Sodium Alginate Toughening of Gelatin Hydrogels

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Sodium Alginate Toughening of Gelatin Hydrogels Michael Anthony Samp, Nicolae C Iovanac, and Adam John Nolte ACS Biomater. Sci. Eng., Just Accepted Manuscript • DOI: 10.1021/acsbiomaterials.7b00321 • Publication Date (Web): 13 Nov 2017 Downloaded from http://pubs.acs.org on November 23, 2017

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ACS Biomaterials Science & Engineering

Sodium Alginate Toughening of Gelatin Hydrogels Michael A. Samp, Nicolae C. Iovanac, Adam J. Nolte* Department of Chemical Engineering, Rose-Hulman Institute of Technology, 5500 Wabash Avenue, Terre Haute, Indiana 47803, United States

Abstract

Gelatin is a popular material for the creation of tissue phantoms due to its ease-of-use, safety, low relative cost, and its amenability to tuning physical properties through the use of additives. One difficulty that arises when using gelatin, especially in low concentrations, is the brittleness of the material. In this paper, we show that small additions of another common biological polymer, sodium alginate, significantly increases the toughness of gelatin without changing the Young’s modulus or other low-strain stress relaxation properties of the material. Samples were characterized using ramp-hold stress relaxation tests. The experimental data from these tests was then fit to the Generalized Maxwell (GM) model, as well as two models based on a fractional calculus approach: the Kelvin-Voigt Fractional Derivative (KVFD) and Fractional Maxwell (FM) models. We found that for our samples, the fractional models provided better fits with fewer parameters, and at strains within the linear elastic region, the linear viscoelastic parameters of the alginate/gelatin and pure gelatin samples were essentially indistinguishable. When the same ramp-hold stress relaxation experiments were run at high strains outside of the linear elastic region, we observed a shift in stress relaxation to shorter time scales with increasing

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sodium alginate addition, which may be associated with an increase in fluidity within the gelatin matrix. This leads us to believe that increasing amounts of sodium alginate acts to enhance the viscosity within the fluidic region of the gelatin matrix, providing additional energy dissipation without raising the modulus of the material. These results are applicable to anyone desiring independent control of the Young’s modulus and toughness in preparing tissue phantoms, and suggest that sodium alginate should be added to low-modulus gelatin for use in biological and medical testing applications.

Keywords: Gelatin, alginate, toughness enhancement, force relaxation, viscoelastic modeling, fractional calculus Introduction Due to the difficulty or impracticality of using biological tissue samples in medical procedure testing and training, in many instances the tissue is substituted with a phantom whose specific physical properties may be tuned to match those of the biological sample.1 The decision on which material to use for a phantom is highly dependent on the type of testing to be performed, as the nature of the tests dictates which material properties are important. For applications involving the physical deformation of the sample, ranging from ultrasonic imaging to tissue palpations, the viscoelastic properties of the material are of particular importance. While many soft materials will exhibit some degree of viscoelastic behavior, biological tissues are often complex heterogeneous structures, and as such may display viscoelastic characteristics that cannot easily be reproduced in many testing materials.1 As such, a number of material candidates have been proposed, each with its own advantages and disadvantages.2 Despite their


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relative simplicity, gelatin-based phantoms have long provided a good starting material for basic tissue modeling.3 The primary advantages of using gelatin over other casting materials lie in its low cost, ease of use4, and safety.5 Additionally, the use of gelatin allows for easy tuning of material parameters, such as the Young’s modulus and/or density of the sample, by simply varying the concentration of gelatin in solution6, or by including additives.3 Despite their advantages, one significant drawback found in gelatin based phantoms is their frangibility, particularly in low modulus samples.7 The toughness of the samples can be increased by raising the concentration of gelatin, but this comes at the cost of increasing the Young’s modulus as well.6 As others have previously noted, in order to accurately mimic low-modulus tissues, it is often necessary to develop procedures that allow one to independently control certain mechanical properties within the phantom material.8 Others have shown that blending sodium alginate into gelatin can provide an increase in toughness for a dried film.9 In this paper we demonstrate that the addition of minority amounts of sodium alginate to aqueous-based low-modulus gelatin hydrogels can lead to an increase in the toughness of the material without appreciably affecting the Young’s modulus, thus suggesting these materials may serve as facile, low-modulus tissue phantom systems. Linear viscoelastic modeling is a tool that can shed light on the mechanisms that give rise to the particular mechanical characteristics of the material. These models typically make use of several fitting parameters that are conceptually associated with arrangements of springs and dashpots, which reflect respectively the elastic (through Young’s moduli, Ei) and viscous (through viscosities, ηi) response mechanisms within the material.10 These responses are captured


