Data-Driven Optimization of Bioink Formulations for Extrusion-Based Bioprinting: A Predictive Modeling Approach

Abstract

The field of tissue engineering has significantly advanced with the development of extrusion-based bioprinting. This technique utilizes shear forces to generate filaments for fabricating intricate structures. The printability and structural integrity of bioprinted constructs rely heavily on the rheological properties of bioinks, particularly viscosity, which varies with the shear rate for non-Newtonian materials. Since the shear rate at the nozzle tip fluctuates during extrusion, it is essential to understand how bioink composition influences this behavior. This study investigates the rheological behavior of ALGEC bioinks, a novel formulation composed of ALginate, GElatin, and 2,2,6,6-Tetramethylpiperidine 1-oxyl (TEMPO)-oxidized nanofibrillated cellulose (TO-NFC). The bioinks were prepared with varying concentrations: alginate (0–5.25%), gelatin (0–5.25%), and TO-NFC (0–1.5%), with a maximum total solid content of 8%. Viscosity was conducted over shear rates ranging from 0.1 to 100 s⁻¹, with 252 viscosity data points used 80% for training and 20% for validation. To predict viscosity, polynomial fit and interaction-based multiple regression models were developed. Experimental data were used to estimate viscosity based on bioink composition and shear rate, with the best-performing model achieving an R² of 0.98 and an mean absolute error (MAE) of 0.12. These predictive models were further utilized to optimize ALGEC formulations to achieve targeted viscosity ranges. Constructs were bioprinted using a random and an optimized composition, demonstrating the effectiveness of model-driven bioink optimization. These findings enhance tissue engineering by improving bioink printability, leading to structurally stable bioprinted constructs for regenerative medicine applications.

1 Introduction

The widespread adoption of extrusion-based 3D bioprinting, predominantly using synthetic semi-solid polymers [1,2] or natural hydrogels [3,4] with shear-thinning properties, has highlighted the importance of understanding rheological properties, especially viscosity, in hydrogel-based bioinks. Current research is focused on hybrid hydrogels, combining multiple bioinks to enhance scaffold shape fidelity and cell viability in 3D bioprinting [5,6]. Effective bioink performance depends on (a) printability, (b) shape fidelity, and (c) biocompatibility [7,8]. In extrusion printing, printability is defined by (1) the bioink's extrudability, determined by rheological properties, and (2) filament formability postextrusion, indicated by the filament and structure shape fidelity [9,10]. Rapid gelation or recovery is essential to prevent immediate deformation after extrusion, impacting scaffold geometry. Gelation can be controlled by adjusting the hydrogel's rheological properties through the solid content, viscosity modifiers, temperature, and cross-linkers [11–13].

While recent studies aim to improve bioink printability and shape fidelity [14], biocompatibility is not always enhanced. For example, increased viscosity enhances printability but often requires higher extrusion pressures, potentially compromising cell viability [15]. Conversely, reducing viscosity to improve cell viability may reduce geometric accuracy [16]. Thus, bioinks must balance geometric fidelity with cell viability. Traditional efforts to improve viscosity rely on extensive trial and error, which is resource intensive.

In response, computational approaches, particularly machine learning (ML), have emerged as promising solutions [17,18]. This article examines a polynomial fit (PF)-based ML model to predict bioink viscosity based on solid content and extrusion shear force, offering a systematic, data-driven alternative to experimental optimization. We employed a PF model with a fourth degree for predicting bioink viscosity with respect to the solid content and shear rate due to its effectiveness in capturing the nonlinear relationships inherent in complex bioink compositions. Unlike linear models, PF models can handle curvilinear relationships [19,20], allowing for a nuanced prediction of viscosity as it varies with both the solid content and the shear rate of ALGEC bioinks. Additionally, interaction-based multiple regression (MR) analysis will be utilized alongside the polynomial fit model to predict viscosity by incorporating changing values of material compositions and shear rate. The results obtained from both methods will be compared to evaluate their accuracy and applicability in real-world bioprinting applications. The interpretability of both models [21,22] also offers insights into how each component—alginate, gelatin, and TEMPO-oxidized nanofibrillated cellulose (TO-NFC)—independently and collectively affects viscosity under different extrusion conditions. This makes predictive models an ideal choice for understanding the viscosity dynamics in bioinks, reducing the dependency on exhaustive experimentation, and enabling more precise bioink formulation for 3D bioprinting applications.

This study employed a comprehensive rheological analysis of a novel bioink, referred to as ALGEC, which is composed of ALginate, GElatin, and Cellulose derivative. We selected alginate, gelatin, and TO-NFC as key components for our novel bioink due to their complementary properties that enhance printability, cell viability, and structural integrity. Alginate, a natural polysaccharide from brown algae, is widely used in bioprinting for its excellent printability and ability to form stable gel structures, providing the necessary scaffold framework [4,23,24]. Gelatin, a denatured collagen derivative, contributes to a cell-friendly environment by promoting cell adhesion, viability, and proliferation, making it ideal for tissue engineering applications [25,26]. TO-NFC, derived from cellulose gel, is modified through TEMPO oxidation to introduce negatively charged carboxylate ions, which significantly improves uniformity, dispersibility, and homogeneity within the bioink. This modification enhances the bioink's overall printability, contributing to precise structural fidelity in printed constructs [23,27,28]. We hypothesize that these three materials will create a bioink that will support both the mechanical demands of 3D bioprinting and the biological requirements for cell viability and function. ALGEC bioinks were prepared with varying concentrations of alginate (0–5.25%), gelatin (0–5.25%), and TEMPO-mediated nanofibrillated cellulose (0–1.5%) were tested across shear rates from 0.1 to 100 s⁻¹. Maximum 8% total solid content was maintained, and 252 rheological measurements were split for model training (80%) and validation (20%).

2 Materials and Methods

2.1 Hybrid Hydrogel Preparation and Data Collection.

Dry TEMPO-oxidized nanofibrillated cellulose (TO-NFC) [(C₆H₁₀O₅) × (C₆H₉O₄CO₂Na)y], with a carboxylate content between 0.2 and 2 mmol/g of solids, was sourced from the Process Development Center at the University of Maine. Medium-viscosity alginate (A), with a viscosity of ≥2000 cps at 2% in water, and gelatin (G), with a gel strength of around 300 g Bloom and 100–200 μg/cm², were obtained from Sigma-Aldrich (St. Louis, MO). These components were combined with TO-NFC as outlined in Fig. 1.

Fig. 1.

This schematic illustrates the process of developing and optimizing the ALGEC bioink. (a) Composition selection (alginate, gelatin, and TO-NFC), (b) rheological testing across shear rates, (c) predictive modeling using polynomial fitting, and (d) the final extrusion-based bioprinting of constructs.

In the previous work, hybrid hydrogel compositions with alginate (2–5.25%), gelatin (2–5.25%), and TO-NFC (0.5–1.0%) were primarily studied [29]. The shear rate has varied from 0.1 to 100 s⁻¹. However, in this study, the experimental dataset was expanded from 168 to 252 compositions by incorporating pure alginate and pure TO-NFC formulations to better capture the material transition from single component to hybrid compositions. Accordingly, the concentration ranges in the ALGEC bioink formulation were adjusted to alginate (0–5.25%), gelatin (0–5.25%), and TO-NFC (0–1.5%). A total of 8% solid content for each combination of the experiment is maintained.

