}$$ (4)

Momentum balance for skin and expander is expressed as

$$\begin{array}{cc}{\mu }_b{h}_b({\lambda }_b-{\lambda }_b^{-3})+{\mu }_s{h}_s({\lambda }_s^e-{\lambda }_s^{e-3})=p_A{R}_c\hfill & \hfill \\ {\mu }_b{h}_b({\lambda }_b-{\lambda }_b^{-3})=p_A{R}_b\hfill & \hfill \end{array}$$ (5)

where mu_b and h_b are shear modulus and thickness of the expander, mu_s and h_s are the corresponding shear modulus and thickness of skin. The new unknown p_A, the pressure inside of the expander, has been introduced. Note that momentum balance of skin needs the elastic stretch.

Lastly, the volume of the expander can be determined from the geometry to be

$$t_b\left(R_b^2(\pi -{\theta }_s)+2R_b{c}_b+(c+c_b)R_b\hspace{0.17em}\text{cos}\hspace{0.17em}{\theta }_s+\frac{R_c^2}{2}(2{\theta }_s-\text{sin}(2{\theta }_s\left)\right)\right)=V_b\hspace{0.17em}$$ (6)

The volume V_b is actually the control variable. Growth is an internal variable ${\lambda }_s^g$. If V_b and ${\lambda }_s^g$ are known at a given time, then one can use Eqs. (3)–(6) to solve for the total stretch of the skin, λ_s, along with the other unknown variables θ_s, c, and p_A. For the simulations, we set the initial condition ${\lambda }_s^g(0)=1$, and impose $V_b(t)$ according to the protocols in Table 1. We solve then for ${\lambda }_s(t)$ and ${\lambda }_s^g(t)$.

2.3 Finite Element Model of Tissue Expansion.

The finite element model of tissue expansion follows from our previous work [23]. The skin is discretized with hexahedral C3D8 elements in Abaqus (Dassault Systems, Waltham, MA), the expander is discretized with shell elements CPS4, and fluid cavity elements are considered for the expander. Expander-skin contact is assumed frictionless. The same split of deformation gradient into growth and elastic parts is considered, analogous to the semi-analytical model,

$$\mathbf{F}={\mathbf{F}}^e{\mathbf{F}}^g$$ (7)

The growth deformation is a second-order tensor, which can be further constrained to a given kinematic family of deformations that reflect the underlying mechanobiological assumptions. We assume area growth [25] and condense the growth process to a single scalar field

$${\mathbf{F}}^g={\vartheta }^g{\mathbf{I}}_s+\mathbf{N}\otimes \mathbf{N}\hspace{0.17em}$$ (8)

with N being the outer normal of skin and the surface identity ${\mathbf{I}}_s=\mathbf{I}-\mathbf{N}\otimes \mathbf{N}$. The scalar variables ${\vartheta }^g$ are the area growth. Note that this variable is the same as ${\lambda }_s^g$ in the semi-analytical model. The difference in notation comes from the fact that the semi-analytical model used a plane strain assumption, and thus, λ_s in the semi-analytical model is actually the area growth. In the finite element model the deformation is in three-dimensions, and area deformation and growth is calculated accordingly. Rate of change of growth follows exactly the same as before

$${\dot{\vartheta }}^g=k({\vartheta }^e-{\vartheta }^{\text{crit}})\hspace{0.17em}$$ (9)

where, again, the parameter k is the rate constant for growth, exactly the same parameter as in Eq. (2), and the parameters ${\vartheta }^{\text{crit}}={\lambda }_s^{\text{crit}}$ are also equivalent. The elastic area change in this case is also similarly obtained from the total area change and the growth ${\vartheta }^e=\vartheta /{\vartheta }^g$. The total area change is $\vartheta =\left|\right|\text{cof}\mathbf{FN}\left|\right|$.

For the finite element model, a neo-Hookean model is used (similar to the semi-analytical model), but in this case, the 3D formulation is employed

$$\sigma ={\mu }_s{\mathbf{b}}^e+p\mathbf{I}$$ (10)

with ${\mathbf{b}}^e={\mathbf{F}}^e{\mathbf{F}}^{e\mathrm{\top }}$, the left Cauchy-Green deformation tensor with F is the deformation gradient and $J=\det \mathbf{F}$. The pressure comes from a volumetric strain energy

$$p=\frac{\partial W_{\text{vol}}}{\partial J}=D(J-1)\hspace{0.17em}.$$ (11)

