An artificial intelligence-assisted physiologically-based pharmacokinetic model to predict nanoparticle delivery to tumors in mice

Abstract

The critical barrier for clinical translation of cancer nanomedicine stems from the inefficient delivery of nanoparticles (NPs) to target solid tumors. Rapid growth of computational power, new machine learning and artificial intelligence (AI) approaches provide new tools to address this challenge. In this study, we established an AI-assisted physiologically based pharmacokinetic (PBPK) model by integrating an AI-based quantitative structure-activity relationship (QSAR) model with a PBPK model to simulate tumor-targeted delivery efficiency (DE) and biodistribution of various NPs. The AI-based QSAR model was developed using machine learning and deep neural network algorithms that were trained with datasets from a published “Nano-Tumor Database” to predict critical input parameters of the PBPK model. The PBPK model with optimized NP cellular uptake kinetic parameters was used to predict the maximum delivery efficiency (DEmax) and DE at 24 (DE24) and 168 hours (DE168) of different NPs in the tumor after intravenous injection and achieved a determination coefficient of R² = 0.83 [root mean squared error (RMSE) = 3.01] for DE24, R² = 0.56 (RMSE = 2.27) for DE168, and R² = 0.82 (RMSE = 3.51) for DEmax. The AI-PBPK model predictions correlated well with available experimentally-measured pharmacokinetic profiles of different NPs in tumors after intravenous injection (R² ≥ 0.70 for 133 out of 288 datasets). This AI-based PBPK model provides an efficient screening tool to rapidly predict delivery efficiency of a NP based on its physicochemical properties without relying on an animal training dataset.

Graphical abstract

1. Introduction

Nanomedicine is a rapidly growing area of biomedical research that may overcome the intrinsic limits of conventional cancer chemotherapies for more effective and safer cancer treatment [1, 2]. Yet, the challenge remains in the development of cancer nanomedicine owing to the inability for nanoparticles (NPs) to efficiently deliver therapeutic agents into solid tumors [3, 4]. This is, in large part, due to the poor understanding of the roles of physicochemical properties (e.g., size, shape, and zeta potential) in pharmacokinetics and target tissue delivery and a lack of computational models to accurately predict pharmacokinetics and tissue delivery of NPs based on their physiochemistry [5].

Recent advances in computational power and the increase in the amount of biological data on NP–biological interactions have made it possible to develop rigorous computational models to systematically identify various physicochemical properties and tumor characteristics that influence tumor delivery efficiency of NPs [6]. Wu et al. [7] summarized a series of mathematical equations that can be used to quantitatively describe the delivery process of NPs in several kinetic compartments including the administration site, target cell vicinity, target cell interior, and off-target sites. These equations can be a basis for the development of more holistic whole-body computational models to facilitate optimal designs of cancer nanomedicine. In this regard, physiologically based pharmacokinetic (PBPK) modeling is an mechanistic-based computational approach that is capable of describing absorption, distribution, metabolism, and excretion (ADME) of chemical substances in the whole body based on physiologically relevant mechanisms using mathematical equations. Over the last decade, significant efforts have been devoted to developing PBPK models for various NPs in rodents and humans [8–17]. Recently, we developed a PBPK for NPs in healthy mice [12, 18] and extrapolated it to tumor-bearing mice, which was calibrated with 376 tumor datasets for different NPs obtained from 200 studies [4]. This model was used to predict maximum delivery efficiency (DEmax), and delivery efficiency at 24 hours (DE24) and 168 hours (DE168) of NPs in tumors following intravenous (IV) injection in tumor-bearing mice. The PBPK model-derived tumor delivery efficiency data at different time points for different NPs and different tumor types with different pharmacokinetic metrics (DEmax, DE24, and DE168) were compiled in to a “Nano-Tumor Database” [4] to facilitate further applications by other studies. However, this earlier PBPK model [4] has a major limitation. In this model, several critical kinetic parameters of NPs in tumor microenvironment, such as maximum uptake rate constant (KTRES_max), release rate constant (KTRES_rel), Hill coefficient (KTRES_n) and time reaching 50% maximum uptake rate (KTRES_50) of NPs are difficult to measure experimentally and are usually obtained by fitting to in vivo animal experimental data. As a result, the model construction still relies heavily on animal studies and the model simulation is limited to one particular type of NP based on each training dataset, leaving the earlier model with limited capability to extrapolate from one NP to other types of NPs.

Recent advancements in machine learning and artificial intelligence (AI) may potentially address the above mentioned limitation in PBPK modeling of NPs [19]. Recently, using chemical structure and dose for small molecule organic drugs, machine learning models have been trained based on the AstraZeneca clinical data to predict several rat and human pharmacokinetic parameters (e.g., Cmax, AUC and volume of distribution) as well as time-concentration pharmacokinetic profiles [20, 21]. Machine learning models have been built to predict ADME properties and to solve complex pharmacokinetic profiles in small molecular drugs [19, 22–24], though they have not been commonly applied to NPs. Theoretically, a machine learning and AI model can generate NP-specific pharmacokinetic parameters based on the NP physicochemical properties as input to the PBPK model to predict pharmacokinetics, tissue distribution and tumor delivery without relying on animal experimentation. If successful, this AI-assisted PBPK approach can serve as an alternative non-animal method to facilitate nanomedicine research and development.