The concentration ranges for alginate, gelatin, and TO-NFC were determined based on a combination of the previous research and the literature support. In earlier studies [5,30,31], the impact of hybrid compositions on bioink viscosity, printability, and structural integrity in extrusion-based bioprinting was extensively investigated. Favorable shear-thinning behavior, structural fidelity, and biocompatibility were demonstrated by these formulations. By broadening the experiment horizon to include pure alginate and TO-NFC formulations, a systematic analysis of the material transition from pure components to hybrid hydrogels was conducted, further refining the understanding of their role in bioprinting applications. In this article, composition percentages are labeled sequentially as A–G–T for alginate–gelatin–TO-NFC.

Figure 1 illustrates how different input variables—alginate (0–5.25%), gelatin (2–5.25%), TO-NFC (0–1.5%), and shear rate (0.1–100 s⁻¹)—were systematically varied to create a diverse set of bioink compositions ranging from pure to hybrid hydrogels. These formulations were experimentally tested to evaluate their effects on viscosity and printability, resulting in a dataset of 252 measurements. The hydrogel compositions were strategically selected to span lower, extrusion-suitable, and higher shear rate conditions, capturing both material and process variability.

This experimental dataset formed the foundation for supervised machine learning models that predict viscosity as a function of composition and shear rate: $\eta =f(\dot{\gamma },A,G,T)$. Two modeling approaches—PF and MR—were applied to capture nonlinear and interaction-based dependencies. The framework also demonstrates the model selection process based on R² performance, where PF was chosen for optimization due to its superior accuracy across test and training datasets. These models enable in silico optimization of new bioink formulations, reducing reliance on exhaustive experimental trials. Model predictions were further validated through actual bioprinting trials, confirming their applicability to real-world conditions.

By clearly showing the integration of data generation, modeling, optimization, and validation, Fig. 1 serves as a visual and functional summary of our approach to accelerating and refining bioink development for extrusion-based bioprinting.

2.2 Rheological Analysis.

Rheological measurements were conducted using a rotational rheometer (MCR 102 Anton Paar, Graz, Austria) with a parallel plate geometry (25.0 mm flat plate) and a consistent plate gap of 1.0 mm. All tests were performed at room temperature (25 °C) to facilitate rapid filament solidification during extrusion. To analyze flow characteristics, a steady sweep test is conducted across shear rates ranging from 0.1 to 100 s⁻¹. Data for viscosity in relation to weight percentage and shear rate were plotted using originpro 2023b (Originlab, Northampton, MA).

2.3 Biocompatibility.

Human adipose-derived mesenchymal stem cells (hMSCs) were cultured with pure gelatin, pure TO-NFC, and combinations of TO-NFC and gelatin to assess biocompatibility and cell behavior. These formulations were incubated and imaged on days 1, 4, and 7. Across all groups, an increase in cell number was observed over time, indicating cell proliferation. Figure 10(a) presents the 7-day cell viability results. Additionally, hMSCs were cultured with the A₅G₂T₁ bioink formulation and similarly imaged over the same incubation period. As shown in Fig. 10(b), cells in the presence of A₅G₂T₁ exhibited clear attachment and spreading behaviors, suggesting favorable interaction with the substrate and confirming the biocompatibility of this composition. Notably, cell density continued to increase from day 1 to day 7, further supporting the potential of A₅G₂T₁ to support cell growth. Moving forward, we plan to use this formulation for 3D bioprinting of scaffolds and evaluate cell viability, proliferation, and migration within a printed environment targeting specific tissue regeneration applications.

Fig. 10.

The porous microstructure and the fibrous morphology of the A₅G₂T₁ materials, indicating a rough and interconnected (a) surface and (b) cross section favorable for cell attachment. Scale bar = 500 µm. (c) hMSCs cultured with A₅G₂T₁, where cells are visibly adhered and spread around the scaffold, confirming its biocompatibility and cell-supportive environment. Scale bar = 100 µm.

2.4 ALGEC Modeling and Optimization.

Both PF and MR-based machine learning models will be utilized to predict the viscosity of ALGEC bioinks. These models will be applied to analyze the relationships between viscosity and key formulation variables, including shear rate $(\dot{\gamma })$, A, G, and T compositions. By comparing their predictive accuracy and interpretability, the most suitable approach for capturing the nonlinear and interactive effects of these components on bioink viscosity will be determined, providing a data-driven framework for optimizing bioink formulations.

2.4.1 Polynomial Fit.

Given the often nonlinear rheological behavior of bioinks, particularly with varying concentrations and shear rates, a polynomial fit was employed to capture more complex, curvilinear relationships that MR might miss. This approach allows for the inclusion of higher-order terms and interaction effects, providing a more accurate representation of the bioink's complex rheological response across the experimental domain. The degree of the polynomial was determined through (R² analysis) to optimize the model's fit while avoiding overfitting.

To model the complex relationship between viscosity and the constituent weight of hybrid hydrogels, as well as the shear rate, a polynomial regression approach has been employed. A series of 252 experimental data was used for this modeling. The viscosity $({\eta }_{\mathrm{pred}})$ is expressed as a function of the weight percentages of A, G, T, and $\dot{\gamma }$, where the governing equation is an nth-order polynomial model for viscosity, which is expressed as follows:

$${\eta }_{\mathrm{pred}}={\displaystyle \sum _{i=0}^n}{\displaystyle \sum _{j=0}^{n-i}}{\displaystyle \sum _{k=0}^{n-i-j}}{\displaystyle \sum _{l=0}^{n-i-j-k}}{\beta }_{ijkl}{A}^i{G}^j{T}^k{\dot{\gamma }}^l$$ (1)

where ${\beta }_{ijkl}$ are the coefficients of the polynomial, and the indices i, j, k, and l range from 0 to n, such that $i+j+k\le n$. This equation accounts for all possible combinations of variables up to the nth power, capturing the intricate dependencies and interactions among the weight of alginate, gelatin, TO-NFC, and shear rate with viscosity. The Akaike information criterion has been used to balance the model complexity and goodness of fit. The polynomial fit is validated by comparing predicted values against experimental data, with the mean absolute error (MAE) used as a metric for model performance measures. The robustness of the model is further assessed through cross-validation, a randomly sampled 80% of the data for training and the remaining 20% of the data for testing the model over 100 iterations. This polynomial regression approach provides a comprehensive framework for predicting bioink viscosity across a range of compositions and shear rates, facilitating the optimization of 3D bioprinting parameters.

2.4.2 Multiple Regression.

To model the complex relationship between viscosity and the constituent weight of hybrid hydrogels, as well as the shear rate, an interaction-based MR approach has been employed. This linear modeling technique was chosen as a foundational approach to identify the individual and combined linear effects of the input parameters on viscosity. Its interpretability allows for a clear understanding of the contribution of each bioink component and shear rate to the overall rheological behavior. Specifically, MR helps to identify significant linear relationships and quantify the extent to which each independent variable influences viscosity [32].