The shear modulus, μ_s, is also shared between the finite element model and the semi-analytical one. Standard momentum balance $\nabla ·\sigma =0$ is sought. The volume of the expander is the other constraint V_b, and the growth is the internal variable. Similar to the semi-analytical case, here the initial condition is ${\vartheta }^g(0)=1$, and we impose the volume $V_b(t)$ to follow the protocol; we impose $V_b(t)$ through a user subroutine and solve for momentum balance and also growth update Eq. (9). The result is a spatio-temporal deformation and growth field $\vartheta (\mathbf{x},t)$ and ${\vartheta }^g(\mathbf{x},t)$, see Fig. 3. However, we are interested in four scalar outputs of interest, the maximum deformation and growth, ${\vartheta }_{\text{max}}(t)$ and ${\vartheta }_{\text{max}}^g(t)$, and the average deformation and growth, ${\vartheta }_{\text{avg}}(t)$ and ${\vartheta }_{\text{avg}}^g(t)$.

Fig. 3.

Finite element simulations of each of the protocols in Table 1. Six expanders were simulated. Only the initial and final time points are shown. The initial skin is considered as a flat tissue with the expander underneath. The volume of the expander is controlled with a user subroutine in Abaqus to follow the protocols in Table 1. A user material subroutine is also used to model skin growth in response to stretch. Shown on the final state of expansion are the contours of area growth.

2.4 Multi-Fidelity Gaussian Process Regression.

Gaussian process is a probability measure over function spaces such that the function values at a subset of input points have a joint Gaussian distribution [27]. Based on the fact that the observations and additional new points together consist of a joint Gaussian distribution, and the distribution of any subset conditioned on the rest is also Gaussian; it enables the separation of the posterior distribution over any unknown points from the joint Gaussian distribution. In other words, the GP posterior distribution is constructed according to the prior state of one's knowledge about an unknown function conditioned on the observed data. The GP posterior distribution is Gaussian, i.e., defined by mean and variance values.

Let us denote that a latent function $f(\mathbf{x})$ describing input and output pairs is trained with N observations such that inputs ${\mathbf{x}}_n$ in ${ℝ}^d$ and corresponding outputs y_n are a training dataset, ${\mathcal{D}}_t={({\mathbf{x}}_n,y_n)}_{n=1}^{n=N}$. Here, we assume that observation error is also assigned in y_n, namely, $y_n=f({\mathbf{x}}_n)+{\mathrm{ϵ}}_n$ where ϵ_m is sampled from Gaussian distribution with zero mean and unknown variance ${\sigma }_G^2$, which is learned by observations. Subsequently, the function f is defined by a GP under the prior state of knowledge such that

$$f(·)|\mathbf{\theta }\sim \mathcal{G}\mathcal{P}(m(·;\mathbf{\theta }),k(·,·;\mathbf{\theta }))$$ (12)

where $m(·)$ and $k(·,·;\mathbf{\theta })$ are the mean and covariance functions, respectively and $\mathbf{\theta }$ (to be learned) includes all hyperparameters that influence the covariance functions. If there is no particular knowledge about the function f, we can pick a zero mean function, which is common in GP applications [22]. Correlation between data points is encoded by the covariance function, In addition, the covariance function also encodes the differentiability of f. Considering that f is infinitely differentiable, we select the squared exponential covariance function

$$k(\mathbf{x},\mathbf{x}\prime ;\mathbf{\theta })=v\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-\sum _{i=1}^d\frac{{\left(x_i-x_i^{\prime }\right)}^2}{2{\ell }_i^2}\right\}$$ (13)

where $\mathbf{\theta }$ includes the positive hyperparameters $\{v,{\ell }_1,\dots ,{\ell }_d\}$, and v and $\ell$ denote the variance of the process and the length scales of each input dimension, respectively. The hyperparameters are learned from observation such that $\mathbf{\theta }$ is selected at a value to maximize the log-marginal likelihood

$$\text{log}\hspace{0.17em}p\left(\mathbf{y}\right|\mathbf{X},\mathbf{\theta })=-\frac{1}{2}{\mathbf{y}}^{\mathrm{\top }}{(\mathbf{K}+{\sigma }_G^2\mathbf{I})}^{-1}\mathbf{y}-\frac{1}{2}\text{log}\hspace{0.17em}|\mathbf{K}+{\sigma }_G^2\mathbf{I}|-\frac{N}{2}\text{log}\hspace{0.17em}2\pi \hspace{0.17em}$$ (14)

where X and y are samples from the training dataset ${\mathcal{D}}_t$ and K is the N × N symmetric and positive definite matrix ${\mathrm{K}}_{ij}=k({\mathbf{x}}_i,{\mathbf{x}}_j;\mathbf{\theta })$ described in Eq. (13). Recalling that any subset from function values is also jointly Gaussian, the prior of the observed data ${\mathcal{D}}_t$ together with a test pair $({\mathbf{x}}_{*},f_{*})$ satisfy