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through the use of constitutive relationships that relate the stress ( ) and strain ( ) on each model element as follows:10

for an elastic, or "Hookean" material response

for a viscous, or "Newtonian" material response

(1) (2)

Three modeling schemes were considered for this study. The first was the Generalized Maxwell (GM) model, shown in Figure 1 (a), which comprises series and parallel arrangements of spring and dashpot elements. Depending on the number of elements provided to the model, the GM model uses varying numbers of elastic moduli and viscosities, corresponding to the numbers of springs and dashpots respectively, to model stress relaxation. We additionally considered the Kelvin-Voigt Fractional Derivative (KVFD) and Fractional Maxwell (FM) models, which consist of a single spring connected either in parallel or in series, respectively, with a fractional-order element displaying properties of an intermediate character between pure elastic and viscous behavior as shown in Figure 1 (b) and (c).

Figure 1. (a) n-armed Generalized Maxwell (GM) model with n viscosities and n+1 elastic moduli. (b) Three term fractional Kelvin Voigt (KVFD) model. (c) Three term fractional Maxwell (FM) model. The diamonds represent fractional-order viscoelastic elements characterized by parameters and .


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Such elements utilize a mathematical operator known as a fractional derivative, where the order of the derivative, α, is permitted to take on real number values between 0 and 1. The stress on the fractional-order element (σf) is related to the strain (εf) by Equation 3.11

(3)

(4)

By introducing a time constant for the fractional-order element, τ, as: . Equation 3 can be rewritten as:

The fractional elements, represented by the diamonds in Figure 2, therefore have two associated parameters: the viscosity, η, and the derivative order, α. Note that for an α value of 0, Equation 4 reduces to Equation 1, indicating a purely elastic (Hookean) response, whereas for α equal to 1, Equation 4 reduces to Equation 2, indicating a purely viscous (Newtonian) response.12 Intermediate values of α describe a hybrid state of viscoelastic behavior; the nature of Equation 4 thus allows it to capture a wide array of response behaviors. Analyzing a material’s viscoelastic response is typically accomplished by straining the sample and comparing the theoretical model and experimental responses. Theoretical stress responses were computationally generated via a convolution integral of each model’s relaxation modulus, (which is equivalent to the material’s stress relaxation for a unit step strain deformation), with the strain rate as follows:11

−

′

(5)

In this paper, a ramp-hold function was used for strain, as has previously been recommended13, where:


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, ,

< ≥

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(6)

Functional forms of have been previously given for each of the models that we

examined11, so that closed form solutions for the predicted stress relaxation could be obtained and fit to the experimental data by adjusting the model parameters in order to minimize the sum of squared errors (SSE), as described further in the following sections.

Materials and Methods Materials. 250 Bloom Type B gelatin powder was obtained from Custom Collagen (Addison, IL). Alginic acid sodium salt was purchased from Sigma-Aldrich. Germall Plus Liquid was purchased from Crafter’s Choice (Cleveland, OH). Ultrapure (Type 1) DI water was obtained from a Direct-Q® 3UV water purification system.

