# Machine Learning-Based Prediction of Immunomodulatory Properties of Polymers: Toward a Faster and Easier Development of Anti-Inflammatory Biomaterials

Aghilas Akkache, Lisa Clavier, Oleh Mezhenskyi, Kateryna Andriienkova, Thibaut Soubrié, Philippe Lavalle, Nihal Engin Vrana, Varvara Gribova

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Aghilas Akkache, Lisa Clavier, Oleh Mezhenskyi, Kateryna Andriienkova, Thibaut Soubrié, et al. Machine Learning-Based Prediction of Immunomodulatory Properties of Polymers: Toward a Faster and Easier Development of Anti-Inflammatory Biomaterials. Advanced NanoBiomed Research, 2024, 4 (3), pp.2300085. 10.1002/anbr.202300085. [https://hal.science/hal-04487706v1](https://hal.science/hal-04487706v1)

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# Machine Learning-Based Prediction of Immunomodulatory Properties of Polymers: Toward a Faster and Easier Development of Anti-Inflammatory Biomaterials

Aghilas Akkache, Lisa Clavier, Oleh Mezhenskyi, Kateryna Andriienkova, Thibaut Soubrié, Philippe Lavalle, Nihal Engin Vrana,* and Varvara Gribova*

In biomaterials development, creating materials with desirable properties can be a time-consuming and resource-intensive process, often relying on serendipitous discoveries. A potential route to accelerate this process is to employ artificial intelligence methodologies such as machine learning (ML). Herein, the possibility to predict anti-inflammatory properties of the polymers by using a simplified model of inflammation and a restrained dataset is explored. Cellular assays with 50 different polymers are conducted using the murine macrophage cell line RAW 264.7 as a model. These experiments generate a dataset which is used to develop an ML model based on Bayesian logistic regression. After conducting a Bayesian logistic regression analysis, two ML models, K-nearest neighbors (KNN) and Naïve Bayes, are employed to predict anti-inflammatory polymers properties. The study finds that the probability of a polymer having anti-inflammatory properties is multiplied by three if it is a polycation, and that nitric oxide secretion is a good indicator in determining the anti-inflammatory properties of a polymer, which in this work are defined by tumor necrosis factor alpha expression decrease. Overall, the study suggests that with appropriate dataset design, ML techniques can provide valuable information on functional polymer properties, enabling faster and more efficient biomaterial development.

## 1. Introduction

Implants and prostheses are now widely used to treat multiple conditions, replace damaged tissues or in aesthetic surgery. Inflammation is a defense mechanism of the body, and in response to implantable materials, an inflammatory reaction called foreign body reaction usually occurs.[1] Although this is a normal reaction of the body to a foreign material, a chronic

response can happen, resulting implant's degradation and dissolution of surrounding tissue, leading to implant's failure.[2]

To address this issue, new immunomodulatory materials are being developed by the researchers.[3] Polymer-based biomaterials are the most common and include scaffolds, coatings, and hydrogels with anti-inflammatory properties.[4-8]

Development of biomaterials with desired properties usually takes a long time and requires significant financial and human resources, unless an unexpected discovery happens. To make new materials development faster and more robust, we decided to turn to artificial intelligence-derived methodologies such as machine learning (ML) for material properties prediction.

ML approach, which is now used as a tool in many areas, including biomedical (mostly in the field of genomics[9]), is now emerging as a revolutionary tool for faster biomaterials development.[10,11] Recently, we demonstrated, for the first

time, utilization of ML for the prediction of polymer-based coating properties.[12] In a more recent work, new algorithms allowing even better prediction were developed.[13]

In our new work, we decided to move from materials properties to their functions, for which we selected immunomodulation. Here, two issues arise. The biggest problem at this stage is the lack of available data to build robust ML models, or their heterogeneity, as we have seen from our previous work.[12] Before

A. Akkache, P. Lavalle, N. E. Vrana
SPARTHA Medical
1 Rue Eugène Boeckel, 67000 Strasbourg, France
E-mail: evrana@sparthamedical.eu

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/anbr.202300085.