A dataset consisting of 252 samples was utilized for this analysis. In this model, the viscosity $({\eta }_{\mathrm{pred}})$ is expressed as a combination of the weight percentages A, G, T, γ, and their relevant interactions terms. The governing equation for the multiple regression model can be expressed as follows:

$$\begin{array}{rl}{\eta }_{\mathrm{pred}}& ={\beta }_0+{\displaystyle \sum _{a=1}^N}{\beta }_a{X}_a+{\displaystyle \sum _{a=1}^N}{\displaystyle \sum _{b=1}^N}{\beta }_{ab}{X}_a{X}_b\\ & +{\displaystyle \sum _{a=1}^N}{\displaystyle \sum _{b=1}^N}{\displaystyle \sum _{c=1}^N}{\beta }_{abc}{X}_a{X}_b{X}_c+\dots +{\displaystyle \sum _{a=1}^N}{\beta }_{1,\mathrm{2..},N}{\displaystyle \prod _{b=1}^N}{X}_b+ϵ\end{array}$$ (2)

where ${\beta }_0=\mathrm{interception}$, X_a = independent variables (A, G, T, and $\dot{\gamma }$), β_a = coefficients for individual predictors, β_ab = coefficients for two-way interaction terms, β_abc = coefficients for three-way interaction terms, ${\beta }_{1,2,\dots N}=\mathrm{coefficients}\mathrm{for}N\text{-}\mathrm{way}\mathrm{interaction}\mathrm{terms}$, $ϵ=\mathrm{error}\mathrm{term}$, N = number of independent variables (four in this case: A, G, T, and $\dot{\gamma }$). This equation captures both the linear and nonlinear relationships between viscosity and the primary independent variables, as well as their pairwise interactions, allowing for a more refined prediction of bioink behavior under various conditions. To ensure robustness, the multiple regression model was trained using an 80/20 split, with 80% of the data randomly selected for training the predictive model and the remaining 20% for testing the obtained model, repeated over 100 iterations to improve estimation. Model selection and performance evaluation were conducted using metrics such as R² and MAE to assess the trade-off between model complexity and goodness of fit. The results of the MR model will be compared against the PF model, evaluating their predictive capabilities and real-world applicability for optimizing bioink formulations in 3D bioprinting.

2.5 Optimization of Viscosity for Better Shape Fidelity.

Our previous research established that hybrid hydrogels with a viscosity range of 300–750 Pa·s can maintain defined structural integrity [5,33,34]. On the basis of this finding, we will optimize the concentrations of alginate, gelatin, TO-NFC, and the shear rate within this viscosity range using the predictive model results. Since the PF model demonstrated better performance than the MR model in terms of R² value, we justify using the PF model results for this optimization, ensuring more accurate and reliable predictions. To identify the optimal values of $\dot{\gamma }$, A, G, and T for achieving the target viscosity, we defined an objective function, $f(A,G,T,\dot{\gamma })$ that minimizes the deviation of predicted viscosity $({\eta }_{\mathrm{pred}})$ from the desired range of 300–750 Pa·s. The objective function is defined as follows:

$$f(A,G,T,\dot{\gamma })=\{\begin{array}{c}{(300-{\eta }_{\mathrm{pred}})}^2\mathrm{if}{\eta }_{\mathrm{pred}}<300\\ {({\eta }_{\mathrm{pred}}-750)}^2\mathrm{if}{\eta }_{\mathrm{pred}}>750\\ 0\mathrm{if}300\le {\eta }_{\mathrm{pred}}\le 750\end{array}$$ (3)

This piecewise function penalizes viscosity values outside the target range, with the penalty increasing as the predicted viscosity deviates further from the acceptable limits. We applied the L-BFGS-B (limited-memory Broyden–Fletcher–Goldfarb–Shanno with Box constraints) algorithm, a bounded quasi-Newton optimization method, to minimize the objective function $f(A,G,T,\dot{\gamma })$ with respect to $\dot{\gamma }$, A, G, and T. Although single-component bioinks were investigated, hybrid hydrogels are more relevant for bioprinting applications due to their enhanced mechanical properties and structural integrity. Therefore, our optimization focuses on hybrid formulations, excluding single-component bioinks from the final analysis. The search is restricted to realistic bounds for each parameter: $\dot{\gamma }$ (0.1–100 s⁻¹), A (2–5.25%), G (2–5.25%), and T (0.5–1.0%). The algorithm iteratively adjusted the values of $\dot{\gamma }$, A, G, and T to find the combination that minimizes f, effectively finding the conditions under which the predicted viscosity falls within the desired range.

2.6 Three-Dimensional Printing of Scaffolds.

By using an extrusion-based 3D bioprinter (BioX, CELLINK, Boston, MA), we fabricated filaments and scaffolds from hybrid hydrogels prepared as outlined in Sec. 2.1. Hydrogels were loaded into a 3.0 ml disposable nozzle and pneumatically extruded onto a stationary build plane.

Scaffold design and toolpath were created in Rhino 6.0² and Slicer,³ which generated BioX-compatible files with toolpath coordinates and process parameters. The bioprinting process followed a layer-by-layer approach, producing three filaments per measurement. Filaments were imaged within 1–2 min under a CK Olympus bright-field microscope (Tokyo, Japan), and their width is measured using imagej (NIH). Table 1 presents the process parameters used for printing scaffolds using our proposed materials.

Table 1.

Process parameters used to 3D print scaffolds

Process parameters	Values with units
Number of layers	5
Print temperature	55 °C
Applied pressure	40–140 kPa
Print speed	5 mm/s
Nozzle diameter	1000 µm

3 Results and Discussion

3.1 Effect of Alginate, Gelatin, TO-NFC, and Shear Rate on the Viscosity.

The plot shown in Fig. 2 reveals the unique flow behavior of eight bioink compositions, each composed with varying concentrations of alginate (A), gelatin (G), and TEMPO-oxidized nanofibrillated cellulose (TO-NFC, T) to tailor their extrusion and structural properties. As the shear rate increases from 0.1 to 100 s⁻¹, all compositions demonstrate a significant decrease in viscosity, exhibiting a classic shear-thinning behavior essential for smooth extrusion for bioprinting. This property allows the bioinks to maintain high viscosity when at rest—supporting shape integrity—while reducing resistance during extrusion, enabling easier flow through the nozzle.

Fig. 2.

The viscosity response of various ALGEC bioink compositions under increasing shear rates from 0.1 to 100 s⁻¹, illustrating the shear-thinning behavior characteristic of non-Newtonian fluids. Among the tested formulations, higher TO-NFC content results in steeper viscosity drops, while lower solid content samples exhibit more pronounced viscosity reduction under shear, indicating composition-dependent flow behavior.

