Data-based regression models for predicting remifentanil pharmacokinetics

Abstract

Background and Aims:

Remifentanil is a powerful synthetic opioid drug with a short initiation and period of action, making it an ultra-short-acting opioid. It is delivered as an intravenous infusion during surgical procedures for pain management. However, deciding on a suitable dosage depends on various aspects specific to each individual.

Methods:

Conventional pharmacokinetic and pharmacodynamic (PK-PD) models mainly rely on manually choosing the parameters. Target-controlled drug delivery systems need precise predictions of the drug’s analgesic effects. This work investigates various supervised machine learning (ML) methods to analyse the pharmacokinetic characteristics of remifentanil, imitating the measured data. From the Kaggle database, features such as age, gender, infusion rate, body surface area, and lean body mass are extracted to determine the drug concentration at a specific instant of time.

Results:

The characteristics show that the prediction algorithms perform better over traditional PK-PD models with greater accuracy and minimum mean squared error (MSE). By optimising the hyperparameters with Bayesian methods, the performance of these models is significantly improved, attaining the minimum MSE value.

Conclusion:

Applying ML algorithms in drug delivery can significantly reduce resource costs and the time and effort essential for laboratory experiments in the pharmaceutical industry.

INTRODUCTION

Remifentanil is a practical option for therapeutic procedures where quicker pain relief is desired due to its fast onset of action post-administration.[1] The typical characteristic of remifentanil is its short half-life, which enables accurate control and rapid recovery even after the medication is discontinued. Its primary application is intravenous analgesia and sedation in spontaneously breathing patients facing painful procedures.

Advanced algorithms in the design of drug delivery models have been treated as effective tools for clinicians to optimise medication administration and enhance treatment efficacy. Conventional approaches for predicting pharmacokinetic and pharmacodynamic characteristics require manual selection of the model parameters, which is challenging and may result in inaccuracies.[2,3] Moreover, subjects may respond differently to the same drug dosage, demanding advanced models for precise predictions.[4,5] Numerous machine learning (ML) algorithms have been proposed to report these challenges.[6,7] Regression analysis aims to automatically learn and quantify the association between the output variables and a set of independent input variables with insights derived from a collection of samples.[8]

This study employs five distinct regression algorithms, including fine tree, bagged tree, fine Gaussian support vector machine (SVM), wide neural network, and exponential Gaussian process regression (GPR),[9,10,11] to forecast the concentration of remifentanil administered through intravenous infusion, utilising various attributes. The model with the least mean squared error (MSE) and highest R-squared value can predict the remifentanil concentration in new datasets.

METHODS

Using various ML algorithms, the pharmacokinetics of a drug like remifentanil can be predicted, which includes different steps such as data collection, feature selection, model selection, training, testing, and performance evaluation. The dataset extracted from the Kaggle database involved administering an intravenous infusion of the potent analgesic remifentanil to 65 individuals at varying rates and for varying lengths.[12,13] Remifentanil was infused at 1–8 µg/kg/min for 4–20 minutes to 38 men and 27 women (age range: 20–85 years) in the study from which the dataset is used. Frequent arterial blood samples were collected from these and assayed for remifentanil concentration.[12] The remifentanil concentration was measured with several variables, resulting in a data frame of 2085 rows and 12 columns in the selected dataset. The features of the dataset include time in minutes since infusion started, concentration of remifentanil in ng/mL, infusion rate in µg/min, remifentanil quantity indicated in µg for the current time interval, age of the patient in years, gender, height in cm, weight in kg, body surface area, and lean body mass. Exploring the available data to identify the missing values and outliers confirms that the data used for predictions is unbiased to reduce the flaws in the predictions. The overall process is explained in Figure 1.

Figure 1.