© 2023 The Authors. Advanced NanoBiomed Research published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/anbr.202300085

L. Clavier, P. Lavalle, V. Gribova
Biomaterials and Bioengineering Laboratory
INSERM UMR 1121
1 rue Eugène Boeckel, 67000 Strasbourg, France
E-mail: gribova@unistra.fr

L. Clavier, P. Lavalle, V. Gribova
Faculté de Chirurgie Dentaire
Centre de Soins Dentaires
Université de Strasbourg
1 place de l'Hôpital, 67000 Strasbourg, France

O. Mezhenskyi, K. Andriienkova, T. Soubrié
Preste
242 boulevard Voltaire, 75011 Paris, France
data libraries become available, the data must be searched in the scientific literature or produced in-house. The second difficulty is to define what to consider “anti-inflammatory,” inflammation being a very complex process involving multiple spatial and temporal events, different cell types, various molecule secretion, and regulation of many intracellular signaling pathways.[14] However, in the context of implantable devices, the first line of defense is innate immunity, particularly macrophages;[15] thus we initially focus on macrophage response.

Here, we explore the possibility to predict anti-inflammatory properties of the polymers by using a simplified model of inflammation and a restrained dataset that we generated in-house. We defined the initial inflammatory reaction to implants in a simplified way, as a capacity of the macrophages to secrete proinflammatory markers such as nitric oxide (NO)[16] and tumor necrosis factor alpha (TNF-α).[17] To collect the data, we performed cellular assays using 50 different polymers and then analyzed their contribution to anti-inflammatory phenotype using RAW 264.7 murine macrophage cell line as a model. Then, we analyzed contribution of the polymers to anti-inflammatory phenotype using Bayesian logistic regression, which is used to estimate the probability of a binary outcome. Bayes theorem works particularly well in a small sample, and its power extends beyond the actual dataset which is the basis of our selection. We analyzed the probability that a polymer expresses an anti-inflammatory phenotype and determined the best predictors to create a robust model to fit it and then interpret the resulting model parameters. Bayesian models quantify uncertainty, so that our predictions and decisions consider how limited or imperfect our knowledge is. We specified statistical models and determined parameter probability estimates using a family of sampling algorithms called Markov Chain Monte Carlo.

At the end, we were able to predict if yes or no a particular molecule has anti-inflammatory potential, which we defined as its ability to decrease TNF-α (proinflammatory molecule) expression in lipopolysaccahide (LPS)-treated macrophages (Figure 1). Our work shows that even a modest dataset with proper design can provide new insights into the functional properties of the polymers, thus allowing to select more promising candidates for the future materials development. This opens completely new perspectives to the way the functional materials are developed: instead of an accidental lucky finding or a long and costly screening, we suggest an “express” test to select the best candidates for a particular application. We believe that an appropriated ML approach where the difficulties in large dataset generation can be overcome with proper output set selection can improve biomaterials design, making their development easier, faster, and cheaper, thus contributing to patients’ health and well-being

with multifunctional medical device and regenerative medicine solutions.

**2. Experimental Section**

**2.1. Data Collection**

As there were no homogeneous datasets available which we could use for our goals, we produced the data to build our dataset. We have therefore used cell biology experimentation and biochemical assays, and the data we obtained were next used to develop ML models for anti-inflammatory properties prediction. The complete dataset is available in Table S1, Supporting Information.

**2.2. Materials**

Polyvinylpyrrolidone-iodine complex and 2-[(carbamoylmethyl) (carboxylatomethyl) azaniumyl] acetate were purchased from Acros Organics; λ-carrageenan, i-carrageenan, chitosan, dopamine hydrochloride, L-arginine, fucoidan, chondroitin sulfate, dextran sulfate sodium salt, poly(2-hydroxyethyl methacrylate), poly(styrene sulfonic acid sodium salt), and heparin were obtained from Sigma; i-carrageenan type II, κ-carrageenan, and chitosan were purchased from Glentham Life Sciences; polyarginine, polylysine, polyornithine, and poly-ε-lysine were bought from Alamanda Polymers; and hyaluronic acid was purchased from Lifecore Biomedical. High glucose Dulbecco’s modified Eagle medium (DMEM), phosphate buffer saline, 100× penicillin-streptomycin, and ultralow endotoxin fetal bovine serum (FBS) were purchased from Dominique Dutscher (France). CellTiter-Glo Luminescent Cell Viability Assay was obtained from Promega. LPS was purchased from Sigma. TNF-α enzyme-linked immunosorbent assay (ELISA) was purchased from Abcam.

**2.3. Cell Culture**

RAW 264.7 murine macrophages (ATCC TIB71) were cultured at 37 °C in DMEM with 5% of ultralow endotoxin FBS and 1% of penicillin-streptomycin. The cells were passaged 2-3 times a week and used at passages 15-25.

