Iterative calibration of medical digital twins via adaptive estimators

While iterative calibration of computational models is a fundamental aspect of digital twins, it has been largely overlooked. Instead of focusing on parameter identification for static models, the implementation of digital twins requires not only high-resolution computational models but also the ability to assimilate patient-specific data continuously. Here, we review adaptive estimator algorithms and how this can address iterative calibration of digital twins, enabling the estimation of unmeasurable states while continuously adapting model parameters. Integrating adaptive estimators into digital twins offers a paradigm shift: transforming them from static representations into living, evolving systems that advance personalized medicine.

1 Introduction

Since their conceptualization in the early 2000s, Digital Twins (DTs) have been built on three fundamental pillars: (i) a virtual model, (ii) a physical counterpart, and (iii) dynamic updating mechanisms that synchronize the first two pillars through continuous data exchange [1–3]. The concept of digital twins was initially implemented for physical industrial products and services [4]. Nowadays, the aforementioned approach is widely applied in many sectors that assist forecasts and minimize emergency behavior by modeling complex systems under a wide range of real-world conditions [3, 5]. For instance, NASA created virtual replicas of spacecraft for design, maintenance, and monitoring [6]. Some companies employ digital twins to enhance energy production [7] or diagnose and anticipate real-world issues in their facilities [8]. Additionally, local and state governments are employing digital twins technology for urban planning and disaster management [9, 10].

Developing digital twins from scratch is a complex task that requires precise modeling and the integration of various data sources. To address these challenges, both industry and academia are focusing on creating comprehensive digital twin platforms. These platforms serve as repositories of ready-to-use or reusable models, data, functions, and tool assets, significantly increasing accessibility for organizations looking to implement digital twin technology on a large-scale and efficient manner [11, 12]. Consequently, numerous digital twin platforms have been proposed in the literature [13], with some aiming to be provided as open-source software [14].

In the medical field, Medical Digital Twins (MDTs) have been envisioned as patient-specific computational models capable of supporting diagnosis, prognosis, and therapy design [15–17]. MDTs offer promising solutions for testing drug efficacy, optimizing treatment plans, and predicting disease progression [17–19]. Some recent applications include modeling Tuberculosis infection [20], modeling virtual patient cohorts during viral infections [21], creating immune system simulation platforms [22, 23], developing pneumonia MDTs to support decision-making processes in intensive care units [24], developing breast cancer MDTs [25], and developing MDTs for leukemia [19].

The workflow for developing MDTs, as illustrated in Figure 1, involves utilizing disease-relevant biological data to establish a generic computational model. This model is then validated using both population and patient-specific data, resulting in a personalized model. This personalized model can then be applied to the prediction and simulation of diseases [19].

Figure 1

Figure 1. Medical digital twin framework. The essential parts for the development of a digital twin, according to the National Academies of Sciences, Engineering and Medicine. A crucial part is that information flows bidirectionally between the physical and virtual counterparts. The human to digital direction allows the recalibrating of the model to make accurate predictions. This allows valuable information for the decision-making process of practitioners to become available. Created in BioRender. Sereno, J. (2025) BioRender.com/zt4sx9x.

1.1 Bottlenecks in calibrating medical digital twins

The boundaries that govern current MDT applications are particularly evident in their dependence on static mechanistic models. These models are typically parameterized once at the beginning of the model identification process using a predefined dataset. However, these computational models remain static for a fixed period of time until a new and sufficiently large dataset becomes available, at which point the model parameter identification procedure is triggered once more. Therefore, while beneficial in providing a foundational understanding of complex systems, current digital twins often fail to keep up with ever-evolving data. Digital twins require continuous model updating that evolves at the same pace as new data becomes available, in order to reflect their physical counterparts accurately.