The objective of this study is to integrate PBPK modeling with machine learning and AI approaches to build an AI-assisted PBPK model to predict pharmacokinetics and tumor delivery of different NPs based on their physicochemistry in mice. A flowchart showing the step-by-step procedure of the methodology of this study is illustrated in Fig. 1. We hypothesize that tumor-related critical kinetic parameters (e.g., KTRES_max and KTRES_rel mentioned above) can be predicted by a machine learning model based on the NPs’ physicochemical properties and tumor physiological information. Then AI-predicted parameters can be used as input into the PBPK model to accurately predict tumor delivery efficiency of NPs. In the present study, we built a machine learning model to predict tumor-related critical kinetic parameters (e.g., KTRES_max and KTRES_rel mentioned above) of NPs and then integrated it with our earlier PBPK model to yield an AI-assisted PBPK model for NPs in tumor-bearing mice. The AI-assisted PBPK model was trained and validated with 378 tumor datasets from our Nano-Tumor Database [4]. Compared to our earlier traditional PBPK model [4], this AI-PBPK model provides a substantially improved computational platform to predict tumor delivery efficiency of NPs based on their physicochemical properties (without relying on an animal training dataset), which can greatly aid the design of nanomedicines with desired optimal tumor targeting efficiency for different types of NPs. This AI-based PBPK model can be applied to different types of NPs and tumors and serves as a non-animal alternative approach to facilitate nanomedicine development.

Fig. 1.

Overview of the computational workflow to integrate machine learning and deep learning models with physiologically based pharmacokinetic (PBPK) modeling to predict delivery efficiency of nanoparticles (NPs) to the tumor site in tumor-bearing mice. (A) Step 1: Nano-Tumor Database, (B) Step 2: Development of AI-QSAR model, (C) Step 3: AI-assisted PBPK model. Abbreviations: DNN, deep neural network; RF, random forest; Adj-R², adjusted coefficient of determination; RMSE, Root mean square error; KTRES_max, maximum uptake rate constant of tumor cells; KTRES_50, time reaching half maximum uptake rate of tumor cells; KTRES_n, Hill coefficient for the uptake of tumor cells; KTRES_rel, release rate constant of tumor cells.

2. Materials and Methods

2.1. Description of the original PBPK model for NPs in tumor-bearing mice

The PBPK model in tumor-bearing mice was developed based on our previous studies [4, 18]. Briefly, the model consisted of eight compartments including plasma, lungs, liver, kidneys, spleen, brain, muscle, and remaining tissues (i.e., pooled other tissues) (Figs. 2A–2B). Each compartment (except the brain and plasma) was divided into three sub-compartments: capillary blood, interstitium, and endocytic or phagocytic cells (PCs). This was described with a permeability-limited model by considering an uneven distribution between capillary blood and tissue to account for membrane-limited transcapillary transport (Fig. 2C), as well as nonlinear endocytic uptake and first order exocytic release of administered NPs. Similarly, in order to simulate the tumor microenvironment, the tumor compartment was specifically divided into capillary blood, tumor interstitium, and tumor cells (TCs) (Fig. 2C). Based on our earlier studies [4, 18], uptake of NPs by PCs or TCs was described using a modified Hill function, whereas the release of NPs from PCs or TCs back to the blood or interstitium was described using a linear first-order equation. Key mathematical equations describing uptake to and release from TCs are provided and further explained in the Supplementary Material. Details for the rest of the equations in the PBPK model are described in our earlier studies [4, 18].

Fig. 2.

Schematic diagram of physiologically based pharmacokinetic (PBPK) model for NPs in (A) tumor-bearing mice intravenously (IV) administrated with AuNPs and various inorganic and organic nanomaterials. (B) This PBPK model consists of eight compartments including plasma, lungs, liver, kidneys, spleen, brain, muscle, remaining tissues (i.e., pooled other tissues) and tumors. (C) Except plasma and brain, each compartment was divided into three major parts: capillary blood, interstitium, and endocytic or phagocytic cells (PCs), or tumor cells (TCs).

2.2. Development of the AI-QSAR model to predict key PBPK kinetic parameters

2.2.1. Datasets and data preprocessing

All datasets used for model training and validation are from our published “Nano-Tumor Database” [4]. This database consists of 378 tumor datasets from 200 studies after IV administration of different types of NPs in tumor-bearing mice, with 376 datasets reported in the original publication [4] and 2 datasets [25, 26] that were missed in the initial data collection and were collected in the present study. Each tumor dataset is accompanied with the physicochemical properties of a NP, experimental design information (i.e., tumor therapy strategies) of a particular study, and data-driven variables from our original PBPK model [4]. NPs’ physicochemical properties included hydrodynamic diameter (HD), Zeta potential (Zeta), surface charge (charge), core material (MAT) and type of NPs; while the tumor therapy strategies included the variables related to the targeting strategies (TS), cancer types (CT), tumor model (TM), tumor weight (TM) and tumor size (TSiz). The data-driven variables included tumor-related PBPK model parameters (i.e., KTRES_max, KTRES_50, KTRES_rel, and KTRES_n) and tumor delivery efficiencies (i.e., DE24, DE168 and DEmax) derived using the PBPK model. The tumor delivery efficiency (i.e., delivery efficiency of NPs to tumors) was defined as “the percentage of administrated NPs that can be accumulated in a tumor [3, 27]”, which can be estimated as $DE=\left(AU{C}_{tumor}\times M_{tumor}\right)/t$ where DE is the delivery efficiency (%ID), $AU{C}_{tumor}$ is the area under curve of NPs’ biodistribution profile in tumor (%ID*h/g), $M_{tumor}$ is the weight of the tumor (g), and t is the time-series endpoint (h) examined for NPs’ biodistribution profile. In our previous study [4], the $DE$ at 24 h (i.e., DE24 [an indicator of short-term $DE$]), 168 h (i.e., DE168 [an indicator of long-term $DE$]), and the maximum (i.e., DEmax) NPs tumor delivery efficiency after IV injection were estimated and used as data-driven variables. The symbols and further explanation for all the variables in the Nano-Tumor Database are provided in Table 1.