Sample Preparation. Gelatin powder was mixed with DI water at room temperature and heated in a 70°C water bath for one hour, stirred every five minutes. After the hour, makeup water was added to achieve a final gelatin concentration of 5 wt%, as well as the desired amount of sodium alginate (from 0-1 wt%) and a small amount of Germall (0.1 wt%) to prevent microbial growth. 20 g of the final solution was then poured in a cylindrical PVC mold, the inner walls of which had been sprayed with mold release spray (MR 311 by Sprayon) to facilitate release of the samples. The molds were then covered with parafilm and left to cure at room temperature for 24 hours. Samples were removed immediately before testing to minimize any water loss. Final sample dimensions were, on average, 35 mm in diameter and 21 mm tall.


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Toughness and Stiffness Testing. All testing was done with a Shimadzu EZ-SX Texture Analyzer equipped a 200 N max load cell and a cylindrical testing head of larger diameter than the samples. Samples to be tested were removed from the molds and their exact dimensions were measured. The bottom plate was covered with water and additional water was placed on top of the sample in order to facilitate free-slip conditions at both plates. The initial point of contact between the sample and the plate was determined as follows. Before the start of testing, the testing head was cleaned of any debris and dried, and the force reading was tared. In order to ensure good contact with the surface, the sample was then compressed at 0.02 mm s-1 until a force of 0.05 N was read, which corresponded to a maximum strain of approximately 2%. This method also allowed us to ensure the plates were in good contact with both surfaces of the cylinder, which typically had a small (< 1 mm) meniscus on one side due to the casting procedure. The sample was then decompressed at the same rate until reading of 0 N was achieved. After this pre-cycle, the sample was allowed to recover for two minutes before the start of the actual experiment. In order to determine the toughness, samples were compressed at a strain rate of 0.5 % s-1 until failure was observed. The strain rate for toughness testing was chosen to coincide with the rate used for relaxation tests, as described below. Since these tests were performed using parallel plate compression, after the material fractured, it did not slide out from under the testing head and continued to be compressed. Because of this, the maximum stress on the stress-strain curve did not always correspond to the fracture strength of the material. Instead, the failure strain was determined manually as the strain where the stress suddenly deviated downward from its smooth upward trend, which often coincided with the visual observation of cracks forming on the sample surface.

Because it was difficult to guarantee that we could always visualize the onset of


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cracking, deviations in the stress-strain curve were taken as the primary indicator of the failure strain. The toughness was then calculated by integrating the area under the stress-strain curve up to the failure strain. The elastic modulus of the sample was determined by performing a linear regression on the stress-strain curve between 0.5% and 3% strain, and the strain-to-break was determined as the strain where the sample began to fracture, determined as described in the previous paragraph.

Relaxation Testing. Two different testing methods were used for force relaxation experiments (the same sample was never re-used for more than one test). The first type of experiment was run up to a lower overall strain (5%) in order to ensure we were probing the linear viscoelastic properties of the sample. The sample was compressed at a constant strain rate of 0.5 % s-1 for 10 s, followed by a five minute holding period. These testing parameters were chosen to be similar to previous protocols that have been described in the literature.13 The second type of experiment was run at a higher overall strain, and was compressed much more quickly in order to approximate a step change in the strain. In this type of test, the sample was compressed at a constant strain rate of 5 % s-1 for 5 s, followed by the same five minute holding period. These latter samples reached a strain of 25%, which is outside the linear viscoelastic region. Nevertheless, comparable strains can be reached in tissue phantoms developed for palpation models,15 and in our case these tests allowed us to empirically model the relaxation of samples closer to their fracture point in order to gain some insight into the mechanisms leading to enhanced toughness at high strains.


