Personalization of Optimal Chemotherapy Dosing Based on Estimation of Uncertain Model Parameters Using Artificial Neural Network

Abstract

Background/Objectives: The effectiveness of chemotherapy in cancer treatment is often compromised by inter-patient variability, leading to suboptimal outcomes. Traditional dosing protocols rely on population-based models that do not account for individual patient responses and the cancer phenotype. This study aims to develop a personalized chemotherapy dosing strategy by estimating uncertain model parameters using artificial neural networks, ensuring an optimal and individualized treatment approach. Methods: A dynamical model of tumor growth, immune response, and chemotherapy effects is used as the foundation for personalization. A training dataset is generated by simulating state responses across a diverse population of virtual patients, capturing inter-subject variability. The state responses are parameterized (approximated) using the sum of exponential functions to reduce dimensionality, and a multilayer perceptron artificial neural network is trained to estimate patient-specific model parameters based on response data from a single chemotherapy dose. Results: The proposed method effectively estimates patient-specific model parameters, significantly reducing uncertainty compared to conventional population-based models or the nonlinear least squares method. Numerical experiments demonstrate that personalized chemotherapy dosing, optimized using the estimated model parameters, achieves fast tumor remission while minimizing the total drug amount. Conclusions: By integrating the artificial neural network as the parameter estimator with model-based optimization, this study presents a novel approach to personalized chemotherapy dosing.

1. Introduction

Cancer is one of the most complex diseases, and as such, scientists are deeply involved in studying not only the biological nature and interactions of the processes involved in cancer but also in creating mathematical models of these processes. Chemotherapy remains a cornerstone of cancer treatment but is often associated with significant drawbacks, including severe toxicity, adverse side effects, and a lack of specificity in targeting tumor cells while sparing healthy tissues. Conventional chemotherapy protocols do not account for individual patient characteristics. As a result, patients may experience suboptimal treatment outcomes, either due to under-dosing, which fails to eradicate the tumor, or excessive dosing, which leads to unnecessary toxicity and immune suppression.

Therefore, the identical cancer treatment applied to a large group of patients can effectively cure only some of them [1,2]. Often, we are unable to fully explain the mechanisms why the treatment fails in some patients [3]. Due to a complex interplay of factors such as the type, aggressiveness, and stage of cancer, patient genetics, diet, immune system status, body weight, and gender, the efficiency of cancer therapies can vary significantly [4,5]. Another important factor affecting the response of the organism is the chemotherapy substance chosen for the treatment [6].

The design of chemotherapy dosing protocols for cancer patients often relies on empirical determination or systematic design based on population models [7,8,9,10]. These models typically derive their parameters from clinical data across large groups of patients. As a result, chemotherapy doses and treatment outcomes have a mean-population character, failing to account for individual patient variability [11]. Consequently, the application of such therapy carries risks, potentially hindering the success of chemotherapy [12]. To compensate for inter-subject variability by aligning the model parameters with the observed patient’s response to the treatment, advanced techniques are necessary [13].

These issues have given rise to the concept of personalized (precision) medicine (PMed), a new healthcare approach that seeks to address why patients with similar clinical symptoms often exhibit different responses to the same treatment [14]. Personalized medicine focuses on tailoring treatment to the individual characteristics of each patient [15]. This approach leverages a set of patient-specific information or characteristics to design customized therapeutic strategies [16]. Unlike the “one-size-fits-all approach”, which aims to develop treatments for the average person, precision medicine focuses on the differences between individuals [17]. This shift towards a patient-centered approach has the potential to improve survival rates, enhance quality of life, and reduce the burden of chemotherapy-related complications.

In the case of model-based personalization, an adequately parsimonious mathematical model is required [18,19]. Numerous mathematical models have been developed to describe tumor growth, its interactions with the immune system, and responses to various cancer therapies, including chemotherapy [9,20,21,22,23,24], immunotherapy [25,26,27], radiotherapy [28], anti-angiogenesis [29], and their combinations [8,30,31]. A comprehensive review of cancer modeling can be found in [9,32,33].

Current efforts in the field are focused on two distinct approaches to developing personalized therapies:

• Iteratively and heuristically adjusting the therapy during treatment. This suggests an adaptive, trial-and-error approach where therapy is modified based on patient response. This is common in clinical practice but may lack a formal mathematical framework.
• Rigorous approach using a mathematical model to guide therapy, followed by model-based optimization to refine treatment strategies.

The latter can be split into two steps [34]:

• Personalization of the model: This step involves identifying the model parameters that correspond to the individual patient, transforming the population model into a personalized one.
• Personalization of the treatment: This step involves determining the optimal chemotherapy dosing protocol tailored to the patient based on the personalized model.

Several studies have explored the integration of machine learning (ML) into oncological treatment, particularly in pharmacogenomics and chemotherapy personalization or by integrating medical imagining [35]. The studies [36,37] demonstrated the potential of ML in analyzing large-scale genomic, proteomic, and other omics datasets to identify genetic markers associated with drug responses. However, a drawback of these studies is that the methods used remain relatively simple, relying on heuristic or data-driven models without a strong mechanistic foundation.

The studies [38,39] focus on the widely used Gompertz tumor growth model, employing methods such as least squares minimization, maximum likelihood estimation, and the Extended Kalman Filter (EKF). While these methods offer robust estimation techniques, they require precise knowledge of process noise and measurement noise parameters to ensure optimal results. This presents a major limitation, as these noise parameters are often unknown in practical applications.

The studies [40,41] employ nonlinear function optimization techniques to estimate the parameters of a fourth-order tumor growth model. These methods rely on cost functions based on the least-squares criterion, where parameter values are chosen to minimize the sum of squared differences between simulated and measured tumor volumes. However, the issue is that fitting the model response to the data does not ensure that parameters are estimated accurately.

The studies [42,43] explore the combination of state and parameter estimation techniques, comparing the Extended Kalman Filter (EKF) with Moving Horizon Estimation (MHE). These approaches augment the state vector with the parameter vector and use optimization techniques to minimize the root-mean-square error between predicted and measured tumor volumes. Both methods require accurate noise modeling, which remains an open problem in practical applications.

The study in [44] applies Monte Carlo Approximate Bayesian Computation (ABC) to simultaneously perform model selection and parameter estimation for five different tumor growth models. This approach provides a probabilistic framework for quantifying uncertainty in both model choice and parameter values. However, a key drawback is the high computational cost, which may limit scalability.

Finally, the most relevant studies [45,46,47] employ artificial neural networks to estimate tumor model parameters using sparse measurement data and prior information about parameter intervals. These approaches leverage in silico-generated training data spanning large parameter intervals and incorporate randomized parameters to improve generalization. Two different neural networks were trained, to estimate the pharmacodynamic and the pharmacokinetic parameters. The most significant limitation is that the inputs of the neural networks are directly the measurements of the tumor volume, which hinders scalability and necessitates the use of a four-layered neural network structure. Besides that, these studies did not develop any personalized treatment strategies.

Therefore, to reflect the current gaps in the state of the art, the strategy proposed in this paper is to obtain personalized model parameters by evaluating the patient’s response to an initial dose of chemotherapy with a multilayer perceptron artificial neural network (MLP NN) with only one hidden layer. The proposed approach does not utilize an MLP NN as a replacement for the original biological dynamic model or as a digital twin [48]. Instead, it employs the neural network as a mapping function that directly relates the subject-specific parameters to their effects on the individual state responses. This method effectively bypasses the need for numerically solving the model differential equations when estimating model parameters and does not require making assumptions about the noise characteristics. Moreover, we shifted from pure model personalization addressed in the literature towards treatment personalization, thus completing the whole clinical scheme.

In the context of the state-of-the-art review presented, the most significant novel contributions of this paper are as follows:

• A methodology to personalize (estimate) uncertain parameters of the dynamic model of tumor growth, chemotherapy and the immune system based on analyzing the clinical response of an oncology patient to a single chemotherapy dose using the pretrained MLP NN.
• A parametrization technique to approximate the full sampled state responses with an analytical parametric function, resulting in a significant reduction in input data dimension and training complexity of the MLP NN, allowing for MLP NN with only one hidden layer.
• Combination of the model personalization with the model-based optimization of chemotherapy dosing protocol ensuring personalized optimal cancer treatment.

In contrast to traditional parameter identification techniques, which typically rely on numerical optimization of a quadratic criterion to minimize residuals between the measured and simulated model responses, or a likelihood function [49,50], the proposed approach ensures faster parameter estimation through the evaluation of the MLP NN output.

Finally, the proposed approach focuses on fitting the parameter values themselves rather than fitting the state responses, which will ultimately result in better general validity of the estimated models than in traditional approaches.

2. Materials and Methods

2.1. Model of Cancer and Chemotherapy

In this section, a mathematical model of tumor growth dynamics, chemotherapy, and the interactions with the immune system will be presented. The model itself and the values of its nominal (mean-population) parameters will be adopted from [23].

The adopted model is not limited to a particular chemotherapeutic drug or cancer subtype. It focuses on the dynamics of a single solid tumor, without accounting for further proliferation or metastatic development. This standard phenomenological model describes the dynamics of four state variables representing the following quantities in a patient with cancer:

• $N\left(t\right)$ natural killer cell population
• $L\left(t\right)$ cytotoxic T cells population
• $T\left(t\right)$ tumor cell population
• $u\left(t\right)$ amount of anticancer drug in the tumor site

Natural killer (NK) cells are a key component of the innate immune system, responsible for detecting and eliminating abnormal cells, including tumor cells and virus-infected cells, without prior sensitization. Unlike adaptive immune cells, NK cells do not rely on antigen recognition but instead use a balance of activating and inhibitory receptors to distinguish between healthy and malignant cells. When NK cells recognize a target cell with reduced major histocompatibility complex (MHC) class I expression—a common feature of tumor cells—they release cytotoxic granules containing perforin and granzymes, leading to cell lysis.