Table 1:

Overview of variables and their symbols used in the machine learning models

Variable	Unit	Symbol	Levels or Ranges
Nanoparticle’s properties
Type of nanoparticles	-	Type	Inorganic; Organic; Hybrid
Core materials of nanoparticles	-	MAT	Gold, Dendrimers, Liposomes, Polymeric, Hydrogels, Other Organic Material, Other Inorganic Material
Shape of nanoparticles	-	Shape	Spherical, Rod, Plate, Others
Hydrodynamic diameter	nm	HD	[5, 456]
Zeta potential	mV	ZP	[0, 274]
Charge	-	Charge	Positive; Negative; Neutral
Tumor therapy strategies
Targeting strategy	-	TS	Passive, Active
Tumor model	-	TM	Allograft Heterotopic, Allograft Orthotopic, Xenograft Heterotopic, Xenograft Orthotopic
Cancer type	-	CT	Brain, Breast, Cervix, Colon, Liver, Lung, Ovary, Pancreas, Prostate, Skin
Tumor weight	g	TW	[0.02, 5.09]
Tumor size	cm	TSiz	[0.02, 1.8]
Dosing regimen
Dose	mg/kg	Dose	[0.001, 1220]
Body weight	g	BW	[16, 35]
Administrated route	-	AR	IV
PBPK model parameters
Release rate constant of tumor cells	1/h	KTRES_rel	[0.0001, 14]
Maximum uptake rate constant of tumor cells	1/h	KTRES_max	[0.001, 25]
Hill coefficient of tumor cells	-	KTRES_n	[0.01, 10]
Time reaching half maximum uptake rate of tumor cells	h	KTRES_50	[0.00001, 180]
Tumor delivery efficiency
DE at 24 hours	%ID	DE24	[0.008, 50]
DE at 168 hours	%ID	DE168	[0.003, 23.8]
Maximum DE	%ID	DEmax	[0.008, 56.9]

Abbreviations: DE, delivery efficiency; ID, injected dose; NPs, nanoparticles; PBPK, Physiologically based pharmacokinetic.

For data preprocessing, we first checked the dataset and removed the rows with missing values because the majority of machine learning methods require all feature values to be present. Two hundred eighty-eight datasets out of 378 datasets were used in the development of machine learning and deep learning (DL) models (The model datasets are available in Supplementary Material). Since there were different types of variables included in this database, two different preprocessing algorithms including feature-scaling and one-hot encoding were applied to process the numerical and categorical datasets, respectively. In brief, the numerical variables were normalized to the values between 0 and 1 by using feature scaling algorithms [28] to aid the model optimization efficiency, while the categorical data were encoded by splitting variables into different columns with its own encoded binary string by one-hot encoding algorithms. All the preprocessing algorithms were implemented by Python library Scikit-Learn [29].

2.2.2. Machine learning model

Five different machine learning algorithms including linear regression (LR), support vector regression (SVR), random forest (RF), XGBoost (eXtreme Gradient Boosting) and LightGBM (Light Gradient Boosting Machine) were used for the machine learning model development. These algorithms were implemented using the ski-learn package in Python 3.6.7 [30].

A deep neural network (DNN) or DL model was developed based on open-source Python library TensorFlow v.2.2 and Keras v2.0. The basic structure of a DNN model includes three types of layers: input layer, middle hidden layers and output layer. The input layer is responsible for receiving the inputs, performing the calculation via nodes (computational units), and transmitting the results to the subsequent layers. Each hidden layer contains a number of nodes, from which the output propagates as input to each node of the subsequent layer. The output layer is responsible for producing the results through the calculation of inputs which are passed in from the layers before it. The architecture or topology of the DNN was determined by hyperparameters (e.g., number of layers and the number of nodes), which can be tuned by the Bayesian optimization algorithm to achieve optimal fitting.

2.2.3. Hyperparameters tuning

To get the best-performance model to predict the target variable, the hyperparameters in the machine learning and DL model need to be optimized. In this study, the Bayesian hyperparameter optimization algorithm was used to tune the machine learning and DL model [31]. Briefly, the Bayesian optimization algorithm constructs a probability model which can map hyperparameters to a probability of a score on the objective function: Pr (score| parameters), where the score represents the selected performance metrics (Please refer to Section 2.2.4) and the parameters indicate the generated hyperparameters by the algorithm. The algorithm can automatically search for the best hyperparameters that perform best on the object function. In this study, a KerasTuner library, a hyperparameter tuning library that is incorporated in the Keras DL library [32], was implemented to optimize the hyperparameters. The search space for the DL model included hidden layers [1–3 layers], Neurons [random uniform distribution between 38 and 512 for each of layers], optimization methods [Adam or SGD], dropout rate [0–0.5] and learning rate [random uniform distribution between 0 and 1]. For all machine learning models, the default search space was used as default settings.

2.2.4. Performance metrics and benchmarks of the AI-QSAR model

The model performance was evaluated via internal and external nested 5-fold cross validation strategies based on earlier studies [20, 33]. Briefly, the original dataset was randomly split into training (80%) and test sets (20%). For the internal validation, the training set was further split into inner and outer sets using nested 5-fold cross-validation setting. Inner cross-validation sets were used to train the model on the machine learning algorithm to determine the optimized sets of hyperparameters achieving the highest scores of selected performance metrics. The hyperparameter tuning was conducted by Bayesian optimization with Gaussian processes. After training and evaluating the models with the 5-fold cross-validation process, the best-performed models were further validated by external validation with test set data.