$$\left[\begin{array}{c}\mathbf{y}\hfill \\ f_{*}\hfill \end{array}\right]\sim \mathcal{N}\left(\left[\begin{array}{c}\mathbf{0}\hfill \\ 0\hfill \end{array}\right],\left[\begin{array}{cc}\mathbf{K}+{\sigma }_G^2\mathbf{I}\hfill & \mathbf{k}(\mathbf{X},{\mathbf{x}}_{*};\mathbf{\theta })\hfill \\ \mathbf{k}({\mathbf{x}}_{*},\mathbf{X};\mathbf{\theta })\hfill & k({\mathbf{x}}_{*},{\mathbf{x}}_{*};\mathbf{\theta })\hfill \end{array}\right]\right)\hspace{0.17em}$$ (15)

where $\mathbf{k}({\mathbf{x}}_{*},\mathbf{X};\mathbf{\theta })={\left(k({\mathbf{x}}_{*},{\mathbf{x}}_1;\mathbf{\theta }),\dots ,k({\mathbf{x}}_{*},{\mathbf{x}}_N;\mathbf{\theta })\right)}^{\mathrm{\top }}$ is the cross covariance between ${\mathbf{x}}_{*}$ and X and $f_{*}$ is the value of the function at the unknown input ${\mathbf{x}}_{*}$. This joint Gaussian distribution can be split in a Gaussian distribution of the observed data and a conditional Gaussian distribution of $f_{*}$ given X and y. Accordingly, the GP posterior of $f_{*}$ is

$$f_{*}|\mathcal{D},{\mathbf{x}}_{*},\mathbf{\theta }\sim \mathcal{N}({\mu }_{*}({\mathbf{x}}_{*};\mathbf{\theta }),{\sigma }_{*}^2({\mathbf{x}}_{*};\mathbf{\theta }))\hspace{0.17em}$$ (16)

where

$${\mu }_{f_{*}}({\mathbf{x}}_{*};\mathbf{\theta })={\mathbf{k}}^{\mathrm{\top }}({\mathbf{x}}_{*},\mathbf{X};\mathbf{\theta }){(\mathbf{K}+{\sigma }_G^2\mathbf{I})}^{-1}\mathbf{y}\hspace{0.17em}$$ (17)

and

$${\sigma }_{*}^2({\mathbf{x}}_{*};\mathbf{\theta })=k({\mathbf{x}}_{*},{\mathbf{x}}_{*};\mathbf{\theta })-{\mathbf{k}}^{\mathrm{\top }}({\mathbf{x}}_{*},\mathbf{x};\mathbf{\theta }){(\mathbf{K}+{\sigma }_G^2\mathbf{I})}^{-1}\mathbf{k}(\mathbf{x},{\mathbf{x}}_{*};\mathbf{\theta })\hspace{0.17em}$$ (18)

are the predictive mean and variance, respectively.

So far the GP regression accounts for single fidelity information, i.e., from only one model relating inputs x and outputs y. Now, let us extend the concept to exploit the availability of different fidelity information, that is, more than one model relating the inputs and outputs. In this study, our problem is composed of two different models, semi-analytical and finite element models, indicating low and high fidelity information, respectively (Fig. 4). Therefore, we focus on two levels of fidelity model although there can be S levels of increasing fidelity each corresponding to an unknown function f_t with $t=1,\dots ,S$, for the interested reader, a good overview and more details of multifidelity GP regression can be found in Refs. [26], [28], and [29].

Fig. 4.

A schematic to build multifidelity Gaussian process (GP) surrogate in skin expansion: Two levels of fidelity model are composed of semi-analytical solution (f₁) and finite element analysis (f₂), respectively, and all fidelity information is combined together in multifidelity GP framework. Training input dataset ${\mathbf{X}}_2$ for high fidelity model is nested in ${\mathbf{X}}_1$ for low fidelity model. Once the multifidelity GP surrogate is created, new input point (shear modulus ${\mu }_{*}$ and growth rate $k_{*}$) or distribution of them can be evaluated in terms of predictive mean and variance.

