Finite Element Modeling of Human Placental Tissue

Abstract

Motor vehicle crashes account for a large portion of placental abruption and fetal losses. To better understand the material properties of the human placenta, a Finite Element (FE) model of human placenta tissue was created and verified using data from uniaxial tension tests. Sixty-four tensile tests at three different strain rates of 7% strain/s, 70% strain/s, and 700% strain/s from six whole human placentas were used for model development. Nominal stresses were calculated by dividing forces at the grips by the original cross-sectional area. Nominal strains were calculated by dividing cross-head displacement by the original gauge length. A detailed methodology for interpreting experimental data for application to material model development is presented. A model of the tension coupon was created in LS-DYNA and stretched in the same manner as the uniaxial tension tests. The behavior of the material was optimized to the uniaxial tension test using a multi-island genetic algorithm. The results demonstrate good correlation between experiments and the model, with an average difference of 2% between the optimized FE and experimental first principal stress at the termination state. The material parameters found in this study can be utilized in FE models of placental tissues for behavior under dynamic loading.

INTRODUCTION

Motor vehicle crashes are the largest cause of death for pregnant females and the leading cause of traumatic fetal injury mortality in the United States (Weiss, 2001a, Weiss, 2001b, Agran et al., 1987). Specifically, placental abruption accounts for up to 70% of these fetal losses. Therefore, the purpose of this study is to characterize human placental tissues in order to ultimately develop FE models of pregnant occupants.

While there have been many computational efforts on the pregnant occupant (Moorcroft et al., 2003, Duma et al., 2004, Duma et al., 2006, Manoogian et al., 2008b, Delotte et al., 2006), computer modeling of the placenta has not had much precedent. Moorcroft’s computer model of the pregnant occupant had a simple linear elastic placenta material, but the placenta material was not the focus of the study (Moorcroft, 2003).

There have been some experimental tests of placental tissue. Pearlman (Pearlman, 2000) performed tensile testing of the placenta at a 100 mm/min displacement rate and found the average Young’s modulus to be 32 kPa. Benson-Martin (Benson-Martin et al., 2006) performed tensile testing of the placenta at a 10 mm/min displacement rate for 5 cm x 2 cm samples and found the Young’s modulus to be 2.29 MPa. The discrepancy in the Young’s moduli between these two studies shows that further investigation into the material is needed, as well as exploration of the effects of differing strain rates, since these two studies did not use comparable strain rates.

In any FE simulation, one of the most significant inputs which affects the realism of the results is the material model. While linear elastic models have fairly robust material formulations and are often used, they represent an ideal that is not physically realistic for placental tissue. The material model created should not be overly simple and it should be validated with experimental tests.

Correlating experimental data with FE material behavior becomes especially important when simulating soft tissues such as the placenta. While most FE material models are widely used for structural analysis of inorganic objects and have been well validated, the behaviors of biological soft tissues are not understood as well. As a result, many of the biological soft tissue material models used in simulations are based upon hyperelastic models that have been fitted to experimental data. The usage of these hyperelastic materials is justified often with the fact that the material parameters are fitted to the real experimental stress vs. strain curve. However, the material parameters for the stress vs. strain curves are only one aspect of the behavior of the model. Effects of strain rate, stress relaxation, and compressibility must also be accounted for. Therefore, an iterative approach that optimizes the error between the fitted model behavior and the actual experimental behavior has the potential to generate more accurate material parameters.

A material model for the placenta was created by attempting to match the nominal stress and nominal strain in a coupon tension test in the FE simulation with the same measurements in the actual experiment. In addition to optimizing the model parameters to reduce error between experimental tests and the FE model, this study also used a material model that employs a direct table lookup method to calculate the stress vs. strain curve.

METHODS

The study is composed of three parts: experimental material testing, FE mesh creation, and optimization of material parameters. The methods described below detail each of these steps.

1. Experimental material testing

The experimental data is based on the work of Manoogian (Manoogian et al., 2008a) where the placenta was tested without the chorion attached. The experimental methods given here represent a brief overview of that work. The focus of this study is the interpretation and utilization of that data in the development of an FE material model, not the actual experimental procedures.