As stated in Wright and Davidson [26], the ability to dynamically update or adjust the model as new data becomes available is a crucial component that differentiates a digital twin from a conventional mathematical model. Traditional mathematical and statistical models train on available data to produce a fitted curve that can be used for predicting some outcome, but with the addition of new data, the majority of these models need to be refitted on the whole data set. There are a few nuanced exceptions to this, with recursive least squares regression (RLS), which updates model parameters upon the addition of a single data point without refitting against the entire data set, being one such example [27]. However, despite having the ability to update model parameters without refitting the whole model when new data is added, there comes a point when RLS can no longer reliably update model parameters without being refit to the entire dataset again [27]. In essence, if a model needs to be refit to the data at any point in time, it is not an adaptive model, and thus, not a digital twin.

Digital twins applications need to consider data assimilation frameworks that allow for continuous updates and refinements of the mechanistic model. This means going from static to self adaptive systems, which are capable of responding to real-time scenarios. In control theory, an observer is an algorithm that provides an estimate of the state variables of a real system by combining information from system inputs, system outputs, and a model of the system's dynamics [28–30]. Adaptive estimators, on the other hand, offer the ability to estimate unmeasured states while enabling real-time parameter recalibration [31]. Adaptive observers differ from traditional observer designs, such as the Luenberger observer or Kalman filter, by extending the observer paradigm to joint state-parameter estimation. While traditional observers assume that parameters are fixed and known [32], adaptive observers integrate an online parameter adaptation law into their dynamics [33].

Although there are proposals in the literature for extended versions of the Luenberger oberver or Kalman filter aimed at jointly estimating states and parameters [34, 35], these proposals often lack certain desired features, such as the ability to explicitly track unknown parameters [36]. This limitation makes the adaptive observer design more suitable for digital twins. Integrating mechanistic computational models with adaptive observer algorithms would enable the transition from static computational systems to digital twins.

In this review, we present a framework for calibrating digital twins using adaptive observers. The next section focuses on classical observers and their limitations. Then, fundamental concepts for adaptive observers, stochastic filters, and the adaptive digital twin framework are presented. Next, we discuss the applicability of digital twins to viral infections, drug-resistant pathogens, and co-infections. We end with conclusions and relevant insights.

2 Observers and estimators

Reconstructing unmeasured states from partial measurable outputs of a system is a central problem in control theory [30]. Classical observers, like the Luenberger case, ensure asymptotic convergence of the estimated states to their actual values through the design process. It is worth mentioning that an estimator is a more general concept—any algorithm or procedure that produces an estimate of an unknown quantity. This could be a state, a parameter, or even a noise signal. For example, the Kalman filter case is based on a probabilistic estimator that minimizes the mean square estimation error by incorporating both the process and measurement noise statistics [29].

While the Kalman filter was initially developed for linear systems, the Extended Kalman Filter or Extended Kalman Observer extends these concepts to nonlinear systems by linearizing the state dynamics at each time step [30]. In general, the Luenberger observer has a simple implementation and low computational cost, whereas the Kalman filter is more computationally demanding, but offers good performance under noise and uncertainty. The above-mentioned observers typically assume full knowledge of system parameters. When this assumption is not satisfied, their performance can degrade significantly [32].

2.1 Adaptive observers

The idea of combining state observation with parameter adaptation dates back to early works, such as Luders and Narendra [37] and Kreisselmeier [33], where convergence under restrictive conditions was established. Later, in Bastin and Gevers [38], a systematic design approach for single-output nonlinear systems was introduced, employing canonical transformations to simplify the parameter dependence. With these foundational results, works such as Besancon et al. [31] and Zhang [36] extended the framework to time-varying systems and state-affine nonlinear systems, overcoming many of the structural limitations of the earlier approaches. Formally, adaptive observers consider state-affine systems of the form:

$\begin{array}{ll}\dot{x}(t)=A(t)x(t)+B(t)u(t)+\Psi (t)\theta & (\mathrm{1a})\end{array}$