Each composition demonstrates a unique rheological character with respect to its shear-thinning behavior. Compositions with higher alginate content, e.g., A₅G₂T₁, start with high initial viscosity, yielding a pronounced viscosity drop at low shear rates, indicating robust shear-thinning. This strong initial resistance is ideal for maintaining structural integrity in printed constructs. Medium compositions, such as A_4.25G_3.25T_0.5, blend alginate, gelatin, and TO-NFC in balanced proportions, showing a more gradual viscosity decline. Here, TO-NFC adds uniformity, helping to moderate the shear-thinning effect, which promotes the stable flow while preserving print fidelity. Compositions with lower alginate and higher gelatin levels, like A_2.25G_5.25T_0.5, exhibit lower initial viscosity and a controlled thinning response across the shear spectrum. This balance supports consistent extrusion, reducing the risk of over-thinning, which could affect print resolution.

Interestingly, compositions with a higher concentration of TO-NFC, such as A₂G₅T₁, display moderate shear-thinning due to the carboxylate-modified NFC structure that provides gel integrity and controlled flow. Each formulation, with its unique blend of alginate, gelatin, and TO-NFC, is carefully characterized to balance viscosity, shear response, and printability. Together, they demonstrate how subtle adjustments in composition can create bioinks optimized for 3D bioprinting, offering controlled flow during extrusion and reliable structural stability upon deposition. Figure 2 also illustrates that while both alginate and TO-NFC exhibit shear-thinning behavior as pure components, TO-NFC has a more pronounced effect on increasing viscosity compared to alginate.

While the graph reveals a general shear-thinning trend across the different compositions of A, G, and TO-NFC at varying shear rates, it lacks the precision required to define numerical relationships between these variables and viscosity. Visual patterns alone cannot quantify how changes in A, G, and T percentages specifically influence viscosity at each shear rate.

Although the power law and cross models can predict viscosity as a function of shear rate for each individual composition, they require a separate model for each of the eight compositions, necessitating the use of eight distinct power law or cross models to cover all cases, and this is computationally costly. Therefore, a best-fitted ML model becomes essential, as it provides a mathematical model that captures the complex, nonlinear relationships among these variables to predict viscosity.

3.2 Polynomial Fit to Predict Viscosity as a Function of Weight of Constituents and Shear Rate.

To determine the optimal model complexity for predicting viscosity based on constituent weights (A, G, T) and shear rate $(\dot{\gamma })$, we evaluated polynomial regression models ranging from the first to fifth order. The R²-training values for the first, second, third, fourth, and fifth order are 85.02%, 95.1%, 97.30%, 98.48%, and 99.06%, respectively. On the other hand, the R²-test values for first, second, third, fourth, and fifth order are 82.53%, 95%, 97.06%, 97.97%, and 99.02%, respectively. As evident that R² values for both training and test datasets improved progressively with the increasing polynomial order, with test R² increasing from 82.53% (first order) to 99.02% (fifth order) and training R² increasing from 85.07% to 99.06% over the same range. However, the incremental gain in R² between the fourth and fifth orders was minimal—only 0.58% for training data. To avoid overfitting and maintain model parsimony, we adopted a cutoff criterion based on diminishing returns: when the increase in R² was less than 1%, we considered the higher-order term negligible. Therefore, the fourth-order polynomial model was selected as the optimal balance between model accuracy and complexity, and it was used for viscosity estimation across all compositions and shear rates. This approach provides an analytical and justifiable basis for model selection within the framework of classical regression.

The polynomial fit model utilized is a fourth-degree polynomial regression for predicting viscosity as a function of constituent weights (A, G, T) and shear rate $(\dot{\gamma })$. This model achieved a high coefficient of determination (R² = 0.9848), explaining 98.48% of the variability in viscosity data, and an improved mean squared error (MSE = 0.12), indicating a lower average deviation between predicted and actual values compared to previous results. The model's performance is shown in Fig. 3. Figure 3(a) presents the true versus predicted viscosity values, illustrating the model's overall accuracy. The subsequent plots illustrate 3D surfaces depicting the relationship between shear rate and different bioink components—Fig. 3(b), TO-NFC; Fig. 3(c) alginate; Fig. 3(d) gelatin—with viscosity in logarithmic scale where dots represent experimental data points, and the surface represents the model predictions.

Fig. 3.

(a) Scattered plot comparing the predicted and true viscosity in the logarithmic scale demonstrating a strong correlation along the reference line. (b)–(d) 3D surface plots depicting the relationship between shear rate and different bioink components: (b) TO-NFC, (c) alginate, and (d) gelatin, with viscosity in the logarithmic scale where dots represent experimental data points and the surface represents the model predictions.

Figure 4 presents scatter plots comparing the true and predicted values of viscosity on a logarithmic scale across different independent variables. Figure 4(a) depicts the relationship between viscosity and shear rate, while Figs. 4(b)–4(d) illustrate viscosity variations with Fig. 4(b) alginate, Fig. 4(c) gelatin, and Fig. 4(d) TO-NFC. The close overlap between the true and predicted points demonstrates the model's strong predictive capability, accurately capturing viscosity behavior across different bioink compositions and shear rates.

Fig. 4.

Scatter plots comparing the true and predicted values of viscosity in the logarithmic scale across different independent variables. (a) The relationship between viscosity and shear rate, while (b)–(d) illustrates viscosity trends with (b) alginate, (c) gelatin, and (d) TO-NFC. The close alignment of true and predicted points indicates the model's accuracy in capturing the viscosity behavior across varying compositions and shear rates.

Compared to the previous analysis, where results were obtained using 168 data points (R² = 0.95, MSE = 0.108) [29], the current dataset of 252 samples has improved the model's predictive accuracy. This enhancement suggests that incorporating additional data points, including pure alginate and TO-NFC formulations, provides a more comprehensive representation of viscosity variations across different bioink compositions.

Gelatin is a thermo-sensitive biopolymer, and therefore, its incorporation can be affected by the working temperature [25,35]. At our working temperature, alginate has been shown to have a greater impact on viscosity compared to gelatin due to its high molecular weight and strong polyelectrolyte nature, which enables extensive ionic interactions and network formation in aqueous solutions [25]. While TO-NFC is known to influence viscosity more significantly than alginate at similar concentrations due to its high aspect ratio and entangled nanofiber structure [5], the present study used a substantially higher concentration of alginate compared to TO-NFC. Consequently, this higher alginate content dictated the overall viscosity of the bioink by forming a dominant polymer network, overriding the viscosity contribution of TO-NFC. This explains why alginate had the most significant impact on viscosity in our study. Overall, this fourth-degree polynomial model provides an effective tool for predicting bioink viscosity based on component ratios and shear rate, achieving high accuracy as indicated by the R² value. The visualizations reveal the complex interplay between constituents, shear rate, and viscosity, offering valuable insights for optimizing bioink formulations in 3D bioprinting applications.

