# Expert-guided multi-objective optimization: An efficient strategy for parameter estimation of biological systems with limited data

Léa da Costa-Fernandes, David Bernard, François Pérès, Paul Monsarrat, Béatrice Cousin, Sylvain Cussat-Blanc

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Léa da Costa-Fernandes, David Bernard, François Pérès, Paul Monsarrat, Béatrice Cousin, et al. Expert-guided multi-objective optimization: An efficient strategy for parameter estimation of biological systems with limited data. BioSystems, 2026, 260, pp.105686. 10.1016/j.biosystems.2025.105686. hal-05488553

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# Expert-guided multi-objective optimization: An efficient strategy for parameter estimation of biological systems with limited data

Léa Da Costa Fernandes<sup>a,1</sup>, David Bernard<sup>b,1</sup>, François Pérès<sup>d</sup>, Paul Monsarrat<sup>a,c</sup> <sup>ID,*2</sup>, Béatrice Cousin<sup>a,2</sup>, Sylvain Cussat-Blanc<sup>b,e,2</sup>

## ARTICLE INFO

Dataset link: https://gitlab.com/isicell-irit/expert-guided-multi-objective-optimization

**Keywords:**

Multi-objective optimization

Soft constraints

Model calibration for biology

## ABSTRACT

Calibrating biological models is challenging due to high-dimensional parameter spaces and the limited availability of reliable experimental data. In this study, we propose a hybrid calibration framework that integrates expert knowledge into a multi-objective optimization process. We have evaluated three multi-objective optimization algorithm (NSGA-III, MOEA/D and MO-TPE) with our framework to combine hard constraints derived from biological measurements with soft constraints encoding qualitative domain expertise, such as expected curve shapes or event timing. This dual-constraint strategy guides the search toward biologically plausible parameter sets while preserving flexibility and interpretability. We demonstrate the effectiveness of our method on a benchmark model of skin wound healing, comparing it to standard and unconstrained optimization strategies. Results show that the framework reduces the risk of overfitting to sparse time-course data by favoring dynamically plausible trajectories that satisfy expert-guided soft constraints, increasing the proportion of biologically plausible solutions generated from 1.8% ± 1.3 to 24.3% ± 8.6 for NSGA-III without constraint to NSGA-III with 6 constraints, respectively (p < 0.0001). The framework is flexible, iterative, and generalizable, offering a principled way to leverage domain knowledge for model calibration in complex biological systems.

## 1. Introduction

Computational or in-silico modeling in life sciences employs a diverse array of strategies to capture the complexity of biological systems across multiple scales. They can be broadly categorized into three main families of strategies (Puniya et al., 2024). First, mechanistic or determinist models use mathematical equations — such as ordinary or partial differential equations — to represent and predict the dynamic behavior of biological systems based on known biological mechanisms (Puniya et al., 2024). Second, data-driven models leverage statistical, machine learning, or other artificial intelligence techniques to uncover patterns and make predictions from large-scale biological data, such as genomics or proteomics datasets (Noordijk et al., 2024). Third, hybrid and multi-scale models combine mechanistic and data-driven methods, or integrate different types of models (e.g., agent-based, Boolean, or constraint-based), to capture biological phenomena

across multiple organizational levels-from molecules to whole organisms (Puniya et al., 2024; Noordijk et al., 2024). This enables a more comprehensive and realistic simulation of biological systems and their emergent properties. Together, these general modeling strategies provide complementary perspectives for understanding, predicting, and manipulating complex life science phenomena.

Calibration is a critical step in the development of mechanistic and hybrid models, as it involves estimating the values of unknown parameters to ensure the model accurately reproduces biological data (Palgen et al., 2022). The robustness and predictive accuracy of such models depend strongly on the quality of this calibration process. While some parameter values can be extracted from existing literature, many remain unknown due to model specificity, experimental limitations, or biological variability, some of which are not directly accessible. Moreover, even when literature-derived values are available, they often

<footnote><sup>a</sup> Université de Toulouse, CNRS UMR 5070, INSERM U1301, EFS, ENVT, Institut RESTORE, Toulouse, France</footnote>

<footnote><sup>b</sup> Université Toulouse Capitole, Institute of Research in Informatics (IRIT) of Toulouse, CNRS – UMR5505, Toulouse, France</footnote>

<footnote><sup>c</sup> Dental Faculty and Hospital of Toulouse, CHU de Toulouse, France</footnote>

<footnote><sup>d</sup> Laboratoire de Génie de Production, Ecole Nationale d'Ingénieurs de Tarbes, Tarbes, France</footnote>

<footnote><sup>e</sup> Institut Universitaire de France, France</footnote>

<footnote>* Correspondence to: RESTORE Research Center, 4bis Avenue Hubert Curien, 31100 Toulouse, France.
E-mail addresses: paul.monsarrat@inserm.fr (P. Monsarrat), beatrice.cousin@inserm.fr (B. Cousin), sylvain.cussat-blanc@irit.fr (S. Cussat-Blanc).</footnote>

<footnote n="1">¹ These authors contributed equally to this work.</footnote>

<footnote n="2">² These authors contributed equally to this work.</footnote>
depend on specific experimental conditions or biological contexts that
may not align with the system under study. As a result, direct parameter
estimation from biological data becomes necessary.