$\begin{array}{lll}y(t)& =C(t)x(t),& (\mathrm{1b})\end{array}$

where $x(t)\in {ℝ}^{n_x}$, $u(t)\in {ℝ}^{n_u}$, and y(t)∈ℝ^m are the states, inputs, and outputs of the system, respectively; matrices A(t), B(t), and C(t) are known time varying matrices; θ∈ℝ^p is the unknown vector of parameters that is assumed constant; and $\text{Ψ}(t)\in {ℝ}^{n_x\times p}$ is a matrix of known signals. Unlike the discussed traditional observers, the adaptive observer design takes advantage of the state-affine form with respect to θ and constructs a coupled scheme. This results in a stabilizing state observer, coupled with a parameter adaptation law [36]. The convergence of adaptive observers rests on two key assumptions: (i) the existence of a stabilizing observer gain K(t) for the nominal system and (ii) persistent excitation (PE) of the regressor signals, which ensures identifiability of the parameters. Under these conditions, adaptive observers achieve global exponential convergence in noise-free settings and robustness in the presence of bounded disturbances.

Adaptive observers differentiate from the classical observers in different aspects as presented in Table 1. First, it is more suitable for both state and parameter estimation. Second, it has achievable design requirements. Third, it has global and exponential convergence when noise is not considered and bounded mean when noise is considered. Fourth, it is robust against bounded disturbances and parameters. Lastly, it has a moderate-to-high computational complexity, depending on the problem dimension, i.e., the considered number of states-parameters to be estimated. Table 1 summarizes the main applications for each type of observer.

Table 1

Table 1. Comparison between Luenberger, Kalman, and Adaptive observers.

Although the adaptive observer was originally developed for joint state-parameter estimation, it is not the sole proposal available in the literature for addressing this task. For instance, in Afri et al. [34], the authors present the theoretical formulation of a nonlinear Luenberger observer approach for state and parameter estimation. This proposal offers a simpler, globally stable alternative to the adaptive observers; however, in this case, parameter estimates are not directly produced, limiting its role to cases in which tracking parameters is not the primary task. In Deng et al. [35] and Van Der Merwe R, Wan et al. [39], extended formulations of the Unscented Kalman Filter (UKF) can be found for joint state-parameter estimation; in these approaches, the augmented state method is implemented.

The aforementioned approaches come with high computational costs, and, moreover, the augmented state methodology increases system dimensions and leads to tuning issues that could result in biased parameter estimates or even filter divergence. For a comprehensive discussion on Gaussian filters used in state and parameter estimation, readers may refer to Afshari et al. [40]. Joint state-parameter estimation algorithms are summarized in Table 2.

Table 2

Table 2. Summary of joint state-parameter approaches.

The adaptive observer, while successfully applied in real settings [41], has two notable drawbacks: (i) its computational demand—this method necessitates the calculation of matrix inverses, and the computational cost escalates as the problem dimension increases; and (ii) its reliance on a continuous form formulation—this limitation becomes evident when implementing this strategy in situations where sparse sensor measurements are prevalent. These challenges highlight the need for alternative approaches that can maintain efficiency and effectiveness in environments with limited data. Exploring new methods that reduce computational complexity or adapt to discrete measurement scenarios could enhance the applicability of observer designs in diverse real-world contexts.

Figure 2 describes our proposal of adaptive observers for digital twins, i.e., the adaptive digital twin framework. In this case, adaptive observer algorithms are used for dynamical updating of model parameters (see the parameter-updating block) at the same pace that new data becomes available and, more importantly, without triggering complete retraining cycles of complex optimization methods for parameter identification—Static vs Continuous. This approach has the potential to enable better monitoring of key biological markers, as well as enhance predictions for virtual drug and therapy testing. This would enable a new paradigm in precision medicine.

Figure 2

Figure 2. Adaptive digital twin framework. The dynamical update of the virtual representation is performed through an adaptive observer algorithm that perform, in a continuous manner, the estimation of unmeasurable states (virtual sensors, state observer block) and the model parameter recalibration (dynamical model update from physical twin data, parameter updating block). Created in BioRender. Sereno, J. (2025) BioRender.com/up1avn5.