**2.4. Cytotoxicity Assay**

Cell viability was evaluated using CellTiter-Glo Luminescent Cell Viability Assay according to manufacturer's instructions. Briefly, RAW 264.7 macrophages were seeded at $1 \times 10^5$ into 96-well

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Schematic presentation of the work on prediction of anti-inflammatory properties of the polymers using ML.</figcaption></figure>
plates and incubated for 24 h at 37 °C. After that, the cells were treated with different quantities of polymers in solution (+/- LPS) for another 24 h at 37 °C. Finally, polymer-containing medium was replaced by a mix of culture medium and CellTiter-Glo reagent (1/1 v/v) and incubated for 25 min, and then luminescence was measured using SafasGenius XC spectrofluorimeter (Monaco).

## 2.5. NO Quantification

NO secretion was measured using Griess-Saville method. For this, $5 \times 10^5$ of RAW 264.7 cells were seeded into 24-well plate for 24 h at 37 °C. For inflammation induction, the cells were treated with $10 \text{ ng mL}^{-1}$ LPS for 24 h, and different polymers at variable concentrations were added at the same time as LPS and also allowed to act for 24 h. After the incubation, $100 \text{ µL}$ of the supernatants were transferred into 96-well plate, and $100 \text{ µL}$ of 10% v/v of acetic acid (pH 2.5) were added. Then, $40 \text{ µL}$ of sulfanilamide $6 \text{ mg mL}^{-1}$ were added, and the plate was incubated for 3 min at room temperature (RT) under agitation. Finally, $10 \text{ µL}$ of N-(1-naphtyl)ethylenediamine were added, and the plate was incubated for 20 min at RT under agitation. The absorbance at 570 nm was measured using SafasGenius XC spectrofluorimeter (Monaco). The results are presented as averages of absorbance at 570 nm normalized to the control (untreated cells).

## 2.6. TNF-α Quantification

For ELISA test, RAW 264.7 macrophages were seeded at $1 \times 10^5$ into 96-well plates and incubated for 24 h at 37 °C. After that, the cells were treated with different quantities of polymers in solution (+/- LPS) for another 24 h at 37 °C. The supernatants were used to measure TNF-α according to manufacturer's instructions.

## 2.7. Data Analysis

The objective of the dataset was to create a binary classification model that predicts whether a polymer is anti-inflammatory or not after a series of experiments conducted by our research team, based on several indicators (Table 1). The target variable (anti-inflammatory properties) takes a value of 1 if the polymer is anti-inflammatory and 0 otherwise. Figure 2 describes training and validation process used in supervised learning.

We examined predictive performance of two prediction models based on Bayesian logistic regression using the rstan package. The performance of the models was evaluated by calculating the accuracy (1)

$$ \text{Accuracy} = (\text{TP} + \text{TN})/(\text{TP} + \text{TN} + \text{FP} + \text{FN}) \quad (1) $$

In Equation (1), we calculate the accuracy of our classification model. TP (true positives) represents the count of polymers that were correctly predicted as having anti-inflammatory properties. FP (false positives) corresponds to polymers incorrectly classified as having anti-inflammatory properties, while TN (true negatives) represents polymers correctly classified as not having anti-inflammatory properties, and FN (false negatives) represents polymers incorrectly classified as lacking anti-inflammatory properties. The accuracy is computed by summing the number of TP and TN and dividing it by the total number of predictions, including FP and FN (TP + TN + FP + FN).

The Python package Scikit-learn was used to implement the K-nearest neighbors (KNN) and Naïve Bayes (NB) models. For validation, 25% of the training part of the dataset was used. We used cross-validation to train the models, which were evaluated at each iteration on curated data to ensure the best performance. In addition, the GridSearchCV algorithm was used to find the best hyperparameters for the model, exploring different parameter combinations to optimize the accuracy and robustness of the model. This process resulted in better results and refined