In the process of model development, the root means square error (RMSE) and adjusted determination coefficient (R²) were selected as performance metrics and defined as below:

$$RMSE= \sqrt{\frac{1}{n}\cdot \left(\sum {\left(y-\widehat{y}\right)}^2\right)}$$ (1)

$$R^2=1- \left(\sum {\left(y-\widehat{y}\right)}^2/\sum {\left(y-\overline{y}\right)}^2\right.$$ (2)

where y is the observed response variable value derived from the Nano-Tumor Database based on a PBPK model, $\overline{y}$ is its mean, $\widehat{y}$ is the corresponding predicted value, and n is the number of data sets. R² was used to evaluate the goodness-of-fit of the model, and RMSE was used to evaluate the error between observed and predicted values. The lower value of RMSE indicates the higher predicted accuracy of the model, while a higher value of R² is considered desirable.

2.3. Integration of the PBPK with the AI-QSAR model to become an AI-assisted PBPK model

2.3.1. AI-assisted PBPK model

In order to develop a generic PBPK model that is applicable to different types of NPs, the critical input parameters to the PBPK model including KTRES_max, KTRES_50, KTRES_rel, and KTRES_n, were computed by AI-QSAR models (Section 2.2). Among the QSAR models developed by the machine learning and AI algorithms, the best-performance AI-QSAR model was chosen to integrate with the PBPK model. Following the integration of the PBPK and AI-QSAR models to become the AI-PBPK model, the critical parameters were predicted and used in a system of differential equations (Eqs. S1-S4) to simulate the endocytic uptake and release of different types of NPs in the tumors of mice.

2.3.2. Evaluation of the AI-assisted PBPK model

To verify the AI-assisted PBPK model, the observed time-course profiles for various types of NPs in tumors were used to compare with the model-predicted pharmacokinetic profiles. The 288 time-course pharmacokinetic profiles of NPs in tumors were collected from our previous Nano-Tumor Database [4] after IV administration of different types of NPs in tumor-bearing mice (The time-course pharmacokinetic profiles can be found in the GitHub repository: https://github.com/UFPBPK/Nano-AI-PBPK-G1). All datasets at different time points were used in the evaluation of the predictive performance of the final AI-assisted PBPK model by visualized check and statistical metrics (i.e., RMSE and adjusted R-squared values).

2.4. Implementation environment

The machine learning models were developed in Python 3.8 using Jupyter Notebook in the environment: 8 Intel(R) Xeon(R) CPUs E5-2698 v3 (3.0 GHz), with 96 GB of RAM, using the University of Florida HiPerGator 3.0 (computing cluster environment). Other important libraries included Scikit-Learn 0.19.0, Keras 2.1.4, Tensorflow 1.9, and Hyperas 0.4. The simulations using the final AI-assisted PBPK model were performed on an Intel laptop with i9-10900X CPU at 3.7 GHz and 64 GB RAM using R program (Version 4.1.3) with the libraries “mrgsolve” [34], “FME” [35], “ggplot2” [36] and “dplyr” [37].

2.5. Data and code availability

The full dataset that was used to train and validate the AI-PBPK model can be accessed from the Supplementary Material (file name: “Model_dataset”). All data files and source code files are available at the GitHub repository: https://github.com/UFPBPK/Nano-AI-PBPK-G1.

3. Results

3.1. Study workflow

The study framework is illustrated in Fig. 1. All data which were used to build the machine learning and DL models, including the physicochemical properties of NPs, tumor therapy strategies and data-driven variables, were obtained from our Nano-Tumor Database [4] (Fig. 1A). The conventional machine learning algorithms (e.g., LR and SVM), ensemble learning models (e.g., RF, LightGBM and XGBoost) and DNN were used to develop AI-based QSAR models based on the input features consisting of the physicochemical properties of NPs (Type, Size, ZP, shape, MAT), and the features related to the description of tumor studies (TM, TS, and CT) to predict the data-driven tumor-related kinetic parameters (i.e., KTRES_max, KTRES_n, KTRES_50 and KTRES_rel) (Fig. 1B). During model training, the 5-fold cross-validation was conducted to identify optimal hyperparameters and evaluate the model with training and test data sets with several model performance indicators such as R² and RMSE. The best-performance QSAR model was integrated into the PBPK model in tumor-bearing mice (the model schematic is showed in Fig. 2) as the final AI-assisted PBPK model to predict the tumor delivery efficiency and biodistribution of NPs to various tissues (Fig. 1C).

3.2. Nano-Tumor Database

The variables included in our database are summarized and presented in Table 1 and Figs. S1–S2. All variables can be categorized by physicochemical properties of the NPs (e.g., Type, Size, ZP, shape, MAT), tumor therapy strategies (e.g., TS, TM, CT, TW, TSiz), data-driven model parameters (e.g., KTRES_max, KTRES_n, KTRES_50 and KTRES_rel), tumor delivery efficiency (e.g., DE24, DE168 and DEmax) and dose regimen (Dose, BW and AR). For the physiochemical properties, the majority of NPs’ types were organic (71%) followed by inorganic and hybrid (Fig. S1). The Zeta potential of NPs was separated into two variables in this database, including Zeta and Charge. The Zeta variable, which was defined as the magnitude of zeta potential, ranged from 0 mV to 274 mV. The Charge variable was defined as surface charge of NPs with categories of negative, neutral and positive. The HD variable, which described the hydrodynamic diameters of the NPs, ranged from 5 nm to 456 nm. The database contained a wide range of cancer cell types originated from different organs, including breast (30%), liver (17%), colon (8%), cervix (7%), and lung (6%), and others (32%) (Fig. S2A). The mouse tumor models included Allograft Heterotopic (AH, 38%), Allograft Orthotopic (AO, 12%), Xenograft Heterotopic (XH, 38%), and Xenograft Orthotopic (XO, 12%) (Fig. S2B). The variable of targeting strategies was defined as categorical variables including “passive” and “active” groups, in which most datasets were passive targeting (67%) (Fig. S2C). Data-driven variables including PBPK model parameters (KTRES_max, KTRES_n, KTRES_50, and KTRE_rel) and tumor delivery efficiency (DE24, DE168 and DEmax) were estimated from our previous study [4], which were used to verify the PBPK model simulations. Other variables such as dosing (e.g., BW, Dose, AR) were used for the PBPK model simulation.