Based on the autoregressive approach explained in Ref. [26], multifidelity information can be combined. At the low fidelity model

$$f_1(·)|{\mathbf{\theta }}_1\sim \mathcal{G}\mathcal{P}(m_1(·;{\mathbf{\theta }}_1),k(·,·;{\mathbf{\theta }}_1))$$ (19)

and then modeling high fidelity according to

$$f_2(\mathbf{x})={\rho }_1{f}_1(\mathbf{x})+{\delta }_2(\mathbf{x})\hspace{0.17em}$$ (20)

where ρ₁ is a scaling factor that quantifies the correlation between output data at consecutive fidelity levels and ${\delta }_2(\mathbf{x})$ is a correction term with the prior modeled as a GP, i.e., ${\delta }_2(·)\sim \mathcal{G}\mathcal{P}({\mu }_2(·),k_2(·,·;\mathbf{\theta }))$, with ${\mu }_2(·)$ and $k_2(·,·;\mathbf{\theta })$ the mean and covariance functions for high fidelity model priors. Note that the input data points are nested; for evaluation of Eq. (20) for the high fidelity model, the points x need to be evaluated in the low fidelity model.

If the number of training dataset or fidelity levels is large, it can be unfeasible to implement the autoregressive scheme [26]. To overcome the computational cost, replacing the prior of the lower fidelity model with its posterior, $f_{*1}$, has been suggested [28], and further improvement have been presented to not only maintain the same philosophy of Ref. [28] but also capture nonlinear correlations between different levels of fidelity [29].

The approach of Ref. [29] is a kind of deep-GP in which the GP posterior from lower level of fidelity is utilized as one more dimension in the training input dataset of next level of fidelity. In other words, the training dataset ends up having d + 1 dimensionality for all except the lowest level of fidelity. Considering two levels of fidelity, for example, the formulation of this approach substitutes for Eq. (20)

$$f_2(\mathbf{x})=g_2(\mathbf{x},f_{*1}(\mathbf{x}))\hspace{0.17em}$$ (21)

where the prior for g₂ follows $\mathcal{G}\mathcal{P}({\mathbf{f}}_2|\mathbf{0},k_{g_2}((\mathbf{x},f_{*1}(\mathbf{x})),({\mathbf{x}}^{\prime },f_{*1}({\mathbf{x}}^{\prime }));{\mathbf{\theta }}_{g_2}))$ with zero mean function and covariance function $k_{g_2}$ associated high fidelity model [29]. Here, the covariance function correlates information from the lower fidelity into the high fidelity model via

$$\begin{array}{cc}{k}_{g_2}(\mathbf{x},{\mathbf{x}}^{\prime };{\mathbf{\theta }}_{g_2})=\hfill & k_{{\rho }_2}(\mathbf{x},{\mathbf{x}}^{\prime };{\mathbf{\theta }}_{{\rho }_2})·k_{f_2}(f_{*1}(\mathbf{x}),f_{*1}({\mathbf{x}}^{\prime });{\mathbf{\theta }}_{f_2})\hfill \\ \hfill & +k_{{\delta }_2}(\mathbf{x},{\mathbf{x}}^{\prime };{\mathbf{\theta }}_{{\delta }_2})\hspace{0.17em}\hfill \end{array}$$ (22)

where $k_{{\rho }_2},\hspace{0.17em}{k}_{f_2}$, and $k_{{\delta }_2}$ denote covariance functions with hyperparameters ${\mathbf{\theta }}_{{\rho }_2},\hspace{0.17em}{\mathbf{\theta }}_{f_2}$, and ${\mathbf{\theta }}_{{\delta }_2}$, respectively. We point out that $k_{f_2}$ makes use of the posterior function value and not the input point at the lower level of fidelity.

We emphasize that the posterior distribution of $f_{*2}({\mathbf{x}}_{*})$ from multifidelity GP regression is no longer Gaussian since the GP of a GP is not a GP anymore. Therefore Monte Carlo integration can be used to calculate the predictive mean and the corresponding variance of the posterior distribution. To carry out the GP regression in this study we make use of the open source library GPy [30], and the multi-fidelity GP regression library from [29].

3 Results

The goal is to make predictions of the amount of skin growth resulting from the tissue expansion protocols in Table 1. As explained before, our previous work has identified the corresponding parameters for the biological adaptation of skin in response to stretch in the porcine model [19]. Additionally, we have knowledge of the mechanical properties of skin in human patient populations [23,31,32]. Based on the clinical evidence and similarities between the TE protocols in the pediatric population and the porcine animal model [7,17], we anticipate that mechanobiological adaptation in pediatric patients will follow similar dynamics to the porcine model after accounting for the change in material properties. To propagate the uncertainty through the computational model, the multifidelity GP surrogate has to be built.