<table><caption>**Table 1.** Definition of variables and their type of coding. ADA, 2-[(carbamoylmethyl)(carboxylatomethyl)azaniumyl]acetate; CAR-iota, carrageenan iota; CAR-kappa, carrageenan kappa; CAR-lambda, carrageenan lambda; CHI, chitosan; CSA, chondroitin sulfate A sodium salt; DEX, dextran sulfate; DOP, dopamine; FUC, fucoidan; HA, hyaluronic acid; HEP, heparin sodium; L-ARG, L-arginine; PAR, polyarginine; PHEMA, poly(2-hydroxyethyl methacrylate); PLE, poly-ε-lysine; PLL, poly-L-lysine; PLO, poly-L-ornithine; PSS, poly(styrene sulfonic acid sodium salt); PVP, polyvinylpyrrolidone-iodine.</caption><thead><tr><th>Input name</th><th>Description (continuous/categorical/Yes-No) variable</th></tr></thead><tbody><tr><td>Polycation</td><td></td></tr><tr><td>PLO</td><td>DOP</td></tr><tr><td>L-ARG</td><td>PAR</td></tr><tr><td>Polyanion</td><td></td></tr><tr><td>PLE</td><td>HA</td></tr><tr><td>CSA</td><td>HEP</td></tr><tr><td>CAR-iota</td><td>DEX</td></tr><tr><td>Polyanion/Polycation unit MW [kDa]</td><td>Continuous variable</td></tr><tr><td>Polyanion/Polycation MW [kDa]</td><td>Continuous variable</td></tr><tr><td>NO secretion [%]</td><td>Continuous variable</td></tr><tr><td>Charge density (polyanion/polycation)</td><td>Continuous variable</td></tr><tr><td>Polyanion/Polycation concentration</td><td>Continuous variable</td></tr><tr><td>Cell viability [%]</td><td>Continuous variable</td></tr><tr><td>Anti-inflammatory properties</td><td>Yes/No</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** An illustrative schematic for polymers anti-inflammatory properties prediction model. In this study, we utilized both baseline data and experimental data, which were combined to create a dataset of 50 observations with nine variables. Logistic regression modeling was used to develop a predictive model for anti-inflammatory properties, using this dataset. To ensure the robustness of our results, we simulated 20 000 additional data points, as well as divided the dataset into test and train sets for the development of KNN and NB models. Model performance was evaluated using an area under the curve (AUC) metric, with the aim of predicting the anti-inflammatory properties of the dataset.</figcaption></figure>

the performance of the model in predicting the anti-inflammatory effect of polymers.

The number of neighbors to be considered was defined using the hyperparameter *n_neighbors* = 3. Three metrics were used to study the models: sensitivity, precision, and F1 score. The sensitivity is the number of true positives identified by a model, divided by the total number of positives present in the real data. The F1 score is the harmonic mean of the accuracy and sensitivity.

The confusion matrix was used to visualize the performance of the algorithms; the rows represent the predicted class, the columns represent the actual class, and a good model should have a true diagonal close to 1.

## 2.8. Statistical Analysis

In this study, a statistical analysis was conducted to assess significant differences between the data groups. Initially, the data underwent Min-Max normalization to ensure a common scale. The results were presented in the form of bar charts, depicting the mean ± standard deviation (SD) for each group. The sample size for these analyses comprised *n* = 3 subjects. To evaluate the statistical differences between the groups, a one-way analysis of variance was employed. The results revealed a highly significant

statistical difference, denoted by **** (*p* < 0.0001). All of these analyses were performed using R Studio.

# 3. Results and Discussion

## 3.1. Cytotoxicity

Cytotoxicity testing is required for any new biomaterial and medical device development before animal experiments.[18] In our case, it was an important criterion to consider before any cellular experiments in the presence of polymers, to avoid bias related to cell mortality. Thus, we started by evaluating RAW 264.7 cell viability in the presence of each selected polymer to determine a nontoxic concentration for further experiments. First, a concentration of 10 µg mL⁻¹ was used for all the polymers. This number was selected based on our previous work with different polymers. However, for some molecules, this concentration is toxic (i.e., viability lower than 70% compared to the positive control, as defined in ISO-10 993-5 cytotoxicity evaluation norm); therefore, it was lowered to 2.5 µg mL⁻¹ (Figure 3 and S1, Supporting information).

## 3.2. NO Secretion

NO is one of the most versatile molecules in the immune system, produced not only by activated macrophages but also by vascular endothelial cells.[13] Because it is relatively easy to quantify using a colorimetric test, we selected it as a first type of numeric values to indicate the anti-inflammatory potential of 50 polymers. Thus, LPS-activated macrophages were incubated in the presence of the polymers, and NO secretion was quantified after 24 h (Figure 4 and S2, Supporting Information). As an example of comparison, the results show low NO secretion by untreated macrophages, decrease of NO secretion in the presence of PAR30, and an increase in the presence of hyaluronic acid (HA), as compared to LPS-treated macrophages considered as 100% (Figure 4). The results on 50 polymers are available in Figure S2, Supporting Information.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Cell viability in the presence of polymers at different concentrations. RAW 264.7 macrophages were cultured in the presence of different molecules, and the viability was evaluated using CellTiter-Glo test. CNTR+ corresponds to a positive control (untreated cells). Each point was performed in triplicate; the bars correspond to average +/− SD.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 4.** NO and TNF-α secretion in the presence of a polycation and a polyanion. A) NO secretion was evaluated using Griess-Seville method. B) TNF-α secretion was evaluated using commercial ELISA kit. "Control-" corresponds to untreated cells. Each point was performed in triplicate; the bars correspond to average +/- SD. ****p < 0.0001.</figcaption></figure>