However, the acquisition of biological data is often an extensive
and laborious process, constrained by privacy, safety, and ethical con-
siderations. In certain domains, the amount of available data may be
very limited. This is for example the case of medical data, where
the data is generally provided by a medical expert, leading to small
sets of annotated samples (Adadi, 2021). The scarcity of reliable data
significantly impacts the calibration of biological models, reducing
their reliability and predictive capabilities. If not properly addressed,
the conflict between model complexity and data scarcity can yield
physiologically irrelevant parameter values and poor predictions of
internal states. For instance, an experimental animal study on car-
diovascular haemodynamics showed that when models were fitted
to limited measurements (e.g., blood pressure only, without cardiac
output or blood volume data), they exhibited large prediction errors for
unmeasured internal states and greater parameter uncertainty (Tivay
et al., 2020). Similar constraints have also limited the development
of AI models in trauma care to narrow applications such as haemor-
rhagic shock (Peng et al., 2023) and sepsis (Teng and Wilcox, 2020),
which often produce generic rather than patient-specific responses.
In addition, the inherently high dimensionality of parameter spaces
and nonlinear dependencies between parameters and outputs further
compound the difficulty of calibration (Nanda and Kirschner, 2023;
Plank and Simpson, 2024). A poorly identified parameter range not
only increases calibration complexity but also threatens the robustness
and interpretability of the resulting model.

Addressing the limitations imposed by scarce or noisy data requires
the integration of additional sources of information beyond raw mea-
surements. It is important to note that data and expert knowledge
have different statuses and scopes. Experimental data typically retain
validity only within the specific setting in which they were collected,
which limits their generalizability. In contrast, expert knowledge can
often achieve a broader degree of validity and applicability across
contexts (Boissel et al., 2015). Expert knowledge, often qualitative
in nature, can provide critical guidance in parameter estimation. For
example, experts can define expected behaviors, plausible parame-
ter ranges, or characteristic curve profiles for variables. Qualitative
constraints can further compensate for insufficient data by encoding
available biological information, such as trends, causal relationships,
or inequality conditions, when precise quantitative measurements are
lacking. Such qualitative observations — ranging from clinical insights
to phenotypic patterns or genetic knockouts — can be formalized
into logical functions or constraints, enabling parameter identification
even in the absence of complete datasets (Mitra et al., 2018). This
strategy is particularly justified in biology, where complex, dynamic,
and sometimes unquantifiable processes cannot always be captured by
quantitative data alone.

Recent studies have increasingly focused on integrating expert
knowledge to improve biological model calibration. Strategies include
the use of rule-based or logical constraints to encode qualitative trends
or relationships observed in biological systems, hybrid frameworks
combining mechanistic models with domain expertise, and Bayesian
approaches incorporating expert-informed priors to guide parame-
ter inference. These approaches are particularly valuable in contexts
with limited or noisy data, where purely data-driven calibration can
yield non-physiological parameter sets or unreliable predictions. For
example, Mitra et al. formalized qualitative biological observations
into constraints for ODE models (Mitra et al., 2018), Boissel et al.
demonstrated expert-informed parameter estimation in pharmacomet-
rics (Boissel et al., 2021), and Luo et al. applied rule-based constraints
in immune system modeling (Luo et al., 2022).

In this study, we propose a calibration framework that places bi-
ological expertise at the center of the optimization process. Beyond

quantitative agreement enforced through hard constraints, we incor-
porate soft constraints that translate expert knowledge into the search
procedure. These constraints encode qualitative insights such as ex-
pected curve shapes, temporal relationships, or plausible parameter
ranges, ensuring that optimization is guided by biological relevance
rather than data alone. By embedding these expert-driven constraints,
the framework enhances the interpretability and robustness of the
inferred parameter sets, even under conditions of data scarcity. To
demonstrate its flexibility, this approach was applied in combination
with three different multi-objective optimization algorithms (multi-
objective Tree-structured Parzen Estimator— MO-TPE (Ozaki et al.,
2020), Non-dominated Sorting Genetic Algorithm III—NSGA-III (Zhang
and Li, 2007), and Multi-Objective Evolutionary Algorithm based on
Decomposition—MOEA/D (Zhou et al., 2012)) on a literature-based
model formulated as a system of ordinary differential equations, show-
ing how domain expertise can effectively steer parameter calibration in
complex biological systems.