A total of 64 uniaxial tensile tests were performed on six whole human placentas (Manoogian et al., 2008a). Following delivery, the tissue was stored in saline solution and tested within 36 hours. The placenta tissue was prepared into tensile coupons and pulled to failure using a custom designed system of linear motors at one of three strain rates: 7% strain/s, 70% strain/s, and 700% strains/s. These values represent a range of loading rates applicable to a motor vehicle crash. Local strain was calculated for each specimen using dot tracking with high speed video, and nominal stress was determined using original cross-sectional area measurements combined with the load sensor data from each grip. The cross-sectional area was calculated by multiplying the original width by the thickness of the specimen at the center, using the assumption of a rectangular cross-section. The number of specimens for each strain rate is shown in Table 1.

Table 1:

Experiment strain rates and specimen count

Strain Rate (% strain/s)	Specimen count
7% strain/s	22
70% strain/s	22
700% strain/s	20

Engineering strain rates were calculated from a constant displacement velocity divided by the gauge length of the specimen.

The specimens were cut into coupon shapes and stretched to failure from both ends at the specified strain rates. The strain rate was multiplied by the specimen gauge length to get the displacement rate of the cross-heads. The forces in the grips and the displacement of the grips were measured over time. The experimental material testing part of this study was performed by Manoogian et al (2008).

Experimental data processing

After data was collected, the data was processed with MATLAB to prepare it for validation with FE simulations.

First, any data before t=0 was truncated. This was performed to remove data where displacement was not being applied. The data for the displacement of the specimen between the grips was then normalized by the gauge length into the total stretch of the specimen. Strain was calculated from the formula ɛ=λ-1, where ɛ is the strain and λ is the stretch. Engineering stress was calculated by dividing the force at the end point by the original cross-sectional area. Engineering stress was used instead of the force in order to account for geometry. The stress vs. strain characteristic was used as the comparison curve.

To prepare experimental data for input into the FE simulation, the results of all the tests for a specific strain rate were averaged to get a characteristic curve for that strain rate. There were two sets of curves that were needed as input for the FE simulation: the strain vs. time response, and the stress vs. strain response.

Strain vs. time

The strain vs. time response for all the tests, separated into discrete strain rates, are shown in Figure 1. The strain is plotted instead of the displacement so that effects of differing specimen lengths are minimized. Strain rate is more easily controlled at lower strain rates, and was well controlled from 10–90% of the time for the two lower rates and from 40–90% of the time for the higher rate.

Figure 1:

Specimen strain vs. time for all three strain rates

The strain rate vs. time responses for a given strain rate can have differing end points for different specimens, as the specimens break at different times. Each breakpoint is shown as a slight horizontal line that protrudes from the constant slope strain vs. time plot. This effect is magnified in Figure 2.

Figure 2:

Effects of different termination points on strain

It can be observed that the different termination point creates changes in slope at the moment before the specimen breaks, which gives the strain vs. time response a unique shape that depends on the time of failure. A simple average of all the curves would remove this behavior and only show this change in slope for the test with the longest strain before failure. Furthermore, simple averaging would also create discontinuities in strain vs. time response due to the breaking of individual samples. A third problem with simple averaging is that the termination point of the characteristic average would always be the termination point of the test with the longest run time. This is not the best way to create a characteristic average of the strain vs. time curve. Extrapolating the strain vs. time curve would also create problems with the end point. If a linear extrapolation routine was used, the final average would have a linear segment as most of the curves appear linear before the curve ends.

The averaging algorithm developed in this study uses a normalized time scale to scale all the curves first by the break point, so that the final displacement value for all the curves become 1. Then averaging was performed. Not only does this preserve the general shape of the curves, but it also allows flexibility in rescaling the curve to an average termination time, instead of the longest time (Lessley, 2004). The strain rate vs. normalized time of the raw data is shown in Figure 3 (raw data filtered at 60 Hz for 700% strain/s, 6 Hz for 70% strain/s, and 0.6 Hz for 7% strain/s according SAE J211 algorithm). It is seen that the general shape of the strain vs. time curve has been preserved. There is an initial ramp up to the target strain rate, and a decrease in the strain rate before the specimen breaks.

Figure 3:

Strain rate (log scale) vs. time for all three strain rates. The dark black line represents the target rate.

To validate filtering levels were appropriate, the filtered curve was subtracted from the unfiltered curve to obtain the noise. The mean of the noise across all experimental tests is then observed to see if there is significant deviation from zero, indicating significant signal content. This is shown in Figure 4. Note that the filtering is performed on the raw data, not the normalized data, due to the fact that time information is used in the filtering algorithm. It can be seen that there is no significant drift in the mean noise response.

Figure 4:

Noise subtracted from the strain vs. time curve

Stress vs. strain

The stress vs. strain curves were processed in a similar manner to the strain vs. time curve. The curve was first scaled in the x-axis by the maximum strain for the individual specimens, and then it was also scaled in the y-axis by the maximum stress. The end results are shown in Figure 5. Note that normalizing stress or force would produce identical results.

Figure 5:

Normalized stress vs. strain for all strain rates

The resulting curves were also fitted to a fourth order polynomial in order to create a function. This function allows interpolation between points without loss of resolution and also ensured that the curve passes through (0, 0) by not allowing a constant in the equation. The function also offers limited smoothing of the data. After fitting the fourth order polynomial, scaling factors were needed to re-scale the functions of the normalized curves back to stress vs. strain curves. This was done by taking a simple 2D average of all the specimen end points for each strain rate, as shown in Figure 6.

Figure 6:

Individual and average end points for stress vs. strain

2. FE model

Specimen dimensions

The experimental specimens ranged in lengths from 24 mm to 48 mm, thicknesses from 2.68 mm to 7.7 mm, and gauge widths from 4.6 mm to 19 mm. The FE model created used the average of these values. These values are shown in Table 2.

Table 2:

FE model specimen dimensions

Type	Value
Gauge Length	31.9 mm
Gauge Width	7.0 mm
Gauge Thickness	4.2 mm
Grip Width	14.9 mm

The FE model for the placental tissue is shown in Figure 7. The model was hand meshed in an attempt to contour the specimen to the expected deformation. The element sizes were chosen to ensure accuracy and stability (approximately 1 mm element side length). The final dimensions of the FE coupon are shown in Figure 7.

Figure 7:

Specimen dimensions and boundary conditions

Boundary conditions

The applied boundary conditions consisted of applied velocities at the top and bottom layer of nodes. The applied velocity at each point was half of the total velocity, so that the specimen was stretched at both ends according to the target strain rate. Forces were tracked at the top face of the middle layer of elements, while displacement was controlled and based on specimen end (grip) location.

The strain rate applied is shown in Figure 8. Three simulations were set up, each representing a unique strain rate. The strain rate applied was smoothed using a spline-based interpolation to remove irregularities in the curves. This was most important for the highest strain rate, wherein the initial overshoot was to corrected to have a more constant strain rate. This was a necessity to satisfy the assumption of the material model that input curves represent distinct strain rates.

Figure 8:

Strain rate applied

Material properties

The material model used in this study is *MAT_SIMPLIFIED_RUBBER (DuBois, 2003). The material model was chosen after evaluating 14 rubber/foam material models in LS-DYNA. Simulations were run using each of the models and the stress vs. strain response for each strain rate was evaluated against the target (average) stress vs. strain for that strain rate. It was found that the simplified rubber material offered the closest potential match to the target curve and also offered flexibility in the strain rate input.

A list of material parameter inputs into the *MAT_SIMPLIFIED_RUBBER deck is detailed in Table 3. Note that there were some additional properties including simulation parameters, which are not relevant to the current study, and stress vs. strain curves, which are explained separately.

Table 3:

Simplified rubber material properties

Parameter	Value
Bulk Modulus	1.91 MPa
Shear Modulus for Damping	2.65 kPa
Stress Limit for Damping	26.5 Pa
Tension Type	Rate effects in loading only
Poisson’s ratio	0.3
Stress vs strain	(set of curves)

The density chosen was approximately that of water. The bulk modulus was calculated from the average slope of stress vs. strain relationship by using linear elastic relationships.

No damping was necessary for these simulations. The tension type chosen allowed the runs to be more stable as unloading rate effects caused numerical noise in the solution to the simulation. The Poisson’s ratio chosen allowed the most realistic behavior of the model.

The material properties also allowed direct input of a set of curves at discrete strain rates that define the force vs. change in gauge length. During element stress update, the stress calculation is based on an interpolation from this set of curves, or a table lookup. Because the load curves were already defined in terms of engineering stress vs. strain, the specimen gauge length, gauge width, and gauge thickness were all set to one. This allowed the stress vs. strain curve to be inputted into the material deck without scaling.

Stress vs. strain properties utilized for the material were modified to add compression behavior. Since only tension tests were available, compression data were generated from a negated and scaled tangent function so that as the compression reaches 80% strain, the forces approach infinity. This helps to ensure stability in cases of extreme compression of the material.

In addition, due to numerical stability concerns, the curves cannot cross each other except at the origin. They also must start and end at the same level of strain. To achieve curves conforming to these rules, the normalized curves for each strain rate could not be scaled independently. Instead, a global characteristic average for all tests regardless of strain rate must be first created, and this curve scaled to the end termination strain for each strain rate. This ensures that the input curve for each strain rate does not intersect. After the curves were scaled back, the curves were truncated to 30% strain. The 30% strain threshold was used because the two lower strain rate curves began converging after this strain level, a phenomenon that cannot be accounted for by the material model. After the truncation was performed, a tangent function extrapolation was added after the threshold to define behavior past 30% strain.

The results of these operations are shown in Figure 10. While some variation in the stress-strain relationship for each strain rate is lost with these operations, it is necessary to ensure stability of the simulation.

Figure 10:

Stress vs. strain curves input to the material deck

Simulation parameters

In addition to the material specific parameters mentioned in the previous section, some simulation-specific parameters are also required. No hourglass control was used in this simulation as it was stable without hourglass control.

The simulations are run using the SMP version of LS-DYNA (Livermore Software Technology Corporation) explicit code (971 R4.2) on an Intel Pentium 4 (3.4 GHz, 3GB of RAM). The simulation end times for each strain rate are 6.6 s, 0.66 s, and 0.066 s for strain rates 7% strain/s, 70% strain/s, and 700% strain/s, respectively.

The forces were measured with the LS-DYNA section force keyword and the displacements between the crossheads were measured with the node out keyword.

3. Optimization

Three simulations were created, one for each strain rate. An optimization routine was performed on all three simulations in which the difference between the FE output and the experimental stress vs. strain was optimized by varying the material input curves.

The stress in the FE model was calculated by dividing the force in the cross-section in the middle by the initial cross-section area as part of the optimization procedure. Engineering stress was used instead of true stress to ensure the FEM was comparable to the experimental results. The strain of the specimen was calculated by taking the difference between the crossheads and dividing it by the initial gauge length. This was performed also to ensure that the strain was comparable between the FEM and experimental strain.

The optimization algorithm used was a Multi-Island Genetic Algorithm (MIGA). The specifications for the MIGA algorithm are shown below in Table 4.

Table 4:

MIGA parameters

Parameter	Value
Size of subpopulation	10
Number of islands	5
Number of generations	10
Total number of runs	500
Gene size	32
Rate of crossover	1
Rate of mutation	0.01
Rate of migration	0.5
Interval of migration	5
Elite size	1
Relative tournament size	0.5

A total of 500 runs were performed, which took an estimated 10 hours. There were three objective functions that were minimized: one objective function for each strain rate. The objective function for each strain rate was defined as the average absolute difference between the FE stress vs. strain response and the experimental response (stress difference).

$$\begin{array}{l}\text{Equation}\hspace{0.17em}\hspace{0.17em}1:\text{Objective}\hspace{0.17em}\text{function}\hspace{0.17em}\text{for}\hspace{0.17em}\text{each}\hspace{0.17em}\text{strain}\hspace{0.17em}\text{rate}\\ Obj=\frac{\sum _{i=1}^N|F{(i)}_{\text{exp}}-F{(i)}_{FEM}|}{N}\end{array}$$

In Equation 1, Obj is the objective function, F(i)_exp is the experimental value at point i, F(i)_FEM is the FE model’s response at point i, and N is the number of points the curves were discretized to. In this study, curves were discretized to 200 points. The curves were truncated at 30% strain to ensure uniformity between experimental results and the FE model at differing strain rates. The absolute value minimization was chosen to minimize the effects of numerical outliers. The global objective function is used in the optimization to minimize error for each strain rate and to penalize variation between the stress-strain characteristics at different strain rates. This creates sensible stress-strain relationships, by ensuring that small variation between curves generated for two strain rates will not be tolerated at the expense of large variation between curves between another combination of strain rates. A global objective function is created from two sources. The first is sum squared error from each strain rate. The second is a penalty function which minimizes large variation in objective function values for different strain rates. The goal of the first set of error terms is to create material parameters that fit the experimental data well at each strain rate. The goal of the second set of error terms is to ensure material parameters do not fit exceptionally well at one strain rate, and poorly at other strain rates. The global objective function for all the strain rates is shown in Equation 2. Obj_Global is the global objective function value, Obj_7%, Obj_70%, and Obj_700% are objective function values for the individual strain rates. The values are squared to penalize large errors.

$$\begin{array}{l}\text{Equation}\hspace{0.17em}2:\hspace{0.17em}\text{Global}\hspace{0.17em}\text{objective}\hspace{0.17em}\text{function}\\ Ob{j}_{\mathit{\text{Global}}}={(Ob{j}_{7\%})}^2+{(Ob{j}_{70\%})}^2+{(Ob{j}_{700\%})}^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{(Ob{j}_{7\%}-Ob{j}_{70\%})}^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{(Ob{j}_{70\%}-Ob{j}_{700\%})}^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{(Ob{j}_{700\%}-Ob{j}_{7\%})}^2\end{array}$$

The independent variables that were optimized were scaling factors for the stress vs. strain curves. Since the simplified rubber material uses a table lookup, where the stress is interpolated from the material model input curves, to calculate the stress vs. strain property at any given point, there were no coefficients of fit that could be used as independent variables. The material properties given in Table 3 are computational parameters that did not have as much of an effect on the stress vs. strain behavior of the material. Therefore, it was decided that by scaling the stress vs. strain curve of the input deck, the material properties could be matched to the experimental stress vs. strain curves since the shape should be similar.

The scale factors were initially set to 0.9 for all strain rates. This was used to give a good initial estimate in terms of the stress vs. strain behavior. The scale factors were constrained between 0.6 and 1.4 for the optimization, as this allowed a 0.3 lower range and a 0.5 upper range from the initial guess. If any run terminated due to instability, the objective function was assigned an extremely high value of 10²⁰.

RESULTS

This section will first describe the results of the final optimized run, which was run 429 (hereafter called the ‘optimal run’) out of 500. The strain rates and stresses, with the behavior of the specimen are compared between the FE model and the experimental tests. Then the optimization results will be described, comparing the errors between the FEM and the experimental results, the progression of the runs, and the stability of the runs.

Strain rate characteristics – Optimal Run

Figure 12, Figure 13, and Figure 14 show the force of the model at the center cross-sectional area and relative velocity of the model between the top and bottom grips. The force vs. time curve has the same shape as the stress vs. time curve of the experimental data. The velocities for the lower two strain rates were within 5% of the target velocity of 2.23 mm/s for the 7% strain/s model and 22.3 mm/s for the 70% strain/s model. Initially, the velocity for the highest strain rate was less than the target velocity of 223 mm/s by more than 10%, but gradually increased to the target velocity around 0.05 seconds into the simulation. The final force for all the strain rates were less than 0.3 N for the time simulated.

Figure 12:

Force and velocity between the crossheads for 7% strain/s

Figure 13:

Force and velocity between the crossheads for 70% strain/s

Figure 14:

Force and velocity between the crossheads for 700% strain/s

The strain rates at mid and top elements in the FE model are recorded by taking the derivative of the mid-plane maximum principal strain.

It can be seen that the elements near the grip (end) consistently undershoot the target strain rate while the elements near the middle consistently overshoot the target strain rate. In addition, strain rate curves for the 700% strain/s simulation show more oscillation in the early portion of the simulation.

Stresses – Optimal Run

The maximum stresses at the end points are shown in Table 5. The longitudinal stresses along the specimen were within 1% of the first principal stresses.

Table 5:

Maximum stress at final state

Strain Rate	First principal stress	Longitudinal stress
7% strain/s	10.14 kPa	10.08 kPa
70% strain/s	12.28 kPa	12.21 kPa
700% strain/s	14.43 kPa	14.34 kPa

The stress vs. strain comparison between the experimental and the FE simulation is shown in Figure 18 and Figure 19. The 7% strain/s and 700% strain/s FE simulations initially followed the behavior of the experimental stresses, but dropped below them at high strains of 31% strain and 20% strain, respectively. However, the 70% strain/s simulation increased above the experimental stress at 22% strain.

Figure 18:

Experimental and FE stress vs. strain

Figure 19:

Experimental and FE stress vs. strain, all three strain rates

Optimization results

The most optimal run, the run with the least difference in stress between experimental results and FE model, is shown below along with the progression of the runs.

Optimal Run

The average difference in stresses between the experimental results and the FE model is shown in Table 6. The global average difference is higher than the 7% strain/s and the 70% strain/s difference, while the 700% strain/s difference is higher than the global average difference.

Table 6:

Average difference in stress from 0–30% strain

Strain Rate	Value
7% strain/s	51.41 Pa
70% strain/s	65.15 Pa
700% strain/s	133.25 Pa
Average of all strain rates	83.27 Pa

The optimal scale factors determined are shown in Table 7. Note that the scale factors presented here are not the overall scale factors for the global average stress vs. strain curve, but additional scale factors for the stress vs. strain curve input for each strain rate.

Table 7:

Optimized scale factors

Strain rate	Scale factor
7% strain/s	0.867
70% strain/s	0.920
700% strain/s	0.835

The scaled stress vs. strain curves based on the optimized scale factors are shown in Figure 20. The scale factors did not change by more than 0.1 from the initial guess.

Figure 20:

Optimized input stress vs. strain curves

Run progression

There were no failed runs in the optimization. This is shown in Figure 21. The top shows the exact globally averaged stress difference for each run, the bottom shows stress difference with a 10 point running average to show the trend better.

Figure 21:

(a) Global stress difference vs. run number (b) 10-point running average of global stress difference vs. run number

The progressions of the stress difference, the objective function value, for each of the three strain rates are shown in Figure 22. The 700% strain/s stress difference has a higher difference on average than the other two strain rates.

Figure 22:

(a) Individual stress difference vs. time (b) 10-point running average of global stress difference vs. run number

The progressions of the scale factors are shown in Figure 23. The 7% strain/s scale factor had large oscillations in its magnitude. None of the scale factors reach the upper bound of 1.2 or lower bound of 0.6.

Figure 23:

(a) Scale factor vs. run number (b) 10-point running average of scale factor vs. run number

DISCUSSION

The experimental stress vs. strain curves show a stiffening of the response as the strain rate increased. This behavior is also seen in the FE model. The experimental stress vs. strain curves also showed that the curve of the two lower strain rates started to converge past 30% strain, which was not captured in the FE material model. The converging of the lower two strain rates suggests that there could be a lower bound to the stiffening behavior such that stiffening only occurs if a certain strain rate limit is exceeded.

The 30% threshold captured most of the initial stiffening behavior of the specimen, but did not include any behavior that could contribute to failure. Since failure was not defined for the material model, the threshold prevented inclusion of any behavior in the input curve that could be a result of failure. However, failure can be defined for the material model as a threshold, and the simulation run out to much higher strains to see the effect of failure. To ensure a stable model, sometimes it is desirable not to define failure.

Previous studies of the placenta utilized linear elastic material models or were experimental studies. Although a hyperelastic model was used in this study, an approximate Young’s modulus can be calculated from the stress vs. strain curve by taking the maximum stress and dividing it by the maximum strain. The comparison results with this Young’s modulus are shown in Table 8. This approximated modulus for this study is slightly softer than the Young’s modulus from Moorcroft (2003), who chose a linear elastic model. Pearlman’s modulus (2000) is approximately twice that of this study while Benson-Martin et al (2006) had a much stiffer modulus of 2.29 MPa for term placenta. Since this study’s material model attempts to mimic the stress vs. strain behavior of the experimental test performed, the deviation from literature value is most likely due to variation in experimental procedures. There is high confidence in the material parameters presented here since they are derived from well-controlled experimental tests with a robust approach to material model parameter determination for a FE model.

Table 8:

Values of placenta properties from the literature

Study	Material	Young’s Modulus (kPa)	Poisson’s ratio
Moorcroft (2003)	Linear Elastic	20	0.45
Pearlman (2000)	Experimental	32.7 +/− 18.6	not reported
Benson-Martin (2006)	Experimental	2290	not reported
Current Study	Simplified Rubber	∼16.7	0.3

Both Benson-Martin and Pearlman used a 10 N load cell. While Pearlman’s specimens were well within the 10 N ranges, Benson-Martin had a much higher maximum force that exceeded 10 N if the force based on their reported peak stresses is calculated by multiplying the peak stress by the cross-sectional area. This seems to indicate that Benson-Martin may have used an inadequate load cell. While Pearlman’s modulus is more than twice that of this study’s approximated modulus, the standard deviation of Pearlman’s modulus is 18.6 kPa, which means there is a high variation in the modulus. This study’s modulus is within one-standard deviation of Pearlman’s modulus.

The average difference between the experimental stress vs. strain curve and the FE stress vs. strain curve was 83.27 Pa; this is less than 2% of the maximum stress experienced by the specimen. The accuracy of the FE model appears to be very good for the lower strains, but the behavior for all the strain rates show that the stress vs. strain curves started to diverge significantly from the experimental data past a strain of 30%. This phenomenon was expected as all the stress difference calculations were performed with the strain truncated at 30% for comparable averaging. Future studies could implement an averaging algorithm that can discretize to variable end strains.

The averaging algorithm used in this study seems to preserve the general shape of the curves. However, due to the requirement of having non-intersecting curves, a globally averaged (averaging all the strain rate curves) stress vs. strain curve had to be used to generate the table of curves. Different scaling factors were used to create the curves for different strain rates, but they all share the same shape. This means that any individual shape differences that were a result of the strain rates were lost. The stiffening behavior was still preserved, but this is one of the drawbacks of using this material model for implementing strain rate behavior. However, no other material model in LS-DYNA can account for this behavior directly, so it is not a limitation of this specific material model as that of the current FE implementations. The averaging algorithm developed was able to capture the general shape for multiple strain rates and preserve that shape for good comparison to the experimental tests.

Compression behavior was not tested in this study, but if experimental compression tests were available, they could be implemented into the material model.

The optimization technique gave a better result than could be achieved with curve fitting. With scale factors of 0.9 for each of the stress vs. strain curves, the stress difference was almost 5 times that of the optimized stress difference. The improvements to the scale factors tended to decrease after 150 runs. This suggests 200 runs would probably have been sufficient to achieve an optimal solution that is not much worse than the current solution. The optimization used in this study demonstrated a robust way of concurrently accounting for multiple objectives.

CONCLUSION

A FE model for placental tissue was created in LS-DYNA using geometry from experimental tension coupon tests. The velocity vs. time of the crossheads in the experimental tests were used as boundary conditions of the FE simulations. The stress vs. strain of the tension coupon was used as the input into the simplified rubber material model. The FE model stress vs. strain curves was optimized using a multi-island genetic algorithm to minimize the average difference in stress between the FE output and the experimental results for the same strain. This was performed by using scale factors for the curves as the independent variable.

It was found that the FE stress vs. strain output matched those of the experiments within 2% for strains up to 30%. This study demonstrated a sophisticated method of processing experimental data for multiple strain rates and multiple tests to create a characteristic average. It also demonstrated a method for concurrently optimizing simulation and experimental data to create a FE model that can be well-correlated to experimental data.

This material model can be used in FE models of the placenta as the run times are low for a complex material model, and the resulting model should approximate the behavior of placental tissues well at a wide variety of strain rates, from quasi-static to dynamic.

Figure 9:

Average experimental stress vs. strain (unmodified)

Figure 11:

Minimization of the error (between experimental and FEM)

Figure 15:

Maximum principal strain rate (strains/s) for top and middle for 7% strain/s

Figure 16:

Maximum principal strain rate (strains/s) for top and middle for 70% strain/s

Figure 17:

Maximum principal strain rate (strains/s) for top and middle for 700% strain/s