The results for PAR30-treated cells are in accordance with previous results obtained by our group.[6] It is also not surprising that HA also regulates NO secretion, as its role in inflammation is well known, with low molecular weight HA being considered as proinflammatory and high molecular weight HA as anti-inflammatory.[19] To confirm the results, we also quantified TNF-α secretion using commercial ELISA kit, and the results showed that TNF-α secretion correlated with NO secretion in LPS-activated macrophages. It is important to note that in our analysis, we took steps to avoid potential bias when using a variable as both the outcome and predictor. Specifically, we used the results of TNF to represent the outcome variable, while NO secretion was used as the explanatory variable in our model.

### 3.3. Data Analysis

In the first part, we explored the correlation between the different variables to see how they are related. In the second part, we applied Bayesian logistic regression. In our case, the question was whether the polymer is anti-inflammatory, with two answers (yes or no), so we chose this algorithm.

Bayesian logistic regression is more efficient than traditional methods when working with small datasets. This is because Bayesian methods can incorporate prior information, which allows for more accurate estimates even when data are limited. In addition, Bayesian models can also provide meaningful uncertainty estimates using Bayesian logistic regression; we can better exploit the information in our small dataset and obtain more robust and informative predictions. Finally, two prediction models were tested: a KNN model and NB.

#### 3.3.1. Correlation Analysis between TNF-α and NO Secretion as Indicators of Cellular Inflammatory Response

As a first step, we examined the correlation between TNF-α and NO secretion as indicators of cellular inflammatory response.

Initially, we used Pearson correlation analysis to assess the linear relationship between these two variables. The results of the Pearson correlation revealed a relatively weak linear correlation of 0.37 between TNF-α and NO secretion. This indicates a moderate positive relationship between the two variables (Figure 5). These results suggest that these two variables may be driven by distinct inflammatory mechanisms or represent different aspects of the cellular inflammatory response.

However, considering the possibility of a nonlinear correlation, we also performed a polynomial regression analysis to explore more complex relationships. The results of the polynomial regression showed relatively low R-squared scores, indicating that the first-degree and third-degree polynomials explained the variance in the data to an equivalent extent. But it is important to note that these scores were low, suggesting a limited

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Scatter plot of TNF-α and NO secretion with linear regression line.</figcaption></figure>
ability of the polynomial model to capture the relationships between TNF-α and NO secretion. Based on these results, we decided not to present the scatter plot with polynomial regression lines, as it did not contribute significantly to our analysis.

Following this analysis, NO secretion was selected as an input predictor due to its relatively easy quantification method by a col- orimetric test, while TNF-α, evaluated using ELISA method, served as the target variable (i.e., the variable to be predicted). Polymers with TNF-α expression lower than 100% (correspond- ing to LPS-treated cells) were considered anti-inflammatory, while TNF-α expression higher than 100% was considered as an indicator of the increased inflammatory response.

### 3.3.2. Correlation between Variables: Pearson Correlation Matrix

The relevance of our set of variables to apply logistic regression can be measured by the level of correlations between them. The correlation matrix below illustrates these relationships between the variables (Figure 6).

We have nine variables for 50 observations, among which some are correlated (blue squares, example: concentration and NO secretion) and others are not (green and yellow squares, example: viability and MW). In such a context, the logistic regression will not be disturbed; therefore, all variables will be used to build a model.

According to the correlation matrix showing the magnitude of the correlation of different parameters in all polymers, the

strongest positive correlations were between the charge density with the unit of molecular weight (r = 0.73), and then the secre- tion of NO and cell viability (r = 0.46). The biggest negative cor- relation was found between the concentration and polyelectrolyte (r = -0.56).

The high degree of interaction of NO secretion with viability and concentration reveals that this parameter could have a sig- nificant effect on the model results in combination with the other parameters.