3.3 Multiple Regression to Predict Viscosity as a Function of Weight of Constituents and Shear Rate.

To predict the viscosity of ALGEC bioinks, an interaction-based multiple regression model is employed to analyze its dependence on the weight of individual components (alginate, gelatin, and TO-NFC) and shear rate. This approach provides a framework for capturing the relationship between these variables, offering a simpler yet interpretable alternative to polynomial regression. In addition to MR, we explore the PF method to predict viscosity based on component weight percentages and shear rate, comparing their performance to determine the most effective predictive model. By evaluating these approaches, we aim to determine the most effective method for accurately predicting viscosity while balancing model complexity and real-world applicability.

Figure 5 presents the performance evaluation of the multiple regression model used to predict viscosity as a function of A, G, T, and $\dot{\gamma }$. The bar chart in Fig. 5(a) displays the coefficients of the multiple regression model, highlighting the contribution of each independent variable and its interactions. TO-NFC and alginate have the highest coefficient values of 1.75, followed by gelatin (1.396). In contrast, the shear rate shows lower coefficient values with a coefficient of 0.02. Interaction terms such as A–G (−0.2529) and A–T (−0.3092) exhibit negative coefficients, suggesting a potential interaction between components. The regression equation provided summarizes how each variable and interaction term contributes to predicting viscosity.

Fig. 5.

Multiple regression model performance evaluation: (a) feature importance and regression equation displaying the contribution of shear rate, alginate, gelatin, TO-NFC, and interaction terms to viscosity prediction (b) Q-Q plot assessing the normality of residuals, showing alignment with the theoretical distribution (c) Residual plot analyzing homoscedasticity, indicating some deviation at higher viscosity values and (d) Predicted versus actual viscosity values, demonstrating a strong correlation between the model's predictions and experimental data.

The Q-Q plot in Fig. 5(b) compares the standardized residuals against a theoretical normal distribution. The points closely follow the reference line, indicating that the residuals are approximately normally distributed, a key assumption of multiple regression. The residuals versus actual values plot in Fig. 5(c) assesses the assumption of homoscedasticity. The residuals appear randomly scattered for lower viscosity values, but some curvature is observed at higher viscosity values, suggesting potential nonlinearity or model misspecification in this range. Finally, the scatter plot in Fig. 5(d) compares predicted viscosity values against actual values. The strong alignment along the reference line suggests that the model effectively captures the relationship between input variables and viscosity although some deviations are observed at higher viscosity values.

Overall, the multiple regression model achieves an R² value of 0.88, indicating that 88% of the variation in viscosity is explained by the model, and a mean squared error of 0.8, reflecting the average squared difference between predicted and actual viscosity values. These results suggest that the MR model provides a predictive framework though potential improvements could be explored to further capture viscosity behavior.

Figure 6 presents pairwise scatter plots and histograms illustrating the relationships between viscosity and all independent variables. Each off-diagonal scatter plot represents the interaction between two variables, while diagonal elements show the distributions of individual variables. (i) The shear rate (SR) versus Vis scatter plot shows a strong inverse relationship, indicating shear-thinning behavior, where viscosity decreases with the increasing shear rate. (ii) The A versus Vis and G versus Vis plots reveal distinct groupings, suggesting that viscosity is influenced by discrete alginate and gelatin concentrations rather than a continuous trend. (iii) The T versus Vis scatter plot displays a more dispersed pattern, indicating a less-pronounced relationship between TO-NFC content and viscosity compared to other components. Overall, this figure provides an exploratory analysis of how individual bioink components and shear rate interact with viscosity, guiding the development of multiple regression and polynomial fit models for viscosity prediction.

Fig. 6.

Pairwise scatter plots of bioink components, shear rate, and viscosity: pairwise scatter plots illustrating the relationships between viscosity, shear rate, alginate, gelatin, and TO-NFC are shown. The scatter plots highlight trends and interactions among variables providing insights into the dataset's structure and potential correlations.

Multiple regression showed a significant improvement in R², increasing from 0.60 to 0.88 as the dataset size grew, demonstrating better predictive accuracy with data ranging from 168 to 252. However, its MSE also increased from 0.74 to 0.80, suggesting slightly greater error variability. Overall, polynomial fit consistently outperforms multiple regression, achieving a much higher R² and lower MSE, making it the superior model. While increasing the dataset size improved both models, multiple regression benefited more in R² improvement, whereas polynomial fit remained the more reliable approach for accurate viscosity prediction.

Despite its lower predictive accuracy compared to polynomial fit, MR still has advantages in real-world applications due to its simplicity, interpretability, and computational efficiency. MR provides a straightforward relationship between variables and captures reasonable interaction terms, making it easier to understand how each component—namely, A, G, T, and $\dot{\gamma }$—influences viscosity without requiring complex transformations. Additionally, MR models are less prone to overfitting to generalize well, making them more practical for real-time applications and industrial bioink formulations where quick, reliable estimations are needed [29,36].

3.4 Multiple Regression to Predict Viscosity as a Function of Weight of Constituents and Shear Rate.

To predict the viscosity of ALGEC bioinks, an interaction-based multiple regression model is employed to analyze its dependence on the weight of individual components (alginate, gelatin, and TO-NFC) and shear rate. This approach provides a framework for capturing the relationship between these variables, offering a simpler yet interpretable alternative to polynomial regression. In addition to MR, we explore the PF method to predict viscosity based on component weight percentages and shear rate, comparing their performance to determine the most effective predictive model. By evaluating these approaches, we aim to determine the most effective method for accurately predicting viscosity while balancing model complexity and real-world applicability.

Figure 5 presents the performance evaluation of the multiple regression model used to predict viscosity as a function of A, G, T, and $\dot{\gamma }$. The bar chart in Fig. 5(a) displays the coefficients of the multiple regression model, highlighting the contribution of each independent variable and its interactions. TO-NFC and alginate have the highest coefficient values of 1.75, followed by gelatin (1.396). In contrast, the share rate shows a lower coefficient value with a coefficient of 0.02. Interaction terms such as A–G (−0.2529) and A–T (−0.3092) exhibit negative coefficients, suggesting a potential interaction between components. The regression equation provided summarizes how each variable and interaction term contributes to predicting viscosity.

The Q-Q plot in Fig. 5(b) compares the standardized residuals against a theoretical normal distribution. The points closely follow the reference line, indicating that the residuals are approximately normally distributed, a key assumption of multiple regression. The residuals versus actual values plot in Fig. 5(c) assesses the assumption of homoscedasticity. The residuals appear randomly scattered for lower viscosity values, but some curvature is observed at higher viscosity values, suggesting potential nonlinearity or model misspecification in this range. Finally, the scatter plot in Fig. 5(d) compares predicted viscosity values against actual values. The strong alignment along the reference line suggests that the model effectively captures the relationship between input variables and viscosity, although some deviations are observed at higher viscosity values.

Overall, the multiple regression model achieves an R² value of 0.88, indicating that 88% of the variation in viscosity is explained by the model, and a mean squared error of 0.8, reflecting the average squared difference between predicted and actual viscosity values. These results suggest that the MR model provides a predictive framework though potential improvements could be explored to further capture viscosity behavior.