To build the multifidelity GP surrogate, we first create training and validations datasets for both low and high fidelity models. Here, we make use of Latin hypercube sampling (LHS) to generate training and validation datasets independently because LHS can make randomly but evenly distributed data points. We first consider the effect of limited training data leading to the so-called epistemic error. It is expected that larger training datasets lead to smaller epistemic error. In principle, the more high fidelity evaluations, the better the performance. However, we have a limited computational budget. Only high fidelity model evaluations are a concern for computational cost. Correspondingly, we evaluate the performance of high fidelity GP surrogates as we vary the size of training dataset from 20 to 100 at 20 intervals. The performance of high fidelity GP surrogate is evaluated in terms of the relative l₂-norm of error for both ${\vartheta }_{\text{max}}^g$ and ${\vartheta }_{\text{avg}}^g$, which are represented in each column in Fig. 5. Overall, the high fidelity GP surrogate shows acceptable l₂-norm of error lower than 1%, and the error decreases as the size of the dataset increases. Not surprisingly, the best performance of the high fidelity GP surrogate occurs with 100 training data points.

Fig. 5.

Investigation on the performance of high fidelity surrogate model: Increasing the number of high fidelity data points, the performance of surrogate model improves in terms of l₂-norm of ${\vartheta }_{\text{max}}^g$ and ${\vartheta }_{\text{avg}}^g$ at the end and all training time points. Each six expansion protocol is plotted with different lines.

For the multifidelity surrogate, let ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ be input points for the low fidelity and high fidelity models. Recalling that the training dataset at lower fidelity model is a superset of the higher fidelity model dataset, the semi-analytical model is hence evaluated at a total of $150(=100+50)$ input points. Note also that for the sampled inputs we run all five protocols. Considering a computational budget of 100 high fidelity evaluations, we run a total of 900 low fidelity evaluations and 600 finite element simulations across all five expanders. For example, for the expander in the lower face, the first protocol in Table 1, the results of the low fidelity model are shown in Fig. 2(b). Results from the finite element simulation for one of the parameters are shown in Fig. 3. The outputs are denoted ${\mathbf{y}}_1$ and ${\mathbf{y}}_2$ and include the average total deformation, maximum total deformation, average area growth, and maximum of area growth.

Figure 6(a) illustrates the performance of the multifidelity surrogate compared to the single fidelity GP trained with FE simulation data alone. Similar to Fig. 5, the performance is investigated as a function of the number of high fidelity training data points. As opposed to Fig. 5, however, the l₂-norm of error is computed across all time points rather than just the final time point. The multifidelity surrogate improves the results modestly compared to the single, high fidelity model.

Fig. 6.

(a) Performance of the high fidelity surrogate compared to the multifidelity surrogate in terms of l₂-norm of the error computed over all time points as a function of high fidelity evaluations. Multifidelity scheme leads to lower errors. (b) Posterior distribution of area growth ${\vartheta }_{\text{max}}^g$ for high fidelity, low fidelity, and multifidelity GP regression for five tissue expanders plotted as a function of time. Low and high fidelity GP surrogates are trained with 150 and 100 training points, respectively, while the multifidelity GP surrogate makes use of all points in the training dataset, combining the different fidelity models. Predictive mean and corresponding variance are shown for each type of model. Multifidelity GP surrogates outperform the single fidelity schemes.

Figure 6(b) illustrates the results of ${\vartheta }_{\text{max}}^g$ over time from GP regression for each of the five expanders of patient 1 for the case of 100 high fidelity evaluations. Growth increase over time ${\vartheta }^g(t)$ is shown as a result of the TE protocols. The low fidelity model captures the mean data from the model evaluations. However, the predicted variance (shaded area) is large. The excessive variance could be due to outliers in the low fidelity model evaluations.

For completeness, results from the high fidelity surrogate are also shown in Fig. 6. In general, the high fidelity model is much better than the low fidelity counterpart. There are only a few regions in which the high fidelity model still shows undesirable variance due to epistemic uncertainty. The multi-fidelity model. leverages the low fidelity information as well as the high fidelity model evaluations and shows the best response, with the mean that closely matches the finite element simulation data, and with decreased variance compared to using high fidelity data alone. Therefore, in the following only the multifidelity scheme is used for predictions. In all five expanders, growth increases over time as a result of the increasing inflation of the expander.

Once the multifidelity GP surrogate is available, we can use it to make predictions about the amount of area growth in each of the expanders. Results are depicted in Fig. 7. In this figure, we summarize the posterior distributions for the total deformation and area growth for patient 1. There are four quantities of interest. Both the total deformation and growth vary spatially, see Fig. 3, but the information can be condensed into four scalar variables: average total deformation ${\vartheta }_{\text{avg}}$, maximum total deformation ${\vartheta }_{\text{max}}$, average area growth ${\vartheta }_{\text{avg}}^g$, maximum area growth ${\vartheta }_{\text{max}}^g$. We emphasize that uncertainty in the predictions arises from two sources. On the one hand, we consider uncertain material properties based on available knowledge of skin mechanics and uncertainty in the mechanobiological adaptation of skin to stretch based on our porcine experiments (Table 2). These uncertainties lead to the main trends that define the posteriors (solid lines in Fig. 7). The second source of uncertainty is epistemic uncertainty from using the GP instead of the true model. This uncertainty leads to the uncertain posterior (shaded area in Fig. 7).