3.3. Training and Testing Results of AI-QSAR Models

The R-squared and RMSE values for tumor-related parameters (i.e., KTRES_rel, KTRES_n, KTRES_max, and KTRES_50) evaluated from the developed machine learning and DL models based on the 5-fold cross validation and testing dataset are summarized in Table 2. Among the selected machine learning algorithms, the RF model demonstrated better predictions for each of model parameters with the higher R² and lower RMSE values by comparing with other machine learning models. The range of R² and RMSE values in the test dataset for the RF model was from 0.16 to 0.43 and from 1.83 to 30.68 across all the PBPK parameters, while the values from the 5-fold cross validation were from 0.03 to 0.14 and from 1.96 to 31.9, respectively. The LR and SVR models showed the relatively weakest prediction with the R² values for all the parameters lower than 0.1 or showing negative values. Although, the RF, XGB and LightGMS models showed similar prediction performances for KTRES_rel, KTRE_50 and KTRES_max, the results may not be reliable because distinct differences of R² values were observed between the results from test set and 5-fold cross validation. For the DL model, the prediction demonstrated superior performance among all methods with the highest R² values and lowest RMSE values. The R² values were 0.47, 0.53, 0.91 and 0.23 for KTRES_rel, KTRES_n, KTRES_max, and KTRES_50 in testing set, respectively, while these values from 5-fold cross validation were 0.67 ± 0.34, 0.47 ± 0.23, 0.44 ± 0.29, and 0.22 ± 0.37. Additionally, the similar ranges of R² and RMSE between the results from 5-fold cross validation and test set in the DL model indicated that the DL model was reliable and there were no overfitting problems.

Table 2:

Five-fold cross-validation and testing results for tumor-related parameters using different machine learning and deep learning models

	KTRES_rel		KTRES_n		KTRES_max		KTRES_50
Model	5-fold CV	Test	5-fold CV	Test	5-fold CV	Test	5-fold CV	Test
LR
R2	−0.08 ± 0.16	0.04	−0.03 ± 0.03	0.03	−0.04 ± 0.18	0.06	−0.04 ± 0.04	0.06
RMSE	2.05 ± 0.45	2.49	2.25 ± 0.15	2.32	3.59 ± 1.03	2.25	33.37 ± 4.86	35.28
SVR
R2	−0.05 ± 0.06	−0.01	−0.08 ± 0.06	−0.05	−0.02 ± 0.01	0.001	−0.18 ± 0.15	−0.02
RMSE	2.03 ± 0.48	2.54	2.29 ± 0.19	2.4	3.62 ± 1.16	2.31	35.4 ± 5.16	36.8
RF
R2	0.03 ± 0.13	0.18	0.138 ± 0.08	0.4	0.035 ± 0.28	0.43	0.11 ± 0.06	0.16
RMSE	1.96 ± 0.49	2.3	2.05 ± 0.07	1.83	3.37 ± 1.05	1.75	31.9 ± 5.33	30.68
XGBoost
R2	0.001 ± 0.13	0.13	0.085 ± 0.103	0.07	0.07 ± 0.24	0.36	0.079 ± 0.05	0.15
RMSE	1.99 ± 0.53	2.37	2.1 ± 0.11	2.27	3.35 ± 1.19	1.84	32.7 ± 4.74	30.78
LightGBM
R2	0.045 ± 0.09	0.2	0.025 ± 0.09	0.11	0.15 ± 0.43	0.31	0.131 ± 0.07	0.16
RMSE	2.03 ± 0.49	2.28	2.17 ± 0.147	2.22	3.58 ± 1.00	1.92	31.7 ± 4.79	30.6
DNN
R2	0.67 ± 0.34	0.47	0.47 ± 0.23	0.53	0.44 ± 0.29	0.91	0.22 ± 0.37	0.23
RMSE	0.74 ± 0.46	1.85	1.51 ± 0.34	1.6	2.65 ± 1.47	0.71	28.6 ± 6.52	32.15

Abbreviations: LR, linear regression; RF, random forest; SVM, regular support vector machine; DNN, deep learning neural network; CV, cross-validation; XGBoost, eXtreme Gradient Boosting; LightGBM, light gradient-boosting machine.