Figure 6 presents pairwise scatter plots and histograms illustrating the relationships between viscosity and all independent variables. Each off-diagonal scatter plot represents the interaction between two variables, while diagonal elements show the distributions of individual variables. (i) The SR versus Vis scatter plot shows a strong inverse relationship, indicating shear-thinning behavior, where viscosity decreases with the increasing shear rate. (ii) The A versus Vis and G versus Vis plots reveal distinct groupings, suggesting that viscosity is influenced by discrete alginate and gelatin concentrations rather than a continuous trend. (iii) The T versus Vis scatter plot displays a more dispersed pattern, indicating a less-pronounced relationship between TO-NFC content and viscosity compared to other components. Overall, this figure provides an exploratory analysis of how individual bioink components and shear rate interact with viscosity, guiding the development of multiple regression and polynomial fit models for viscosity prediction.

The performance of PF and MR models was evaluated on both training and test datasets using 168 and 252 data points, with results presented in Table 2. In terms of model performance on test data, PF consistently outperforms MR in explaining viscosity variations. With 168 data points, PF achieved an R² of 0.931, significantly higher than MR's 0.599, indicating that PF explains 93.1% of the variance in viscosity compared to only 59.9% for MR. As the dataset size increased to 252 data points, PF further improved to an R² of 0.979, while MR also showed improvement, reaching 0.881. Although the MR model benefits from additional data, it still lags behind the PF model, which remains the more accurate and reliable model. Additionally, the MSE for PF (0.136) is significantly lower than MR (0.796) in the 252-data test set, demonstrating that PF produces predictions with substantially smaller errors, further justifying its use for bioink viscosity prediction.

Table 2.

Performance comparison of polynomial fit and multiple regression with respect to R² and MSE

	Polynomial Fit		Multiple Regression
Test data
	168 data	252 data	168 data	252 data
R 2	0.931	0.979	0.599	0.881
MSE	0.101	0.136	0.737	0.796
Training data
	168 data	252 data	168 data	252 data
R 2	0.955	0.984	0.728	0.887
MSE	0.108	0.125	0.662	0.928

In terms of model performance on training data, PF consistently demonstrates superior predictive accuracy compared to MR. With 168 data points, PF achieved an R² of 0.955, significantly outperforming MR's 0.728, highlighting its ability to better capture viscosity trends. As the dataset increased to 252 data points, PF further improved to an R² of 0.984, while MR also showed growth, reaching 0.887. Despite this improvement, PF continues to maintain superior predictive power. Additionally, MSE remains lower for PF (0.125) compared to MR (0.928) in the 252-data training set, confirming that PF not only provides more precise predictions but also reduces errors more effectively, making it the more reliable model for bioink viscosity optimization.

Given its higher predictive accuracy, lower error rates, and superior generalization with increasing data, PF consistently outperforms MR in all cases, making it the preferred model for bioink viscosity optimization. PF consistently explains more variance in viscosity than MR, ensuring more precise optimization of bioink formulations. Additionally, MSE for PF remains lower, minimizing deviations between predicted and actual viscosity values, which is crucial for maintaining precision in bioprinting control. While both models improve as data increases, PF benefits more, achieving near-optimal accuracy at R² = 0.979 (test data) and R² = 0.984 (training data) with 252 samples. Moreover, PF effectively captures nonlinear interactions, making it better suited for modeling complex bioink behaviors, where viscosity is influenced by multiple interacting factors such as alginate (A), gelatin (G), TO-NFC (T), and shear rate $(\dot{\gamma })$. Given the high precision required in viscosity-sensitive bioink formulations, PF's superior performance further justifies its use in defining optimal material compositions and printing conditions for 3D bioprinting applications.

Despite its lower predictive accuracy compared to polynomial fit, MR still has advantages in real-world applications due to its simplicity, interpretability, and computational efficiency. MR provides a straightforward relationship between variables and captures reasonable interaction terms, making it easier to understand how each component—A, G, T, and $\dot{\gamma }$—influences viscosity without requiring complex transformations. Additionally, MR models are less prone to overfitting to generalize well, making them more practical for real-time applications and industrial bioink formulations where quick, reliable estimations are needed [29,36]. Therefore, while PF remains the superior choice for accuracy and precision, MR still offers valuable insights and practical advantages in situations where computational simplicity and rapid decision-making are prioritized.

3.5 Analysis of Optimized Parameters.

The optimization results indicate that the model successfully identified parameter values that yield a predicted viscosity within the desired target range of 300–750 Pa·s. The optimized parameters are as follows: $\dot{\gamma }$, 10 s⁻¹; A, 3.19%; G, 3.23%; and T, 0.78, and corresponding ${\eta }_{\mathrm{pred}}$: 408 Pa s. These optimized values result in a predicted viscosity of 408 Pa·s, which falls well within the target range of 300–750 Pa·s, indicating that the model effectively balanced the contributions of each component (A, G, T) along with the shear rate $(\dot{\gamma })$ to achieve the desired viscosity for structural integrity and printability. A low shear rate of 10 s⁻¹ is selected, indicating that slower extrusion is optimal for achieving the target viscosity, likely enhancing control over filament formation during bioprinting.

Both alginate (A) and gelatin (G) concentrations are set at approximately 3.2%, contributing to the bioink's structural stability and cell compatibility while maintaining a workable viscosity range. The addition of 0.78% TO-NFC improves dispersibility and homogeneity without significantly increasing viscosity. This optimized composition achieves a balance between viscosity, structural integrity, and biocompatibility, making it well suited for extrusion-based bioprinting applications that require shape fidelity postextrusion. With a resulting viscosity of 408 Pa·s at the set extrusion conditions, this formulation supports smooth extrusion and provides sufficient viscosity to maintain stable scaffold structures.

3.6 Three-Dimensional Printing of ALGEC.

Figure 7 demonstrates the effect of varying extrusion pressures on the fabrication of square and bilayer constructs using the composition A₅G₂T₁. Figures 7(a)–7(f) depict square constructs printed under increasing pressures from 40 kPa to 140 kPa. At lower pressures, such as 40 kPa as shown in Fig. 7(a), the structure appears incomplete with the discontinuous filament, suggesting insufficient extrusion force.

Fig. 7.

A square construct fabricated with A₅G₂T₁ at different applied pressure: (a) 40 kPa, (b) 60 kPa, (c) 80 kPa, (d) 100 kPa, (e) 120 kPa, and (f) 140 kPa; bilayer fabricated with (g) 60 kPa, (h) 80 kPa (microscopic pore geometry); (i)–(j) a freeform complex construct printed with A₅G₂T₁. Scale bar = 10 mm.

As pressure increases to 60 kPa and 80 kPa (Figs. 7(b) and 7(c)), the filament definition and boundary formation improve significantly. With 80 kPa, the construct achieved a well-defined square structure with balanced filament consistency. At 100 kPa as shown in Fig. 7(d), the construct shows enhanced structural integrity and sharper edges, indicating a stable deposition. However, at higher pressures, such as 120 kPa and 140 kPa (Figs. 7(e) and 7(f)), there is slight deformation and excessive material deposition, with rounded edges that compromise shape fidelity. Bilayer constructs, shown in Figs. 7(g) and 7(h), were fabricated at 60 kPa and 80 kPa, respectively. The bilayer printed at 60 kPa as shown in Fig. 7(g) shows compromised layer integrity with discontinuous filament deposition, while the bilayer at 80 kPa as shown in Fig. 7(h) maintains clear internal architecture and stable layer stacking, indicating optimal pressure for multilayered constructs. Overall, these results underscore the importance of selecting appropriate extrusion pressures to achieve precise filament deposition, structural fidelity, and overall integrity in bioprinted constructs with the A₅G₂T₁ bioink formulation.