A flowchart explaining the procedure

Regression tree

Regression trees have become increasingly popular in predictive applications due to their high prediction accuracy and easily interpretable structure.[14] Ensembling methods such as bagging, boosting, and random trees can combine the weaker regression tree models and produce a better overall model. Bagging, commonly called bootstrap aggregating, uses random sampling with replacement, creating multiple training sets.[15] The random tree, also called the random forest method, works by selecting a fixed number of features at random and using them to train multiple weak regression trees.[16] The focus of the regression tree algorithm is to construct a function ‘f’ such that the MSE converges to the minimum value. The expressions for MSE and average are appended in Annexure A. Decision trees are hierarchical structures that recursively divide the data into subsets depending upon the values of different features [Annexure A]. To address overfitting, approaches such as pruning and setting constraints on tree depth or minimum samples per leaf can be used, and to further enhance the predictive performance, ensemble techniques such as random forests and gradient-boosted trees are often employed to associate the predictions of multiple trees.[8]

Gaussian process regression

It is a non-parametric, stochastic Bayesian method of probabilistic supervised regression, gaining popularity in ML.[17,18] Gaussian process (GP) uses the mean function to indicate the predicted value of the function modelled at every input point.[19] In GP, the covariance function, also called the kernel function, quantifies the degree of similarity between the input data points[20] The covariance function calculates pair-wise similarities to determine the correlation between function values.[21] The prior distribution is characterised by mean and standard deviation. The posterior enables the probabilistic predictions, quantifies the uncertainty, and hence swings between the prior distribution and the information conveyed by the input and the response variable.[22] The mathematical representation of the Kernel function and the procedure for GPR is described in Annexure B.

Support vector regression

A SVM is a non-parametric technique that aims to predict a continuous output target variable with minimum error while staying within the definite margin. The hyperplane needs to be defined, and the user can determine the width of the strip [Annexures C and D]. Support vector regression (SVR) models combined with real-time monitoring systems can regulate remifentanil dosing depending on the physiological responses through closed-loop feedback.[23]

Neural network

A neural network-based regression is an effective tool for prediction based on input features.[24] It consists of multiple layers, each with its activation function, allowing the network to learn the complex relationships between characteristics and targets.[25] This network produces the predicted response values as its output.[26] The first phase of an artificial neural network (ANN) is called forward propagation and is characterised by the multiplication and addition of weights with each feature, with the bias.[27] The second phase is termed backward propagation and updates the model’s weights through optimisation and loss functions.[28] In regression, a loss function is the difference between the predicted and actual outputs to train the ANN.[29] To lessen the loss, the network’s weights are adjusted using a suitable optimisation method, such as stochastic gradient descent [Annexure E].[30]

Feature selection

The F-test is a statistical method to calculate the ratio of variances to provide the F-score. The F-test assesses the significance of all the input data features separately and ranks the predictors by comparing the P values of the F-test statistics. The F-test assesses whether the variances or means of various groups are significantly different. A higher F-score indicates lesser differences between the groups. The features are selected with a higher value of F-score expressed in Annexure F.[31]

Principal component analysis

Principal component analysis is a dimensionality reduction technique that transforms a higher-dimensional dataset into a smaller representation while retaining as much of the original variance as possible.[32] It can be helpful for visualisation, noise reduction, and improving the efficiency of ML algorithms [Annexure G].

Optimisation

Bayesian optimisation is a valuable method that leverages probabilistic models to optimise black-box functions.[33,34] By iteratively updating the model, optimising the function, and evaluating the actual objective function, Bayesian optimisation efficiently explores the search space, ultimately converging to the optimum of the true objective function.[35]

Performance evaluation

Performance indices compare the various models’ performances. It assesses how well a regression model predicts the target (dependent) variable based on the input features (independent variables) [Annexure H].[36]

Hold-out validation

The hold-out validation method divides the dataset into training and testing (or validation) sets assessing the model’s performance.[37] Specifically, the dataset is split into a training set and a validation (or test) set, with 80% of the data used for training and 20% reserved for testing in this study. The model is then trained using the training dataset and evaluated based on its performance using the validation set, which the model has not encountered during the training process.[38]

RESULTS

F-Test for feature selection

The dataset containing the remifentanil concentration, infusion rate, remifentanil quantity indicated for the current time interval, patient age, gender, height, weight, body surface area, and lean body mass was used for modelling and analysis. However, according to the F-test scores [Table 1], the patient’s gender was found to have an insignificant impact on the response. Therefore, it was excluded as a feature in the input matrix.

Table 1.