Fig. 7.

At the end of expansion protocol, total deformation ϑ and growth ${\vartheta }_g$ are summarized in four quantities of interest: average total deformation ${\vartheta }_{\text{avg}}$, maximum total deformation ${\vartheta }_{\text{max}}$, average area growth ${\vartheta }_{\text{avg}}^g$, maximum area growth ${\vartheta }_{\text{max}}^g$. Predictions of the posterior distributions using the multifidelity GP surrogate for the four quantities of interest and for each of the expanders are shown. Uncertainty propagation over input space is quantified over output space and illustrated by 95% predictive intervals. Uncertainty comes from two sources. On the one hand, the input parameters are assumed to follow a uniform distribution according to Table 2. This uncertainty leads to the main trends in the posteriors. Additionally, there is epistemic uncertainty from using the GP instead of the true model, which leads to the shaded area surrounding the posteriors.

Table 2.

Ranges for mechanical and biological behavior of skin to generate training dataset using Latin hypercube sampling [19]

Parameter	Range
μ (MPa)	[0.1, 1.0]
k ( ${\text{hr}}^{-1}$)	[0.02, 0.1]

Because of the uncertainty in mechanical properties and mechanobiological response, the predictions show considerable variance. For example, for the scalp anterior expander, the maximum total deformation ${\vartheta }_{\text{max}}$ can range between 3.3 and 3.8. This means that a region of skin of 1 × 1 cm² is deformed to a square of dimensions ranging from $1.81\times 1.81$ to $1.97\times 1.97$ cm². Not all the values are equally likely, distributions are skewed toward lower values of deformation and growth. The expectation and variance summarizing the results of Fig. 7 are reported in Table 3. Note that the original distributions in Table 2 are assumed uniform. The posteriors in Fig. 7 reflect the nonlinearity of the TE process: the uniform distribution of the inputs leads to skewed unimodal distributions of the output. We decided to propagate the uniform distributions in the absence of better information. In other words, we aimed at making conservative predictions that can be used to guide treatment given our limited knowledge of the mechanical and mechanobiological properties of the individual patient. However, if more information became available, we could use the same GP, trained on the LHS dataset of Table 2, but propagate narrower distributions for μ and k, see our previous work [21].

Table 3.

Summary of Fig. 7 in terms of expectation value and standard deviation: Multifidelity Gaussian process surrogate is used to sample ${\vartheta }_{\text{max}},\hspace{0.17em}{\vartheta }_{\text{avg}},\hspace{0.17em}{\vartheta }_{\text{max}}^g$, and ${\vartheta }_{\text{avg}}^g$ over the range of input shown in Table 2

		${\vartheta }_{\text{max}}$		${\vartheta }_{\text{avg}}$		${\vartheta }_{\text{max}}^g$		${\vartheta }_{\text{avg}}^g$
Location	Volume (ml)	E(x)	S(x)	E(x)	S(x)	E(x)	S(x)	E(x)	S(x)
Scalp anterior	243	3.59	0.13	3.05	0.07	2.76	0.11	2.26	0.06
Scalp posterior	221	2.03	0.05	1.81	0.03	1.67	0.04	1.48	0.02
Forehead	33	1.65	0.03	1.35	0.01	1.43	0.03	1.17	0.01
Anterior face	38	1.82	0.04	1.44	0.01	1.58	0.04	1.25	0.01
Lower face	38	1.82	0.04	1.44	0.01	1.58	0.04	1.25	0.01
Left clavicle	133	2.09	0.06	1.86	0.03	1.81	0.05	1.64	0.03

Uncertainty is quantified over the posterior distribution in terms of expectation value and standard deviation.

The anterior scalp expander leads to the greatest total deformation, reflected in both the maximum value ( ${\vartheta }_{\text{max}}$ at the apex) and the average ( ${\vartheta }_{\text{avg}}$). The scalp posterior produces the second highest total deformation across all four expanders, followed by the anterior face and the forehead expanders. Growth follows the total deformation. Since only part of the total deformation is growth, the posterior distributions for ${\vartheta }_{\text{max}}^g$ and ${\vartheta }_{\text{avg}}^g$ are shifted toward lower values compared to ${\vartheta }_{\text{max}}$ and ${\vartheta }_{\text{avg}}$ in each of the panels of Fig. 7.

