The nonlinear mechanical properties of engineered soft biological tissues determined by finite spherical indentation

Transcription

1 The nonlinear mechanical properties of engineered soft biological tissues determined by finite spherical indentation BMTE Bart van Dijk March 2007 Internship at Eindhoven University of Technology Supervisors: Cees Oomens Martijn Cox Debby Gawlitta Debbie Bronneberg Eindhoven University of Technology Department of Biomedical Engineering

2 Abstract The very low stiffness of soft biological tissues excludes them from most conventional tensile tests. Indentation tests can be used to determine mechanical characteristics of soft planar biological tissues, but are often limited to linear elastic behaviour. This study uses a nonlinear Neo-Hookean model to determine the typical nonlinear material behaviour of soft biological tissues. To test the feasibility of this approach, two model systems used in the research to examine the development of pressure ulcers are investigated. Bio-Artificial Muscles (BAMs) and tissue engineered epidermis samples (EpiDerm) are deformed using finite indentation experiments. The force on the spherical indenter was measured as a function of the indentation depth. A three dimensional finite element model assuming nonlinear incompressible Neo-Hookean material behaviour was applied to determine their initial stiffness (G 0 ) and the degree of nonlinearity (n). The thickness of the samples was determined from the experimental data. Two groups of BAMs were examined, either with or without the addition of C2C12 myoblast cells. The EpiDerm samples where either modeled as a whole, or their tissue engineered epidermis layer only. Both the samples exhibit nonlinear Neo-Hookean behaviour under large deformation. The BAMs with C2C12 myoblasts have a higher initial stiffness than the BAMs without these cells, and their degree of nonlinearity is higher as well. The BAMs with cells are thinner than the BAMs without cells. The approach makes it possible to determine the material properties of the epidermis layer of the EpiDerm samples without damaging the sample. In conclusion, a method for determining the nonlinear mechanical properties of very soft and very thin biological materials was applied successfully. The used method is not limited to investigate mechanical characteristics of the two model systems. It is expected to be a feasible approach for characterization of the nonlinear mechanical behaviour of soft planar biological tissues in general. 1

3 Contents 1 Introduction 3 2 Materials and Methods Experiments Bio-Artificial Muscle samples EpiDerm samples Indentation experiments Numerical Model Sample dimensions Constitutive model Parameter estimation Results Sensitivity analysis Bio-Artificial Muscle samples Tissue engineered epidermis samples Discussion 12 A All fits 14 A.1 BAMs with cells A.2 BAMs without cells A.3 EpiDerm samples including membrane A.4 Epidermis layer only B Estimated parameters 18 B.1 BAMs B.2 EpiDerm samples

4 1 Introduction The very low stiffness and strength of soft biological tissues in general and early stage engineered tissues in particular limit the feasibility of conventional tensile tests for their mechanical characterization. Furthermore characterization of local mechanical properties is not possible with tensile tests. Indentation tests may provide an alternative. In this study, a spherical indentation test is proposed in which large deformations are applied to determine the mechanical properties of two model systems of engineered soft biological tissue. A bioartificial muscle model (BAM) has been developed by D. Gawlitta et al [5]. These samples have a very low stiffness, which excludes tensile tests. EpiDerm is an in vitro model of the epidermis and used by D. Bronneberg et al [2]. These are very thin, and consist of a membrane and an epidermis layer. Quantifying the effect of this membrane on the material properties, without damaging the sample, can only be tested by indentation. Both model systems are used in the research for pressure ulcer development. Pressure ulcers are areas of soft tissue breakdown that result from loading of the skin and the underlying tissues [1]. The role of deformation in the development of pressure ulcers is not fully understood. Knowledge of the mechanical properties of these model systems is necessary for studying deformation induced damage pathways. In a previous study, BAMs have been mechanically characterized using a linear Neo-Hookean model [8]. This approach is only valid for small deformations, and not for the large deformations that were applied in the experiments in that study. To quantify the behaviour at large indentations, a nonlinear Neo-Hookean model is used on these experimental results. Furthermore, the material parameters of the EpiDerm samples are estimated as well, using the same model. A parameter estimation algorithm is used to compare the experimental data with the numerical data, and estimate the material parameters of the samples. The present study will focus on the mechanical characterization of BAMs with and without C2C12 cells. Because matrix remodeling only occurs in the BAMs with cells, it is expected that they are stiffer than the BAMs without cells. The EpiDerm samples, which consist of a layer of engineered epidermis tissue and a stiff membrane, will be modeled in two different ways. Either the whole sample or the engineered epidermis layer will be modeled. Because the membrane is expected to be more stiff than the engineered epidermis layer, modeling of the whole sample will overestimate the actual stiffness of the epidermis layer. 3

5 2 Materials and Methods 2.1 Experiments Two different biological samples were examined in this study, bio-artificial muscles and tissue engineered epidermis samples. In order to investigate their compressive material properties, they were indented with an indentation device. In the following subsections, the two different samples, as well as the indentation experiments are described Bio-Artificial Muscle samples Bio-Artificial Muscles (BAMs) were designed and cultured by D. Gawlitta et al [6] by modifying the protocols of Vandenburgh et al [9]. Two different groups of BAMs were grown in 35mm culture dishes, between two pieces of velcro as anchoring points. A mixture of collagen I gel and Matrigel [BD Biosciences, Alphen a/d Rijn, The Netherlands] was mixed either with or without cells. C2C12 myoblasts, a mouse skeletal myoblast cell-line, were used. After one day of incubation in growth medium, the medium was changed to differentiation medium, to initiate maturation of the cells into myotubes. This process occurs only in the samples that contain the C2C12 myoblasts [5]. In total, 23 BAMs with cells and 14 BAMs without cells were tested EpiDerm samples Tissue engineered epidermis samples called EpiDerm samples (EPI-200, Mat- Tek Corporation, Ashland, MA, USA) are an in vitro model of the epidermis ( = 9mm, thickness = ± 150 µm). They consist of human-derived epidermal keratinocytes, which have been cultured on porous, flexible membranes of cell culture inserts at the air-liquid interface to form a multilayered, differentiated model of the human epidermis [2]. The thickness of the epidermis layer of the samples is ± 130 µm and of the membrane layer is ± 16 µm. After overnight culture at 37 and 5% CO 2, the four samples were directly used in the loading experiment. Because the membrane part of the sample is significantly stiffer than the epidermis layer, the material properties of the samples were investigated in two different manners, using the same experimental data. First, the thickness of the whole sample was used as the thickness of the mesh. Then, the hypothesis that the membrane part of the samples was infinitely stiff was taken into account. The thickness of the epidermis layer was then used as the thickness of the mesh. This enabled the determination of the material properties of the tissue engineered epidermis layer only, instead of the properties of the whole EpiDermis sample Indentation experiments For both the samples, the experiments were performed in a similar manner. The samples were placed in a sample holder, with a glass surface. The EpiDerm samples were placed with the stiff membrane on the bottom of the glass surface. A spherical indenter was used to compress the samples slowly and the axial 4

6 force and the position of the indenter were measured. An indenter with a diameter of 2 mm was used. A confocal microscope was used to ensure that the indenter was in position. The indentation device contained two leaf-springs, but only the vertical leaf-spring was used. This leaf-spring could move with speeds from 0.05 to 1500 µm/s. The samples were indented for approximately 80% at a velocity of 50 µm/s. The force on the indenter was measured and plotted against the indentation. 2.2 Numerical Model Sample dimensions The finite element model required the dimensions of the samples to determine the material properties. The length and width were measured during the indentation test. The data from the indentation experiments were used to determine the thickness. First, the indenter was moved till it reached the glass plate, giving the position of the bottom of the sample. Then the experiment was performed, and by using the force data, the position where the indenter touches the top of the sample was clearly visible Constitutive model For the mechanical characterization, a numerical model was fit to the forceindentation curves [4]. In absence of inertia and body forces, the balance of momentum is given by σ = 0 (1) where σ represents the Cauchy stress tensor. Conservation of mass is preserved by the incompressibility condition J 1 = 0 (2) with J = V V 0 = det(f) (3) The volume in the current and reference configuration are denoted by V and V 0 respectively and F is the deformation gradient tensor. For incompressible materials the stress tensor σ is divided in an extra stress part τ and a hydrostatic pressure p. σ = pi + τ (4) where I represents the unity tensor. The material behaviour is described as an incompressible nonlinear Neo-Hookean material, therefore the extra stress tensor τ is linearly related to the left Cauchy-Green deformation tensor: τ = G(B-I) (5) where G is the shear modulus. The left Cauchy-Green deformation tensor B is defined by B = F F T (6) 5

7 The shear modulus is described for this nonlinear Neo-Hookean material [4] as ( ) n tr(b) G = G 0 (7) 3 where G 0 is the initial stiffness. The parameter n is used to control the degree of nonlinearity of the constitutive equation: n > 0 indicates stiffening of the material with increasing stretches and n < 0 indicates softening. If n equals 0, G equals G 0, and the material exhibits linear Neo-Hookean behaviour. The finite element package SEPRAN is used for solving the balance equations. The samples were meshed with = 200 elements. Under symmetry assumptions, a quarter tissue block was modeled. The element type is a Taylor-Hood triquadratic brick element [3], which has continuous pressure interpolation and satisfies the Babuska-Brezzi condition. Mesh refinement is applied in x and y direction; the elements at the site of indentation are fifteen times smaller than the largest elements in both directions as can be seen in Figure 1. The indentation is applied in vertical direction in 60 steps, resulting in a maximal global indentation of 60% of the thickness of the sample. Figure 1: Top view (left) and front view (right) of the 3D model, showing indenter and boundary conditions Parameter estimation To determine the material parameters, G 0 and n, a Gauss-Newton estimation algorithm is used [3]. The parameters are estimated iteratively, by perturbing the parameters one by one. Measurements are stored in a column m = [m 1,..., m N ] T, where N is the number of measurement points. The material parameters are stored in a column θ = [θ 1, θ 2 ] T. The finite element model is used to calculate the response h to a given set of parameters θ and input ũ. The input ũ may for example consist of prescribed forces and/or displacements. 6

8 The estimation algorithm is aimed at minimizing the difference between h (ũ, θ ) and m. The quadratic objective function J(θ ), is defined to quantify the goodness of fit, J(θ ) = [m h (ũ, θ )] T V [m h (ũ, θ )] (8) where V is a positive definite symmetry weighing matrix and in this case:v = I. The estimation is continued until parameter changes are smaller than a critical value. A value of n smaller than 0 represents softening of the material with increasing stretches. This is not common in biological samples, therefore the estimated value of n has to be positive. 7

9 3 Results 3.1 Sensitivity analysis A sensitivity study is performed on one of the samples, as described by Hendriks [7]. For a number of combinations of the two parameters, the objective function J(θ ) is calculated. In Figure 2, a sensitivity plot is shown for one of the BAMs with cells. This sensitivity plot is a contour plot of calculated values of Figure 2: Sensitivity plot of the objective function for various combinations of n and G0. J(θ ) for these combinations of the two parameters. It shows that there is only one combination of parameters for which the value of the objective function is minimal for this sample. This means that the two parameters that are calculated by the estimation algorithm, are the only correct parameters. 8

10 3.2 Bio-Artificial Muscle samples Force indentation curves of the BAMs are shown in Figure 3. The best and the worst fits are shown for both the BAMs with and without cells. For the BAMs with cells, both the best and worst experimental results are fit very well by the model. For the BAMs without cells, the best experimental data is fit very well, but the worst is fit reasonably. 3.1: Best fit of the BAMs with cells 3.2: Worst fit of the BAMs with cells 3.3: Best fit of the BAMs without cells 3.4: Worst fit of the BAMs without cells Figure 3: Representative force indentation curves of the BAMs. The best and worst numerical fits are shown for the BAMs with and without cells. The results show that the average initial stiffness (G 0 ) of the BAMs with cells is 390 Pa. For the BAMs without cells, this value is 93 Pa, about 4 times smaller than with cells. So the BAMs with cells have a higher initial stiffness than the BAMs without cells (p < 10 8 ). Furthermore, the average degree of nonlinearity (n) is 2.75 for the BAMs with cells and 0.52 for the BAMs without cells. The box plot (Figure 4.2 on page 10) shows that the BAMs without cells have a skewed With Cells Without Cells G 0 (Pa) 390 ± ± 40 n 2.75 ± ± 0.72 Average thickness (mm) 0.75 ± ± 0.30 Table 1: Average estimated parameter values and standard deviations of the BAMs. 9

11 distribution for n. Because the standard statistical tests are only valid for normal distributed data, a nonparametric (Wilcoxon rank-sum) test was used. This test shows that the degree of nonlinearity is significantly higher for the BAMs with cells than for the BAMs without cells (p < 10 5 ), and the box plot confirms this as well. The cause of this skewed distribution is that the nonlinearity parameter for the samples without cells is close to zero in 8 of the 14 samples, which means these can be described with linear Neo-Hookean behaviour. As can be seen from Figure 3.4, the fit suggests that a negative nonlinearity value (n) would give a better fit. Because softening under increasing stretches is not 4.1: Box plot G 0 of BAMs 4.2: Box plot n of BAMs Figure 4: Box plots of G 0 and n of BAMs with and without cells common in biological samples, an explanation for this behaviour could be that the mechanical integrity of the samples is too low. The average thickness of the samples with cells is 0.75 mm. The average thickness of the samples without cells is significantly larger and around 1.3 mm (p < 10 5 ). All the data is summarized in Table 1 on page 9. 10

12 3.3 Tissue engineered epidermis samples In Figure 5, two representative force indentation curves of the EpiDerm samples are shown. Both the best and the worst experimental data are fit very well by the model. The average initial stiffness (G 0 ) of the whole EpiDerm samples is 5.1: Best fit of the EpiDerm samples 5.2: Worst fit of the EpiDerm samples Figure 5: Force indentation curves of the EpiDerm samples. Both the best and worst experiments are fit very well by the model. 40 kpa. For the tissue engineered epidermis layer only, this parameter is 33 kpa. The average nonlinearity parameter (n) for the whole sample is For the epidermis layer only, this parameter is When comparing these groups, they are not significantly different from each other, for both G 0 and n. This data is summarized in Table 2. But the effect of the membrane on the material parameters can be demonstrated when the difference between with and without membrane is calculated, for every individual sample: G 0 = G + 0 G 0 (9) n = n + n (10) with G + 0 the initial stiffness of the whole EpiDerm sample, G the initial stiffness 0 of the epidermis layer only, n + the degree of nonlinearity of the whole EpiDerm sample and n the degree of nonlinearity of the epidermis layer only. The average difference of the initial stiffness ( G 0 ) is significantly larger than zero (p < 0.01) and the average nonlinearity difference ( n) is not (p = 0.56). This demonstrates the feasibility of modeling the membrane as an infinite stiff material. Additionally, modeling the membrane layer as an infinite stiff material reduces the value of G 0, for every individual sample. Whole sample Epidermis layer only G 0 (kpa) 40 ± ± 11 n 1.52 ± ± 0.6 Average thickness (mm) ± ± 0.02 Table 2: EpiDerm samples 11

13 4 Discussion Finite indentation experiments are applied successfully to determine material properties of both very soft and very thin biological tissues. Additionally, the examined materials can be described by the proposed nonlinear Neo-Hookean material model, and their mechanical characteristics can be investigated. Furthermore, indentation tests make it possible to determine the material properties of the epidermis layer of the EpiDerm samples, without damaging the sample. Addition of cells to BAMs, increases their initial stiffness with a factor of 4 and increases their nonlinear behaviour as well. The BAMs with cells are significantly thinner than the BAMs without cells. Although the EpiDerm samples are very thin, it is still possible to determine their material properties using an indentation device. When modeling the samples as a whole, instead of the engineered epidermis layer only, there is a significant increase in the initial stiffness for every sample, but not for their degree of nonlinearity. Preliminary studies have shown that when a sample is indented three times, the second and third measurement are more similar than the first one. The BAM samples are indented only once, instead of three times, because this was not known at the time of these experiments. In future studies, it is highly recommended to indent the samples three times. There is more noise on the experimental data of the BAMs without cells, than on the other samples, because these samples are very soft. Nevertheless, for most of the samples, their material properties can be examined. Using a more accurate setting on the indenter or reducing speed of indentation might reduce this noise. In some of the data, especially the samples without cells, it is not always visible when the indenter touches the sample exactly, and for these samples the thickness might be inaccurate. When the samples are dry, this is not clearly visible. An over- or underestimation of the thickness will result in incorrect material parameters, so immediately testing the samples when they are placed in the indentation device is important. The first discussion point, concerning the EpiDerm samples, is using only four EpiDerm samples will only give a good approximation of the material properties of these samples. Using more samples is highly recommended to get more accurate results. Unfortunately that was beyond the scope of this study. The indenter sometimes deviates from the imposed velocity, which results in deviations in the force signal. In the very thin EpiDerm samples this deviation can be significant, as can be seen in Figure 5.2 on page 11, although the effect on the fit is small. Causes for these irregularities could be an unstable environment during the testing, like disturbances of the experimental setup or a fault in the control of the spring. Reducing speed of indentation and being more careful may help reduce this effect. To determine the material properties of the epidermis layer only, adjusting the model is sufficient. To test this with a tensile test, the membrane layer has to be removed from the EpiDerm sample and this might damage the sample. This approach makes it possible to determine material properties of multilayered planar soft biological tissues. In conclusion, a method for determining the nonlinear mechanical properties of very soft and very thin biological materials was applied successfully. The used approach is a valuable tool to investigate mechanical characteristics of the two model systems in particular and soft planar biological tissues in general. 12

14 References [1] C. V. C. Bouten, C. W. J. Oomens, F. P. T. Baaijens, and D. L. Bader. The etiology of pressure ulcers: Skin deep or muscle bound? Archives of Physical Medicine and Rehabilitation, 84(4): , April [2] D. Bronneberg, C. V. C. Bouten, C. W. J. Oomens, P. M. van Kemenade, and F. P. T. Baaijens. An in vitro model system to study the damaging effects of prolonged mechanical loading of the epidermis. Annals of Biomedical Engineering, 34(3): , March [3] M. A. J. Cox, N. J. B. Driessen, C. V. C. Bouten, and F. P. T. Baaijens. Mechanical characterization of anisotropic planar biological soft tissues using large indentation: A computational feasability study. Journal of Biomechanical Engineering, 128(3): , June [4] N. J. B. Driessen, A. Mol, C. V. C. Bouten, and F. P. T. Baaijens. Modeling the mechanics of tissue-engineered human heart valve leaflets. In Proceedings of ICCB 2005, pages , Lisbon, Portugal, September [5] D. Gawlitta. Compression induced factors influencing the damage of engineered skeletal muscle. PhD thesis, Eindhoven University of Technology, January [6] D. Gawlitta, C. V. C. Bouten, C. W. J. Oomens, and F. P. T. Baaijens. Tissueengineered model for evaluating skeletal muscle damage in pressure ulcers. Internal Poster, April [7] F. M. Hendriks. Mechanical behaviour of human epidermal and dermal layers. PhD thesis, Eindhoven University of Technology, March [8] M. A. A. van Vlimmeren. The mechanical characterisation of tissue engineered skeletal muscle. Internal Report, December [9] H. H. Vandenburgh, M. Del Tatto, J. Ahnsky, J. Lemaire, A. Chang, F. Payumo, P. Lee, A. Goodyear, and L. Raven. Tissue-engineered skeletal muscle organoids for reversible gene therapy. Human Gene Therapy, 7(17): , November

15 A All fits A.1 BAMs with cells Figure 6: All the force-indentation curves of the BAMs with cells and their numerical fits, first part. 14

16 Figure 7: All the force-indentation curves of the BAMs with cells and their numerical fits, second part. A.2 BAMs without cells Figure 8: All the force-indentation curves of the BAMs without cells and their numerical fits, first part. 15

17 Figure 9: All the force-indentation curves of the BAMs without cells and their numerical fits, second part. A.3 EpiDerm samples including membrane Figure 10: All the force-indentation curves of the EpiDerm samples and their numerical fits, first part. 16

18 Figure 11: All the force-indentation curves of the EpiDerm samples and their numerical fits, second part. A.4 Epidermis layer only Figure 12: All the force-indentation curves of the Epidermis layer only and their numerical fits, first part. 17

19 Figure 13: All the force-indentation curves of the Epidermis layer only and their numerical fits, second part. B Estimated parameters B.1 BAMs BAM sample Dimensions [mm] G 0 [Pa] n x 2.0 x x 2.5 x x 2.5 x x 2.5 x x 2.5 x x 3.5 x x 2.5 x x 2.0 x x 3.0 x x 3.0 x x 3.0 x x 3.0 x x 3.0 x x 3.0 x x 3.5 x x 3.0 x x 2.5 x x 2.5 x x 3.0 x x 3.0 x BAM x 3.0 x BAM x 3.0 x BAM x 3.0 x Table 3: Dimensions and estimated parameters of the BAMs with cells. 18

20 BAM sample Dimensions [mm] G 0 [Pa] n x 4.5 x x 5.0 x x 4.5 x x 5.0 x x 5.0 x x 6.0 x x 5.0 x x 4.0 x x 4.0 x zonder x 6.0 x zonder x 4.0 x zonder x 2.5 x zonder x 4.5 x zonder x 5.0 x Table 4: Dimensions and estimated parameters of the BAMs without cells. B.2 EpiDerm samples EpiDerm Thickness [mm] G 0 [kpa] n sample 1a b c a b c a b c a b c Table 5: Thickness and estimated parameters of the whole EpiDerm samples. 19

21 EpiDerm Thickness [mm] G 0 [kpa] n sample 1a b c a b c a b c a b c Table 6: Thickness and estimated parameters of the epidermis layer only. 20