Figure 8 illustrates the fabrication of scaffolds using the optimized bioink composition A_3.19G_3.23T_0.78 under various applied pressures. Figures 8(a)–8(c) display square constructs printed at pressures ranging from 20 to 100 kPa. At 20 kPa as shown in Fig. 8(a), the structure shows uneven filament deposition, indicating that this lower pressure does not provide sufficient force for maintaining shape fidelity. At 40 kPa as shown in Fig. 8(b), the construct achieves improved uniformity with well-defined edges, suggesting that this pressure produces stable square formations with the chosen bioink composition, and this further improves with the increasing pressure (Figs. 8(c) and 8(d)). When the pressure increases to 100 kPa (Fig. 8(e)), the structure becomes more robust with thicker filament deposition; however, slight overdeposition can be observed, which causes minor shape distortion. Additionally, Fig. 8(e) depicts a five-layer grid structures fabricated at 60 kPa.

Fig. 8.

A square construct fabricated with A_3.19G_3.23T_0.78 at different applied pressure: (a) 20 kPa, (b) 40 kPa, (c) 60 kPa, (d) 80 kPa, and (e) 100 kPa; five-layer fabricated with (f) 60 kPa. Scale bar = 10 mm.

Here, the layers are well aligned, with distinct gaps between filaments, maintaining good porosity for potential cell infiltration. Overall, these results suggest that 60 kPa provides an optimal balance of structural integrity and porosity, making it suitable for both single-layer and multilayer scaffolds in tissue engineering applications where precise geometry and scaffold permeability are critical.

Figure 9 presents the surface roughness analysis of two ALGECs to assess their surface morphology and corresponding intensity profile of the A₅G₂T₁ (Figs. 9(a) and 9(c)) and A_3.19G_3.23T_0.78 (Figs. 9(b) and 9(d)) compositions. The roughness average (Ra) values—39.46 ± 1.23 for A₅G₂T₁ and 39.48 ± 1.33 for A_3.19G_3.23T_0.78—are nearly identical, indicating no statistically significant difference in surface texture despite the variation in formulation. This suggests that the inclusion or variation in gelatin and TO-NFC content does not adversely affect surface smoothness. Such consistent roughness values confirm that both compositions exhibit stable extrusion characteristics, making them well suited for layer-by-layer deposition in extrusion-based bioprinting applications. The ability to maintain uniform surface properties across different compositions is critical for ensuring reproducibility, structural integrity, and reliable cell interaction in printed scaffolds.

Fig. 9.

The surface roughness distribution of (a) and (c) A₅G₂T₁ and (b) and (d) A_3.19G_3.23T_0.78. The Ra values indicate no significant difference in surface roughness, suggesting that changes in composition do not affect print smoothness. Scale bar = 5 mm.

Normal hMSCs were cultured in the medium contained 2% fetal bovine serum, 5 ng/ml rh Fibroblast Growth Factor (FGF)-basic, 5 ng/ml rh FGF-acidic, 5 ng/ml rh EGF, 2.4 mM L-alanyl-L-glutamine, and 0.5 ml of penicillin–streptomycin–amphotericin B solution (ATCC, Manassas, VA). Cells were incubated at 37 °C with 5% CO₂, with media changes twice weekly, and passage 3 cells were used for encapsulation. The A₅G₂T₁ composition was tested for cell survivability, and bright-field microscopy confirmed that cells remained viable and proliferated over 7 days of incubation (Fig. 10(c)).

Figure 10 highlights the structural characteristics and biological compatibility of the A₅G₂T₁ bioink. Figure 10(a) displays the surface microstructure, and Fig. 10(b) shows the cross-sectional morphology of the scaffold, both revealing a porous, fibrous network that provides favorable topographical cues for cell attachment and migration. These structural features suggest the material's suitability for supporting cellular integration in 3D bioprinting [37]. Figure 10(c) presents an optical microscopy image of hMSCs cultured with A₅G₂T₁, where visible cell spreading and interaction with the matrix confirm the composition's biocompatibility and potential to support cell viability and growth for tissue engineering applications.

4 Discussion

Both Figs. 7 and 8 show scaffold structures fabricated with different bioink compositions and applied pressures. Figure 8 represents the optimized composition (A_3.19G_3.23T_0.78), and it exhibits a noticeably rougher filament surface compared to Fig. 7, which has a lower concentration of gelatin. The higher gelatin content shown in Fig. 8 contributes to increased filament roughness, which may be due to gelatin's viscosity and gelling characteristics that affect the extrusion smoothness and filament formation [38].

Despite the rougher filaments observed in Fig. 8, this composition demonstrates an optimal balance of structural integrity, viscosity, and printability suitable for bioprinting applications. To mitigate filament roughness while retaining the benefits of the optimized composition, one approach is to slightly reduce the gelatin concentration while maintaining the concentrations of alginate and TO-NFC. Alternatively, introducing a fine-tuning step with lower extrusion pressure or adjusting the extrusion speed could enhance filament smoothness without significantly altering the structural stability or the bioink's biocompatibility. In conclusion, the composition shown in Fig. 8 remains optimal but could benefit from minor adjustments in gelatin content or extrusion parameters to achieve a smoother filament surface, thereby combining structural integrity with improved print fidelity.

The morphological analysis (using imagej software) of the filaments of 3D-printed structures at different applied pressures reveal deviations from the 1 mm reference dimension, highlighting differences in accuracy between the tested formulations (Fig. 11). The structure-labeled A₅G₂T₁ demonstrates a higher dimensional accuracy with a minimal deviation, whereas A_3.19G_3.23T_0.78 exhibits greater variation, suggesting increased shrinkage or expansion during the printing and crosslinking process. The first structure (A₅G₂T₁) exhibited an average deviation of 9%, measuring 1.09 mm, indicating slight overextrusion at 60 kPa. In contrast, the second structure (A_3.19G_3.23T_0.78) demonstrated an overextrusion effect, with an average width of 2.4 mm, resulting in a deviation of 140% from the reference. These results suggest that A₅G₂T₁ maintains better structural integrity but with minor excess material deposition, while A_3.19G_3.23T_0.78 exhibits more controlled extrusion but with slight material shrinkage affecting geometric fidelity.

Fig. 11.