Ranking of features as per F-test scores

Features	F-test score
Rate	494.2142
Amount	478.0917
Time	404.5930
Age	78.9096
Weight	71.3334
Body surface area	53.7115
Lean body mass	49.4118
Height	46.7217
Gender	15.1308

Regression

Math Works, Inc., MATLAB, version 2023b was used as the simulation platform for predicting the drug concentration using the earlier features. With a minimum leaf size of 4, the model was trained using fine-tree regression. The response plot and the predicted versus actual plot are shown in Figure 2.

Figure 2.

Predicted and actual response using fine tree regression

By choosing the minimum leaf size as 8 and the number of learners as 30, bagged tree ensembling was used for regression; the predicted versus actual plot is shown in Figure 3. Using the Gaussian kernel function and a kernel scale of 0.71, a fine Gaussian SVM regressor was used on the selected features to predict the response; the results are projected in Figure 4. A wide neural network was used on the features with one fully connected layer with 100 neurons. A rectified linear unit (ReLU) activation function was used. The actual and response plots are described in Figure 5.

Figure 3.

Predicted and actual response using bagged tree regression

Figure 4.

Predicted and actual response using fine Gaussian support vector machine (SVM) regression

Figure 5.

Predicted and actual response using wide neural network

GPR was used on the features using the exponential kernel function; the actual and response plots are detailed in Figure 6. The summary of the performance comparison of the above five models is described in Table 2. Among the five models described above, GPR yielded the least root mean squared error (RMSE), mean absolute error (MAE), and better R-squared value. Hence, it was optimised further to reduce the RMSE to 5.4003. The convergence of the MSE is plotted in Figure 7.

Figure 6.

Predicted and actual response using exponential Gaussian process regression (GPR)

Table 2.

Summary of the performances of the selected regression models

Model Type	Root Mean Squared Error	Mean Squared Error	R-Squared value	Mean Absolute Error
Fine Tree	13.4816	181.7550	0.7607	6.4410
Bagged Tree	12.5332	157.0820	0.7932	6.4820
Fine Gaussian SVM	10.9246	119.3480	0.8429	5.6617
Wide Neural Network	8.0493	64.7923	0.9147	5.4528
Gaussian Process Regression	6.1365	37.6568	0.9504	2.7524
Optimised GPR	5.4003	29.1639	0.9616	2.5151

Figure 7.

Optimisable Gaussian process regression (GPR) showing the convergence of minimum mean squared error

With these models, one can aim for personalised dosing, especially in patients with varying physiological conditions and comorbidities. By optimising the dosing, the risk of respiratory depression or hypotension can be minimised. In addition, predicting the onset and offset times helps titrate remifentanil infusion rates more accurately. Including more features such as hepatic and renal function, comorbidities, and cardiac output may vary remifentanil pharmacokinetics. Considering non-linear interactions between the multiple drugs may also alter the response. Incorporating analgesia measurement using definite metrics and measuring the area under the curve to capture the drug dynamics will enhance model accuracy. Bootstrap resampling confirms the model’s robustness against variability, and the Bayesian approaches handle prior knowledge about remifentanil’s pharmacokinetics and manage uncertainty in predictions.

DISCUSSION

Exploring and evaluating various potential alternative modelling approaches is crucial in analysing pharmacokinetic models. Non-linear models commonly tackle these challenges with non-uniform time-series measurements. In several studies, ANNs are more accurate than conventional pharmacokinetic models.[39] Tolle et al.[40] compared the prediction of tobramycin concentration in the blood by using a back-propagation neural network and a non-linear mixed effects model in a paediatric population.

Many investigations have demonstrated comparable precision between ANN and SVM models. In contrast to our study, some investigations have proved that SVM’s accuracy is higher than ANN’s.[41]

In a study, remifentanil proved to be an appropriate opioid for subjects with severe hepatic failure, and hepatic impairment did not affect its pharmacokinetics.[42] Minto et al.[43] confirmed that age and lean body mass are critical demographic factors in establishing a dosage regimen for remifentanil. Smith et al.[44] investigated the pharmacokinetic profile of remifentanil in pregnant and non-pregnant women. They found it to be closer, even though there were variations in volume of distribution, clearance, and half-life. However, pharmacokinetic models must be improved to compare different methods, such as ensembling, for more accurate and individualised predictions.