3.4. Evaluation of AI-QSAR Model Predictions with Data-driven PBPK Parameters

Comparisons of the AI-QSAR model predictions and data-driven parameter distributions from the original PBPK model are shown in Fig. 3 [4]. The similarity between the predictive parameter distribution with corresponding distribution of data-driven parameters obtained in the Nano-Tumor Database were evaluated for each parameter (i.e., KTRES_50, KTRES_max, KTRES_n, and KTRES_rel) by a linear regression model. Overall, predicted parameters’ distributions showed a similar range and high correlation with data-driven parameters. The median with 95% confidence interval (CI) of generated parameters were 13.4 (95% CI: 0.83–80.3), 0.47 (95% CI: 0.24–13), 1.8 (95% CI: 0.37–7.42) and 0.18 (95% CI: 0.001–6.16) in KTRES_50 (Fig. 3A), KTRES_max (Fig. 3B), KTRES_n (Fig. 3C) and KTRES_rel (Fig. 3D), respectively, which were close to the ranges of data-driven values of KTRES_50 [9.5 (95% CI: 0.01–118)], KTRES_max [0.31 (95% CI: 0.01–11.6)], KTRES_n [2 (95% CI: 0.05–8)] and KTRES_rel [0.1 (95% CI: 0.001–7.76)]. In addition, the high similarity of parameter distribution between predictions and data-driven values was observed with high Adj-R² values (R² >0.7) for each of parameters.

Fig. 3.

Densities of predicted (yellow bar) and data-driven parameters (purple bar) distributions for (A) KTRES_50, (B) KTRES_max, (C) KTRES,_n and KTRES_rel. The range of predicted and data-drive values were expressed as median (95% CI) in the plots. The adjusted-R2 (Adj-R²) was estimated by linear regression model.

3.5. Evaluation of AI-PBPK Model Predictions with Data-driven Tumor Delivery Efficiency

To further verify the model, the final AI-PBPK model based on the input of DL-predicted model parameters was used to predict tumor delivery efficiency and compared with the data-driven values from the Nano-Tumor Database (Figs. 4A–C). The results showed that the prediction of DE24 (Fig. 4A) and DEmax (Fig. 4C) were better than DE168 (Fig. 4B) based on the higher R² values and higher prediction percentages within two- and three-fold errors. The adjusted R² values were 0.83, 0.56 and 0.82 along with the RMSE of 3.01, 2.27 and 3.51 in DE24, DE168 and DEmax, respectively. In addition, the prediction percentages within two-fold errors were 69.7%, 11% and 74.6%, while these values within three-fold errors were 92%, 19% and 92% for the DE24, DE168 and DEmax, respectively.

Fig. 4.

Evaluation results of AI-PBPK model-predicted tumor delivery efficiency. A global evaluation of goodness of model fit between the data-driven (x-axis) and AI-PBPK model-predicted delivery efficiency (DE) (y-axis) at (A) 24 hours, (B) 168 hours and (C) the maximum DE. Abbreviations: %2e and %3e, represent the percentage of data points within 2-fold and 3-fold errors, respectively; Adj-R² and RMSE represent the adjusted determination coefficients and root mean square, respectively.

3.6. Evaluation of AI-PBPK Model Predictions with Measured Pharmacokinetic Profiles

With the PBPK parameters predicted by the DNN model, the goodness of fit of the final AI-PBPK model was evaluated by comparing model-predicted median concentrations with observed mean values (given in percent of initial dose [%ID or %ID/g]) in tumors at all measured time points for all datasets. Overall, the AI-PBPK predictions were in good agreement with the observed data with the adjusted-R value of 0.67 and the RMSE of 14.7 (Fig. 5A). In addition, most predictions were within the two- (57%) or three-fold (87%) error range, suggesting adequate model predictions (Fig. 5B). To visually confirm predictability, the time-course predictions based on the final AI-PBPK model were generated and compared with measured concentrations of NPs in tumors for each dataset from each study. The AI-assisted PBPK model adequately predicted the kinetic profiles of different types of NPs in tumors after IV administration. Fig. 6 presents representative results for 15 randomly selected datasets of different NPs. Comparisons of the AI-PBPK-predicted pharmacokinetic profile of different NPs in tumor after IV administration with observed data for each individual dataset are provided in the Supplementary Material (Figs. S3–S22). The AI-PBPK model predictions adequately correlated with available 288 observed pharmacokinetic profiles of different NPs in tumors after IV injection (R2 ≥ 0.70 for 133 out of 288 datasets).

Fig. 5.

Evaluation results of AI-PBPK model-predicted time-dependent distribution of nanoparticles (NPs) to tumors. (A) Comparisons between observed and predicted NPs in tumors (%ID/g) for all datasets (i.e., all types of NPs and different tumors) and (B) predicted-to-observed ratio versus model prediction plot. Abbreviations: %2e and %3e, represent the percentage of data points within the 2-fold and 3-fold errors, respectively; Adj-R² represent the adjusted determination coefficients.

Fig. 6.

Representative evaluation results of comparisons between the AI-PBPK model predictions versus experimentally measured pharmacokinetic profiles of NPs in tumors. NP concentration in tumor (%ID/g) predicted from the PBPK model with optimized parameters (dashed line) compared to the observed NPs amount in tumor from experimental data (black closed circles) for 15 randomly selected NPs from the study of (A-B) Cabral et al. (2011) [65], (C-E) Guo et al. (2013) [66], (F-G) Sumitani et al. (2013) [67], (G-M) Bae et al. (2007) [68], (N) Bibby et al. (2005) [69] and (O) Bae et al. (2005) [70]. Tumor tissue concentrations as presented in the y-axis are expressed in the units of percent of the injected dose (%ID/g) according to units used in the original articles.