### 3.3.3. Bayesian Logistic Regression

In this section, we describe how we used the “rstanarm” and “bayesrules” packages to study the anti-inflammatory properties of polymers. We applied a Bayesian logistic regression model to analyze the internally generated data. Bayesian logistic regres- sion with rstanarm is a method that combines the concepts of logistic regression and Bayesian inference to model a relation- ship between the binary output variable (anti-inflammatory prop- erties) and several continuous or categorical explanatory variables (NO secretion, cell viability, concentration, charge density, unit molecular weight, and polyelectrolyte type).

The general formula for logistic regression is (2)

$$p = 1 / (1 + \exp(-(\beta0 + \beta1X1 + \beta2X2 + \dots + \beta k * Xk))) \quad (2)$$

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Correlation matrix showing the Pearson correlation coefficients (r) between different variables in our dataset. Cells are shaded according to the strength of relationships based on the Pearson product moment correlation coefficient (p < 0.001) and colored according to whether the correlations are positive (blue) or negative (green).</figcaption></figure>
where $p, p \in [0,1]$ is the probability of a binary event; the intercept is represented by $\beta0$ and is the value of the dependent variable (probability that $Y=1$) when all independent variables ($X1, X2, ..., Xk$) are equal to zero. The regression coefficients $\beta1, \beta2, ..., \betak$ are associated with the respective predictor variables $X1, X2, ..., Xk$ and indicate the effect that these variables have on the probability of the binary event.

In logistic regression, the link function used is the logit function, which is the log function of the odds ratio (OR). The logit function is used to transform the probability “p” into a continuous value that can be related to the predictor variables (3):

$$ \operatorname{logit}(p) = \log(p / (1 - p)) \quad (3) $$

The OR is a measure used to quantify the association between a binary variable and a predictor variable. The OR is calculated by taking the exponential of the regression coefficient of the predictor variable in the logistic regression model (4). For example, if the regression coefficient for a predictor variable $X1$ is $\beta1$, the OR associated with $X1$ is $\exp(\beta1)$.

$$ \text{odds} = p / (1 - p) \in [0, \infty] \quad (4) $$

And the formula for converting odds to probability is (5):

$$ p = \text{odds} / (1 + \text{odds}) \quad (5) $$

The regression coefficients of these predictor variables represent the effect of each variable on the output variable. These regression coefficients are unknown and must be estimated from the data. In the Bayesian approach, we first specify a probability distribution for each regression coefficient before we see the data. This probability distribution is called the prior. The prior reflects a priori belief about the possible values of the regression coefficients. For example, if we believe that the regression coefficients are likely to be close to zero, we can use a prior distribution centered on zero. Similarly, if we believe that certain regression coefficients are more likely to have specific values, we might use an a priori distribution that reflects that belief. If we do not have strong a priori knowledge or beliefs about the possible values of the regression coefficients, we can use uninformative priors that do not impose a particular bias on the regression coefficients.

After specifying the priors, we observe the data and update our a priori beliefs using Bayes theorem to obtain a posteriori distribution of the regression coefficients. Once we have generated enough samples of the posteriori distribution, we can use these samples to analyze our logistic regression model. We can calculate credibility intervals for the regression coefficients, which allows us to quantify the uncertainty associated with the estimation of these coefficients, taking into account the data and a priori information we have provided.

**Impact of NO Secretion:** To better understand the impact of NO secretion on the probability that the polymer is anti-inflammatory (defined by the measurement of TNF-α), we will begin with the analysis of a single predictor model. In this model, the relationship between NO secretion and anti-inflammatory properties is captured in the parameter $\beta1$, whose approximate posterior model is summarized by the formula (6):

$$ P(y = 1 | x) = 1 / (1 + \exp(-(\beta0 + \beta1 * x))) \quad (6) $$

This formula allows us to calculate the probability that the polymer is anti-inflammatory as a function of NO secretion level. Then, we can add the other predictors to get a more complete view of the influence of each variable on the target probability.

The results of our first model (Table 2) indicate that the intercept is significantly different from zero and has a possible range of values from 51.19 to 2936.41 (Table 2). This means that when the NO secretion value is zero, the probability that the polymer is anti-inflammatory is very low, but this probability can increase significantly with an increase in the NO secretion value.

NO secretion has a significant effect on the probability that the polymer is anti-inflammatory. In particular, for a one-unit increase in NO secretion value, the probability that the polymer is anti-inflammatory increases on average by a factor between 0.91 and 0.95 with a probability of 0.80.

However, it is important to note that these results are based on a single predictor model and that other variables may also have a significant impact on the target probability. Therefore, it is important to add other predictors to obtain a more complete view of the influence of each variable on the target probability.