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Modeling. All modeling was done in MATLAB by varying a given model’s parameters to minimize the sum of the squared errors between experimental data and the model fit. Due to random noise in the force reading, the force was not always zero when data collection began. Before running the fitting algorithm, all data was shifted up or down such that the first point was equal to zero. The magnitude of this shift was never more than ± 0.002 N. Due to the coupled nature of the E and τ parameters in the KVFD and FM models, these parameters are very sensitive to the initial values supplied to the algorithm. The initial value for τ was always set to 150 s (approximately half of the total testing time). The parameters E and α were manually varied to approximate the fit, and these approximations were supplied to the algorithm as the initial values for the parameters. The experimental data was also fit with the GM model, which contains pure Newtonian dashpot elements, rather than the fractional forms present in the previously described models. A GM model with one Maxwell arm (GM-3) is more commonly known as the Standard Linear Solid or Zener model.14 Finally, the data was also fit with a GM model containing two Maxwell arms (GM-5). This latter model was used to observe the relative advantage that could be gained by allowing more fitting parameters (five, as opposed to the three parameters each of the other models contain) into a standard linear viscoelastic model. It should be noted that a different technique was necessary when modeling the data taken from the high strain experiments. At the strain at which these tests were performed the samples no longer displayed a linear stress response in the ramp region, which made fitting the entire ramphold stress responses difficult. Instead, a relatively fast ramp was used to approximate a step change in strain, and the holding period was modeled with a five-term Prony series, which represents a GM-5 model response following a step change in strain. In order to get more


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accurate fits in the region immediately following the ramp, this model was constrained to pass through the first experimental data point of the holding region. Although we chose a ramp speed of 0.5 % s-1 for our low strain relaxation tests and toughness testing, additional relaxation testing and modeling was performed in order to examine our ability to model relaxation data over a range of ramp strain rates, and to examine the sensitivity of the measured Young’s modulus to the ramp strain rate. Further discussion of this work is provided in the Supporting Information.

Results and Discussion

Figure 2. Plots of (a) toughness, (b) elastic modulus, and (c) strain-to-break as a function of alginate concentration. The height of each bar represents the average of all trials and the uncertainty bars represent the standard error in the average measurement. For these results, 7 samples were tested at each alginate concentration. These tests were performed with a strain rate ramp of 0.5 % s-1.


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Our first objective was to quantify the effect of sodium alginate concentration on the mechanical properties of the gelatin samples. As shown in Figure 2a, adding alginate to gelatin samples increases the toughness, which has been shown previously for gelatin/alginate blended (dried) films.9 Adding just 0.1 wt% alginate increased the measured toughness by 52%, and adding more than 0.75 wt% increased the toughness by approximately 150%. It should be noted that these toughness tests were performed using two separate batches of the same type of gelatin powder. In the first batch, the observed toughness increase was larger (a 120% increase at 0.1 wt% and 480% increase at 0.75 wt%), than in the second batch (a 58% increase at 0.1 wt% and a 78% increase at 0.75 wt%). In the second batch of gelatin, however, the neat gelatin samples (no alginate) were naturally tougher (the average toughness of the neat samples in the first batch was 511 J/m3, while the average toughness of the neat samples in batch two was 1598 J/m3). Because gelatin is a natural product, some variations in mechanical properties between batches of the same product are to be expected. For each batch, however, we noted that the addition of alginate provided a relative increase in toughness. Figure 2 shows the averages of all data across both batches. Toughness is calculated by integrating the area under the stress-strain curve to the point of failure; generally speaking, it can be increased by either increasing the elastic modulus (more force required to achieve the same level of strain), or by increasing the strain-to-break of the material. Because the Young’s modulus of gelatin phantoms is often a key parameter in accurately mimicking biological tissue of a certain stiffness, it is often desirable to increase the toughness without changing the modulus of the material (i.e. increase strain-to-break). Figure 2b shows that for the range of alginate concentrations tested in this experiment, the average elastic modulus at each concentration remained relatively constant, varying between