ML models can identify individuals at higher risk of opioid misuse or addiction by analysing patterns and patient histories while also assisting in the prediction of potential adverse reactions or interactions with other medications. Furthermore, ML has the potential to simulate various pharmacokinetic scenarios, thereby aiding in the design of improved opioid formulations and delivery systems. Real-time data from electronic health records and monitoring devices can be processed using ML to track opioid use and patient health, providing early warnings for issues such as overdose or withdrawal symptoms.

This study has various limitations. The authors used multiple input variables to evaluate different ML algorithms to predict remifentanil concentrations. This was based on an analysis of a reference dataset from remifentanil infusions given to 65 healthy volunteers with EEG-adjusted dosages. EEG titration is closely associated with the anti-nociceptive effect-site concentration of remifentanil. However, implementing such dosimetry in clinical settings may be challenging, particularly when employing balanced anaesthesia techniques during major surgeries.

CONCLUSION

Our research involved various ML algorithms to accurately predict remifentanil concentration based on multiple factors, including infusion time, concentration, rate, amount, age, height, weight, body surface area, and lean body mass. We compared several regression algorithms, including fine tree, bagged tree, fine Gaussian SVM, wide neural network, and exponential GPR, to determine their effectiveness based on parameters such as MSE, RMSE, R-squared value, and MAE. GPR proved the most effective, producing superior results on the test dataset and demonstrating convergence to minimum MSE through optimisable GPR and hyperparameter tuning.

Ethics approval and informed consent

This study involved the secondary analysis of previously collected and anonymized data, which was publicly available. The dataset used is anonymized and does not contain identifiable information. Informed consent was originally obtained from all participants by the primary researchers as part of the initial data collection process.

Data availability

The authors may request the data with reasonable justification (email to the corresponding author), and they shall share them upon request.

Informed consent statement

Not applicable.

Conflicts of interest

There are no conflicts of interest.

ANNEXURES

1. Working of regression tree

   1. Splitting: The tree starts with the entire dataset from the root node. The algorithm chooses an input feature and a split point at every node to divide the data into two subsets.

   2. Recursive Process: The splitting process is performed recursively to each subset, generating a tree-like structure. This process lasts until a stopping criterion is seen, like a maximum tree depth or a minimum number of data points in a leaf node stated by the user.

   3. Leaf Nodes: The tree’s terminal nodes are called leaf nodes, and every leaf node comprises the predicted numerical value for the corresponding subset of data.

   4. Prediction: To predict a new occurrence, one has to navigate from the root to the leaf node based on the values of its features, and the expected value at that leaf node is considered the output.

   5. Loss Function: The algorithm minimises the loss function at every node, reducing the variance and the mean squared error in the target data.

      

2. Procedure for GPR

   1. Data Collection : Collect the response variables and the input features related to the regression problem.

   2. Choosing a Kernel Function : Choose a definite covariance function (kernel) to confine the problem statement. The selection of the covariance function impacts the functions of the GPR model.

   3. Hyperparameter Optimisation : Calculate the covariance function’s hyperparameters using the input’s maximum likelihood function. Algorithms such as gradient descent can be used to do the same.

   4. Prediction: Use the trained GPR model to predict the new dataset. GPR can calculate the expected mean and quantify the uncertainty:

      

      where k(x_i,x_j) is the kernel function that estimates the correlation between data points x_i and x_j. σ² is the variance, and ‘l’ is kernel distance or characteristic length to assess the rate at which the kernel function declines as data points are more distant.

      The posterior distribution can be calculated as follows:

      

      where P(y|x) is the marginal probability of the data, the prior distribution of f is P (f), and the likelihood of the data given the function f is (y|x, f).

      Given the training data y and x, the conditional probability of the predicted function values f* at a fresh input point x* is

      

      where p(f^*|x^*,f) is the conditional probability distribution of the expected function values f* at x*, and p(f|y,x) is the conditional probability distribution of the function values f.

