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Multi-scale finite element model of growth plate damage during the development of slipped capital femoral epiphysis

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Abstract

Slipped capital femoral epiphysis (SCFE) is one of the most common disorders of adolescent hips. A number of works have related the development of SCFE to mechanical factors. Due to the difficulty of diagnosing SCFE in its early stages, the disorder often progresses over time, resulting in serious side effects. Therefore, the development of a tool to predict the initiation of damage in the growth plate is needed. Because the growth plate is a heterogeneous structure, to develop a precise and reliable model, it is necessary to consider this structure from both macro- and microscale perspectives. Thus, the main objective of this work is to develop a numerical multi-scale model that links damage occurring at the microscale to damage occurring at the macroscale. The use of this model enables us to predict which regions of the growth plate are at high risk of damage. First, we have independently analyzed the microscale to simulate the microstructure under shear and tensile tests to calibrate the damage model. Second, we have employed the model to simulate damage occurring in standardized healthy and affected femurs during the heel-strike stage of stair climbing. Our results indicate that on the macroscale, damage is concentrated in the medial region of the growth plate in both healthy and affected femurs. Furthermore, damage to the affected femur is greater than damage to the healthy femur from both the micro- and macrostandpoints. Maximal damage is observed in territorial matrices. Furthermore, simulations illustrate that little damage occurs in the reserve zone. These findings are consistent with previous findings reported in well-known experimental works.

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Abbreviations

SCFE:

Slipped capital femoral epiphysis

3D:

Three-dimensional

ECM:

Extracellular matrix

RVEs:

Representative volume elements

MPC:

Multi-point constraint

TM:

Territorial matrix

ITM:

Interterritorial matrix

UEL:

User element

UMAT:

User material

RZ:

Reserve zone

NSNPA:

Neck shaft-neck plate angle

NDA:

Neck-diaphysis angle

ATD:

Articulotrochanteric distance

PhW:

Physeal width

PDA:

Physis-diaphysis angle

PSA:

Physeal sloping angle

NSPSA:

Neck shaft plate shaft

HR:

Head radius

FNW:

Femoral neck width

MCW:

Medullary channel width

DW:

Diaphyseal width

FAL:

Femoral axis length

PT:

Plate thickness

NL:

Neck length

DL:

Diaphyseal length

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Acknowledgments

This research was supported by the Spanish Ministry of Economy and Competitiveness (Grant DPI2012-32880) and the European Commission Seventh Framework Program (FP7/2007-2013) through Grant agreement number 286179. This project is partly financed by the European Union (through the European Regional Development Fund).

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Correspondence to M. J. Gómez-Benito.

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Appendix

Appendix

1.1 Irreversible damage model

The algorithmic form of the constitutive update using an implicit backward Euler scheme is given below (Matous et al. 2008)

$$\begin{aligned} \omega ^{n+1}&= \omega ^n + \frac{\Delta {t\mu }}{1 + \Delta {t\mu }} \left[ G(\psi _{\mathrm{in}}^{n+1}) - \xi ^n \right] \end{aligned}$$
(12)
$$\begin{aligned} \xi ^{n+1}&= \frac{\xi ^n+ \Delta t \mu G(\psi _{\mathrm{in}}^{n+1})}{1+\Delta t \mu } \end{aligned}$$
(13)

where \(n+1\) represents the current time step and \(i\) is the current Newton–Raphson iteration.

1.2 Discretization of equations at the microscale

The microscale and damage models described in Sect. 2 are implemented into finite element method. At the microscale, Eq. 5 is discretized as following to guarantee the incremental accuracy during integration of developed equations (Matous et al. 2008)

$$\begin{aligned} \mathbf{R}(^{mi} \mathbf{U})&= \frac{1}{\mid \varTheta \mid } \underset{e=1}{\overset{N_\mathrm{el}}{\mathbb {A}}} \left[ \int _{\varTheta ^e} \mathbf{H}^T \, {\mathbb {C}} \mathbf{H} \hbox {d}{\varTheta ^e} \, ^{mi} \mathbf{U} \right. \nonumber \\&\quad +\,\left. \int _{\varTheta ^e} \mathbf{H}^T \, {\mathbb {C}} \mathrm {d}{\varTheta ^e} \, ^{ma} \varvec{\varepsilon }\right] = 0 \end{aligned}$$
(14)

where \(\mathbf{R},\,\mathbf{H}\), and \(^{mi}\mathbf{U}\) are the residual vector, gradient matrix and degree-of-freedom vector of the microscale, respectively. \({\mathbb {A}}\) is assembly operator and \(N_\mathrm{el}\) denotes the number of finite elements. In Eq. 14, the second term is the contribution of the macroscale in microscale acting as a force term on microdomain. Eq. 14 is a system of the nonlinear equations, which are solved using the Newton–Raphson iterative procedure to obtain a linearized system of equations as (Matous et al. 2008)

$$\begin{aligned} D\mathbf{R}(^{mi} \mathbf{U}^{n+1}_{i}) \triangle ^{mi} \mathbf{U} = -\mathbf{R}(^{mi} \mathbf{U}^{n+1}_{i}) \end{aligned}$$
(15)