4. Discussion

In the past decade, PBPK modeling and simulation approaches have been proposed for use in the development of nanomedicines [13, 14, 16, 33, 38, 39]. However, existing NP PBPK models are limited to one model for each single NP, thus the scalability and generalization of existing models are limited, in part, due to lack of a large database to build a more robust model. Motivated by the rapid advance in machine learning and AI algorithms and the availability of a large well-curated pharmacokinetic database for different types of NPs, the present manuscript reports on the development of an AI-assisted PBPK model that can be used to predict tumor delivery efficiency of different types of NPs in different types of tumors in mice based on the physicochemical properties of NPs and the study design information. This modeling strategy can be applied to other animals and humans. This AI-PBPK model represents an emerging research paradigm of integrating robust machine learning and AI algorithms with well-curated large pharmacokinetic databases of NPs to build better and more accurate computational models to support design and evaluation of cancer nanomedicines.

The difficulty in the development of cancer nanomedicines stems from the fact that only a small percentage of injected doses of NPs (<1%) are delivered to the solid tumor [3, 4, 40], which is also a major challenge in the clinical translation of nanomedicines. Although NPs preferentially accumulate within a tumor over the normal tissue based on the mechanisms of the enhanced permeability and retention effects, the lack of understanding the interactions between the NPs’ physicochemical properties and tumor microenvironment barriers hinder the improvement of drug delivery to tumors. Our previous study used a predictive model with machine learning and DL methods to predict the tumor delivery efficiency and identified key determinants that played an important role in tumor delivery efficiency of NPs, including core material type (e.g., gold, polymeric, liposomes), surface properties (e.g., Zeta potential) and particle size [33]. While the previous machine learning models were capable of predicting tumor delivery efficiency of NPs by establishing the relationship between input variables and the output endpoint, these models are unable to provide interpretable predictions that are achievable by using mechanistic models (e.g., PBPK models) due to lack of the considerations of mechanistic equations characterizing the pharmacokinetic profiles of NPs. In addition, the intrinsic black-box nature of machine learning models results in the unknown causal relationships between input and output variables. In the present study, instead of predicting delivery efficiencies of NPs in tumor, the machine learning and DL models were implemented to predict the tumor-related parameters that were used to describe the endocytosis of NPs in tumor cells. These tumor-related parameters were then integrated into the mechanistic equations of the PBPK model. The results from the PBPK models are interpretable as they were built based on the physiologically-realistic compartments (e.g., liver, spleen, lung, etc.), biological-plausible parameters (e.g., body weight, organ weight, blood flow, etc.), and available mechanisms on the absorption, distribution, metabolism, and excretion (ADME) of NPs. Our results showed that the predictions of tumor delivery efficiency using present model (R²: 0.83 for DE24, R²: 0.82 for DEmax and R²: 0.53 for DE168) has better performance than our previous models (R²: 0.62 for DE24, R²: 0.81 for DEmax and R²: 0.51 for DE168) [33]. Therefore, this hybrid method of using machine learning models to predict the pharmacokinetic parameters that are difficult to be experimentally measured can not only achieve mechanism-based interpretable results but also provide more efficient nanomedicine development process.

Among these machine learning and DL models, we found that the ensemble learning model (e.g., RF, LightGbm and XGB) had better performance on the predictions of related model parameters compared to traditional machine learning models (e.g., SVM and LR). Compared to all other methods, the DNN model had the best performance across all the parameters. The final DNN-based QSAR model was rigorously validated with internal (R²: 0.22 for KTRES_50, R²: 0.44 for KTRES_max, R²: 0.47 for KTRES_n and R²: 0.67 for KTRES_rel) and external validation (R²: 0.23 for KTRES_50, R²: 0.91 for KTRES_max, R²: 0.53 for KTRES_n and R²: 0.47 for KTRES_rel). Our results suggest that the DNN model is a superior model than the conventional machine learning methods in the prediction of key PBPK parameters related to cellular uptake and release of NPs in tumors. The ability to prevent overfitting [e.g., early stopping, regularization algorithms (L1, L2 and dropout methods)], improvements to optimization algorithms (e.g., Adaptive Moment Estimation) and computational hardware (e.g., GPU and TPU) in DNN models have added more benefits in its predictability [19, 22]. In addition, the feature selections are conducted by the neurons within the hidden layers of a neural network self-learn key features in DNN model rather than selecting by humans as in conventional machine learning to prevent potential loss of predictive information from human-made errors [41]. These above-mentioned features have made DNN models more effective and show stronger predictive capabilities over conventional machine learning models as reported in previous studies [33, 42, 43].

In this study, the final DNN-based QSAR model was subsequently integrated with a whole-body PBPK model to generate an AI-assisted PBPK model for NPs in tumor-bearing mice. Compared to existing PBPK models for NPs that typically requires an animal in vivo pharmacokinetic dataset to train a PBPK model for a certain type of NPs [9, 18, 38], the present AI-PBPK model can be used to predict delivery efficiency to different types of tumors with different NPs based on the physicochemical properties of NPs and study design information (e.g., tumor model and targeting strategy) without relying on new animal in vivo pharmacokinetic data to calibrate the model. This is a major advancement in the field of pharmacokinetic modeling for NPs as it optimizes use of the wealth of existing published data with different experimental designs that until now could not be used to predict the kinetics of new NPs.