First, we developed a Bayesian logistic regression model to better understand the relationship between NO secretion and anti-inflammatory properties of polymers. We showed that NO secretion is significantly associated with the anti-inflammatory properties of polymers, with an 80% credibility interval. This suggests that the higher the rate of NO secretion, the higher the probability that the polymer is proinflammatory.

**Prediction of the Anti-Inflammatory Properties:** We also used Bayesian logistic regression model to predict whether a polymer X with NO secretion rate of 75% has anti-inflammatory properties. To do this, we approximated the a posteriori predictive model for binary outcomes (7):

$$ Y | \beta0, \beta1 \sim \text{Bern}(p) \text{ with } \log(p / 1 - p) = \beta0 + \beta1 * 75 \quad (7) $$

Using this posterior prediction function, we simulated 20 000 Y outcomes, one by each of the 20 000 parameter sets in our Markov chain, to obtain the probabilities and odds of the anti-inflammatory properties of polymer X. In the simulation of 20 000 outcomes, the model will use the 20 000 parameter sets generated by the Markov chain to simulate values of the dependent variable Y, one for each parameter set. This produces a distribution of possible values of Y, which takes into account the uncertainty associated with the model parameters.

The model then uses this distribution of Y to approximate the a posteriori predictive model for the binary outcomes, assuming that Y follows a Bernoulli distribution with a probability of

<table><caption>**Table 2.** Estimation by credible intervals of the logistic regression coefficients for predicting NO secretion in a binary classification model with 80% credibility.</caption><thead><tr><th colspan="3">Exp (Posterior interval (model_1, probability = 0.80))</th></tr></thead><tbody><tr><td></td><td>10%</td><td>90%</td></tr><tr><td>(Intercept)</td><td>51.19</td><td>2936.40</td></tr><tr><td>NO_secretion</td><td>0.91</td><td>0.95</td></tr></tbody></table>
it is crucial to carefully consider the complexity of the algorithm and how it relates to the specific requirements of the problem.

First, we evaluated the performance of the algorithms using the k-fold cross-validation method. This involves dividing the dataset into k equal parts, and then training the model on k-1 parts and evaluating it on the remaining part. This process is repeated k times, with each part used exactly once as the test set, allowing us to maximize the use of data for training and evaluating models.

For our study, we chose to perform a k-fold cross-validation with k = 5 by dividing the data into five equal parts. The model was trained and evaluated on each of these parts. The results of each model are measured in terms of accuracy, sensitivity, and precision and are presented as bar graphs in Figure 7.

The accuracy measures the rate of correct predictions in relation to the total number of predictions (1).

Sensitivity represents the proportion of anti-inflammatory polymers that the model correctly identified among all the actual anti-inflammatory polymers (10):

$$Sensitivity = TP/(TP + FN) \qquad (10)$$

High precision means that the model tends to correctly identify true anti-inflammatory polymers, with few false positives (11):

$$Precision = TP/(TP + FP) \qquad (11)$$

The results presented in Figure 7 show that both algorithms, the KNN and NB, perform similarly in terms of their ability to distinguish between anti-inflammatory and proinflammatory polymers. However, it is important to note that the KNN model performed slightly better in all metrics. Both models had fairly high accuracy, with 94% accuracy for KNN and 91% accuracy for NB, meaning that both models were able to accurately predict anti-inflammatory and proinflammatory polymers. These results

may indicate that the KNN model performs slightly better than the NB model.

The confusion matrix for each model confirmed the similar performances of both algorithms, with relatively low FP and FN rates for both models (Figure 7). However, the slightly superior performances of the KNN model are reflected in a slightly better precision and sensitivity for the positive class, as indicated in the confusion matrix. These results are presented in Figure 8, which shows that the confusion matrix for the KNN model had 28 TP and 21 TN, while the confusion matrix for the NB model had 26 TP and 20 TN. On the other hand, the NB model has lower sensitivity in identifying the two classes, as it detected two FP and FN values, whereas the KNN model has only one FP error; however, this leads to high accuracy, which is useful for eliminating FN.

## 4. Discussion and Conclusion

In this work, we present the results of our analysis where we aimed to predict a binary variable (anti-inflammatory properties: Yes/No) from a dataset with only 50 observations, which we produced in-house using a macrophage cell line. Given the limited amount of data, we faced significant challenges in building an accurate and reliable ML model. Despite this, we continued to explore different algorithms to find those that could offer the best results.