For the constitutive model described in Sect. 2, stiffness matrix can be linearized using the system of nonlinear Eq. 15 as

$$\begin{aligned}&D\mathbf{R}(^{mi} \mathbf{U}^{n+1}_i) \triangle ^{mi} \mathbf{U}\nonumber \\&\quad = \frac{1}{\mid \varTheta \mid } \underset{e=1}{\overset{N_el}{\mathbb {A}}} \left[ \int _{\varTheta ^e} \mathbf{H}^T (1-\omega ^{n+1}) \, ^{mi} \mathbf{C} \mathbf{H} \hbox {d}{\varTheta ^e} \right. \nonumber \\&\qquad - \int _{\varTheta ^e} \mathbf{H}^T \, ^{mi} \mathbf{C} \mathbf{H} \, ^{mi} \mathbf{U}^{n+1}_i \otimes \frac{\partial \omega ^{n+1}}{\partial \, ^{mi} \mathbf{U}^{n+1}_i} \hbox {d}{\varTheta ^e}\nonumber \\&\qquad \left. + \int _{\varTheta ^e} \mathbf{H}^T \, ^{ma} \varvec{\varepsilon }\otimes \frac{\partial \omega ^{n+1}}{\partial \, ^{mi} \mathbf{U}^{n+1}_i}\hbox {d}{\varTheta ^e} \right] \bigtriangleup \, ^{mi} \mathbf{U} \end{aligned}$$
(16)

where each associated term can be described as secant, stiffness correction and macroscale contribution parts, respectively. Derivation of damage can be obtained as

$$\begin{aligned} \frac{\partial \omega ^{n+1}}{\partial \, ^{mi} \mathbf{U}^{n+1}_{i}} = \frac{\partial \omega ^{n+1}}{\partial \psi _\mathrm{in}^{n+1}} \frac{\partial \psi _\mathrm{in}^{n+1}}{\partial ^{mi} \varvec{\varepsilon }^{n+1}} \frac{\partial ^{mi} \varvec{\varepsilon }^{n+1}}{\partial \, ^{mi} \mathbf{U}^{n+1}_{i}} \end{aligned}$$
(17)

where

$$\begin{aligned} \frac{\partial \omega ^{n+1}}{\partial \psi _\mathrm{in}^{n+1}} = \frac{\Delta t \mu }{1+\Delta t \mu } \frac{\partial G}{(\partial \psi _\mathrm{in}^{n+1})} \end{aligned}$$
(18)

and according to Eq. 9

$$\begin{aligned} \frac{\partial G}{\partial \psi _\mathrm{in}^{n+1}} = \frac{p_2}{p_1 \psi _{b}} {\left( \frac{\psi _\mathrm{in} - \psi _{b}}{p_1 \psi _{b}}\right) }^ {p_2 - 1} e^{-{\left( \frac{\psi _\mathrm{in} - \psi _{b}}{p_1 \psi _{b}}\right) }^ {p_2}} \end{aligned}$$
(19)

1.3 Coupling micro- and macroscales

The calculation starts from the macroscale. For each Gauss point from the macroscale, a strain tensor is transferred to the microscale at each increment of the macroscale. As a result of the microscale, for each Gauss point of the macroscale, we look for a unique damage value to transfer to the UMAT to adapt material properties of the growth plate to the macroscale. According to the presented model, the damage of each element in the microscale, \(\omega _i\), is calculated so that the average volumetric damage in the macroscale is as follows:

$$\begin{aligned} D=\frac{\sum \nolimits _{i=1}^{N_\mathrm{el}}\omega _i V_i}{\sum \nolimits _{i=1}^{N_\mathrm{el}}V_i} \end{aligned}$$
(20)

where \(N_\mathrm{el}\) denotes the number of microscale elements and \(V_i\) denotes the corresponding volume of each element in microscale. This damage must be passed to UMAT subroutine (macroscale). Since the damage process is irreversible process, the homogenized damage calculated in microscale is accumulated at each increment of macroscale in UMAT subroutine. Thus, the material properties (stiffness matrix) of the growth plate at the macroscale can be adapted as

$$\begin{aligned} {\mathbb {C}}^{\prime } = (1 - D) \, \mathbf{C}^{\prime } \end{aligned}$$
(21)

where \(\mathbf{C}^{\prime }\) is the intact stiffness matrix of the growth plate at the macroscale.

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Farzaneh, S., Paseta, O. & Gómez-Benito, M.J. Multi-scale finite element model of growth plate damage during the development of slipped capital femoral epiphysis. Biomech Model Mechanobiol 14, 371–385 (2015). https://doi.org/10.1007/s10237-014-0610-8

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