The present AI-PBPK model-predicted maximal and 24-hours tumor delivery efficiencies (DE24 and DEmax) showed a high correlation with the data-driven results (R²>0.8) and vast majority of predictions were within 2-/3-fold error ranges. The predictive performances for DE24 and DEmax were much better than the 168-hours delivery efficiency (i.e., DE168) (Fig. 4), yet there was an overestimation in the prediction of DE168. There were two possible reasons for the overestimation of DE168. First, our model was developed only based on the synthetic physicochemical properties of NPs, thus there was a lack of consideration that NPs’ physicochemical properties may be altered over time in vivo, especially the surface coating. Previous studies have reported that several serum proteins (e.g., cysteine and cystine) are able to disrupt the surface coating (e.g., PEG layer) of NPs, which could trigger the formation of protein coronas on the surface of NPs and change their biodistribution [44, 45]. Therefore, it is also possible that the delivery efficiency of NPs might be driven by NP-protein corona complex during extended exposure to plasma rather than the initial physicochemical properties of NPs [46, 47]. This could partly explain why our short-term predictions (i.e., 24 hours) were consistent with the observations but our model failed in the long-term predictions (i.e., 168 hours). In addition, the data-driven values in the Nano-Tumor Database were generated by fitting and reproducing experimental data through the conventional PBPK model in our early study [4]. Most existing experimental data are limited to the measurement for the short-term NPs accumulation in tumor (e.g., within 24 hours after IV injection) and only a few discussed the long-term retention (e.g., longer than 72 hours after IV injection). Therefore, the data-driven values of long-term NPs accumulation in tumor in the Nano-Tumor Database are quite limited and the AI-PBPK model’s predictions of the long-term retention of NPs in tumor are uncertain, resulting in the discrepancy between predicted DE168 and data-driven values. Additional long-term pharmacokinetic and biodistribution studies for NPs, if available, need to be considered in our model to reduce the uncertainty of prediction of tumor delivery efficiency of NPs in the extended time points (e.g., >72 h) after IV injection.

Even though our AI-PBPK model had great overall performance in the prediction of tumor delivery efficiency, especially at the earlier time points, the model still has several limitations and needs to be improved. First, our machine learning and AI models are limited to a small sample size. In the field of data science, it is common to have overfitting issues when building AI models with a small sample size but a high dimension of data [48, 49]. In this study, to sufficiently control the overfitting, we shuffled our dataset in the internal cross-validation and external validation so that the inputs and outputs were completely random to make sure the potential bias were minimized in our model. For the DL model, we added a penalty to the loss function in proportion to the size of the weights in the model and used other accepted regulation methods such as activity regularization, early stopping and dropout. Since the Nano-Tumor Database was published a few years ago, it is warranted to collect recent relevant data to expand the database and use additional data to optimize our AI-PBPK model in the future.

Second, the focus of our model simulation results is limited to the pre-clinical study because our database is only based on the studies in tumor-bearing mice. One of major challenges in the development of cancer nanomedicine is the cross-species extrapolation from laboratory animals to human clinical trials because of the uncertain comparability of kinetic and physiological processes between species, which can be addressed by extrapolating the PBPK model from tumor-bearing mice to cancer patients. PBPK models are capable of describing kinetic and physiological determinants governing the fate of a chemical based on prior information regarding species-specific parameters, which are particularly suited for extrapolation to a specific species or humans. In this study, our developed AI-QSAR model predicted the critical parameters as the input of the PBPK model, which provides promise that this could be used to perform animal-to-human extrapolation. Thus, human-specific PBPK parameters can be theoretically predicted using our AI-QSAR model to generate a human AI-PBPK model. However, additional human clinical data [50, 51] are needed for further verification of the human model.

Third, this study only includes the NPs’ physicochemical properties and tumor information to predict the tumor delivery efficiency. However, it is possible to improve our model performance by considering additional descriptors in our AI-QSAR model. For example, several studies have shown that the pharmacokinetics and tumor targeting of NPs is determined not only by the surface properties of NPs but also its interactions with biomolecules such as the formation of protein coronas [52–59]. When NPs are administered or exposed to biological systems, the biomolecular corona is immediately formed covering the NPs and can alter the physicochemical properties of NPs and further influence their pharmacokinetics [46, 47]. Thus, future studies should consider other factors such as protein corona to improve our AI-PBPK model for better predictability of delivery efficiencies of NPs to tumors.

Finally, while the prediction performance of the present AI-PBPK model is considered acceptable, its prediction accuracy (R² ≥ 0.70 for 133 out of 288 datasets) is overall less than that of a PBPK model that is developed for a specific chemical or NP, which can achieve a R² value of >0.9 [12, 60]. More advanced machine learning methods should be considered to improve our AI-PBPK model performance in the future. Recently, the SHapley Additive exPlanations (SHAP) methods [61] have been used to identify the key features and estimate the quantitative SHAP values to calculate the contribution of these key determinants to output results of a machine learning model [62–64]. SHAP algorithm is not only able to improve the model performance, but also can control the overfitting of a machine learning model by selecting key features and removing the redundant variables. The SHAP method may further improve our AI-PBPK model, and should be considered in the future studies.

5. Conclusion

In summary, this study significantly advances the field of cancer nanomedicine by developing a computational tool that integrates machine learning and AI technologies with a mechanistic PBPK model to generate an AI-assisted PBPK model to predict the NP tumor delivery efficiency and tumor disposition profile. One important component of this AI-PBPK model is the AI-based QSAR model that can be used to predict NP-specific tumor-related parameters as input to the PBPK model for different NPs solely based on physicochemical properties of the NPs and some study design information, without relying on an additional animal pharmacokinetic dataset to estimate NP-specific parameters. This approach allows for the optimal use of existing yet disparate published in vivo NP studies. Our AI-assisted PBPK model not only provides an early screening tool for estimating tumor delivery efficiency of NPs, but also can reduce the number of animal use at the early-stage preclinical trials to identify a targeted library of new NPs with desired delivery efficiency to tumors. Additionally, our model is possible to extend to other applications of PBPK models such as cross-species extrapolation, which can potentially enhance the successful rate from pre-clinical animal testing to clinical human studies.