Quantification of morphological features and dimensional deviations in 3D-printed structures relative to the 1 mm reference

5 Conclusion

This study presents a comprehensive approach to developing and optimizing a novel bioink formulation, ALGEC, tailored for extrusion-based 3D bioprinting applications in tissue engineering. By leveraging a machine learning-based polynomial model, we predicted bioink viscosity based on varying concentrations of alginate, gelatin, and TO-NFC, along with the applied shear rates. The optimized bioink composition (A_3.19G_3.23T_0.78) demonstrated a viscosity within the desired range of 300–750 Pa·s, providing structural integrity and printability suitable for bioprinting applications. However, higher gelatin content led to slightly rougher filament surfaces, emphasizing the need for further refinement of the formulation or extrusion parameters to achieve smoother filaments. This work highlights the potential of predictive modeling in bioink development, reducing the reliance on extensive experimental trials and advancing the field of 3D bioprinting by enabling the creation of structurally reliable and biologically compatible constructs. The findings contribute valuable insights into bioink optimization, offering a robust foundation for future research aimed at enhancing bioprinting precision and scalability.

Future work will focus on refining the bioink formulation and extrusion parameters to further enhance filament smoothness and print fidelity, particularly addressing the roughness associated with higher gelatin concentrations. Scanning electron microscopy and Fourier-transform infrared spectroscopy will be conducted to further characterize the optimized bioink and fabricated constructs. These tests will provide surface morphology and microstructure of the printed constructs and the chemical composition and interactions between bioink components, respectively. Additionally, we plan to incorporate cell-laden bioinks to evaluate cellular viability, proliferation, and differentiation within the optimized ALGEC constructs, assessing their suitability for various tissue engineering applications. Expanding the machine learning model to include more complex bioink formulations and shear rates could also improve prediction accuracy, offering greater versatility in the bioink development. This direction aims to establish a robust, adaptable platform for creating biocompatible scaffolds with precise structural integrity across a range of tissue engineering applications.

Appendix

PF equation for 168 data:

$$\begin{array}{rl}\eta & =16.6943-0.0001\dot{\gamma }+0.0009A-0.0009G+0.0001*T-0.0003\dot{\gamma }A\\ -& 0.0004\dot{\gamma }G-0.0004\dot{\gamma }T+0.0037\mathrm{A}2-0.0000AG+0.0032AT+-0.0045G^2+-0.0028GT+0.0001T^2\\ +& 0.0001{\dot{\gamma }}^{2}A+0.0001{\dot{\gamma }}^{2}G+0.0001{\dot{\gamma }}^{2}T-0.0005\dot{\gamma }{A}^2-0.0009\dot{\gamma }AG-0.0012\dot{\gamma }AT-0.0006SR{G}^2-0.0014SRGT\\ -& 0.0005\dot{\gamma }{T}^2-0.0104A^3+0.0210A^{2}G+0.0193A^{2}T-0.0232*A{G}^2+0.0017AGT+0.0043A{T}^2+0.0070G^3\\ -& 0.0201G^{2}T-0.0037G{T}^2+0.0001T^3+0.0002{\dot{\gamma }}^2{A}^2+0.0002{\dot{\gamma }}^{2}AG+0.0004{\dot{\gamma }}^{2}AT+0.0002{\dot{\gamma }}^2{G}^2+0.0005{\dot{\gamma }}^{2}GT\\ +& 0.0002\dot{\gamma }2T^3-0.0007\dot{\gamma }{A}^3-0.0012\dot{\gamma }{A}^{2}G-0.0022\dot{\gamma }{A}^{2}T-0.0005\dot{\gamma }A{G}^2-0.0058\dot{\gamma }AGT-0.0016\dot{\gamma }A{T}^2-0.0008\dot{\gamma }{G}^3\\ -& 0.0032\dot{\gamma }{G}^{2}T-0.0041G{T}^3+0.0001T40.0094A{G}^3-0.1859A{G}^{2}T+0.0026AG{T}^2\\ +& 0.0049A{T}^3-0.0065G^4+0.0527G^{3}T-0.0279G^2{T}^2+0.0019\dot{\gamma }G{T}^2-0.0006\dot{\gamma }{T}^3+0.0075A^4-0.0204A^{3}G\\ -& 0.0702T^{2}T-0.0087A^2{G}^2+0.1972A^{2}GT+0.0270A^2{T}^2\end{array}$$

PF equation for 252 data:

$$\begin{array}{rl}\eta & =5.1861-46304.9260\dot{\gamma }+0.3143A-0.0010G+0.0056T+0.0084{\dot{\gamma }}^2+38192.9427\dot{\gamma }A-77.2661\dot{\gamma }G\\ +& 94428.0152\dot{\gamma }T+0.4321A^2-0.0037AG+0.0018AT-0.0023G^2-0.0018*GT+0.1495*T^2-0.0001*{\dot{\gamma }}^3-0.0001*{\dot{\gamma }}^{2}A\\ +& 0.0000{\dot{\gamma }}^{2}G+0.0041{\dot{\gamma }}^{2}T-8386.3748\dot{\gamma }{A}^2-194.2217\dot{\gamma }AG-1159.2190\dot{\gamma }AT-148.6945\dot{\gamma }{G}^2\\ -& 275.2125\dot{\gamma }GT+15729.7068\dot{\gamma }{T}^2+0.2851A^3+0.0020A^{2}G+0.0123A^{2}T-0.0357A{G}^2+0.0037AGT\\ -& 0.0018A{T}^2+0.0329G^3-0.0156G^{2}T-0.0021G{T}^2+0.2951T^3+0.0000{\dot{\gamma }}^4+0.0000{\dot{\gamma }}^{3}A-0.0015{\dot{\gamma }}^2{T}^2+433.0463*\dot{\gamma }{A}^3\\ -& 96.5435\dot{\gamma }{A}^{2}G+854.9014\dot{\gamma }{A}^{2}T-449.6201\dot{\gamma }A{G}^2-1007.6097\dot{\gamma }AGT-9121.0433\dot{\gamma }A{T}^2+79.9689\dot{\gamma }{G}^3\\ -& 819.9048\dot{\gamma }{G}^{2}T-374.1858\dot{\gamma }G{T}^2-38734.5969\dot{\gamma }{T}^3-0.0813A^4+0.1625A^{3}G-0.0841A^{3}T-0.3474A^2{G}^2\\ +& 0.2011A^{2}GT-0.0184A^2{T}^2+0.2407A{G}^3-0.1790A{G}^{2}T+0.0074AG{T}^2-0.0036A{T}^3-0.0539G^4+0.0765G^{3}T\\ -& 0.0222G^2{T}^2-0.0023G{T}^3+0.4784T^4\end{array}$$

MR equation for 168 data:

$$\begin{array}{rl}\eta & =12.6346+-0.0032\dot{\gamma }+0.0897A+-0.1363G+0.0465T+-0.0046\dot{\gamma }A+-0.0033\dot{\gamma }G+-0.0174\dot{\gamma }T\\ +& -0.0734AG+0.2363AT+0.0663GT\end{array}$$

MR equation for 252 data:

$$\eta =3.76-0.02\dot{\gamma }+1.75A+1.19G+2.82T-0.0011SR\dot{\gamma }A-0.0015\dot{\gamma }G-0.02\dot{\gamma }T-0.2529AG-0.3092AT-0.0972GT$$

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper.