A Mechanistic Dynamic Surrogate Framework for Personalized Radioactive Iodine Therapy in Metastatic Thyroid Cancer

Abstract

Background/Objectives: Radioactive iodine (RAI) therapy is widely used to treat metastatic differentiated thyroid cancer. To investigate physiological determinants of treatment response, a mechanistic model was developed, formulated as a system of coupled ordinary differential equations. Methods: The model captures the interactions between tumor burden, thyroglobulin (T_g) production and clearance, and radioactive iodine activity within a pharmacokinetic–pharmacodynamic framework. Model parameters were estimated using the Monte Carlo Stochastic Approximation Expectation–Maximization (MCMCSAEM) algorithm, based on clinical data from a cohort of 50 patients. Results: Tumor radiosensitivity (ρ) and initial tumor burden (N₀) consistently emerged as the most influential factors in both responder and non-responder groups classified by disease doubling time under RAI (T_d). A reduced model using only these two parameters preserved the principal response patterns of the full model. Other parameters influenced transient dynamics but had limited effect on overall T_g variance. Conclusions: These results support the use of a reduced calibration approach focused on ρ, N₀, and the effective doubling time T_d. The findings establish a theoretical foundation for developing tractable dynamic surrogates that reproduce the main treatment kinetics and support model-based clinical decision-making in RAI therapy.

1. Introduction

Thyroid cancer ranks as the most common endocrine malignancy. Its incidence has been increasing in recent years due to the improvement of screening tools (high-resolution ultrasound), sometimes leading to overdiagnosis [1]. Most cases of thyroid cancer have a favorable prognosis. Radioactive iodine (RAI) therapy has long been established as a standard treatment [2]. It plays an important role in the management of thyroid cancer, particularly in patients with metastatic disease [3]. However, each patient response to treatment is driven by specific biological and clinical characteristics, highlighting the need for precision-medicine approaches and model-based decision-support tools to optimize the efficacy–toxicity balance of RAI therapy.

1.1. RAI and Metastatic Thyroid Cancer

Radioactive iodine (RAI), primarily administered as iodine-131 (I-131), is selectively taken up by thyroid cells via the sodium-iodide symporter. Once internalized, I-131 emits beta radiation that induces double-stranded DNA breaks, leading to cellular apoptosis or necrosis in iodine-avid tumor cells. RAI is usually administered after total thyroidectomy and lymph node dissection, according to prognostic indicators of recurrence or mortality [4]. This focused approach works best for well-differentiated thyroid cancers like papillary and follicular types. RAI targets remnant thyroid cells and can reach distant metastases in the lungs or bone. RAI is generally not indicated for low-risk patients (undetectable thyroglobulin, T_g < 1 ng/mL postoperatively) and selectively for intermediate-risk patients [5]. Additionally, a major controversy in RAI therapy concerns optimal dosage. Previous studies [3,6] have shown that higher doses of RAI may lead to salivary gland dysfunction, manifesting as sialadenitis (inflammation of the salivary glands) or xerostomia (dry mouth due to reduced saliva production) [7]. Such radiation-induced toxicities can manifest in the short, medium, or long term [8,9]. Furthermore, for patients who received a high dose of administered activity, the risk of developing leukemia is proven [10]. It is also noted that some patients with refractory thyroid cancer do not respond to RAI treatment [11,12,13]. These limitations emphasize the need for individualized treatment planning that balances efficacy and toxicity based on each patient’s biological characteristics and tumor profile.

1.2. The Personalized RAI

Today, the importance of decision support tools in medicine is recognized, and more particularly in oncology, to offer personalized medicine and to propose a personalized treatment protocol. This kind of therapy should ideally minimize the risk of severe toxicity and avoid transient or partial responses, which may contribute to direct or indirect resistance after repeated RAI administration [14,15]. It is therefore essential to identify the individual determinants of therapeutic response. Studies have thus reported the existence of numerous factors influencing the response to RAI [16,17,18,19]. This work aims to identify those contributing to response variability. The characteristics of the protocol (dosage, number of RAI sessions, interval between sessions) directly depend on the decisions taken during multidisciplinary consultation meetings, considering the information known about the patient and relying on empirical data [20]. Incorporating patient-specific clinical and biological data could enable realistic simulations of the response to RAI. This study focuses on cases of metastatic thyroid cancer where the factors influencing the response to RAI, as already mentioned, are numerous. The doubling time (T_d) of tumor cells is a first important parameter in the response to treatment and thus makes it possible to define responder status more quickly (after only 3 fractions) (for a $\overline{T_d}$= 66.6 months) or non-responder (for a $\overline{T_d}$= 9.8 months) at RAI [11]. Importantly, T_d was not arbitrarily chosen: in our cohort it emerged as the dominant dichotomizing marker among all candidates, with a clear bimodal structure in the mixture-model analysis (responders vs. non-responders). We also stress that the T_d estimated here is an effective doubling time under RAI exposure, not the natural history growth rate of untreated disease. Unlike previous modeling approaches that focused exclusively on responders, the present analysis includes both groups, enabling a comprehensive characterization of the underlying dynamics and supporting personalized therapeutic strategies across the entire patient spectrum.

1.3. Clinical Interest of This Work

To deliver the most appropriate treatment for each patient (minimizing toxicity while maximizing efficacy), this study aims to provide clinicians with a model-based decision-support framework capable of predicting therapeutic response. The approach focuses on identifying a minimal set of physiological parameters that govern RAI treatment dynamics and can be estimated from routine follow-up data. Although previous studies have shown that BRAF mutations can influence patient sensitivity to RAI [21], no genomic analyses were included in the present work. Instead, the analysis centers on measurable or inferable physiological quantities. Among them, the effective tumor doubling time under RAI exposure (T_d) is introduced as a quantitative parameter that naturally distinguishes responders from non-responders and provides a clinically interpretable handle for dose adjustment and treatment scheduling. This modeling strategy lays the foundation for constructing reduced mechanistic representations (or dynamic surrogates) that could ultimately support real-time personalization of RAI therapy.

Section 2 describes the model structure and sensitivity-analysis framework, Section 3 presents the results, and Section 4 discusses clinical implications and perspectives.

2. Material and Methods

The following section describes the methodological framework adopted in this study, including the mathematical formulation of the model, its analytical developments, and the approach used for sensitivity analysis.

2.1. Mathematical Novelty and Originality of This Work

This study extends the Ordinary Differential Equation (ODE) model of Barbolosi et al. (2017) [11] with major advances in mathematical formulation and clinical applicability. The original work relied solely on numerical integration and offered a qualitative exploration focused on T_d. This parameter was not predefined but rather emerged from a bimodal mixture distribution of patient estimates, naturally separating the cohort into two distinct subpopulations corresponding to responders and non-responders. T_d reflects an effective doubling time under RAI exposure, which is not known prior to treatment and differs from the natural growth rate of the untreated disease. In the present study, analytical approximations are developed to better characterize model dynamics and derive explicit relationships between all physiological parameters and clinical outcomes. Because no parameter sensitivity analysis had previously been performed, a systematic quantitative study was undertaken to quantify the clinical impact of each parameter and to identify those requiring accurate measurement or estimation. These improvements turn the model into a decision-support framework for personalized RAI therapy, aimed at maximizing efficacy while limiting radiation-induced toxicity. T_g is an observable biomarker used as the model output; parameter sensitivities were therefore evaluated with respect to the simulated T_g response, providing insight into how physiological parameters influence measurable treatment dynamics. The model includes seven parameters, of which six are investigated here, since T_d has already been validated as a clinically and mathematically discriminant variable. High T_d values are associated with favorable outcomes (responders), whereas low values correspond to poor therapeutic response (non-responders), consistent with both clinical data and model simulations. In the present work, this dichotomy was further supported by the mixture model implemented in Monolix (version 2024R1), which revealed a bimodal distribution of estimated parameters (particularly T_d) corresponding to two statistically distinct subpopulations. This data-driven partition provides a robust basis for modeling responders and non-responders as separate groups in the sensitivity analysis. Accordingly, two patient groups were simulated in parallel: responders ($\overline{T_d}$= 66.6 months) and non-responders ($\overline{T_d}$= 9.8 months).

2.2. Data

Clinical data were collected over three consecutive years, up to 2016. The cohort consists exclusively of patients with metastatic differentiated thyroid carcinoma presenting pulmonary metastases, thereby minimizing tumor heterogeneity. Serum thyroglobulin (T_g) time series were used as the primary observable variable for model fitting. Each of the 50 patients contributed at least six T_g measurements obtained during RAI follow-up, providing over 300 longitudinal data points. Parameter estimation was performed using the updated Monte Carlo Stochastic Approximation Expectation–Maximization (MCMCSAEM) algorithm implemented in Monolix [22], which benefits from recent numerical and convergence improvements. All diagnostic checks (such as “Observed vs. Predicted” (Population vs. Individual), “Visual Predictive Check”) were verified. Although the cohort is identical to that analyzed by Barbolosi et al. [11], all parameters were newly estimated to ensure methodological consistency and provide unbiased input distributions for subsequent sensitivity and simulation studies. Monolix was employed within a nonlinear mixed-effects framework to estimate population means and inter-individual variability, while MATLAB (version R2024a) was used to solve the system of ordinary differential equations describing the coupled dynamics of administered activity $A\left(t\right)$, tumor-cell population $N\left(t\right)$, and thyroglobulin concentration T_g$\left(t\right)$. The two environments were combined in a consistent pipeline: statistical calibration in Monolix provided the parameter distributions used as inputs for deterministic simulations in MATLAB. These simulations explored parameter perturbations and their impact on T_g dynamics under controlled conditions. For comparability across sensitivity analyses, an initial T_g value of 100 ng/mL and a single RAI dose of A₀ $=3.7 \mathrm{G}\mathrm{B}\mathrm{q}$ at $t=0$ were used in all experiments. The MATLAB code (available at: https://github.com/MarieFG49/PROJET-SIMU_RAI, accessed on 8 December 2025) was executed on a 28-node Intel Xeon 6230R cluster (2 × 26 cores @ 2.1 GHz, 192 GB RAM) using 96 parallel workers and the Symbolic Math Toolbox. Each simulation was completed in under 30 s per run. The ODE system was not solved for individual patient fits in this section but rather for representative parameter sets sampled from the calibrated distributions in order to characterize model sensitivity and qualitative behavior under clinically realistic conditions.

2.3. Description of the Reference Therapeutic Response Model

The model used in this study aims to optimize the efficacy–toxicity balance of RAI therapy in metastatic differentiated thyroid cancer. It is formulated under biologically grounded assumptions consistent with pharmacokinetic–pharmacodynamic (PK–PD) modeling principles. The administered iodine activity $A\left(t\right)$ is described by a mono-exponential clearance function reflecting iodine-131 pharmacokinetics. Tumor dynamics follow classical cytotoxic-response principles, where cell loss is proportional to both the viable tumor mass $N\left(t\right)$ and the effective radiation dose. Thyroglobulin (T_g) production is modeled as directly proportional to the number of viable tumor cells, and its clearance follows first-order kinetics. These assumptions are consistent with widely accepted PK–PD frameworks used in thyroid and other cancers [23,24]. Clinical T_g time-series from 50 patients (described in Appendix A.1) were used to calibrate a system of three coupled ordinary differential equations representing (i) iodine decay, (ii) tumor-cell response, and (iii) T_g kinetics (Equations (1a)–(1c)). The analytic solution of this system is provided in Appendix A.2. In this formulation, the administered activity $A\left(t\right)$ influences the number of viable tumor cells $N\left(t\right)$, which in turn determines the T_g concentration. The model introduces an iodine efficacy rate, denoted $\rho$, representing the fraction of administered radioactivity that effectively acts on tumor cells to induce destruction. This parameter captures patient-specific sensitivity to RAI and accounts for inter-individual variability in treatment response and it can be interpreted as an integrated measure of I-131 uptake and radiobiological effectiveness within iodine-avid tumor cells.

$${\displaystyle \frac{dA}{dt}}\left(t\right)=-a·A\left(t\right),\mathrm{with} A(0)=A_0$$

(1a)

$${\displaystyle \frac{dN}{dt}}\left(t\right)=\left({\displaystyle \frac{\mathit{log}\left(2\right)}{T_d}}\right)·N\left(t\right)-\rho ·A\left(t\right)·N\left(t\right),\mathrm{with} N(0)=N_0$$

(1b)

$${\displaystyle \frac{d{T}_g}{dt}}\left(t\right)=-k_e·T_g+\lambda ·N\left(t\right),\mathrm{with} T_g\left(0\right)=T_{g0}$$

(1c)

Each component of the model was formulated to represent a physiologically grounded mechanism relevant to RAI therapy in metastatic thyroid cancer. The exponential decay in Equation (1a) mirrors the pharmacokinetics of iodine-131. Tumor-cell dynamics in Equation (1b) follow classical cytotoxic-response models, where cell loss is proportional to both tumor mass and absorbed radiation dose. Equation (1c) describes thyroglobulin (T_g) kinetics, assuming secretion proportional to viable tumor cells and first-order elimination. Together, these equations form a biologically interpretable and clinically consistent PK–PD model structure. The clearance rate of radioactive iodine ($a$) and the T_g elimination rate (k_e) are illustrated in Figure 1.

Model parameters are detailed in Section 2.4, and validation results are provided in Appendix A.3. Within this framework, patients are classified according to their T_g dynamics. Responders are defined as those whose T_g concentration decreases after RAI administration and remains below 10 ng/mL for at least 12 months, corresponding to a partial response in clinical practice. Non-responders show a T_g rebound indicating treatment failure. This functional definition is consistent with previous studies and corresponds to a tumor doubling time (T_d) above or below a threshold reflecting slower or faster tumor progression. Figure 2 shows two examples of thyroglobulin kinetics for patients from our cohort: (a) responder patient with a T_d estimate of approximately 75 months, (b) non-responder patient with a T_d estimate of approximately 6 months. The model allows early identification of responder status: typically, after two to three RAI fractions and three T_g measurements. In thyroid cancer, this typically means a reduction in tumor size or activity, leading to better clinical outcomes. Factors influencing response to RAI therapy include the type and stage of the thyroid condition, the dose of radioactive iodine administered and individual patient characteristics, physiological parameters that we wish to identify and estimate. Genetic factors may also play a role in determining responsiveness to treatment [25].

2.4. Model Parameters Characteristics

The model integrates seven physiological parameters governing the coupled dynamics of administered activity A$\left(t\right)$, tumor-cell population N$\left(t\right)$, and thyroglobulin concentration T_g$\left(t\right)$ (Equations (1a)–(1c)). Parameter estimation was carried out using the MCMCSAEM algorithm implemented in Monolix, combining stochastic approximation with Markov Chain Monte Carlo sampling to handle latent variables and inter-individual variability. The procedure was executed under strict convergence and reproducibility constraints, with all diagnostic criteria satisfied. The resulting parameter distributions constitute a statistically consistent dataset enabling downstream numerical analyses and operational validation. A structured sensitivity analysis was then applied as a data-driven diagnostic layer of the model, quantifying how parameter variations propagate through the simulated T_g$\left(t\right)$ trajectories. This analysis highlights the parameters that drive model uncertainty and therapeutic outcomes, thereby identifying levers for dose optimization and individualized control strategies. From a computational standpoint, the approach establishes a transparent pipeline linking patient data ingestion, parameter calibration, and system-level response analysis. Model parameters and reference values are summarized in

Table 1. Population parameter estimates related to Equations (1a)–(1c) obtained using MCMC-SAEM algorithm [26] from MONOLIX^® software [27] *.

	N0	a	ρ	Td	λ	ke
Mean	1.12 × 109	0.0169	0.00407	55.23	3.86 × 10−9	0.319
Std	7.87	<10−17	<10−18	27.65	<10−6	0.379

* The small standard deviations reported for ρ and a are not the result of data error but rather reflect constrained variability under the mixed-effects estimation process. These values emerge from limited identifiability, compensatory effects among parameters, and regularization within the estimation algorithm. This behavior is acknowledged and discussed as a limitation of the current model implementation.

.

Where

• A₀: Initial activity dose administered (GBq [28]);
• N₀: Number of initial tumor cells [29];
• a: clearance rate of administered iodine (month⁻¹ [30]);
• ρ: effectiveness rate of iodine (GBq × months)⁻¹ [11]);
• T_d: tumor cell doubling time (months [31]);
• λ: concentration of T_g produced by a tumor cell (µg × L⁻¹ × month⁻¹ [32]). λ is defined as the production rate of thyroglobulin per tumor cell and is expressed in µg·L⁻¹·month⁻¹. While T_g levels are typically measured in ng/mL clinically, λ quantifies the dynamic rate of T_g accumulation in the blood and reflects a distinct, time-dependent biological process;
• k_e: elimination rate of T_g in the blood (month⁻¹ [33]).

Additionally, during the simulations described below, sensitivity analysis was used to identify parameters that are poorly estimated or associated with high uncertainty, indicating where additional data or model refinement would be most beneficial. This step enhances the robustness and interpretability of the model, ensuring that it remains numerically stable and clinically meaningful within realistic parameter domains. From a theoretical perspective, sensitivity analysis is an essential component of any system of coupled differential equations describing biological processes. It provides a quantitative means to assess the relative influence of each parameter on model behavior, to evaluate the conditioning of the system, and to verify the structural identifiability of parameters under real clinical variability. In practice, it allows discrimination between parameters that primarily affect scaling (amplitude of the T_g response) and those that drive temporal dynamics (response rate or rebound behavior). Parameters with minimal influence can be fixed without loss of generality, whereas highly sensitive parameters require individualized estimation to maintain predictive reliability. The model includes seven parameters: the initial administered activity A₀ and six physiological parameters characterized by their population mean and standard deviation estimated from 50 patients. Parameters with low variability (relative standard error < 2%), namely N₀, $a$, and $\rho$, were considered stable and could reasonably be fixed in clinical applications. While T_d remains a discriminating indicator of patient response to RAI, it is equally important to evaluate the influence of the remaining parameters on the evolution of T_g during treatment. Accordingly, a systematic sensitivity analysis was performed to assess the contribution of each parameter to T_g dynamics over time. Simulations were implemented in MATLAB using the differential system previously defined, varying each parameter within physiologically acceptable ranges [34]. This approach provides a rigorous, data-informed framework for quantifying parameter influence and guiding the development of personalized therapeutic strategies.

3. Results

Although clinical definitions of RAI response continue to evolve, T_g remains a relevant surrogate biomarker for assessing disease activity and treatment efficacy, particularly in differentiating refractory and non-refractory patients [35]. Sensitivity analyses were conducted separately for each group. For all simulations, T_g dynamics were modeled in response to a single RAI administration at time zero (t = 0). A one-at-a-time approach was employed, wherein one of the six model parameters varied over a physiologically plausible range while the others were fixed at their mean values, as estimated from the full cohort. A parameter was considered influential when small variations within its physiological range produced marked deviations in the simulated T_g trajectory, affecting key features such as peak amplitude, time to nadir, or overall decay profile. This criterion reflects the model local sensitivity, whereby parameters exerting stronger perturbations on these outputs are deemed more impactful. To aid interpretation, T_g values are expressed as percentages relative to their baseline, for example, T_g (100%) representing the initial concentration while T_g (50%) denotes the time derivative (rate of change) of T_g when its value has decreased to 50% of the baseline level.

3.1. Sensitivity Analysis: Group of Responders ($\overline{T_d}$ = 66.6)

Figure 3 illustrates two distinct categories of parameters: those that have a strong influence on the temporal evolution of T_g and for which patient-specific estimation is likely to be critical in optimizing therapeutic response, and those whose impact is present but more limited. Based on criteria mentioned above, the following parameters were identified as influential:

• a: clearance rate of radioactive iodine from the body, consistent with a first-order decay process rather than any biological delay in effectiveness (in month⁻¹) (Figure 3a);
• k_e: elimination rate of T_g in the blood (in month⁻¹) (Figure 3b);
• λ: concentration of T_g produced by a tumor cell (in µg × L⁻¹ × month⁻¹) (Figure 3c);
• N₀: Number of initial tumor cells (Figure 3d).

In each simulation, one parameter was varied across its full physiological range while the others were fixed at the mean values of the posterior distributions estimated from the clinical dataset. Figure 3a–d illustrate the evolution of T_g for different values of the parameters $a$, k_e, $\lambda$, and N₀. The trajectories clearly differ according to parameter variation, indicating that $a$, k_e, $\lambda$, and N₀ significantly influence the temporal dynamics of T_g. In all cases, T_g initially decreases following RAI administration; however, depending on parameter values, either a subsequent increase occurs at variable rates (Figure 3a) or T_g stabilizes at distinct levels (Figure 3b–d). Each subfigure also reveals that certain parameter values correspond to favorable therapeutic responses (characterized by a sustained or consistently low T_g) whereas others lead to elevated or non-declining T_g profiles, indicative of suboptimal response. This underscores the need for accurate, patient-specific estimation of these parameters to simulate individual therapeutic outcomes and guide clinical decision-making. Notably, each curve exhibits a minimum point (indicated by a cross), after which T_g begins to rise again, to varying degrees depending on the parameter. This inflection point is a key indicator for planning subsequent therapy sessions to prevent loss of tumor control. Parameters whose influence on the therapeutic response is less pronounced but still detectable are:

• ρ: iodine effectiveness rate (in (GBq × months)⁻¹) (Figure 3e);
• A₀: Initial activity administered (in GBq) (Figure 3f).

Following an initial decline phase that is identical across values for both parameters and lasts approximately 10 months, the T_g concentration either continues to decrease gradually (particularly for the highest parameter values) or begins to rise slightly for the lowest ones. Across the entire range of values, T_g levels remain between 5% and 20% of their initial concentration, which may correspond to favorable (lower T_g) or unfavorable (higher T_g) simulated responses. The local minimum of each curve, marked with crosses, indicate the transition point after which T_g begins to rise again.

These observations, across all 6 sub-figures, highlight the crucial importance of precisely determining the values of these parameters for each patient. The measured T_g concentrations and rebound times serve as crucial indicators to determine both the optimal timing of the second RAI administration and the required activity dose (based on the measured T_g value) to effectively manage the disease. The precise determination or estimation of the values of each specific parameter for each patient allows the protocol to be adapted to optimize the response to RAI, considering the variation in responses depending on the parameter values.

3.2. Sensitivity Analysis: Group of Non-Responders ($\overline{T_d}$ = 9.8)

In such cases, evaluating the role of each model parameter in RAI response is essential for defining optimized therapeutic protocols, while respecting recommended constraints on cumulative administered activity and minimum intervals between successive treatments. The analysis relies on the temporal evolution of T_g concentration as a function of parameter variation, under simulation conditions identical to those described in Section 3.1. Parameter estimates remain unchanged, and all simulations correspond to a single RAI administration ($A_0=3.7 \mathrm{G}\mathrm{B}\mathrm{q}$). Figure 4 illustrates the impact of varying each of the six model parameters on the T_g trajectory for a representative non-responder. Across all subfigures, T_g initially declines during the first ten months before diverging according to parameter values, indicating that each parameter exerts a measurable influence on treatment dynamics, even in clinically defined non-responders. Certain parameter combinations nevertheless yield simulated responses characterized by sustained T_g suppression. For instance, lower values of $a$, $\lambda$, and N₀ (Figure 4a,c,d), or higher values of k_e, $\rho$, and A₀ (Figure 4b,e,f), are associated with more favorable T_g trajectories.

These results suggest that certain patients currently labeled as non-responders might benefit from adjusted treatment regimens. This highlights the potential of personalized parameter estimation to guide RAI protocol adaptation. In practical terms, simulations based on estimated parameters could inform clinical decisions, such as advancing or intensifying a second dose in patients with anticipated rebound or redirecting early to alternative therapies when poor response is predicted. While these are theoretical insights, they demonstrate how model-based exploration can support dynamic and individualized treatment strategies. Finally, the similarity in T_g response patterns across different parameters reflects the structure of the model itself: multiple physiological inputs interact to shape a common dynamic output. Clinically, this implies that similar T_g curves can arise from different underlying mechanisms. This reinforces the need for patient-specific parameter profiling to avoid one-size-fits-all treatment strategies and enable personalized therapeutic planning.

3.3. Global Sensitivity Analysis

To evaluate the overall influence of model parameters on treatment dynamics, a Sobol variance-based global sensitivity analysis was performed using synthetic simulations. This approach quantifies how uncertainty in each input parameter contributes to output variability, capturing both main effects and parameter interactions. The analysis considered six physiological parameters: tumor radiosensitivity ($\rho$), initial tumor burden (N₀), thyroglobulin production rate ($\lambda$), T_g clearance rate (k_e), iodine clearance rate ($a$), and administered activity (A₀). Each simulation generated a surrogate measure of treatment response, defined as the area under the T_g curve (AUC), reflecting cumulative tumor activity over time. Parameter combinations were sampled across physiologically plausible ranges using the Saltelli method, and Sobol indices were computed following the standard formulation [36]. The first-order index (S₁) quantifies the direct effect of a parameter on output variance, while the total-order index (ST) captures both its direct and interaction effects.

Table 2. Sobol Sensitivity Indices in Responders and Non-Responders.

Parameters	S1 (Responders)	ST (Responders)	S1 (Non-Responders)	ST (Non-Responders)
N0	0.36	0.53	0.25	0.38
ρ	0.44	0.61	0.12	0.24
ke	0.11	0.22	0.29	0.45
λ	0.07	0.17	0.11	0.21
a	0.02	0.08	0.06	0.15
A0	0.01	0.05	0.05	0.10

summarizes the Sobol indices for responder and non-responder profiles.

In responders, ρ and N₀ dominate both the first-order Sobol indices (direct individual effect on T_g) and the total indices (including parameter interactions). In non-responders, although these indices are less pronounced, ρ and N₀ still exert a measurable direct effect and interact with other physiological parameters influencing T_g. Overall, for both groups, radiosensitivity and initial tumor burden clearly emerge as the main determinants of T_g dynamics. The elimination rate k_e shows a weaker direct and total influence in responders than in non-responders, preventing any generalization of its effect across both populations. In contrast, the parameters λ, a, and A₀ systematically exhibit low sensitivity indices in both cohorts, suggesting a marginal contribution to long-term T_g evolution within the explored parameter space.

These results complement the local sensitivity trends and provide a more rigorous quantitative basis for parameter prioritization during calibration. In clinical applications, such insight supports decisions on which biological variables must be carefully measured and which can be reasonably approximated. While the visual analysis of T_g profiles indicates that k_e, λ, and a influence key trajectory features such as rebound timing and stabilization, variance-based global sensitivity indices confirm that their contribution to cumulative T_g response (e.g., AUC) is limited compared to ρ and N₀. This suggests that although these parameters affect the shape of the response curve, their relative impact on overall treatment efficacy is less pronounced.

3.4. Minimal Parameter Subset for Model Reduction

The combination of local, global, and statistical sensitivity analyses identified ρ and N₀ as the primary drivers of T_g dynamics across both patient groups. These two parameters account for the majority of the model output variance, supporting their role as essential targets for calibration. A reduced model using only ρ and N₀ as free parameters successfully reproduced the key dynamic patterns observed in the full model, including early decay and overall response trends. This conclusion is further supported by ANOVA results, where ρ (p < 10⁻⁴) and N₀ (p = 8.3 × 10⁻⁴) showed statistically significant influence on the model output. However, parameters such as k_e, λ, and a, while contributing less to the total variance, visibly modulate transient features of the T_g trajectory, such as rebound timing and curve steepness. Their ANOVA p-values (k_e: p = 0.12, a: p = 0.41, λ: p = 0.77) confirm their limited statistical effect. Nevertheless, their influence may remain clinically relevant in situations where short-term T_g dynamics impact treatment scheduling. The combined interpretation suggests a hierarchy of importance among the parameters: ρ and N₀ form the core set required for predictive calibration, while k_e, λ, and a influence secondary, short-term features such as rebound timing. However, given their limited impact on global output variance, these latter parameters may be omitted from individual calibration without compromising the model’s predictive utility. This makes the reduced model more amenable for clinical application, particularly when only routine T_g measurements are available.

4. Conclusions and Discussion

This study extended the mechanistic model of Barbolosi et al. [11] for metastatic differentiated thyroid cancer by introducing a quantitative, data-driven framework linking pharmacokinetic–pharmacodynamic modeling, parameter estimation, and numerical simulation. The coupled ordinary differential equations describing iodine decay, tumor burden, and thyroglobulin (T_g) kinetics were solved analytically when possible and numerically using MATLAB. Parameter estimation was performed with the MCMCSAEM algorithm implemented in Monolix, allowing population (and individual) level calibration from a cohort of 50 patients. The combination of these two computational environments ensured consistency between statistical inference and deterministic system analysis.

Local and global (Sobol) sensitivity analyses, complemented by ANOVA testing, identified tumor radiosensitivity ($\rho$) and initial tumor burden (N₀) as the dominant determinants of T_g evolution in both responder and non-responder groups. These two parameters accounted for most of the model variance and define a minimal, clinically actionable set for predictive calibration. A reduced model including only $\rho$ and N₀ successfully reproduced the main T_g dynamics of the full system, while parameters k_e, $\lambda$, and $a$ modulated transient features such as rebound timing and curve steepness with limited global influence. The parameters $\rho$ and N₀ constitute the core calibration set, while k_e, $\lambda$, and $a$ primarily govern short-term response dynamics. In practice, the model can be reduced from the full parameter space θ_full = {T_d, a, $\rho$, N₀, ke, λ, A₀} to a minimal subset θ_reduced = {T_d, $\rho$, N₀} while preserving the main dynamics of the treatment. This reduced formulation defines a tractable dynamic surrogate of the full PK–PD system, retaining the essential therapeutic kinetics.

A priori estimation of these three parameters (ideally by AI-based inference from multimodal patient data) would enable real-time simulation and personalized optimization of RAI treatment. Although the study is limited by the moderate cohort size and the absence of an independent validation dataset, the integrated Monolix–MATLAB workflow demonstrated numerical stability, computational efficiency, and internal consistency. Posterior predictive checks confirmed that simulated T_g trajectories reproduced clinical observations within expected uncertainty bounds, reinforcing the reliability of the calibration procedure. Additional limitations include the assumption of a constant tumor doubling time during treatment and the absence of molecular predictors such as BRAF status in the current model formulation. Despite these constraints, the model captures the essential clinical dynamics of T_g evolution and provides a robust foundation for future extensions toward multi-dose and molecularly informed simulations. Clinically, these findings support the feasibility of patient-specific parameter estimation as a basis for adaptive RAI protocol design, enabling adjustment of administered activity and timing while maintaining compliance with cumulative dose constraints.

Future developments will extend the model to multi-dose regimens, include uncertainty propagation, and integrate the reduced formulation into a clinician-oriented decision-support interface capable of simulating individualized T_g trajectories in real time. Such implementation could ultimately strengthen personalized treatment planning and help achieve an optimal efficacy–toxicity balance in radioactive iodine therapy.