During the testing, we discovered that some algorithms exhibited overlearning or overcomplexity, which reduced their ability to generalize on unseen data. As a result, we had to do our due diligence to select the best performing algorithms that did not suffer from overfitting. After a rigorous evaluation, we finally selected the algorithms that exhibited the best performance on the test data and were best suited for the binary classification problem. Thus, the study suggests that KNN may be a slightly

<figure><img src="image_8.png" /><figcaption>Figure 7. Accuracy, sensitivity, and precision of regression models for datasets.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Confusion matrix for the classification models: A) NB; B) KNN.</figcaption></figure>

better model than NB for this particular task, but both models perform similarly well.

Our findings provide insights into the strengths and weaknesses of each model and highlight the importance of selecting the appropriate algorithm based on the characteristics of the data and the problem at hand. We demonstrated that even a modest dataset with appropriate design can provide new insights into the functional properties of polymers, allowing for the selection of more promising candidates for future material development.

For instance, the results of a Bayesian logistic regression showed that the probability of a polymer having anti-inflammatory properties is multiplied by three if it is a polycation, suggesting that incorporating polycations in material design could potentially improve their anti-inflammatory properties. In addition, our results suggest that, when controlling for the other predictors in the model, the probability that the polymer is anti-inflammatory increases in proportion to the value of the polymer unit molecular weight. Polymer molecular weight is known to be an important parameter defining its immunomodulatory properties. For instance, chitosan (polycation) with higher molecular weight has anti-inflammatory properties,[20] which is also true for hyaluronic acid (polyanion).[19] However, the role of the polymer unit molecular weight has not been described in the literature yet.

The anti-inflammatory biomaterials literature does not have many comparative studies, and most of the available studies focus on in-depth characterization of a set of polymers rather than establishing comparative values. Although these studies are very useful for better understanding the mechanisms and the extent of the anti-inflammatory effect, they are less helpful for medical device design or improvement. Besides, there is a significant number of contradictions in the interpretation of the results regarding the properties of different biomaterials within the context of modulation of inflammation. The current methodology can provide a means to avoid such contradictions and help the decision-making process.

Our work opens up completely new prospects for how functional materials are developed: instead of a lengthy and costly screening process, we propose an approach that, after further development, can be used to select the best candidates for a particular application in advance. This can be facilitated by

creating standardized databases by different laboratories working in the field of biomaterials. We believe that the ML approach can revolutionize the design of biomaterials, facilitating their development, faster and cheaper, thereby contributing to the health and well-being of patients.

The main shortcoming of this study is the relative oversimplification of the definition of pro- and anti-inflammatory. However, the definition of the polymers in this manner is not a final classification but rather an initial decision-making steps for anti-inflammatory system design. After the initial screening with simple tests, the appropriate design procedure should include the verification of the anti-inflammatory properties with other appropriate markers (such as IL1-RA secretion, IL-10 secretion, detection of the effects of the polymer in the relevant inflammation-related pathways such as NF-κB and its inhibitors[21]) and furthermore the effects on the downstream adaptive immunity-related reactions and even the effects on the connective tissue cells, such as fibroblasts. Having an initial screening before embarking on extensive characterization will not only save significant experimental expenses and time, but also it can contribute to the generation of comparable data for better understanding of the specific immune reactions to the polymers which is different from the well-defined pro- and anti-inflammatory pathways.

The next step in our modeling efforts is a proper definition of anti-inflammatory properties as a continuous function, where quantitative comparison of the polymers' ability to induce an anti-inflammatory effect can be achieved. The other area of expansion will be the increasing the number of input parameters, together with the determination of the optimal dataset sizes for more accurate prediction. Finally, another important aspect for proper model development is a better overall definition of the anti-inflammatory or immunomodulatory effect expected for medical devices.

## Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.
## Acknowledgements

This work is funded by the European Union under grant agreement number 101058554 (EIC Accelerator-SPARTHACUS), Horizon Europe Global Challenges and European Industrial Competitiveness N° 190184905, Horizon Europe Framework Programme N° 101058554 (NOVA), and Horizon 2020 Marie Skłodowska-Curie Actions N° 872869. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. SPARTHA Medical also thanks BPI France (French Public Bank of Investment) ADD (Aide Deep Tech programme) for grant N° 3981662. P. Lavalle was supported by a Projet Région Grand Est “ERMES”.

## Conflict of Interest

A Akkache: employed by SPARTHA Medical, P Lavalle, NE Vrana: Stockholders of SPARTHA Medical.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Keywords

Bayesian logistic regression, in silico, inflammation, machine learning, polymers, predictive models

Received: July 27, 2023

Revised: November 3, 2023

Published online:

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