## 2. Materials and methods

As illustrated in Fig. 1, The framework consists of six main steps: (1) collecting expert knowledge about the expected dynamics of the biological process under study and converting it into soft and hard constraints; (2) selecting a range for each parameter to define the search space; (3) defining a cost function to evaluate the error between biological and simulated data; (4) selecting an appropriate optimization algorithm; (5) retaining optimal solutions from the Pareto front that minimize the cost function; and (6) visualizing the dynamics of model simulations. According to the consistency of the obtained dynamics, the process is iterative and soft and hard constraints can be further modified. In summary, this strategy is designed to ensure that a biologically plausible solution is obtained by having the algorithm guided through the incorporation of all available knowledge-data, observations, and hypotheses.

### 2.1. Multi-objective optimization algorithms

We evaluated three state-of-the-art multi-objective optimization al-
gorithms (MOTPE, NSGA-III, and MOEA/D) within our expert-guided
calibration framework. Each algorithm is designed to identify trade-
offs between conflicting objectives by constructing a Pareto front of
non-dominated solutions. MOTPE (Ozaki et al., 2020) is a Bayesian
optimization approach that models the objective functions with prob-
abilistic surrogates, making it well suited for problems with expensive
evaluations. NSGA-III (Zhang and Li, 2007) extends NSGA-II (Deb et al.,
2002) by using a set of reference directions to maintain diversity
across many objectives, improving its scalability. MOEA/D (Zhou et al.,
2012), in contrast, decomposes the multi-objective problem into a set of
scalar optimization subproblems that are solved in parallel, facilitating
efficient exploration of the search space. Each algorithm is run 12
independent times with the different number of constraints defined
in the next section. We used the implementations of these algorithms
provided of the Optuna library (Akiba et al., 2019). The multi-objective
optimization algorithms were configured with the parameters listed
in Supplementary Table 1, which values were selected based on the
recommendations in the Optuna documentation. Any parameters not
specified in the table were left at their default values.

In the context of our framework, all three algorithms optimize
parameter sets with respect to both hard and soft constraints. Hard
constraints quantify the agreement between model simulations and
empirical data, using measures such as mean squared error (MSE) or
mean absolute error (MAE). Soft constraints encode qualitative expert
knowledge of the system, such as expected curve profiles, temporal
dynamics, or biologically plausible parameter ranges. By combining
quantitative and qualitative information, the optimization process is
steered toward parameter sets that are both statistically accurate and
biologically meaningful.
<figure><img src="image_2.png" /><figcaption>Fig. 1. General framework of the expert-guided multi-objective optimization strategy for parameter estimation of biological systems.</figcaption></figure>

## 2.2. Hard constraints definition

Hard constraints represent the target dynamics that the model must reproduce based on biological data specifically acquired for calibration purposes. As these data are considered ground truth, it is crucial for the model to fit the provided measurements as closely as possible. To address multi-objective optimization, we propose an alternative approach inspired from the MSE metric, defining a separate objective function for each kinetic profile. This strategy avoids the need for subjective weighting coefficients, which can be difficult to determine reliably. Furthermore, aggregating multiple kinetics into a single objective requires prior normalization to ensure that all profiles contribute comparably to the optimization process. By treating each kinetic profile as an independent objective, normalization becomes unnecessary, allowing errors to be computed and evaluated within their natural scales:

$$err = \frac{1}{n} \sum_{i=1}^{n} \left| \frac{y_i - \hat{y}_i}{y_{\max}} \right|^2 . \qquad (1)$$

Here, $y_i$ denotes the biological values, $\hat{y}_i$ the simulated values, $n$ the number of measurements and $y_{\max}$ the maximum value observed for each biological dynamic. This customized error metric ensures each error term is normalized by the maximum biological value for its respective dynamic. This normalization allows differences across profiles to be compared on a consistent scale.

## 2.3. Soft constraints definition

In addition to the commonly used hard constraints reported in the literature, we propose improving the optimization process by incorporating soft constraints that encode expert knowledge about the expected system dynamics. These soft constraints may capture features such as the timing and amplitude of peak values, the sequence of events, or phase-specific characteristics of kinetic profiles. As this strategy is intended to be broadly applicable across different biological systems, the corresponding fitness functions must be carefully tailored to reflect the relevant expert insights for each specific context.
Formally, the fitness function $f_{soft}(v_s)$ of a soft constraint can be defined as follows:

$$f_{soft}(v_s) = \begin{cases} 0 & \text{if } v_s \in r_a \\ v_s - u_{r_a} & \text{if } v_s > u_{r_a} \\ l_{r_a} - v_s & \text{if } v_s < l_{r_a} \end{cases} \qquad (2)$$

where $v_s$ is the simulated data and $r_a = [l_{r_a}, u_{r_a}]$ is the “acceptable range” proposed by the biological expert. The objective is to identify parameter sets that best satisfy the soft constraints, while using the hard constraints to fine-tune the parameters.