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approximately 2.8 and 3.2 kPa. There is an apparent spike in the Young’s modulus from 2.86 kPa at 0 wt% alginate to 3.21 kPa at 0.1 wt% alginate. This peak was found to be statistically significant (p = 0.006), however, a second set of trials run with the same conditions and using the same method did not contain this same peak. A rise in modulus at low alginate concentrations could be possible as a result of alginate scavenging Ca2+ ions (which act as ionic crosslinkers) from the gelatin matrix and forming either a biphasic fluid phase within the gelatin matrix or possibly even an interpenetrating alginate matrix. At larger alginate concentrations, this effect would not be observed because there would not be enough Ca2+ ions in the gelatin to crosslink further alginate molecules to an appreciable degree. At any rate, even if a slight rise in Young’s modulus exists at 0.1 wt% alginate, the upward trend does not continue at higher alginate concentrations, and the modulus remains unchanged within statistical certainty up to 1.0 wt% alginate addition. Changes in the modulus do not therefore provide an explanation for the observed increase in toughness. Instead, increases in the hydrogel’s strain-to-break (Figure 2c) can be identified as the cause of the observed toughness increase. Adding only 0.1 wt% of sodium alginate increased the average strain-to-break from 53.6% to 59.2%, and adding 0.5 wt% or more of alginate increases the average strain-to-break to approximately 64%. From a qualitative standpoint, we also noted gelatin/alginate samples were easier to de-mold and handle without fracture than their neat counterparts, which could be a key advantage for such materials in medical testing or training scenarios. It is important to note that when performing toughness experiments, any crack in the sample will create a localized stress increase that could lead to premature fracture. Therefore, only samples with no notable surface defects were used.


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After quantifying the toughness enhancement imparted by alginate addition to gelatin samples, we set about determining whether these mechanical property changes were reflected in the linear elastic properties of the material. As described in the Methods section, stress relaxation experiments were conducted using a ramp-hold strain profile, and the experimental results were modeled using Equation 5 above.

For each of the models being studied, a form of the relaxation modulus, , has been found

previously. For the GM model:13

+ For the KVFD model:13

' ()

∙ exp %− &

*+,- ∙ .1 + For the FM model14,

2 0 1

Γ1 −

4

, ∙ ℳ %− % & &

(7)

(8)

(9)

where ℳ is the Mittag-Lefler function defined by:14 ℳ,6 7

:

'(

7' Γ 8 + 9

(10)

General nomenclature for the Mittag-Leffler function states that if the function only has one

subscript, the value of 9 in the function is equal to one, i.e. ℳ 7 ≡ ℳ,) 7. Note that in this

paper, ℳ is being used for the Mittag-Leffler function rather than the more common E, in order to avoid confusion with the Young’s modulus.


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Taking any of these forms of the relaxation modulus and plugging them into the convolution integral (Equation 5) yields a closed-form solution for the predicted stress response. For the GM model:

@ +

? =

' A()

A A %1 − exp %−

&&B , ≤ A

> − A A Dexp D− E − exp %− &E , > = + A A < A() '

For the KVFD model:

*+,-

For the FM model:

? = =

2 0 1 .1 + 4 , ≤ Γ2 −

− 2 2 > − 0 1 0 1 = . − + 4 , > = Γ2 − Γ2 −
ℳ %− % & & − − ℳ %− % − & & , > ,H ,H = < ? =

(11)

(12)

(13)

Since the GM-3, KVFD, and FM models all have three fitting parameters, a direct comparison was done on these three models to determine their relative effectiveness at fitting a given set of stress relaxation data. Predicted stress data was generated using Equations 11-13, and Figure 3 shows an example of the model fits for a single data set. All three models do a reasonable job of fitting the long-term relaxation behavior of the samples (Figures 3a, b, and c). The two fractional models, however, do a much better job of fitting the region of the data immediately following the ramp (inset of Figures 3a, b, and c), which results in a much lower overall sum of squared errors.


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This indicates that three parameter fractional-element models have a much greater ability to model the ramp-hold stress relaxation in these systems.

Figure 3. Model fits of a single data set using the (a) KVFD, (b) FM, (c) GM-3, and (d) GM-5 models. Inset images are zoomed in on the end of the ramp portion of the data. Data in plots are downsampled by a factor of 10 in the ramp region and 100 in the hold region for visualization purposes.

The ability of three-parameter fractional calculus models to better fit the stress relaxation data suggests that stress relaxation in gelatin-based hydrogel systems benefit from the flexibility of models with a tunable degree of viscoelastic character, reflected here in the parameter α.


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Fractional α values in such models can be thought of as generating a stress relaxation response with a continuous distribution of time constants, instead of discrete values.10 Gelatin samples, in particular, comprise a random network of hydrogen-bonded crosslinks. The addition of sodium alginate, which may be electrostatically crosslinked through the presence of residual divalent cations in the gelatin, would presumably yield an interpenetrating or biphasic network structure capable of viscous relaxations over a spectrum of time scales, as has similarly been observed in other polymeric systems.7 Given these considerations, it is no surprise that the GM-3 model, which predicts an exponential stress relaxation at a single time constant, τ = η/E, provided a comparably worse fit for the experimental force relaxation data (Figure 3c). Adding additional “arms” to the GM model can provide a better fit for the data through the inclusion of additional parameters (each arm adds another Young’s modulus and viscosity term, and hence another relaxation time parameter). Figure 3d shows the model fit when using a GM-5 model. Despite a comparable SSE between this five-parameter GM-5 model and the two three-parameter fractional models, the inset image reveals that the GM-5 model still underestimates the amount of relaxation happening on small time scales (a trend that was usually true across multiple samples), again suggesting the superiority of the fractional models for capturing dynamic stress relaxation in the gelatin material. Despite observed success of the fractional models in modeling stress relaxation in our samples, we found that the computational fitting routines were not straightforward to implement. Without careful selection of starting parameters, the fitting algorithms were capable of converging to yield widely different parameter sets with similar SSE values. Care must be taken to avoid local minima traps and solutions with unrealistic values (very large or small values of E or τ).13 Some


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consistency was gained by initially adjusting the fitting parameters manually, then using these values as the initial inputs to the fitting algorithm. In some cases, the function tolerance needed to be relaxed to keep the algorithm from iterating past a reasonable solution.

Figure 4. (a) Average value of α using the KVFD model and (b) average ratio of relaxation moduli to the total modulus using the GM-5 model. Uncertainty bars represent the standard error in the sample mean. For these results, 5-8 samples were tested at each alginate concentration. These tests were performed with a strain rate ramp of 0.5 % s-1. In an effort to determine whether the increase in the toughness of gelatin/alginate samples was reflected in a corresponding change in the viscoelastic model parameters, we examined the fit values of α (for the fractional models) as a function of alginate concentration, as this parameter is tied to the degree of viscous relaxation occurring within these samples. As seen in Figure 4a, we found no statistical difference between the values of the α parameter for the different alginate


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concentrations (p = 0.174). The value of α does spike at low alginate concentrations, but we believe that spike to be due to outliers in the data. A similar metric can be calculated for the GM5 model fits by calculating the ratio of the sum of the coefficients in front of the exponential terms to the total modulus, which can be calculated by adding up all the “E” terms. This plot is shown in Figure 4b, and once again, there appears to be no statistical difference across the different alginate concentrations (p = 0.112). In fact, we could find no statistically relevant trends between any of the modeling parameters across the different alginate concentrations we tested. These results suggest that for the low strains (up to ~5%) and strain rates (~0.5% s-1) utilized in our initial ramp-hold tests that the addition of sodium alginate, while enhancing the toughness of the samples, does not appreciably affect the viscoelastic mechanical properties of the gelatin, which could be of interest for designing test phantoms for particular applications. In order to further explore the observed toughness enhancement observed in the gelatin samples upon addition of sodium alginate, additional force relaxation experiments were run at a much higher ultimate strain (25%). As noted in the Methods section, at larger strain the gelatin samples deviate from linear elastic behavior, precluding modeling the entire ramp-hold strain profile. Instead, a faster strain rate was utilized in the ramp region to provide an abrupt increase in strain to εmax = 25%, and the relaxation in these samples was modeled with a five-term Prony series, which is the analytical solution to the GM-5 model for a step change in strain.