3. Key features of SVR

   1. Kernel Function: The choice of kernel function is essential in SVR. Standard kernel functions include: Linear Kernel: K(x,y)=x^T

      • Polynomial Kernel: K(x,y)=(x^T.y+c)^d

      • Radial Basis Function Kernel (RBF Kernel):

   2. Cost (C) Parameter: Controls the trade-off between achieving a low training error and a smooth decision function.

   3. Epsilon (∈) Parameter: Determines the width of the margin. It represents a tube around the predicted values within which no penalty is incurred.

4. The Training Process of SVR

   1. Input Data: SVR takes a set of input-output pairs (x_i, y_i).

   2. Kernel Matrix: Computation of Kernel matrix K_ij = K (x_i, x_j)

   3. Optimisation Algorithm: Formulate and solve an optimisation problem to find the weights αi that minimise the error, with the condition that the predictions are within a margin element of the actual values.

   4. Prediction: Once the weights α_i are determined, the prediction for a new input x is made by computing the weighted sum of kernel evaluations.

5. Steps for Neural Network Modelling

   1. Data Preparation: The dataset is pre-processed and cleaned. The data is later split into training and testing datasets.

   2. Input and Output Layers: The input layer has neurons equal to the number of input features in the dataset. The output layer should have a single neuron as the algorithm predicts the continuous data.

   3. Hidden Layers: The number of hidden layers and the number of neurons in every hidden layer should be decided using hyperparameter searching. The structure of the hidden layers relies on the complexity of the problem. The frequently used activation functions in the hidden layers are rectified linear unit (ReLU) or hyperbolic tangent (tanh).

   4. Activation Function for Output Layer: The output layer uses a linear activation function for the regression process. It allows the model to output unbounded values.

   5. Loss Function: A suitable loss function, such as MSE, is used. MSE calculates the average squared difference between the predicted and true response.

   6. Optimiser: A suitable optimiser is chosen to minimise the loss function during training.

   7. Training: The model is trained with the training data. The weights and biases are adjusted during training to optimise the loss.

   8. Hyperparameter Tuning: Experiment with hyperparameters such as the learning rate, the number of hidden layers, and the number of neurons to decide the best configuration.

   9. Validation: A validation set screens the model’s performance during training and avoids overfitting.

   10. Evaluation: The model is assessed on the test set to evaluate its generalisation to new, unseen data.

   11. Fine-tuning: The model architecture or hyperparameters are altered based on their performance on the test dataset. Hyperparameter tuning and experimentation with different architectures are critical to achieving the top performance of the model.

6. F-Test

Hypothesis formulation

Null hypothesis: This indicates that the regression coefficients are equal to zero, and the independent variables do not have any relationship with the dependent variable.

Alternative hypothesis: At least one of the regression coefficients is not zero, implying some relationship exists between the independent and dependent variables.

Comparison with critical value

The F-score is then compared to a critical value from the F-distribution table. If the F-score > critical value, the null hypothesis should be rejected, and if the F-score ≤ critical value, one cannot reject the null hypothesis.

• G. Steps in principal component analysis

  1. The covariance matrix computation provides the relationship between different features.

  2. Eigen-decomposition is performed on the covariance matrix to obtain its eigenvectors (direction of maximum variance) and eigenvalues (magnitude of variance).

  3. The eigenvectors become the principal components, and eigenvectors are sorted.

  4. After choosing the top ‘k’ vectors, a new transformed matrix with fewer dimensions is formed.

• H. Performance evaluation of the models

  1. MSE calculates the squared difference between predicted and actual values of each data point. It penalises the more significant errors more heavily and can calculated using the following expression:

     

  2. RMSE: The square root of the mean squared error. It measures the average magnitude of errors in the same units as the target variable.

  3. MAE: It uses absolute differences instead of squared differences. It is less sensitive to outliers compared to mean squared error.

  4. R-squared value or coefficient of determination: Measures the proportion of the variance in the dependent variable that is predictable from the independent variables. Ranges from 0 to 1, where 1 indicates a perfect fit and is expressed as follows:.

     

Funding Statement

This work was supported by the [Indian Council of Medical Research] under Grant [letter No. 5/3/8/81/2020-lTR].