Soft constraints are often difficult to define precisely. An iterative approach is proposed, in which new constraints are progressively refined between expert-guided successive optimization loops. It is argued that the analysis of the Pareto front obtained after each loop of optimization can be used to inform the refinement of soft constraints, thereby guiding the optimization process toward increasingly biologically plausible solutions. In our case study, the transition from no constraints to 2 and 6 constraints illustrates a typical iterative refinement step, where expert knowledge is progressively translated into additional soft constraints. By continuously refining or introducing additional soft constraints, this mitigates the tendency of genetic algorithms to become trapped in local optima, allowing more biologically relevant parameter sets to be identified. To avoid overly constraining the optimization process — which could reduce the algorithm’s ability to explore the parameter space — we recommend defining soft constraint functions using acceptable value ranges, considering that (i) when simulated values fall within the “acceptable range” defined in collaboration with the biological expert, the corresponding fitness value is minimal (as required by minimization algorithms); (ii) as the simulated values deviate from the nearest boundary of this range, the fitness value increases linearly.

## 2.4. Define the search space for each parameter

During the optimization step, it is essential that constraints with upper and lower bounds be established for each parameter. These constraints are used to restrict the search space to reasonable ranges, thereby reducing computational time. They should ideally be defined based on available knowledge so that the algorithm is effectively guided toward biologically plausible solutions and prevented from exploring unrealistic regions.

## 2.5. Case study

To evaluate the performance of the proposed strategy, we considered a benchmark model from the literature (de Oliveira et al., 2021). This model consists of five coupled ordinary differential equations (ODE) describing the skin wound healing process in rats over 50 days. The dynamic variables include overtime active neutrophils ($a(t)$), pro-inflammatory mediator (IL-6, $c(t)$), anti-inflammatory mediator (IL-10, $g(t)$), macrophages ($m(t)$), and apoptotic neutrophils ($n(t)$). The simulation was performed under the experimental condition involving 10% oil-resin. The mathematical equations are given by the following ordinary differential equations:

$$\left\{
\begin{aligned}
\frac{dn}{dt} &= \frac{\chi_n c}{1 + \frac{g}{\beta_{gc}}} - v \frac{1 + \frac{g}{\beta_g}}{1 + \frac{c}{\beta_c}} n, \\
\frac{da}{dt} &= vn \frac{1 + \frac{g}{\beta_g}}{1 + \frac{c}{\beta_c}} - \gamma_a a - \phi ma, \\
\frac{dm}{dt} &= \chi_m c - \gamma_m m, \\
\frac{dc}{dt} &= \alpha F(t) + k_n \left( \frac{n^2}{\beta_n^2 + n^2} \right) + \gamma_a k_a \left( \frac{a^2}{\beta_a^2 + a^2} \right) - \gamma_c c + k_m \left( \frac{m^2}{\beta_m^2 + m^2} \right), \\
\frac{dg}{dt} &= k_g \phi ma - \gamma_g g.
\end{aligned}
\right.$$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Benchmark model of skin wound healing: reproduction of simulation results from de Oliveira et al. (2021). The model simulates the kinetics of active neutrophils ($a(t)$), IL-6 ($c(t)$), IL-10 ($g(t)$), macrophages ($m(t)$) and apoptotic neutrophils ($n(t)$) under the experimental condition of 10% oil-resin.</figcaption></figure>

For each variable, at least one validated parameter set is available, allowing us to reproduce the original simulations and use them as a reference for evaluation (Fig. 2). Supplementary Table 2 lists all model parameters to be optimized together with the original values from de Oliveira et al. (2021) used to define the ranges of the parameters for the optimization algorithms.

The objective of this case study is to assess whether present calibration method can recover biologically plausible parameter sets capable of reproducing the benchmark simulation dynamics. Among the variables, only IL-6 ($c(t)$) and IL-10 ($g(t)$) are associated with real biological data, each comprising four time points. The optimization thus focuses on fitting these two dynamics. Thus the optimization process used the error function previously defined in Eq. (1), computed independently for $c(t)$ and $g(t)$. The goal was to minimize the error for each kinetic profile.

Each soft constraint was implemented as an independent objective function, resulting in a multi-objective optimization problem: two for fitting biological data ($c(t)$ and $g(t)$, hard constraints), and several qualitative biological constraints (soft constraints). For soft constraints, only solutions that exactly satisfied the constraint (i.e., with an error equal to zero) were retained. This approach efficiently guides the optimization toward biologically valid parameter sets.

Final validation of the estimated parameters was based on the model’s ability to reproduce the full dynamics of the benchmark simulation, not just the data-associated variables. This ensures that the solution is not only data-compliant, but also biologically interpretable and dynamically consistent.

# 3. Results

## 3.1. Defining the constraints based on expert knowledge and literature

Soft constraints were derived from the dynamics described for model 1 in de Oliveira et al. (2021) and Gushiken et al. (2017), and grouped into two categories: time-based and amplitude-based constraints. Time-based constraints are the following:

T1: The peak of macrophage recruitment ($m(t)$) must occur between day 9 and 11;

T2: The peak of active neutrophil recruitment ($a(t)$) must occur before the macrophage peak;

T3: The active neutrophil peak must be transient and occur between day 3 and 8.5.

Amplitude-based constraints are the following:
A1: The macrophage peak must range between $0.9 \times 10^4$ and $2.0 \times 10^4$ cells/mm³;

A2: The neutrophil peak must represent approximately 20%–80% of the macrophage peak;

A3: The inflammatory response should be resolved by day 35, with the mean value of all dynamic variables remaining below 10 beyond this point.

These constraints are based on established biological knowledge of wound healing. The inflammatory phase involves a well-characterized sequence of immune cell recruitment, with neutrophils being the first responders, followed by macrophages (Gurtner et al., 2008; Wilkinson and Hardman, 2020; Raziyeva et al., 2021). Neutrophil recruitment is known to be driven by pro-inflammatory cytokines such as IL-6 (Gushiken et al., 2017), which reaches its peak around day 7 in de Oliveira's model. Macrophage infiltration follows, becoming the dominant cell population during the transition to tissue remodeling (Koh and DiPietro, 2011; Kotwal and Chien, 2017; Shook et al., 2016; Joshi et al., 2020). The end of the inflammatory response is marked by a reduction of the number of pro-inflammatory cells, such as neutrophils, a decrease in the concentration of pro-inflammatory mediators, and a production of anti-inflammatory mediators such as IL-10. These biological insights justify the soft constraints applied in our framework. However, the experimental data on which de Oliveira's model is based is limited to the quantification of pro- and anti-inflammatory cytokines (Gushiken et al., 2017). This may be indicative of why the soft constraints on cell type amplitude and timing do not correspond to the literature.

## 3.2. Description of the benchmark model under study

The benchmark model was used to evaluate the performance of the proposed constraint-guided optimization strategy under three conditions: no soft constraints (WoC), two out of the six defined soft constraints (2C) and six soft constraints (6C). In the latter case, the two selected constraints corresponded to the timing of peak recruitment for macrophages (constraint T1) and neutrophils (constraint T2). Temporal features are well established in the literature on wound healing. To assess the influence of the parameter search space, experiments were conducted with different levels of constraint relaxation, simulating conditions of increasing uncertainty. The initial search space was thus defined by applying a multiplicative factor of 2, 10, 50 or 100 around the nominal values reported in Oliveira et al. (e.g., for a parameter $v = 0.3$, the interval was [0.003, 30] with a multiplicative range factor of 100). This corresponds to widening log-scaled supports, a standard practice in systems-biology calibration where parameter values commonly span multiple orders of magnitude (Renardy et al., 2019). No prior knowledge on the true parameter values was assumed.

With this experimental setup, we compared three multi-objective optimization (MOO) algorithms: NSGA-III, MOEA/D, and MO-TPE. For each configuration (combination of constraint level, range factor, and optimization algorithm), 12 independent runs were performed, leading to 432 runs. To limit the computation cost, a 3-hours time limit was used for each run.

## 3.3. NSGA-III is the most efficient MOO algorithm of the benchmark

NSGA-III outperformed the both MOEA/D and MO-TPE in terms of the number of explored solutions (Supplementary Figure 1). It was able to explore a broader range of solutions and demonstrated scalability with respect to the number of constraints, showing no significant reduction in the number of trials as the number of constraints increased. In addition, both NSGA-III and MOEA/D proved robust to variations in parameter range size, maintaining their exploration capacity even as the search space expanded. In contrast, MO-TPE produced significantly fewer trials than NSGA-III and MOEA/D, indicating lower efficiency in optimizing the parameters of this model. Nevertheless, MO-TPE may

<caption>**Table 1**</caption>

Statistical comparison of soft-constraint validation across experimental settings. Proportion of biologically plausible solutions obtained with NSGA-III with six constraints (6C) versus either no constraints (WoC) or two constraints (2C), across different parameter-bound settings. An analysis of variance (two-way ANOVA) followed by Tukey's post-hoc multiple comparison test was performed to test for statistical differences. Statistical Shared letters in superscript within a same row indicate significant difference between conditions at the *p*-value 0.01 level. 12 independent runs were performed.

<table><thead><tr><th>Range factors</th><th>NSGA-III-WoC</th><th>NSGA-III-2C</th><th>NSGA-III-6C</th></tr></thead><tbody><tr><td>2</td><td>5.7 ± 3.4<sup>A</sup><br>(CV 0.59)</td><td>21.8 ± 10.3<sup>A</sup><br>(CV 0.47)</td><td>28.1 ± 7.2<sup>A</sup><br>(CV 0.26)</td></tr><tr><td>10</td><td>1.8 ± 1.3<sup>A</sup><br>(CV 0.69)</td><td>11.3 ± 6.6<sup>A</sup><br>(CV 0.58)</td><td>24.3 ± 8.6<sup>A</sup><br>(CV 0.36)</td></tr><tr><td>50</td><td>0.50 ± 0.67<sup>B</sup><br>(CV 1.35)</td><td>3.9 ± 2.1<sup>A</sup><br>(CV 0.54)</td><td>15.3 ± 3.4<sup>A,B</sup><br>(CV 0.22)</td></tr><tr><td>100</td><td>0.17 ± 0.58<sup>B</sup><br>(CV 3.46)</td><td>3.0 ± 3.1<sup>A</sup><br>(CV 1.03)</td><td>8.7 ± 4.2<sup>A,B</sup><br>(CV 0.48)</td></tr></tbody></table>

still offer advantages in contexts where the computational cost of model evaluation is high, such as in agent-based modeling. NSGA-III also show higher performance in terms of proportion of satisfied constraints (Supplementary Figure 3). Consequently, all main results presented in the paper were generated using NSGA-III, while the results obtained with MOEA/D and MO-TPE are provided in the Supplementary Material (Supplementary Figures 1-6).

## 3.4. A trade-off is necessary between data-oriented optimization and expert-guided optimization

A Pareto fronts obtained under the NSGA-III-WoC condition outperform those generated with NSGA-III-2C and NSGA-III-6C in terms of data-oriented hard constraints (Fig. 3A). This highlights the need for a trade-off between solutions that optimize data-oriented objectives (hard constraints based on biological data) and those that satisfy expert-driven soft constraints.

The Pareto fronts obtained under different constraint settings and parameter ranges were then evaluated in terms of soft-constraint validation. The goal was to assess how well the parameter solutions agreed with expert knowledge. Fig. 3B. shows the number of constraints satisfied by each point on the Pareto front across the 12 independent runs. The results clearly indicate that incorporating expert constraints leads to a higher proportion of solutions consistent with the theoretically modeled phenomena, regardless of the parameter search ranges (Table 1). Overall, our expert-guided approach increased the proportion of biologically plausible solutions generated by NSGA-III from 1.8% ± 1.3 to 24.3% ± 8.6 for NSGA-III-WoC to NSGA-III-6C, respectively (*p* < 0.0001).

## 3.5. Data-only optimization often produces biologically implausible dynamics

An essential step in parameter selection involves visualizing the kinetics of each variable to ensure biological consistency. To this end, the best solutions from the Pareto front of each replicate run were retained, representing the optimal trade-offs between the calibration objectives $c(t)$ and $g(t)$.

When the dynamic profiles were evaluated against the expert constraints, most of the solutions were found to be inconsistent with the benchmark simulations presented in Oliveira's work for NSGA-III-WoC (Fig. 4). The corresponding parameter sets failed to accurately reproduce the wound healing dynamics. This discrepancy was especially evident in the expanded search space setting, where overfitting to the limited biological data ($c(t)$ and $g(t)$) was likely to have occurred. In particular, oscillatory behavior was observed across both
<figure><img src="image_4.png" /></figure>

B

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** A: Pareto front minimizing the two objective functions *c*(*t*) and *g*(*t*). The performance of NSGA-III with 2 and 6 constraints (NSGA-III-2C and NSGA-III-6C) was compared to the optimization strategies NSGA-III without constraints (NSGA-III-WoC) on different search spaces. A range factor 2, 10, 50 and 100 was applied to the parameter ranges. B: Number of constraints satisfied by final solutions of the Pareto front. NSGA-III-6C find more solution that satisfy all six constraints in comparison to NSGA-III-WoC and NSGA-III-2C. 12 independent runs were performed.</figcaption></figure>
restricted and expanded search spaces (Fig. 4, Supplementary Fig. 4). Such oscillations are biologically implausible and suggest instability in the simulated system. These anomalies may result from challenges in estimating parameters in the presence of numerically stiff terms and a high-dimensional parameter space. Oscillatory dynamics suggested the presence of attractive local minima in a rugged fitness landscape. Although earlier results showed that NSGA-III-WoC yielded better Pareto front solutions in terms of objective values, the present findings confirm that the corresponding parameter sets produced dynamics inconsistent with biological expectations. The issue was exacerbated in the largest search space condition (bounds 100), which produced a significantly higher number of activated neutrophils compared to macrophages—an outcome that contradicts established biological understanding. Interestingly, these oscillatory behaviors were eliminated with the inclusion of only two soft constraints and were further refined when all six expert-defined soft constraints were applied. If such behaviors had not been captured, it would have been possible to design an additional soft constraint specifically aimed at penalizing oscillations—for example, by counting the number of peaks in the dynamic profiles and minimizing this number.

Taken together, these results show that integrating expert-guided soft constraints into the calibration reduces the risk of overfitting to sparse data by favoring dynamically plausible trajectories rather than merely minimizing error at a few sampling times.

## 4. Discussion

Calibrating mechanistic models in biology remains a challenging task, particularly in data-limited settings where parameter identifiability is compromised and overfitting to sparse measurements can lead to physiologically implausible solutions. In this study, we demonstrated that integrating expert-guided soft constraints into a multi-objective optimization framework improves the biological plausibility of simulated dynamics. While solutions optimized solely against biological data often achieved better scores on data-oriented objectives, they frequently generated unrealistic system behavior, such as oscillatory dynamics or incorrect relative abundances of key cell populations. These findings underscore that optimal agreement with available measurements does not necessarily guarantee biologically meaningful solutions—a trade-off that is especially pronounced in high-dimensional models under limited data availability.

The integration of expert-guided qualitative constraints for model calibration offers substantial benefits that enhance the calibration process beyond purely quantitative optimization (MacNamara et al., 2012; Voit, 2023). Expert knowledge provides qualitative constraints or constraints — such as expected behavioral patterns, or known system dynamics — that help steer the NSGA-III search toward more meaningful regions of the parameter space. This guidance reduces the risk of converging to parameter sets that fit data numerically but lack biological realism (Efstratiadis and Koutsoyiannis, 2010; Mai, 2023). Constraint specification may also be formalized through structured expert-consensus procedures (e.g., Delphi panels, RAND/UCLA appropriateness methods) where appropriate, tailoring the process to the specific biological question and context. Our results further indicate that expert-derived constraints can act as an effective regularization mechanism, steering the search process toward parameter sets that respect fundamental biological principles even in the presence of wide and uncertain parameter ranges. This effect was observed consistently across different multi-objective algorithms, demonstrating that the benefits of expert guidance are not tied to a particular optimization strategy. This multi-constraints approach also facilitates the detection of prominent behavioral solutions that satisfy both quantitative and qualitative expectations, leading to more robust and interpretable calibrated models, narrow the Pareto front to solutions that are both statistically sound and scientifically plausible thus improving model generalizability, reduce uncertainty and facilitates interpretability (Efstratiadis

and Koutsoyiannis, 2010; Mai, 2023). In our setting, overfitting is therefore assessed at the level of system dynamics: by enforcing expert-guided soft constraints, the framework reduces the risk of overfitting to sparse measurements by favoring trajectories that remain biologically plausible over time, rather than relying on formal cross-validated generalization metrics.

In this study, we employed NSGA-III, an efficient algorithm for multi-objective optimization, better suited to capture a broader and more diverse Pareto front in more than 4 objectives (Deb and Jain, 2014), decomposition-based approaches such as MOEA/D (Multi-Objective Evolutionary Algorithm Based on Decomposition which divides problems into scalar subproblems for parallel optimization Zhang and Li, 2007), and MOTPE (Ozaki et al., 2020). Other algorithms would have been possible such as GDE3 (Generalized Differential Evolution 3, combining differential evolution with Pareto dominance Vargas et al., 2021), hybrid stochastic-gradient methods such as spMODE-II (integrating differential evolution with sensitivity analysis Reynoso-Meza et al., 2014), particle swarm and metaheuristic variants such as MOPSO (Multi-Objective Particle Swarm Optimization, using particle swarms for exploration Coello Coello and Lechuga, 2002). Although NSGA-III exhibited better exploration of the Pareto front and higher compliance with expert constraints compared to MOEA/D and MOTPE, the improvements in biological plausibility (consistency with existing biological knowledge and scientific understanding) were primarily driven by the presence of soft constraints rather than by the choice of the optimization algorithm. Therefore, we report computational overhead in terms of algorithm-agnostic quantities (number of trials and constraint satisfaction) which are comparable metrics between all three tested algorithms. While this strategy enables the generation of calibrated models that capture key biological dynamics, model validation and structural identifiability remain crucial. A formal global sensitivity analysis of soft-constraint ranges and weights will be an important next step to quantify the robustness of optimization outcomes to these user-defined choices (Raue et al., 2009; Fröhlich et al., 2017), but this lies beyond the scope of the present work. These analyses may also inform the design of new biological experiments aimed at constraining the most sensitive parameters. To effectively explore the range of solutions that a model can produce, flexible constraint definitions are essential. The parameters obtained should be further validated *in vitro* with new biological data.

These findings position our framework within a growing body of work aimed at incorporating expert knowledge into model calibration. Bayesian approaches have traditionally been used prior distributions informed by expert opinion to regularize parameter estimation, offering a probabilistic framework but requiring extensive prior elicitation and being sensitive to prior misspecification. Fuzzy logic and rule-based approaches provide a natural means of encoding qualitative trends and inequality constraints but can be difficult to integrate with multi-objective optimization and often lack flexibility in adapting to new data or different models. In addition, they typically require strong assumptions on the form of prior distributions or ad hoc rule encoding. By contrast, our approach formalizes expert knowledge as soft constraints within a multi-objective framework, balancing data fidelity and biological plausibility while remaining agnostic to the choice of optimization algorithm. This flexibility is particularly valuable in applications where expert knowledge is inherently qualitative, sparse, or system-agnostic. Future work could include hybridizing our approach with Bayesian or fuzzy methods to combine their respective advantages.

Despite these advantages, several limitations of the proposed framework should be acknowledged. First, the formulation of soft constraints relies on the availability of expert knowledge and may introduce biases if such knowledge is incomplete or uncertain. Second, the evaluation of constraints during optimization adds computational overhead, particularly in high-dimensional parameter spaces or when model simulations are expensive. It remains to be seen how well the strategy
A

<figure><img src="image_6.png" /></figure>

B

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Predicted wound healing dynamics under each optimization strategies (NSGA-III-WoC, NSGA-III-2C and NSGA-III-6C). These plots show the dynamics of parameter sets on the pareto front. A range factor 2 (A) and 100 (B) was applied to the parameter ranges. Factors 10 and 50 can be found as Supplementary Fig. 4. Dynamics obtained with MOEA/D and MO-TPE are available in Supplementary Figures 5 and 6 respectively. 12 independent runs were performed.</figcaption></figure>

scales to more complex systems involving dozens of equations, intri-
cate regulatory interactions, feedback loops, or time delays—features
commonly encountered in biomedical field. Third, while the framework
improves the plausibility of inferred dynamics, it does not guarantee
global identifiability of the underlying parameters. Addressing these
challenges will be important for extending the framework to more
complex biological systems.

Future work could focus on three main directions. One is to de-
velop systematic methodologies for eliciting and formalizing expert
knowledge, for example through structured interviews or probabilistic
rule extraction, to reduce potential biases and ensure reproducibility.
Another is to integrate adaptive constraint weighting strategies that

balance the influence of data-oriented objectives and expert knowledge
as optimization progresses. A more systematic formalization — through
probabilistic encoding or expert-driven annotation — could enhance
the reproducibility and robustness of the approach. Future extensions
could include the integration of automatic validation tools, such as
dynamic pattern classifiers (Gilpin et al., 2020), to objectively evaluate
the plausibility of simulated trajectories. Finally, coupling the proposed
approach with surrogate modeling or emulator-based methods could
substantially reduce computational costs, enabling its application to
larger-scale and more computationally demanding biological models.
Overall, this study demonstrates that expert-guided soft constraints
provide a principled and practical means of improving the biological
relevance of calibrated model dynamics, complementing existing approaches and mitigating a key limitation of data-driven optimization in biology: the risk of overfitting to sparse or noisy data at the expense of theoretical and biological consistency.

## 5. Conclusion

We introduced an expert-guided multi-objective optimization framework that combines quantitative data with qualitative biological knowledge to calibrate mechanistic models. Using NSGA-III, our approach encodes expert constraints as soft constraints, improving convergence towards biologically plausible parameter sets, even under data scarcity or wide search spaces. Comparative evaluations showed that unconstrained optimization, while achieving lower numerical errors, often produced biologically implausible dynamics. In contrast, expert-informed calibration acted as an effective regularizer, filtering out unrealistic behaviors. The iterative integration of qualitative constraints allowed for progressive refinement of the search space, offering a practical strategy to guide the optimization towards coherent solutions. While demonstrated on a moderately complex benchmark, the framework is extensible to more intricate models. Future work will focus on scaling the method, formalizing expert input, and integrating it into semi-automated calibration pipelines. By leveraging expert knowledge in addition to sparse data, this approach supports the development of reliable models in contexts where data alone is technically complex to acquire.

### CRediT authorship contribution statement

**Léa Da Costa Fernandes:** Writing – review & editing, Writing – original draft, Visualization, Methodology, Investigation, Conceptualization. **David Bernard:** Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **François Pérès:** Writing – review & editing, Funding acquisition. **Paul Monsarrat:** Writing – review & editing, Writing – original draft, Validation, Formal analysis. **Béatrice Cousin:** Writing – review & editing, Validation, Supervision, Funding acquisition, Conceptualization. **Sylvain Cussat-Blanc:** Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Formal analysis, Conceptualization.

### Code availability

https://gitlab.com/isicell-irit/expert-guided-multi-objective-optimization

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work was supported by the Occitanie Region, the Federal University of Toulouse Midi-Pyrénées (grant ADI, N° ALDOCT-000689/19008724). This study has been partially supported by the Agence Nationale pour la Recherche, through the grant EUR CARe N° ANR-18-EURE-0003, the grant N° ANR-24-CE45-5810, the grant N° ANR-22-CE18-0020, the grant N° ANR-24-CE45-5810 and the national infrastructure “ECELLFrance: Development of mesenchymal stem cell based therapies” (PIA-11-INBS-0005) in the framework of the Programme des Investissements d'Avenir.

## Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.biosystems.2025.105686.

### Data availability

https://gitlab.com/isicell-irit/expert-guided-multi-objective-optimization.

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