A main clinical variable that is adjusted by the surgeon is the volume of expansion V_b. Here, the volume was considered as a deterministic input based on the protocols that were performed clinically, see Table 1. In general, smaller volumes lead to smaller total deformation and growth compared to larger volumes, see Fig. 8. For the smaller rectangular expanders, the average total deformation has mean average values of 1.35 and 1.44 for the 33 and 38 ml expanders, and mean maximum values of 1.65 and 1.82. For the larger volume of 133 ml, the mean average deformation is higher, ${\vartheta }_{\text{avg}}=1.86$, yet, for the next larger volume V_b = 221 ml, this metric decreases to ${\vartheta }_{\text{avg}}=1.81$. Finally, for the largest volume of V _b = 243 ml, the mean prediction of the total deformation from the multifidelity GP surrogate is ${\vartheta }_{\text{avg}}=3.05$, the largest amongst all expanders. Therefore, volume alone does not determine the amount of skin deformation and growth. For the same expander shape, greater volume leads to greater deformation and growth. Yet, changing the expander shape will alter this relationship. This underscores the need for the finite element simulations to predict the spatial variations in deformation due to expander volume and shape [25].

Fig. 8.

At the end of expansion protocol, total deformation ϑ and growth ${\vartheta }_g$ are compared together with respect to volume of the expander. There are five volumes considered 33, 38, 133, 221, and 243 ml. Expectation value and two standard deviation of the mean probability density function from Fig. 7 is shown for each volume. Even though in general we would expect that increasing volume would lead to larger deformation and growth, the relationship also depends on the shape of the expander.

A clear illustration of the biological adaptation of skin in response to stretch can be seen in Fig. 9, in which we compare the total deformation to the area growth. The plots in Fig. 9 bypass the volume and shape of the expander. Even though volume and shape of the expander are the variables that can be controlled clinically [7], our experiments have shown that the important variables to predict skin growth are the total deformation field and time [9]. Figure 9 clearly depicts this relationship. Greater deformation induces greater growth. Figure 9 also illustrates the fact that the skin deforms heterogeneously upon expansion. The apex undergoes the largest deformation and growth, with less deformation away from the apex of the expander. Thus, even though the maximum deformation at the apex has the mean 3.59 for the anterior scalp expander, leading to an expected maximum growth of 2.76 at this location, the skin over the anterior scalp expander deforms on average 3.05 with an average growth 2.26. Figure 9 also allows visualization of the uncertainty in the predicted deformation and growth. The scatter plots show the variation in the predictions from propagating the uniform distribution in Table 2.

Fig. 9.

At the end of expansion protocol, growth ${\vartheta }_g$ with respect to total deformation ϑ are paired in terms of mean and max values of ${\vartheta }_g$. The scatter plots reflect the sampling from the uniform distribution of the inputs in Table 2, with the marginal distributions shown to the right and on top of the plots.

The values reported in Figs. 8 and 9 are relative area changes. To put the results into perspective, consider that for patient 1, the area of the initial skin patches are approximately $18.00-19.84$ cm² for the lower face, anterior face, and forehead expanders. The range is deduced based on the patient photographs and the fact that the skin over the expander does not exactly match the 3 × 6 cm² area of the expander but that there is an additional region of skin that is deformed adjacent to the expander. Our simulations predict an average area gain of $11.00-14.43$ cm² in total from these expanders compared to their combined initial area of $54.00-59.52$ cm². For the anterior scalp, the initial area is approximately 48.96 cm², while for the posterior expander the initial area of skin over the expander is $72.00-75.64$ cm². Our simulations predict that on average, the total area gain from the anterior expander would be $55.83-67.33$ cm², while for the posterior expander it would be $31.73-39.25$ cm². In summary, the total area gained expected from these five TE protocols is $98.5-120.9$ cm². From the patient photographs, it is estimated that the total defect area that needed to be resurfaced was approximately $93.7-114.6$ cm². Thus, the TE protocols chosen should lead to enough skin gain to resurface the majority of the defect.

4 Discussion

We simulate skin growth in response to stretch for five clinical protocols of pediatric TE. The predictions are based on our knowledge of skin mechanics in human patients and our understanding of skin mechanobiology from porcine TE protocols [19,31–33]. We report probabilistic outcomes that reflect our uncertainty in the knowledge of skin mechanics and mechanobiology. To do the uncertainty analysis, we leverage multifidelity GP surrogates.