+ ) ∙ exp %− & + H ∙ exp %− & IJK ) H

(14)

By utilizing the GM-5 model to fit the experimental stress relaxation (or equivalently, modulus relaxation) data for these high-strain samples, we are able to observe the relative contribution of shorter-term and longer-term relaxation effects by restricting the fitting parameters to include


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only two time constants: τ1 (on the order of 3 – 5 s), which represents short-term relaxation effects, and τ2 (on the order of 90 – 150 s), which represents long-term relaxation effects.

Figure 5. Average value of E1/E2 for high strain force relaxation experiments. Uncertainty bars represent the standard error in the sample mean. For these results, 6-8 samples were tested at each alginate concentration.

Figure 5 shows the results of plotting the metric E1/E2, which represents the ratio of the magnitudes of the short-term and long-term modulus relaxation parameters, versus alginate concentration. As can be seen in Figure 5, E1/E2 increases as the concentration of alginate in the sample increases. Also, the value of E1/E2 for samples without alginate was smaller than for all alginate-containing samples. This indicates that for samples under large strains, adding alginate to gelatin results in a shift in stress relaxation to shorter time scales. This trend is consistent with the observed toughness enhancement when adding alginate to gelatin samples. Mechanistically, if a sample has the ability to relax more quickly, it can more quickly dissipate energy under strain, which would result in an increase in the sample’s toughness. Furthermore, the observed trend is very similar to the trend observed in the increasing strain-to-break of alginate-containing


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samples (Figure 2c), which further supports the hypothesis that such samples have an enhanced fluidic character. Others have shown previously that, for pure gelatin systems, short-term relaxation can be associated with fast moving fluid flows within the hydrogel matrix, and longer-term relaxation can be associated with the restructuring of the matrix itself.1 An increase in the short-term relaxation modulus therefore suggests that sodium alginate increases the viscous character of the hydrogel at high strains. In regards to the observed toughness increase, the enhanced viscous character of the sample would presumably also be responsible for energy dissipation at higher strains, which would offset chain relaxation and broken crosslinks, and lead to a higher strain-tobreak.

Conclusions We have demonstrated that adding sodium alginate to aqueous-based gelatin samples can increase their toughness without changing their Young’s moduli, making them useful candidates for tissue-mimicking phantoms where specific mechanical properties are desired. The linear viscoelastic properties of gelatin and gelatin/alginate samples were modeled using both the Generalized Maxwell model and two models based on fractional calculus. In all cases, the fractional models provided better three-parameter fits for stress relaxation in the linear elastic region.

The models suggest that at low strains, there is no discernible difference in the

viscoelastic properties between gelatin and gelatin/alginate samples.

At high strains, the

observed toughness increase is correlated with a relative increase in short-term relaxation in the samples, suggesting that alginate modifies short-term fluid flow within the gelatin matrix, allowing for more effective stress dissipation. Although previous studies have indicated that the


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addition of sodium alginate to gelatin can increase the toughness of dried films,9 our study is the first to demonstrate that small additions of alginate to gelatin hydrogels can provide a significant enhancement in toughness for low-modulus gelatin hydrogels, while leaving the Young’s modulus unchanged. Our results could be useful in improving the durability of low-modulus tissue phantom materials for biological and medical diagnostic techniques.

Corresponding Author *E-mail: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ACKNOWLEDGMENT The authors acknowledge Dr. Michael Insana and Yue Wang of UIUC, and Matthew Billingsley, Matthew Conrad, Emily Cottingham, Caitlin Douglas, and Haley O’Neil of RHIT for their assistance. Funding for this research was provided by the NSF under award number ECCS1306808. Supporting Information. Stress relaxation modeling with ramps having different strain rates, and analysis of ramp speed on the measurement of Young’s modulus. This material is available free of charge via the Internet at http://pubs.acs.org.

ABBREVIATIONS


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GM, Generalized Maxwell model; KVFD, Kelvin-Voigt Fractional Derivative model; FM, Fractional Maxwell model; SSE, Sum of Squared Errors REFERENCES 1.

Sridhar, M.; Liu, J.; Insana, M. F., Elasticity imaging of polymeric media. Journal of

Biomechanical Engineering-Transactions of the ASM 2007, 129 (2), 259-272. DOI: 10.1115/1.2540804. 2.

de Carvalho, I. M.; De Matheo, L. L.; Costa, J. F. S.; Borba, C. D.; von Kruger, M. A.;

Infantosi, A. F. C.; Pereira, W. C. D., Polyvinyl chloride plastisol breast phantoms for ultrasound imaging. Ultrasonics 2016, 70, 98-106. DOI: 10.1016/j.ultras.2016.04.018. 3.

Madsen, E.; Zagzebski, J.; Banjavie, R.; Jutila, R., Tissue mimicking materials for

ultrasound phantoms. Journal of Medical Physics 1978, 5 (5), 391-394. 4.

Sisney, G. A.; Hunt, K. A., A low-cost gelatin phantom for learning sonographically

guided interventional breast radiology techniques. American Journal of Roentgenology 1998, 171 (1), 65-66. 5.

Young, S.; Wong, M.; Tabata, Y.; Mikos, A. G., Gelatin as a delivery vehicle for the

controlled release of bioactive molecules. Journal of Controlled Release 2005, 109 (1-3), 256274. DOI: 10.1016/j.jconrel.2005.09.023. 6.

Czerner, M.; Fasce, L. A.; Martucci, J. F.; Ruseckaite, R.; Frontini, P. M., Deformation

and fracture behavior of physical gelatin gel systems. Food Hydrocolloids 2016, 60, 299-307. DOI: 10.1016/j.foodhyd.2016.04.007.


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Page 23 of 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7.

Sun, J. Y.; Zhao, X. H.; Illeperuma, W. R. K.; Chaudhuri, O.; Oh, K. H.; Mooney, D. J.;

Vlassak, J. J.; Suo, Z. G., Highly stretchable and tough hydrogels. Nature 2012, 489 (7414), 133136. DOI: 10.1038/nature11409. 8.

Kong, H. J.; Wong, E.; Mooney, D. J., Independent control of rigidity and toughness of

polymeric hydrogels. Macromolecules 2003, 36 (12), 4582-4588. DOI: 10.1021/ma034137w. 9.

Xiao, C. B.; Liu, H. J.; Lu, Y. S.; Zhang, L., Blend films from sodium alginate and

gelatin solutions. Journal of Macromolecular Science-Pure and Applied Chemistry 2001, 38 (3), 317-328. DOI: 10.1081/ma-100103352. 10. Lakes, R. S., Viscoelastic Solids. CRC Press: Boca Raton, FL, 1999; p 476. 11. Magin, R., Fractional Calculus in Bioengineering. Begell House: 2006. 12. Zopf, C.; Hoque, S. E.; Kaliske, M., Comparison of approaches to model viscoelasticity based on fractional time derivatives. Computational Materials Science 2015, 98, 287-296. DOI: 10.1016/j.commatsci.2014.11.012. 13. Zhang, H. M.; Wang, Y.; Insana, M. F., Ramp-hold relaxation solutions for the KVFD model applied to soft viscoelastic media. Measurement Science and Technology 2016, 27 (2), 11. DOI: 10.1088/0957-0233/27/2/025702. 14. Mainardi, F.; Spada, G., Creep, relaxation and viscosity properties for basic fractional models in rheology. European Physical Journal-Special Topics 2011, 193 (1), 133-160. DOI: 10.1140/epjst/e2011-01387-1.


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Li, M.; Konstantinova, J.; Xu, G. H.; He, B.; Aminzadeh, V.; Xie, J.; Wurdemann, H.;

Althoefer, K., Evaluation of stiffness feedback for hard nodule identification on a phantom silicone model. PLOS ONE 2017, 12 (3), e0172703. DOI: 10.1371/journal.pone.0172703.


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Sodium Alginate Toughening of Gelatin Hydrogels

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