Cytotoxic T lymphocytes (CTLs), also known as CD8⁺ T cells, are key players in the adaptive immune system and play a vital role in tumor immunosurveillance. Unlike natural killer (NK) cells, CTLs must be activated through antigen presentation by antigen-presenting cells (APCs) via MHC class I molecules. Once activated, they recognize and attach to tumor cells displaying tumor-specific or tumor-associated antigens. This leads to the induction of apoptosis in target cells, either through the release of perforin and granzymes or by engaging death receptor signaling pathways.

The considered mathematical model is defined by differential equations [23]

$$\begin{gathered} {\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}\left(t\right)=N\left(t\right)\left(a\left(1-bN\left(t\right)\right)-{\alpha }_{1}T\left(t\right)-k_{N}u\left(t\right)\right), \\ {\displaystyle \frac{\mathrm{d}L}{\mathrm{d}t}}\left(t\right)=rN(t)T(t)-L(t)\left(\mu +{\beta }_{1}T\left(t\right)+k_{L}u\left(t\right)\right), \\ {\displaystyle \frac{\mathrm{d}T}{\mathrm{d}t}}\left(t\right)=T(t)\left(c\left(1-dT\left(t\right)\right)-{\alpha }_{2}N\left(t\right)-{\beta }_{2}L\left(t\right)-k_{T}u\left(t\right)\right), \\ {\displaystyle \frac{\mathrm{d}u}{\mathrm{d}t}}\left(t\right)=v\left(t\right)-\omega u\left(t\right). \end{gathered}$$

(1)

Input $v\left(t\right)\in \mathbb{R}$ represents the rate of chemotherapy administration.

The state variables form the state vector $x\left(t\right)\in {\mathbb{R}}^4$ defined as

$$x\left(t\right)={ \left(\begin{array}{cccc}N\left(t\right)& L\left(t\right)& T\left(t\right)& u\left(t\right)\end{array}\right)}^{\mathrm{T}}.$$

(2)

The adopted model is based on the following assumptions [23]:

• Tumor cell populations are reduced by both immune effector cells and chemotherapy.
• Effector cell numbers decline due to natural degradation, their consumption in tumor cell destruction, and the impact of chemotherapy.
• Chemotherapeutic agents influence both tumor cells and immune effector cells through a mass-action process.
• Administering the drug at a higher constant rate may lead to a greater reduction in both tumor cells and immune effector cells.

The growth of natural killer cells $N\left(t\right)$ follows a logistic model as $aN\left(t\right)\left(1-bN\left(t\right)\right)$, while their suppression occurs through interactions with tumor cells, defined by ${\alpha }_{1}T\left(t\right)N\left(t\right)$. Cytotoxic T cells $L\left(t\right)$ are activated only in response to tumor presence while being generated via a bilinear term $rN\left(t\right)T\left(t\right)$. Their natural decline follows a linear process $-\mu L\left(t\right)$, and they are further inactivated upon interaction with tumor cells, represented by $-{\beta }_{1}T\left(t\right)L\left(t\right)$. Tumor cell dynamics $T\left(t\right)$ is modeled using a logistic growth function $cT\left(t\right)\left(1-dT\left(t\right)\right)$. Tumor elimination occurs through the action of both NK cells and cytotoxic T cells, with respective terms $-{\alpha }_{2}N\left(t\right)T\left(t\right)$ and $-{\beta }_{2}L\left(t\right)T\left(t\right)$ accounting for their effects.

The chemotherapy drug $u\left(t\right)$ follows first-order pharmacokinetics and influences all three cell populations—NK cells, cytotoxic T cells, and tumor cells—via a mass-action mechanism. Each cell type exhibits a specific susceptibility to the drug, characterized by distinct mortality rates with coefficients $k_N$, $k_L$, and $k_T$, respectively.

The parameters of the model defined by (1) can be gathered into the parameter vector $\mathit{\theta }\in {\mathbb{R}}^{14}$ as

$$\mathit{\theta }={\left(\begin{array}{cccccccccccccc}a& b& c& d& r& \mu & {\alpha }_1& {\alpha }_2& {\beta }_1& {\beta }_2& \omega & k_N& k_L& k_T\end{array}\right)}^{\mathrm{T}}.$$

(3)

The nominal values of the model parameters and their biological interpretations are summarized in

Table 1. Nominal parameters of the model and their biological interpretation [23].

Parameter	$\overline{\mathit{a}}$	$\overline{\mathit{b}}$	$\overline{\mathit{c}}$	$\overline{\mathit{d}}$	$\overline{\mathit{r}}$	$\overline{\mathit{\mu }}$	${\overline{\mathit{\alpha }}}_{\mathbf{1}}$
Nominal value	$1$	$3.17\times {10}^{-6}$	$5.14\times {10}^{-1}$	$1.02\times {10}^{-9}$	$1.1\times {10}^{-7}$	$2.0\times {10}^{-2}$	$1.0\times {10}^{-7}$
Meaning	Growth rate of natural killer cells	Inverse of natural killer cells’ capacity	Growth rate of tumor	Inverse of tumor capacity	Activation rate of cytotoxic T cells	Cytotoxic T-cell death rate	Natural killer cell death rate
Parameter	${\overline{\mathit{\alpha }}}_{\mathbf{2}}$	${\overline{\mathit{\beta }}}_{\mathbf{1}}$	${\overline{\mathit{\beta }}}_{\mathbf{2}}$	$\overline{\mathit{\omega }}$	${\overline{\mathit{k}}}_{\mathit{N}}$	${\overline{\mathit{k}}}_{\mathit{L}}$	${\overline{\mathit{k}}}_{\mathit{T}}$
Nominal value	$6.41\times {10}^{-11}$	$3.42\times {10}^{-10}$	$3.5\times {10}^{-7}$	$9.0\times {10}^{-1}$	$6.0\times {10}^{-1}$	$6.0\times {10}^{-1}$	$8.0\times {10}^{-1}$
Meaning	Tumor death rate of natural killer cells	Cytotoxic T-cell death rate	Rate of cytotoxic T cells induced tumor death	Drug decay rate	Natural killer cell death rate by drug	Cytotoxic T-cell death rate by drug	Tumor cell death rate by drug

. Considering these nominal parameters, we define the nominal parameter vector $\overline{\mathit{\theta }}\in {\mathbb{R}}^{14}$ as

$$\overline{\mathit{\theta }}={\left(\begin{array}{cccccccccccccc}\overline{a}& \overline{b}& \overline{c}& \overline{d}& \overline{r}& \overline{\mu }& {\overline{\alpha }}_1& {\overline{\alpha }}_2& {\overline{\beta }}_1& {\overline{\beta }}_2& \overline{\omega }& {\overline{k}}_N& {\overline{k}}_L& {\overline{k}}_T\end{array}\right)}^{\mathrm{T}}.$$

(4)

Each patient developing cancer is unique, hence the parameter vector $\mathit{\theta }$ typically varies significantly between the subjects within the population. This makes the evolution of state variables hard to predict without model personalization for the particular patient and their cancer phenotype.

Because model (1) is relatively broadly applicable to various chemotherapeutic agents and cancer subtypes, if the focus is on a single solid tumor, this motivates personalization by tailoring the model parameters to reflect the specific characteristics of the cancer type, the patient’s physiological conditions, and the properties of the administered chemotherapeutic drug. Key characteristics of the chemotherapeutic agent are encapsulated in the coefficients $k_N$, $k_L$, $k_T$, which quantify its selectivity in targeting specific cells. The aggressiveness of the cancer is characterized by parameter $c$, representing the tumor growth rate. Meanwhile, the immune system’s effectiveness in combating the tumor is determined by a complex interplay of additional model parameters such as the growth rate of natural killer cells $a$, the activation rate of cytotoxic T cells $r$, capturing the dynamic interactions between the tumor and the immune response.

The roles of this mathematical model are the following:

• To substitute for in vivo data by providing the simulated state responses for various parameters, dose sizes, and initial conditions when creating the training and validation datasets.
• To provide the simulated state responses when evaluating the cost function of the chemotherapy dosing protocol optimization.

Since the model in (1) is nonlinear, the state responses $N\left(t\right), L\left(t\right), T\left(t\right)$ cannot be obtained analytically. Because of this, the only way to obtain the state responses $N\left(t\right), L\left(t\right), T\left(t\right)$ is by numerical solution of the differential equations given by (1), using Runge–Kutta methods [51] for numerical integration.

2.2. Statistical Properties of the Model Parameters

Consider that the parameter vector $\mathit{\theta }$ given by (3) is a random variable following a multivariate normal distribution, which describes the statistical properties related to the inter-subject parametric variability in the population of patients with cancer.

The mean value is given by vector $\overline{\mathit{\theta }}$ as

$$\overline{\mathit{\theta }}=E\left\{\mathit{\theta }\right\}.$$

(5)

The second statistical characteristic is the covariance matrix $Q\in {\mathbb{R}}^{14\times 14}$ defined as

$$Q=\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathit{\theta },\mathit{\theta }\right)=E\left\{{\left(\mathit{\theta }-\overline{\mathit{\theta }}\right)\left(\mathit{\theta }-\overline{\mathit{\theta }}\right)}^{\mathrm{T}}\right\}.$$

(6)

The diagonal elements of $Q$ represent the variances of the individual parameters, hence defining the measure of parametric uncertainties. The off-diagonal elements represent the covariances between pairs of parameters, which quantify the degree to which these parameters change together.

To generalize the theoretical analysis, we will introduce a new dimensionless random vector $\mathit{\xi }\in {\mathbb{R}}^m$ following the standard normal distribution, which can be noted as

$$E\left\{\mathit{\xi }\right\}=\mathbf{0}, \mathrm{c}\mathrm{o}\mathrm{v}\left(\mathit{\xi },\mathit{\xi }\right)=E\left\{{\mathit{\xi }\mathit{\xi }}^{\mathrm{T}}\right\}=I.$$

(7)

Then, the parameter vector $\mathit{\theta }$ will be related to random vector $\mathit{\xi }$ via matrix $R\in {\mathbb{R}}^{14\times m}$ and the nominal vector $\overline{\mathit{\theta }}$ as

$$\mathit{\theta }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R\mathit{\xi }+\overline{\mathit{\theta }}.$$

(8)

Dimension $m\in \mathbb{N}$ represents the number of random factors affecting the parameter vector $\mathit{\theta }$. According to (8), the covariance matrix $Q$ defined by (6) obtains

$$Q=E\left\{{\left(\mathit{\theta }-\overline{\mathit{\theta }}\right)\left(\mathit{\theta }-\overline{\mathit{\theta }}\right)}^{\mathrm{T}}\right\}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R{R}^T\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right).$$

(9)

Considering the confidence level $\alpha$, the random vector $\mathit{\xi }$ will lie in the symmetrical confidence interval

$$-\mathbf{1}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\le \mathit{\xi }\le \mathbf{1}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(10)

with $1-\alpha$ probability, where $r_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is the$1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ quantile of the standard normal distribution and $\mathbf{1}={\left(\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right)}^{\mathrm{T}}$ is the vector of ones.

Based on (8), we can claim that the parameter vector $\mathit{\theta }$ lies in the confidence interval

$$-\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R\mathbf{1}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\overline{\mathit{\theta }}\le \mathit{\theta }\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R\mathbf{1}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+\overline{\mathit{\theta }}.$$

(11)

Then, the absolute interval parametric uncertainty obtains

$$∆\mathit{\theta }=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R\mathbf{1}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$$

(12)

Equation (12) will be used to determine the entries of matrix $R$ based on the assumptions about the relative interval parametric uncertainties given in percent by ${\displaystyle \frac{∆{\mathit{\theta }}_i}{{\overline{\mathit{\theta }}}_i}}\times 100$ as

$$R_{ij}={\displaystyle \frac{∆{\mathit{\theta }}_i}{{\overline{\mathit{\theta }}}_i}}{\displaystyle \frac{1}{r_{\raisebox{1ex}{$1-\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}},$$

(13)

where $i=1,2,\dots ,14$ is the index of parameter and $j=1,2,\dots ,m$, which increments if ${\displaystyle \frac{∆{\mathit{\theta }}_i}{{\overline{\mathit{\theta }}}_i}}\ne 0$.

2.3. Parametrization of the State Responses

It is an important fact that the sequences of sampled state responses $N\left(k{T}_s\right),L\left(k{T}_s\right), T\left(k{T}_s\right)$ for $k=1,2,\dots ,n$ and the sample time $T_s$ cannot be effectively used as inputs to the MLP NN. Although theoretically possible, it is very unpractical due to the dimensions, i.e., the large number of inputs to the MLP NN. Besides that, it is very likely that the MLP NN would not be able to sufficiently generalize the information comprised in such raw data and thus provide a reliable parameter estimate.

Therefore, the proposed strategy will be to extract the features of the sampled state responses $N\left(k{T}_s\right),L\left(k{T}_s\right), T\left(k{T}_s\right)$ by parametrizing them. The parametrization technique will be based on an approximation of the state responses by a parametric analytical function while determining its parameters in terms of linear regression. After the parametrization, the sampled state responses are described by vectors ${\mathit{p}}_{\mathit{N}}\in {\mathbb{R}}^{n_p}$, ${\mathit{p}}_{\mathit{L}}\in {\mathbb{R}}^{n_p}$, ${\mathit{p}}_{\mathit{T}}\in {\mathbb{R}}^{n_p}$ of $n_p$ parameters as

$$\begin{gathered} N\left(k{T}_s\right), k=1,2,\dots ,n\stackrel{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\to }{\mathit{p}}_{\mathit{N}}={\left(\begin{array}{cccc}{p_N}_1& {p_N}_2& \cdots & {p_N}_{n_p}\end{array}\right)}^{\mathrm{T}}, \\ L\left(k{T}_s\right), k=1,2,\dots ,n\stackrel{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\to }{\mathit{p}}_{\mathit{L}}={\left(\begin{array}{cccc}{p_L}_1& {p_L}_2& \cdots & {p_L}_{n_p}\end{array}\right)}^{\mathrm{T}}, \\ T\left(k{T}_s\right), k=1,2,\dots ,n\stackrel{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\to }{\mathit{p}}_{\mathit{T}}={\left(\begin{array}{cccc}{p_T}_1& {p_T}_2& \cdots & {p_T}_{n_p}\end{array}\right)}^{\mathrm{T}}. \end{gathered}$$

(14)

This approach will lead to a significant reduction in the training complexity and the memory footprint since $n_p\ll n$.

Consider a parametrizable weighted sum of exponential functions $\tilde{X}\left(t\right):\mathbb{R}\to \mathbb{R}$ as

$$\tilde{X}\left(t\right)=p_1{e}^{a_{1}t}+p_2{e}^{a_{2}t}+p_3{e}^{a_{3}t}\dots p_{n_p-1}{e}^{a_{n_p-1}t}+p_{n_p},$$

(15)

where $p_1$, $p_2$, …, $p_{n_p}$ are the function parameters to be fitted and $a_1,a_2,\dots ,a_{n_p-1}<0$ are appropriately chosen tuning coefficients.

This function will approximate signal $X\left(t\right)$, which substitutes for the state responses $N\left(t\right)$, $L\left(t\right)$, $T\left(t\right)$ available in the form of finite sampled sequences $N\left(k{T}_s\right),L\left(k{T}_s\right), T\left(k{T}_s\right)$ for $k=1,2,\dots ,n$. The absolute term $p_{n_p}$ in (15) is involved to reflect nonzero steady states of model (1), allowing for a nonzero finite limit $\underset{t\to \infty }{\lim }\tilde{X}\left(t\right)=p_{n_p}$.

Equation (15) can be noted in linear form

$$\tilde{X}(t)=\left(\begin{array}{cccccc}{e}^{a_{1}t}& e^{a_{2}t}& e^{a_{3}t}& \cdots & e^{a_{n_p-1}t}& 1\end{array}\right)\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\\ ⋮\\ p_{n_p-1}\\ p_{n_p}\end{array}\right)={\mathit{h}}^{\mathrm{T}}\mathit{p},$$

(16)

where $\mathit{h}\in {\mathbb{R}}^{n_p}$ is the regression vector.

Based on (16), we can formulate the linear regression problem

$$\begin{gathered} \left(\begin{array}{cccccc}{e}^{a_1{T}_s}& e^{a_2{T}_s}& e^{a_3{T}_s}& \cdots & e^{a_{n_p-1}{T}_s}& 1\\ e^{a_1{2T}_s}& e^{a_2{2T}_s}& e^{a_3{2T}_s}& \cdots & e^{a_{n_p-1}{2T}_s}& 1\\ e^{a_1{3T}_s}& e^{a_2{3T}_s}& e^{a_3{3T}_s}& \cdots & e^{a_{n_p-1}{3T}_s}& 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ e^{a_1{nT}_s}& e^{a_2{nT}_s}& e^{a_3{nT}_s}& \cdots & e^{a_{n_p-1}{nT}_s}& 1\end{array}\right)\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\\ ⋮\\ p_{n_p-1}\\ p_{n_p}\end{array}\right)=\left(\begin{array}{c}X\left(T_s\right)\\ X\left(2T_s\right)\\ X\left(3T_s\right)\\ ⋮\\ X\left(n{T}_s\right)\end{array}\right), \\ H\mathit{p}=\mathit{X}, \end{gathered}$$

(17)

where $H\in {\mathbb{R}}^{n\times n_p}$ is the regression matrix and $\mathit{X}\in {\mathbb{R}}^n$ is the vector of samples.

The parameter vector $\mathit{p}$ will be obtained in terms of the least squares method analytically as [52]

$$\mathit{p}={\left(H^{\mathrm{T}}H\right)}^{-1}{H}^{\mathrm{T}}\mathit{X}.$$

(18)

2.4. Artificial Neural Network

2.4.1. Structure of the Network

We chose the structure of multilayer perceptron artificial neural network having the output layer with linear activation function and one hidden layer comprising $n_n$ neurons with hyperbolic tangent sigmoid activation function [53]. The structure of this MLP NN is depicted in Figure 1.

The learning algorithm for training this MLP NN was the classical Levenberg–Marquardt error backpropagation [54]. The functionality of the MLP NN, including the output calculation, and the weights and biases adaptation (training) are implemented by the functions of Matlab Deep Learning Toolbox (R2023b, https://www.mathworks.cn/products/deep-learning.html) [55].

2.4.2. Inputs and Outputs

The input vector $\mathit{U}$ to the MLP NN will be formed by vectors of parameters ${\mathit{p}}_{\mathit{N}}$, ${\mathit{p}}_{\mathit{L}}$, ${\mathit{p}}_{\mathit{T}}$ obtained by the parametrization of the state responses $N\left(t\right),L\left(t\right),T\left(t\right)$ respectively, the initial state vector $x\left(0\right)$ and the administered leading dose size $D$ as

$$\mathit{U}=\left(\begin{array}{c}{\mathit{p}}_{\mathit{N}}\\ {\mathit{p}}_{\mathit{L}}\\ {\mathit{p}}_{\mathit{T}}\\ x\left(0\right)\\ D\end{array}\right).$$

(19)

If $n_p$ is the dimension of ${\mathit{p}}_{\mathit{N}}$, ${\mathit{p}}_{\mathit{L}}$, ${\mathit{p}}_{\mathit{T}}$ according to (14), then $\mathit{U}\in {\mathbb{R}}^{3n_p+5}$.

The reason for providing the information on $x\left(0\right)$ is to ensure parameter estimation under the assumption of an arbitrary initial state. By taking the initial state into account explicitly, the MLP NN will be capable of separating its effect on the state responses from the effect of parametric variations.

It is important to mention that by providing the dose size $D$ and the initial state $x\left(0\right)$, which involves the initial drug amount $u\left(0\right)$, the information on the chemotherapy administered within the current and previous dosing periods (cycles) also enters the personalization. A significant (nonzero) dose $D$ is mandatory to properly capture the effects of chemotherapy. This enables the parameters $k_N$, $k_L$, $k_T$ to be estimated.

The model personalization itself consists of evaluating vector $\mathit{U}$ by the MLP NN to provide vector $\widehat{\mathit{\xi }}\in {\mathbb{R}}^m$ representing the estimate of $\mathit{\xi }$. Subsequently, the estimate of model parameter vector $\widehat{\mathit{\theta }}$ is determined from $\widehat{\mathit{\xi }}$ according to (8).

The MLP NN can be seen as a nonlinear multivariate vector function $\mathcal{N}:{\mathbb{R}}^{3n_p+5}\to {\mathbb{R}}^m$ [53].

$$\widehat{\mathit{\xi }}=\mathcal{N}\left(\mathit{U}\right)=\left[W^o\tanh \left(W^h\mathit{U}+{\mathit{b}}^h\right)+{\mathit{b}}^{\mathit{o}}\right],$$

(20)

where $\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\mathit{x}\right): {\mathbb{R}}^{n_n}\to {\mathbb{R}}^{n_n}$is a vectorized hyperbolic tangent function, $W^h\in {\mathbb{R}}^{n_n\times 3n_p+5}$, $W^o\in {\mathbb{R}}^{m\times n_n}$ are the matrices of weights in the hidden and the output layer, respectively, and ${\mathit{b}}^{\mathit{h}}\in {\mathbb{R}}^{n_n}$, ${\mathit{b}}^{\mathit{o}}\in {\mathbb{R}}^m$ are the vectors of biases in the hidden and the output layers, respectively. The number of neurons in the hidden layer $n_n$ was determined based on the heuristic rule of thumb $m<n_n<3n_p+5$, which is satisfactory for $n_n={\displaystyle \frac{m+3n_p+5}{2}}$.

2.4.3. Generating the Training Dataset

The dataset for training of the MLP NN was generated according to the procedure described as follows. The parameters given by (3) corresponding to virtual oncology subjects were generated randomly while considering their statistical distribution as defined before. The relative interval of parametric uncertainties of the individual model parameters was chosen as

$${\displaystyle \frac{∆\mathit{\theta }}{\overline{\mathit{\theta }}}}={\left(\begin{array}{cccccccccccccc}0.8& 0.05& 0.05& 0& 0& 0.4& 0.8& 0& 0& 0& 0.25& 0.85& 0.25& 0.25\end{array}\right)}^{\mathrm{T}}.$$

(21)

Notice that in (21), only $m=9$ of a total of 14 parameters in the parameter vector $\mathit{\theta }$ are considered uncertain, while the others remain nominal, i.e., fixed across the population. In particular, parameters $a,b,c,\mu ,{\alpha }_1,\omega ,k_N,k_L,k_T$ are considered uncertain and vary among the subjects. This choice is strongly supported by analyzing the magnitudes and the linear dependencies of the corresponding sensitivity functions of the state variables with respect to the parameters as presented in Appendix A.

The sensitivity analysis allowed us to understand the percentual changes in which parameters have an observable effect on the state variables. Then, we could adjust the relative interval parametric uncertainties (21) according to the magnitudes of the sensitivities (see Figure A1, Figure A2 and Figure A3) to ensure that their effects are realistically balanced. It means that if the state sensitivity to the $i$-th parameter is relatively low, the corresponding parametric uncertainty should be higher than that of a parameter with much greater sensitivity.

However, it is important to acknowledge that the uncertainties given by (21) were chosen without the support of clinical results or data, as these are not available in the literature or in the public domain. Experimentally determining the actual statistical distribution of the parameters is beyond the scope of this paper, as it requires large studies with real subjects.

Considering the $1-\alpha =0.95$ confidence level, the entries of matrix $R$ were determined according to (13) as

$$R=0.1531\times \left(\begin{array}{ccccccccc}0.8& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0.05& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0.05& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0.4& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0.8& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0.25& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0.85& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0.25& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0.25\end{array}\right).$$

(22)

The structure of matrix $R$ implies that the covariance matrix $Q$ defined by (9) will be diagonal, which means that the random variations in the model parameters are not mutually correlated within the population.

To create the training dataset, there will be generated a total of 1500 different virtual patients. More specifically, for each of these subjects in the virtual population, a random vector $\mathit{\xi }$ will be generated considering standard normal distribution (zero mean and unit covariance matrix). The corresponding parameter vector $\mathit{\theta }$ will be calculated according to formula (8), considering the mean $\overline{\mathit{\theta }}$ given by the values in Table 1 and matrix $R$ given by (22). After that, the states of each virtual patient will be simulated by numerically solving the differential Equation (1) using the Runge–Kutta 4-th order numerical integration method [51] to obtain the sampled state responses $N\left(k{T}_s\right),L\left(k{T}_s\right), T\left(k{T}_s\right)$ for a given $T_s$ and the dosing period (cycle duration) $T_D$. This simulation (numerical solution) will be repeated for 100 different initial conditions and leading dose sizes $D$. The initial conditions $N\left(0\right),L\left(0\right), T\left(0\right)$ will also be randomly generated, assuming the normal distributions with the mean values equal to the corresponding stable equilibrium states $N_0,L_0,T_0$ and the standard deviations are given as ${\displaystyle \frac{N_0}{2}},{\displaystyle \frac{L_0}{2}},{\displaystyle \frac{T_0}{2}}$, respectively. The leading dose size $D$ for each entry of the dataset was generated as a uniformly distributed random number from the interval 0.1–1.5. The training dataset thus includes $M=\mathrm{150,000}$ entries. After obtaining the simulated state responses, their parametrization will be performed to obtain the individual input vectors $\mathit{U}$ defined by (19).

The full training dataset was divided randomly into three subsets within the Matlab Deep Learning Toolbox: 70% for training, 15% for validation, and 15% for testing.

2.4.4. Validation by the Covariance Matrix of the Estimate Error

The aim of the validation is to assess the personalization performance of the trained MLP NN using new data. The validation dataset with $M=\mathrm{150,000}$ entries was generated randomly by following the same procedure as in the case of the training dataset but with a different seed for the random number generator.

By evaluating the output of the MLP NN for all entries of the validation dataset, the estimates ${\widehat{\mathit{\xi }}}_i$ will be determined according to (20) for $i=1,2,\dots ,M$ and the estimated residuals will be calculated as ${\widehat{\mathit{\xi }}}_i-{\mathit{\xi }}_i$.

The validation and performance assessment will be carried out in a statistical framework by analyzing the covariance matrix of the estimated residuals. The covariance matrix of the estimated residual $P\in {\mathbb{R}}^{m\times m}$ will be defined as

$$P=\mathrm{c}\mathrm{o}\mathrm{v}\left(\widehat{\mathit{\xi }}-\mathit{\xi },\widehat{\mathit{\xi }}-\mathit{\xi }\right)=E\left\{{\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)}^{\mathrm{T}}\right\}.$$

(23)

From the statistical perspective, personalization should minimize the variances of the residuals $\widehat{\mathit{\xi }}-\mathit{\xi }$, while ensuring statistical unbiasedness.

In practice, $P$ will be obtained as the sample covariance matrix $\widehat{P}\in {\mathbb{R}}^{m\times m}$ considering $M$ entries of the validation dataset as [56]

$$\widehat{P}=\mathrm{c}\mathrm{o}\mathrm{v}\left(\widehat{\mathit{\xi }}-\mathit{\xi },\widehat{\mathit{\xi }}-\mathit{\xi }\right)={\displaystyle \frac{1}{M-1}}{\sum }_{i=1}^M{\left({\widehat{\mathit{\xi }}}_i-{\mathit{\xi }}_i\right)\left({\widehat{\mathit{\xi }}}_i-{\mathit{\xi }}_i\right)}^{\mathrm{T}}.$$

(24)

Since $\mathit{\xi }$ follows the standard normal distribution with $\mathrm{c}\mathrm{o}\mathrm{v}\left(\mathit{\xi },\mathit{\xi }\right)=I$ as defined by (7), the personalization is successful if the variances of the residuals are significantly lower than one. This can be noted as

$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)\ll \mathbf{1}.$$

(25)

The above can be interpreted such that the aim of personalization is not to provide the exact values of the unknown model parameters but to lower their uncertainties. Another condition is that the mean value of the residuals must be close to zero to ensure the estimate unbiasedness, which can be noted as

$$E\left\{\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)\right\}\to \mathbf{0}.$$

(26)

2.4.5. Confidence Interval of the Parameter Estimate

To be able to assess the accuracy of the estimated dimensionless vector $\widehat{\mathit{\xi }}$ or the estimated parameter vector $\widehat{\mathit{\theta }}$, which both represent the point estimates, it is useful to determine their confidence intervals.

Considering the confidence level $\alpha$ and the sample covariance matrix $\widehat{P}$, the confidence interval of $\mathit{\xi }$ obtains

$$\begin{gathered} \widehat{\mathit{\xi }}+\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\le \mathit{\xi }\le \widehat{\mathit{\xi }}+\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}. \\ \widehat{\mathit{\xi }}-\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\le \mathit{\xi }\le \widehat{\mathit{\xi }}+\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}. \end{gathered}$$

(27)

The confidence interval is symmetrical because $r_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=-r_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ for the normal distribution. Then, considering (8), we can find the confidence interval of the model parameter vector $\mathit{\theta }$ as

$$\begin{gathered} \overline{\mathit{\theta }}+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\left(\widehat{\mathit{\xi }}-\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{\mathit{P}}\right)}{r}_{\mathbf{1}-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)\le \mathit{\theta }\le \overline{\mathit{\theta }}+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\left(\widehat{\mathit{\xi }}+\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{\mathit{P}}\right)}{r}_{\mathbf{1}-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right) \\ \overline{\mathit{\theta }}+\widehat{\mathit{\theta }}-\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\le \mathit{\theta }\le \overline{\mathit{\theta }}+\widehat{\mathit{\theta }}+\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \end{gathered}$$

(28)

Based on the above, the interval uncertainty of the estimated parameter vector $\widehat{\mathit{\theta }}$ obtains

$$∆\widehat{\mathit{\theta }}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$$

(29)

It can be claimed that the personalization should provide the parameter estimate with the uncertainty $∆\widehat{\mathit{\theta }}$ satisfying

$$∆\widehat{\mathit{\theta }}\ll ∆\mathit{\theta }.$$

(30)

Substituting $∆\mathit{\theta }$ according to (12) and $∆\widehat{\mathit{\theta }}$ according to (29) yields

$$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right) R\sqrt{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)}{r}_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\ll \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\overline{\mathit{\theta }}\right)R1r_{\mathbf{1}-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

(31)

which confirms (25).

2.5. Nonlinear Least Squares Method to Minimize the Model Response Error

A more conservative approach to the personalization of model (1) is the least squares minimization of the model simulation error with respect to the data in the form of sampled state responses. This approach considers a quadratic cost function to minimize the sum of the squared errors/residuals between the provided sampled states $N\left(k{T}_s\right), L\left(k{T}_s\right), T\left(k{T}_s\right)$ and the fitted model responses $\widehat{N}\left(k{T}_s,\widehat{\mathit{\xi }}\right), \widehat{L}\left(k{T}_s,\widehat{\mathit{\xi }}\right), \widehat{T}(k{T}_s,\widehat{\mathit{\xi }})$ obtained by numerical solution of (1). The corresponding cost function has the form

$$\begin{gathered} J_{\mathrm{L}\mathrm{S}}\left(\widehat{\mathit{\xi }}\right)={\sum }_{k=1}^n\left[{\lambda }_N{\left(\widehat{N}\left(k{T}_s,\widehat{\mathit{\xi }}\right)-N\left(k{T}_s,\mathit{\xi }\right)\right)}^2+{\lambda }_L{\left(\widehat{L}\left(k{T}_s,\widehat{\mathit{\xi }}\right)-L\left(k{T}_s,\mathit{\xi }\right)\right)}^2+ \\ {\lambda }_T{\left(\widehat{T}\left(k{T}_s,\widehat{\mathit{\xi }}\right)-T\left(k{T}_s,\mathit{\xi }\right)\right)}^2\right], \end{gathered}$$

(32)

where ${\lambda }_N>0$, ${\lambda }_L>0$, ${\lambda }_T>0$ are the weights.

It is important to remark that since the model responses $\widehat{N}\left(k{T}_s,\widehat{\mathit{\xi }}\right)$, $\widehat{L}\left(k{T}_s,\widehat{\mathit{\xi }}\right)$, $\widehat{T}\left(k{T}_s,\widehat{\mathit{\xi }}\right)$ are not linear with respect to the estimated vector $\widehat{\mathit{\xi }}$, the linear regression cannot be used to find $\widehat{\mathit{\xi }}$ minimizing (32). This also means that the cost function given by (32) cannot be written as a quadratic form with respect to $\widehat{\mathit{\xi }}$ and minimized analytically, implying a nonlinear least squares problem. Therefore, the estimate $\widehat{\mathit{\xi }}$ must be determined by numerical optimization using some gradient-free algorithm such as the Nelder-Mead [57] or genetic algorithms [58]. It suggests that the main drawback of this conventional approach is the computational complexity of the personalization itself, which is significant due to the need for numerical optimization involving repetitive evaluation of cost function given by (32) and numerical solution of the model differential equations.

This contrasts with the parametrization of the state responses and calculating the output of the MLP NN in the proposed approach. Therefore, it can be concluded that despite the significant burden involved in generating the training dataset and in the process of training itself, the proposed approach is superior in terms of the computational complexity of personalization.

2.6. Personalized Chemotherapy

In this section, the chemotherapy dosing protocol will be optimized considering the mathematical model given by (1) with its parameters estimated by the proposed personalization approach.

2.6.1. Target Steady State

The steady states of the personalized model must be analyzed to find an appropriate target state for the therapy. For $t\to \infty$ and $v\left(t\right)=0$, the state variables will approach their steady values $N_0, L_0, T_0$. Given the fact that we are dealing with a nonlinear system, it is typical to have multiple steady states, each with a specific region of attraction. Convergence of the states to one of the stable steady states depends on the initial state $x\left(0\right)$. Determining the steady states is crucial for the chemotherapy dosing design as we need to specify the target (reference) state representing the cancer in remission. The steady states can be determined as the solutions of a system of algebraic equations resulting from (1) under the assumption of ${\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{d}L}{\mathrm{d}t}}={\displaystyle \frac{\mathrm{d}T}{\mathrm{d}t}}=0$.

Another important aspect is that since the chemotherapy dosing will terminate after completing the treatment protocol, it is necessary to ensure that the state variables remain stable after reaching the vicinity of the target steady state. Therefore, satisfying the criterion of local stability for the chosen target steady state is necessary for the success of therapy and inducing remission. Otherwise, the cancer would relapse, and the tumor would regrow. By studying the Jacobian matrix, we can determine the local stability of a steady state based on its eigenvalues. If all eigenvalues have negative real parts, the equilibrium point is locally stable.

The Jacobian matrix $A(x,\mathit{\theta })\in {\mathbb{R}}^{4\times 4}$ is given by

$$A\left(x,\mathit{\theta }\right)=\left(\begin{array}{cccc}a\left(1-2bN\right)-{\alpha }_{1}T-k_{N}u& 0& -{\alpha }_{1}N& -k_{N}N\\ rT& -\left(\mu +{\beta }_{1}T+k_{L}u\right)& rN-{\beta }_{1}L& -k_{L}L\\ -{\alpha }_{2}T& -{\beta }_{2}T& c\left(1-2dT\right)-{\alpha }_{2}N-{\beta }_{2}L-k_{T}u& -k_{T}T\\ 0& 0& 0& -\omega \end{array}\right).$$

(33)

There are four different clinical scenarios for the steady states. The first is the Dead state represented by all state variables being zero, which means that the tumor is gone, but the immune system is also depleted. This state is undesired from the clinical perspective. The ideal condition is the Cured state having zero tumor cells and enough natural killer cells. However, this state is typically unstable and hence cannot be maintained. The worst scenario is the Grown state manifesting by a large number of tumor cells, representing the final stage of cancer. Therefore, the so-called Coexisting state seems to be the best option as it corresponds to cancer in remission having a small number of $T$ cells and enough $N$ cells. If one of the Coexisting states satisfies the local stability criterion, it will be chosen as the target state for the treatment.

2.6.2. Personalization and Need for the Leading Dose

Before optimizing the chemotherapy dosing protocol tailored for the patient, the model needs to be personalized. However, to personalize the model, we need the sampled state responses from one dosing period (cycle), which means that the first dose has already been administered. This order of operations results in a causality paradox. As a consequence, the first administered dose $D_0$, called the leading dose, cannot be personalized. After the first dosing period elapsed, the sampled state responses are available and thus the model personalization can be carried out. Finally, the remaining doses of the dosing protocol will be optimized based on the personalized model. It can be concluded that the personalization of chemotherapy cannot be achieved without the data including the subject’s response to a single chemotherapy administration.

2.6.3. Optimization Problem

The aim of cancer treatment is to minimize the tumor cell population $T\left(t\right)$, maintain a healthy population of natural killer cells $N\left(t\right)$ while minimizing the total amount of chemotherapy administered during the treatment to reduce its adverse effect on healthy cells.

The objective of designing the chemotherapy dosing protocol is to ensure the transition of the state variables from their initial values $N\left(0\right), L\left(0\right), T\left(0\right)$ to the coexisting state $N_0,L_0,T_0$, which represents cancer remission [59]. The therapy is implemented as a finite sequence of $n_d$ individual (discrete) doses $D_i$, delivered periodically as short, uniformly spaced impulses (boluses) with a period of $T_D$ days. Each dose is represented as a rectangular pulse of the administration rate signal $v\left(t\right)$, lasting $0.1\times T_D$ and having an amplitude of $D_i\times {\displaystyle \frac{10}{T_D}}$, ensuring that the pulse’s area under the curve equals $D_i$.

The aim of the design is to find vector $\mathit{D}\in {\mathbb{R}}^{n_D}$ of $n_D$ positive doses

$$\mathit{D}={\left(\begin{array}{ccccc}{D}_1& D_2& D_3& \cdots & D_{n_D}\end{array}\right)}^{\mathrm{T}},$$

(34)

minimizing the criterion

$$\begin{gathered} J_D\left(\mathit{D}\right)={\sum }_{k=1}^{n_D\times n}\left[w_N{\left(N\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)-N_0\right)}^2+w_L{\left(L\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)-L_0\right)}^2+ \\ {w}_T{\left(T\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)-T_0\right)}^2+w_u{u\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)}^2\right], \end{gathered}$$

(35)

where $w_N>0$, $w_L>0$, $w_T>0$, $w_u>0$ are the weighting coefficients. Since $\mathit{D}$ represents positive quantities of chemotherapy to be administered, the additional constraints are imposed as $\mathit{D}>0$. Because the responses $N\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$, $L\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$, $T\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$ are not linear with respect to the decision vector $\mathit{D}$, the traditional quadratic programming problem cannot be formulated. Therefore, given that our problem is finite-dimensional, we opt for a numerical optimization while $N\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$, $L\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$, $T\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$, $u\left(k{T}_s,\mathit{D},\widehat{\mathit{\theta }}\right)$ are obtained by the numerical solution of differential equations given by (1) using the 4th-order Runge–Kutta method.

To determine the vector $\mathit{D}$ that minimizes (35), numerical optimization was performed using the fmincon function available in MATLAB Optimization Toolbox (R2023b, https://www.mathworks.cn/products/optimization.html) [60]. This function implements the interior-point method to solve constrained numerical optimization problems [61,62].

Interior-point methods represent a class of optimization algorithms that focus on exploring the interior of the feasible region. They use a barrier function (typically logarithmic) to penalize points near the boundary, ensuring iterates remain within the feasible region defined by inequality constraints. The barrier parameter is gradually reduced, allowing the solution to approach the boundary as it converges to the optimum. These methods leverage a central path, a trajectory defined by the solutions of the barrier-augmented optimization problem as the barrier parameter is reduced, to approach the optimal solution satisfying the primal and dual conditions (Karush–Kuhn–Tucker conditions). In the nonlinear least-squares optimization problem considered, where explicit gradients and Hessians are unavailable, the optimizer employs finite-difference approximations for gradients and approximates the Hessian matrix using a quasi-Newton method, specifically the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.

Another aspect of personalized therapy lies in the patient’s state at the beginning of the treatment (initial state) $x\left(0\right)$. However, contrary to unknown parameters $\mathit{\theta }$, the initial state $x\left(0\right)$ can be determined by performing the corresponding medical diagnostic procedures (blood analysis, histology, medical imagining techniques).

The whole process of data acquisition, state response parametrization, model personalization, and personalized optimization of chemotherapy doses is visualized in Figure 2.

3. Results

In this section, numerical experiments will be conducted to demonstrate the effectiveness of the proposed methodology while assessing the accuracy of the model personalization and the performance of the personalized optimal chemotherapy dosing protocol compared to one designed based on the nominal model.

Within this experiment, we will consider the dosing period (cycle duration) $T_D=40$ days and the sample time $T_s=1/4$ of the day or $T_s=1$ day, which implies $n=160$ or $n=40$ samples of the state vector. Evaluating the performance of the proposed methodology with an extended sampling interval reduces the number of data points available for personalization. This infrequent sampling represents a significant practical challenge in clinical applications.

It is also important to mention that changing the sample time $T_s$ does not require re-training of the MLP NN as it affects only the parametrization of the sampled state responses while the number of parameters $n_p$ describing them would remain the same.

The proposed methodology will be compared with the nonlinear least squares-based model response error minimization as described in Section 2.5. The weights in the cost function given by (32) were empirically tuned as

$${\lambda }_N={5\times 10}^4, {\lambda }_L={10}^4, {\lambda }_T={10}^2.$$

(36)

3.1. Personalization of the Model Parameters

For the parametrization of the state responses, the number of parameters was chosen as $n_p=8$, which implies $\mathit{U}\in {\mathbb{R}}^{28}$. The tuning coefficients (exponents) $a_i$ in (15) were chosen as $\begin{array}{ccccccc}{a}_1& a_2& a_3& a_4& a_5& a_6& a_7\end{array}=\begin{array}{ccccccc}-100& -30& -10& -5& -1& -0.5& -0.1\end{array}$.

Processing the validation dataset and evaluating the corresponding statistics yields the mean value of the residuals $\widehat{\mathit{\xi }}-\mathit{\xi }$ and the diagonal entries of the covariance matrix $\widehat{P}$ of the residuals as given in

Table 2. Mean value of the residuals $\widehat{\mathit{\xi }}-\mathit{\xi }$ and the diagonal entries of the covariance matrix $\widehat{P}$ of the residuals evaluated across $M$ entries of the validation dataset for different $T_s$.

		${\mathit{T}}_{\mathit{s}}$
Proposed machine learning method	$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)$	${\displaystyle \frac{1}{4}}$	${\left(\begin{array}{ccccccccc}-0.0046& -0.0113& -0.0047& 0.0009& -0.0009& -0.0013& -0.0019& 0.0009& 0.0008\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)$		${\left(\begin{array}{ccccccccc}0.0371& 0.1350& 0.0370& 0.0080& 0.1330& 0.0399& 0.1222& 0.0426& 0.0633\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)$	$1$	${\left(\begin{array}{ccccccccc}0.0123& 0.0003& 0.0022& -0.0028& -0.0031& -0.0079& -0.0010& -0.0017& 0.0068\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)$		${\left(\begin{array}{ccccccccc}0.1009& 0.1679& 0.0624& 0.0182& 0.1661& 0.0618& 0.2205& 0.0535& 0.1017\end{array}\right)}^{\mathrm{T}}$
Nonlinear least squares method	$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)$	${\displaystyle \frac{1}{4}}$	${\left(\begin{array}{ccccccccc}0.1624& -0.1257& -0.0205& -0.0689& 0.0838& -0.0260& 0.0033& 0.0460& -0.0399\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)$		${\left(\begin{array}{ccccccccc}0.6265& 0.9521& 0.8547& 0.8774& 0.7292& 0.9043& 0.9793& 0.8476& 1.0195\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left(\widehat{\mathit{\xi }}-\mathit{\xi }\right)$	$1$	${\left(\begin{array}{ccccccccc} 0.1940& -0.0534& -0.0025& -0.0679& 0.0779& -0.0357& 0.0001& 0.0473& -0.0056\end{array}\right)}^{\mathrm{T}}$
	$\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\widehat{P}\right)$		${\left(\begin{array}{ccccccccc}0.6280& 0.9626& 0.8560& 0.8456& 0.7436& 0.8792& 0.9462& 0.8357& 1.0461\end{array}\right)}^{\mathrm{T}}$

.

For better interpretation, the relative interval uncertainty of the original parameter vector ${\displaystyle \frac{\pm ∆\mathit{\theta }}{\overline{\mathit{\theta }}}}$ obtained according to (12) and the relative interval uncertainty of the estimated (personalized) parameter vector ${\displaystyle \frac{\pm ∆\widehat{\mathit{\theta }}}{\overline{\mathit{\theta }}}}$ obtained according to (29) are visualized in Figure 3.

To ensure comprehensive validation, the comparison between the state responses of the estimated model $\widehat{N}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)$, $\widehat{L}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)$, $\widehat{T}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)$ and the reference model $N\left(k{T}_s,{\mathit{\xi }}_i\right)$, $L\left(k{T}_s,{\mathit{\xi }}_i\right)$, $T\left(k{T}_s,{\mathit{\xi }}_i\right)$ was repeated for each entry (subject) in the validation dataset. Then, the root mean squared error (RMSE) metric has been evaluated independently for each state variable as

$$\begin{gathered} {\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_N=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{n}}{\sum }_{i=1}^M{\sum }_{k=1}^n{\left(\widehat{N}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)-N\left(k{T}_s,{\mathit{\xi }}_i\right)\right)}^2}, \\ {\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_L=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{n}}{\sum }_{i=1}^M{\sum }_{k=1}^n{\left(\widehat{L}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)-L\left(k{T}_s,{\mathit{\xi }}_i\right)\right)}^2}, \\ {\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_T=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \frac{1}{n}}\sum _{i=1}^M\sum _{k=1}^n{\left(\widehat{T}\left(k{T}_s,{\widehat{\mathit{\xi }}}_i\right)-T\left(k{T}_s,{\mathit{\xi }}_i\right)\right)}^2} \end{gathered}$$

(37)

where $M$ is the number of entries in the validation dataset and $n$ is the number of samples. The obtained results are summarized in

Table 3. RMSE evaluated across $M$ entries of the validation dataset for different $T_s$.

	Proposed Machine Learning Method		Nonlinear Least Squares Method
	${\mathit{T}}_{\mathit{s}}=1/4$	${\mathit{T}}_{\mathit{s}}=1$	${\mathit{T}}_{\mathit{s}}=1/4$	${\mathit{T}}_{\mathit{s}}=1$
${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_N$	$2.2891\times {10}^4$	$2.3747\times {10}^4$	$1.6740\times {10}^4$	$1.8361\times {10}^4$
$\mathrm{R}{\mathrm{M}\mathrm{S}\mathrm{E}}_L$	$4.4498\times {10}^4$	$4.7333\times {10}^4$	$5.9670\times {10}^4$	$5.9306\times {10}^4$
${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_T$	$5.4686\times {10}^5$	$5.8619\times {10}^5$	$4.2857\times {10}^5$	$4.4625\times {10}^5$

.

As a demonstrative case study, consider a particular subject with vector $\mathit{\xi }$ given as

$$\mathit{\xi }={\left(\begin{array}{ccccccccc}0.388& 1.137& 0.711& -0.968& -0.482& -1.183& 0.430& -0.400& 1.132\end{array}\right)}^{\mathrm{T}}.$$

(38)

For this subject, the actual coexisting stable steady state is

$$N_0=2.9035\times {10}^5,L_0=14.940\times {10}^5,T_0=7.6286\times {10}^5.$$

(39)

Considering the initial conditions and the dose size given by (40) for this subject will result in the state responses $N\left(k{T}_s\right),L\left(k{T}_s\right), T\left(k{T}_s\right)$ used for the model personalization.

$$N\left(0\right)=1\times {10}^5,L\left(0\right)=20\times {10}^5,T\left(0\right)=50\times {10}^5, u\left(0\right)=0, D=1$$

(40)

The next step is the parametrization of the state responses. The accuracy of the parametrization is visualized by comparing the original state responses $N\left(t\right),L\left(t\right),T\left(t\right)$ with their parametric approximations $\tilde{N}\left(t\right),\tilde{L}\left(t\right),\tilde{T}(t)$ for $T_s=1/4$ in Figure 4.

Evaluating vector $\mathit{U}$ obtained by parametrizing the state responses by the trained MLP NN according to (20) results in

$$\widehat{\mathit{\xi }}={\left(\begin{array}{ccccccccc}0.5588& 1.2151& 0.5866& -1.0861& -0.3160& -1.1409& 0.1058& -0.3675& 1.0203\end{array}\right)}^{\mathrm{T}}.$$

(41)

The estimated model parameters $\widehat{\mathit{\theta }}$ determined from $\widehat{\mathit{\xi }}$ according to (8) and their corresponding confidence interval $∆\widehat{\mathit{\theta }}$ determined according to (29) are compared with the actual parameter vector $\mathit{\theta }$ in Figure 5.

The coordinates of the target coexisting state were estimated from the personalized model as

$${\widehat{N}}_0=2.9029\times {10}^5,{\widehat{L}}_0=15.015\times {10}^5,{\widehat{T}}_0=8.0630\times {10}^5.$$

(42)

The reliability of the personalized models was assessed by comparing their dynamic responses to the reference state responses used during the personalization process, providing validation within the time domain. A comparison of the reference state responses $N\left(t\right),L\left(t\right),T\left(t\right)$ considering vector $\mathit{\theta }$ and the responses $\widehat{N}\left(t\right),\widehat{L}\left(t\right),\widehat{T}\left(t\right)$ obtained using the estimated vector $\widehat{\mathit{\theta }}$ can be seen in Figure 6.

Considering the identical state responses as in the previous scenarios, numerical minimization of (32) yields

$$\widehat{\mathit{\xi }}={\left(\begin{array}{ccccccccc}0.178& 0.681& 0.005& -1.165& -0.600& -1.822& 0.188& -0.816& 0.575\end{array}\right)}^{\mathrm{T}}.$$

(43)

A corresponding visual comparison of the estimated parameters can be seen in Figure 7.

The coordinates of the target coexisting state were estimated from the personalized model as

$${\widehat{N}}_0=2.9478\times {10}^5,{\widehat{L}}_0=14.677\times {10}^5,{\widehat{T}}_0=7.0081\times {10}^5.$$

(44)

A comparison of the reference state responses $N\left(t\right),L\left(t\right),T\left(t\right)$ considering vector $\mathit{\xi }$ and the simulated responses $\widehat{N}\left(t\right),\widehat{L}\left(t\right),\widehat{T}(t)$ obtained using the estimated vector $\widehat{\mathit{\xi }}$ can be seen in Figure 8.

3.2. Personalized Optimal Chemotherapy

Now, having the model personalized according to the responses obtained after the leading dose within the first dosing period, the optimal chemotherapy dosing protocol will be determined considering $n_D=6$ optimized doses. The weighting coefficients $w_N$, $w_L$, $w_T$, $w_u$ were tuned to ensure that all four terms in (35) have comparable magnitudes. This is essential to avoid significant imbalances and overfitting, especially when one term disproportionately outweighs the others. Ensuring penalties of comparable magnitudes promotes a balanced optimization process, allowing each component’s role to be appropriately weighted in attaining the targeted therapeutic outcome.

$$w_N={10}^5, w_L={10}^2, w_T=1, w_u={10}^{15}$$

(45)

The state of the subject at the beginning of the optimized treatment is equivalent to the state after elapsing the first dosing period (cycle), which is

$$N\left(T_D\right)=3.0357\times {10}^5,L\left(T_D\right)=1.6863\times {10}^5,T\left(T_D\right)=1.3135\times {10}^4, u\left(T_D\right)=0.001.$$

(46)

The optimization of chemotherapy will be performed considering the personalized model parameters (41) while the target steady state will be given by (42). By minimizing criterion (35), the sequence of chemotherapy doses was obtained as

$$\mathit{D}={\left(\begin{array}{cccccc}0.2263& 0.1413& 0.0302& 0.1039& 0.0921& 0.0769\end{array}\right)}^{\mathrm{T}}.$$

(47)

To ensure objectiveness, the designed personalized treatment protocol will be simulated considering the actual model parameters given by (38) and not the estimated ones. The evolution of the states obtained by applying the designed chemotherapy dosing protocol is visualized in Figure 9 and the resulting value of criterion is $J_D\left(\mathit{D}\right)=2.0653\times {10}^{16}$.

Considering the nominal model parameters, by performing the optimization of the treatment by minimization of criterion (35), the sequence of chemotherapy doses was obtained as

$$\mathit{D}={\left(\begin{array}{cccccc}0.7620& 0.5804& 0.4187& 0.4610& 0.2235& 0.2296\end{array}\right)}^{\mathrm{T}}.$$

(48)

Also in this scenario, the designed nominal treatment protocol will be simulated considering the actual model parameters given by (39). The evolution of the states obtained by applying the designed chemotherapy dosing protocol is visualized in Figure 10 while the resulting criterion is $J_D\left(\mathit{D}\right)=3.6283\times {10}^{16}$.

4. Discussion

The first portion of the results dealt with the statistical assessment of the model personalization accuracy. This assessment was substantiated by the mean value and the covariance matrix of the parameter estimate residuals as given in Table 2. Since the mean value of the residuals is relatively close to zero, it can be claimed that the personalization does not produce statistically biased estimates of the parameters. Another important result to highlight is that the diagonal entries of the covariance matrix of the estimate residuals satisfy the variance condition given by (25) with a large margin. This proves that personalization significantly lowers the uncertainties of the model parameters. For better interpretation, the relative interval uncertainty of the original parameter vector ${\displaystyle \frac{\pm ∆\mathit{\theta }}{\overline{\mathit{\theta }}}}$ obtained according to (12) and the relative interval uncertainty of the estimated (personalized) parameter vector ${\displaystyle \frac{\pm ∆\widehat{\mathit{\theta }}}{\overline{\mathit{\theta }}}}$ obtained according to (29) were visualized in Figure 3, demonstrating the shrinking of the uncertainty interval after personalization.

It also implies that the trained MLP NN captures and sufficiently generalizes the relationships between the variations in the parameters, the initial state, dose size, and the state responses to a leading dose of chemotherapy.

Besides comparing the estimated parameters themselves, the personalized models were validated by comparing their responses with the reference state responses used for personalization. This can be seen as a validation of the personalized model in the time domain. Such a comparison is documented in Figure 6 and Figure 8. By studying these figures, it can be concluded that the personalized models provided an almost perfect match of the state responses.

To confirm this observation, the root mean squared error was computed by comparing state responses between the estimated and reference models for each subject in the validation dataset, separately for each state variable $N\left(t\right)$, $L\left(t\right)$, $T\left(t\right)$ as documented in Table 3. These results revealed that for the sample time $T_s=1/4$ the metrics ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_N=2.2891\times {10}^4$, ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_L=4.4498\times {10}^4$, ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_T=5.4686\times {10}^5$ are sufficient, especially if taken relative to the typical magnitudes of the state responses, which have average values $\overline{N}\left(t\right)=2.4965\times {10}^5$, $\overline{L}\left(t\right)=1.4522\times {10}^6$ and $\overline{T}\left(t\right)=3.9565\times {10}^6$. This indicates that the mean error of the state responses provided by the personalized model is only a few percent ${\displaystyle \frac{{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_N}{\overline{N}\left(t\right)}}=0.0922$, ${\displaystyle \frac{{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_L}{\overline{L}\left(t\right)}}=0.0307$, ${\displaystyle \frac{{\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_T}{\overline{T}\left(t\right)}}=0.1382$ and that the best accuracy was achieved for $L\left(t\right)$ and the worst for $T\left(t\right)$.

In contrast, increasing the sample time to $T_s=1$ day resulted in higher ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_N$, ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_L$ and ${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_T$ indicating slightly deteriorated, yet acceptable accuracy of the state responses prediction and good overall model validity under reduced sampling frequency. Increasing the sampling interval to $T_s=1$ day also led to marginally higher variances of the residuals of the estimated parameters and broader confidence intervals in Figure 3. Nonetheless, this analysis remains crucial, as frequent data collection may pose practical challenges in clinical settings.

For the sake of comparison with other relevant parameter identification approaches, the conventional least squares minimization of the model response error with respect to the reference state responses was evaluated for the validation dataset and the obtained results were quantified by statistical metrics as summarized in Table 2 and Table 3. This quantitative comparison demonstrated that the proposed machine learning approach generally achieves much better accuracy of parameter estimation and slightly lower accuracy of state response prediction than obtained by the conventional methodologies that minimize the model response error. Therefore, it can be concluded that the proposed approach provides more accurate parameter estimation, which enhances the general validity of the resulting models, even if it does not achieve the closest fit to the observed state responses. In contrast, although the prediction error methods yield a good fit between the model response and the data, its general validity is limited due to inaccuracies in parameter estimation.

Another important section of the results addresses a detailed case study focused on experimenting with a single particular patient. First, an in silico subject was randomly generated following the considered statistical distribution of the model parameters within the population. Then, the state responses to the suboptimal leading dose were obtained for the duration of one dosing period by simulating the model defined by (1) for the given initial conditions.

A comparison of these state responses with their approximations obtained by the proposed parametrization technique is given in Figure 4. It is evident that the state responses can be approximated very accurately by the function of an exponential nature and an absolute (constant) term. Nevertheless, it is important to remember that the aim of parametric approximation is not to achieve a perfect fit but rather to extract the features of the responses and thus describe them with a small set of parameters.

Another important result of the personalization accuracy analysis is the comparison between the actual instance of vector $\mathit{\xi }$ given by (38) and its estimate $\widehat{\mathit{\xi }}$ given by (41). This is also visualized in Figure 5. for the parameter vector $\widehat{\mathit{\theta }}$. The results demonstrated that the parameters estimated from the finite sequence of state responses to the leading dose were of good accuracy, with all estimates falling within the confidence intervals. The relative (percentual) width of the confidence interval $\pm {\displaystyle \frac{∆{\widehat{\theta }}_{\mathit{i}}}{{\overline{\theta }}_i}}$ is visibly different for each parameter depending on the diagonal entries of the covariance matrix $\widehat{P}$ following (38).

The conventional least squares minimization of the model simulation error yielded the estimates given by (43) and visualized in Figure 7. Based on these results, it can be claimed that the error of personalization is higher than, but still acceptable.

Another important part of the results analysis is the personalization of the steady states, which is ultimately required for the optimal chemotherapy design. Since the steady states are derived from the model parameters, their accuracy results from the accuracy of model personalization. The actual coexisting state is defined by (39), and the personalized models resulted in the estimated coexisting states as given by (42) and (44). We can claim that the most accurate estimate of the coexisting steady state was obtained by the proposed approach.

The remaining results addressed the problem of personalized optimization of the chemotherapy dosing protocol. Based on the model personalized from the patient’s response to the suboptimal leading dose, the sequence of six optimal chemotherapy doses was determined by numeric optimization, and this dosing protocol was applied to the virtual subject, resulting in the state responses plotted in Figure 9. These state responses demonstrate effective and relatively fast stabilization of all three variables to the target coexisting state representing cancer in remission. However, significant fluctuations are present at the beginning of the treatment, which are difficult to manage due to the fixed suboptimal leading dose and adverse initial conditions. It is also important to notice that the tumor responses showed periodic behavior. The observed periodicity means that the tumor size stably oscillates around the target state, which corresponds to the so-called Jeff’s phenomenon [7,20] observed in clinical practice. The personalized optimal treatment was compared with the suboptimal dosing protocol obtained by optimizing the response of the nominal (mean-population) model without tailoring the treatment to the characteristics of the subject. The suboptimal treatment responses in Figure 10 featured highly undesired fluctuations in the tumor cell population $T\left(t\right)$ and the cytotoxic cell population $L\left(t\right)$, resulting in a higher value of criterion $J_D\left(\mathit{D}\right)$. Besides that, the total amount of administered chemotherapy was significantly higher, implying that the personalized optimal treatment ensures the desired therapeutic outcomes with lower chemotherapy doses.

The paper presented a novel methodology for personalizing mathematical models of tumor growth dynamics, immune system interactions, and chemotherapy effects. By leveraging machine learning techniques and computer simulations, this study aimed to address the limitations of traditional approaches in chemotherapy dosing protocols, which often rely on mean-population models that fail to account for individual patient variability.

This study offers practical benefits for clinicians by providing a framework for guiding decision-making. By integrating machine learning, dynamic models, and optimization, clinicians can simply make more informed decisions. By incorporating patient-specific responses into the treatment protocol, it is possible to reduce the trial-and-error approach, ensuring that patients receive the most effective dose for their unique needs. Clinicians could also use this approach to assess how an individual patient is likely to respond to treatment and whether successful remission is even feasible, leading to a more proactive approach to cancer therapy.

The clinical workflow should be as follows:

• Conventionally determine and administer the leading dose $D_0$.
• Determine the sampled patient responses $N\left(t\right)$, $L\left(t\right)$, $T\left(t\right)$ via regular bloodwork and medical imagining tools during one dosing period (cycle).
• Feed the obtained data to the computer implementing the proposed methodology.
• The computer provides the following information to the clinician:
  • Estimated parameters of the tumor model.
  • Optimal sequence of chemotherapy doses.
  • Feasibility of cancer remission and the estimated final tumor size.
• The clinician expertly assesses the results and finally determines the future steps and approves the treatment.

The methodology involved generating training and validation datasets by simulating a diverse population of virtual patients. The parameter vector for each patient was assumed to follow a multivariate normal distribution, capturing inter-subject variability. Sensitivity analysis showed that some parameters had negligible effects on the state variables, narrowing down the number of practically identifiable parameters to 9 out of 14. The trained neural network was then capable of estimating personalized model parameters based on clinical data from a single chemotherapy dose, facilitating more accurate predictions of tumor behavior and treatment outcomes.

Key innovations included the use of machine learning to personalize cancer model parameters and the application of parametric approximation of the state responses to simplify training data. The parametric approximation of sampled state responses by the sum of exponential functions reduced the complexity of the input data, ensuring feasible training of the MLP NN and better generalization capability. These advancements enable optimized chemotherapy dosing protocols tailored to individual patients, improving both treatment efficacy and safety. An important benefit of determining personalized steady states is the ability to predict cancer remission feasibility and estimate final tumor size.

The proposed approach has several limitations that should be acknowledged. First, the model personalization process requires administering an initial chemotherapy dose before estimating patient-specific parameters, which may introduce delays in optimizing the dosing protocol and pose risks to the patient. Additionally, once the neural network has been trained, the chemotherapy dosing period (cycle duration) remains fixed. Another challenge lies in the frequent sampling of tumor and immune cell populations, which may not always be feasible in clinical practice. Moreover, only parameters with a significant impact on tumor dynamics can be effectively personalized, potentially reducing the overall accuracy of the patient-specific model. Furthermore, a distinct neural network must be trained for each variation in the mathematical model, increasing the computational burden and limiting generalizability. The last drawback is that the computational complexity of generating the datasets and training the MLP NN is high. These limitations highlight the challenges that must be addressed for effective clinical implementation of the proposed approach.

Future research should focus on addressing the limitations of the proposed approach and enhancing its clinical applicability. One key direction is the development of adaptive dosing strategies that allow real-time therapy adjustments (optimization) based on continuous patient monitoring providing new data. Improving the feasibility of patient-specific data collection is another critical area, which could involve integrating non-invasive biomarkers or imaging techniques to estimate tumor and immune cell populations more efficiently. Further work is also needed to test different and more complex structures of cancer models. Clinical validation through retrospective and prospective trials will be essential to assess the effectiveness and safety of the personalized dosing protocol in real-world settings.

5. Conclusions

This study represents an important advancement in precision medicine for cancer treatment. The methodology demonstrated that optimized chemotherapy dosing protocols, personalized using an MLP NN, can lead to more effective and safer treatment outcomes by accounting for individual patient variability. The results showed that personalized chemotherapy dosing could effectively stabilize tumor cell populations, achieving cancer remission while minimizing the risks associated with conventional, one-size-fits-all dosing protocols. Furthermore, the findings demonstrated that relying on a nominal, non-personalized model often leads to suboptimal dosing strategies characterized by excessive drug administration. The trained MLP NN successfully captured the complex relationships between the initial tumor stage, chemotherapy dose, and model parameter variations.

A critical implication of this work is its potential to shift the paradigm of cancer treatment toward truly individualized therapy. By integrating computational modeling and machine learning into clinical decision-making, oncologists could develop personalized treatment strategies that adapt to patient-specific tumor responses. This would mark a departure from the traditional trial-and-error approach, leading to more precise and predictable therapeutic outcomes.