The study responds to the need to have quantitative understanding of skin mechanics and mechanobiology to better plan TE and reconstructive procedures [34]. Current protocols are based on surgeon's training and experience. Not only are there no standardized protocols or software tools for the design of TE protocols, which limits the wider use of this powerful technique, but there are also unintended consequences of suboptimal protocol designs. These include the need for recurrent TE if not enough skin was gained [35], or, worse, closure of flaps under excessive tension which can lead to wound dehiscence, infection, and tissue necrosis [36,37]. The difficulty in estimating skin growth is twofold. First, the deformation is not uniform and simple estimates of area gain based on the area or volume of the expander are not accurate. This is clear even in the current study in which the semi-analytical model, based on a simple geometric model but still including mechanics and mechanobiology information, is unable to describe the regional patterns of deformation predicted by finite element simulations and observed experimentally [9,38]. The second challenge for the surgeon is that not all deformation of skin is growth. Growth of skin is not directly measured clinically and it only becomes evident when the expanders are removed [17,39]. Thus, finite element simulations to predict skin growth in TE are needed to guide treatment planning. There have been efforts in this direction by our group and others. In our earlier work we have developed models of pediatric tissue expansion [4]. The predictions from earlier models matched qualitatively the clinical observations, but these models lacked calibrated parameters of skin mechanics and mechanobiology. Since then, our understanding of the range of mechanical properties in human patients and how they vary with age and sex has improved [31,32]. Additionally, we have measured the dynamics of skin growth in response to stretch in the porcine model of TE [19]. The swine is the closest animal model for human TE [40,41]. With this new information, here we revisit the problem of predicting skin growth in clinical TE protocols.

The work presented here also reflects recent advances in ML methods in biomechanics. In particular, ML algorithms such as artificial neural networks or GPs have been effective at replacing standard finite element solvers by inexpensive metamodels [42,43]. In turn, the use of ML metamodels has enabled Bayesian frameworks in which predictions are probabilistic and reflect our uncertainty in the process and parameters [44,45]. The model predictions of skin growth shown here describe a range of outcomes rather than deterministic outputs. We expect that this type of Bayesian framework can be used as an additional input to the surgeon planning a TE protocol even if exact prediction of skin growth in human pediatric patients remains out of reach.

There are a number of limitations in the work presented here, and future model developments and data gathering are still needed. For instance, the geometries considered were idealized since we did not have the patient specific geometry. In previous work, we had access to computer tomography scans that were acquired as part of treatment [4,21]. We have demonstrated the used multiview stereo and 3D photography in the swine model and in the operating room [33,46], but these techniques are not part of standard treatment in TE. We have recently started collecting 3D photos for pediatric TE. The new data will allow us to assess to what extent the idealized geometries are representative of the patient-specific cases. While we expect that the patient-specific geometries will increase the predictive power of the model, we believe that idealized geometries could be used to guide TE when the detailed geometry is not available. In fact, our previous work with the porcine model has shown that the idealized geometries can adequately capture the skin growth process [19]. An alternative to deal with lack of individual geometries but to refine the model beyond the simple geometric shapes, is to collect human patient data and define probability distributions over the patient geometries using a few degrees-of-freedom. The effect of patient geometry could then be considered as an additional source of uncertainty to predict the growth of skin in a new patient for which no 3D geometry is available. In addition to geometry, model refinement in terms of skin mechanical behavior is a clear next step. The model predictions reflect knowledge of skin mechanics measured through suction tests [31,32]. Unfortunately, this type of test cannot be used to inform the more complex mechanical response of skin which includes anisotropy and poroelasticity [47,48]. New noninvasive testing methods are being developed to measure anisotropic properties of skin in vivo [49]. Combination of existing ex vivo data on human skin mechanics with the in vivo suction test data could also be done to refine the model. Despite the simplifications, our methodology can still capture the skin growth in porcine TE, providing confidence in the ability of the model to translate to clinical application. In the longer term, we anticipate that ongoing collection of human patient data in terms of 3D photos and noninvasive mechanical testing will lead to the continuous update of the model through our Bayesian framework.

Conclusions

We present a computational modeling framework to predict skin growth in clinical TE protocols. The model builds upon on our extensive prior work in the modeling and characterization of TE, as well as existing work on noninvasive mechanical testing of skin in vivo. The proposed Bayesian framework uses this information to make predictions of skin growth that reflect our uncertainty in the skin mechanical properties and mechanobiological parameters. We expect that our methodology will be the basis of software tools to aid in the planning of TE protocols in pediatric patients. Given the Bayesian framework, we anticipate a continuous update of the model as we continue to gather data and refine the model.

Funding Data

• Myongji University (2021 Research Fund of Myongji University; Funder ID: 10.13039/501100002538).

• National Institute of Arthritis and Musculoskeletal and Skin Diseases (Award ID: R01AR074525; Funder ID: 10.13039/100000069).

Data Availability Statement

The code and datasets supporting this article are available at online.2