2.2 Stochastic adaptive filters

Consider the following stochastic linear differential equation with an unknown vector parameter θ(t) [42]:

$\begin{array}{ll}dx(t)=(A_0(\theta ,t)+A_1(\theta ,t)x(t))dt+b(t)d{W}_1(t),& (2)\end{array}$

with the observation process

$\begin{array}{ll}dy(t)=(C_0(t)+C_1(t)x(t))dt+B(t)d{W}_2& (3)\end{array}$

where x(t₀) = x₀, x(t)∈ℝⁿ is the state vector, y(t)∈ℝⁿ is the linear observation vector such that the observation matrix A(t) is invertible, θ(t)∈ℝ^p, p ≤ n×n+n is a vector of unknown parameters, W₁(t) and W₂(t) are Wiener processes independent of each other and of the initial condition x₀. The estimation problem consists in finding the best estimates for the state vector x(t) and unknown parameters θ(t) by using the observation measurements y(t).

One of the approaches that estimates the state and parameters is presented in Yaakov [43], where the maximum likelihood estimation method is employed to simultaneously estimate the state and parameters for linear discrete-time dynamic systems driven by additive white Gaussian noises. The parameters to be estimated are considered unknown and constants. The whole problem is separated into two interconnected linear problems: one for the state, and the other one for the parameters. Employing the extended Kalman filter (EKF) and the expectation-maximization (EM) algorithms. Togneri and Deng [44] reported the problem of joint state and parameter estimation for stochastic hidden dynamic models with switching states. The idea is to define a new state vector that is composed of the parameters and the state vector, and then conduct several experiments where the EKF or EM algorithms are employed.

Another approach that jointly identifies the state vector and parameters is proposed in Basin et al. [42] for linear stochastic systems. To cope with the reconstruction of state and parameter identification, the unknown parameters θ(t) are modeled as Wiener processes, and then a new state vector that is composed of the state vector and the unknown parameters is defined, z(t) = [^{x⊤(t)θ⊤(t)]⊤}. Finally, a new set of stochastic differential equations with a polynomial form is obtained and the optimal estimate for the new state vector is computed. Some applications of parameter identification are found in Nguyen et al. [45], where the least square method is employed to identify the parameters of a six-DOF manipulator.

Sun and Yang [46] employed the Kalman filter with the Maximum Likelihood (KF-ML) method to handle the joint parameter identification and state estimation problem for a fault-tolerant space robot system. The problem is divided into two stages: the first one employs the Kalman filter algorithm to obtain an estimation of the states that is dependent on the unknown parameter; and the second stage employs the ML technique to estimate the unknown parameter. Finally, the estimated parameter is substituted into the parameterized state estimation to obtain the desired solution. For a review of different methods of adaptive filters, see [40].

3 Developing medical digital twins for infectious diseases

The immune system plays a critical role in the development and progression of numerous significant diseases [17]. Consequently, the development of a reliable software representation of the immune system, known as an Immune Digital Twin (IDT), could significantly impact healthcare and biological research [47]. As expected, there is a large interdisciplinary and international research community focused on developing an IDT [16, 48, 49]. Yet the immune system response is intricate and diverse for various medical conditions and individuals [50], which presents significant technical challenges that require careful consideration.

Accordingly, the integration of adaptive observers into an IDT represents a paradigm shift from static to self adaptive computational models that will help practitioners toward therapy selection, optimal decision making for specific patients, and allow disease prediction.

Currently, there are promising applications of MDTs, such as the living heart project [51, 52] or the artificial pancreas project [53, 54]. This adaptive digital twin framework is expected to broaden these promising applications by enabling systems capable of continuously updating and iteratively reparameterizing, closing the gap between static models and true digital twins. The following subsections demonstrate how adaptive observers can revolutionize personalized medicine across different infectious disease contexts.

3.1 Viral infections

Viral infections pose unique challenges for immune DT implementation due to their rapid mutations and complex interactions with the host immune response. Mathematical models of viral infections have been instrumental in understanding disease progression, but often lack the adaptability required for personalized treatment strategies [55–57]. Monitoring immunological markers can be expensive, and some tests for certain markers are not feasible to perform. The use of adaptive observers can enable the monitoring of immune system responses by using blood samples [58, 59].

3.1.1 HIV infection

For HIV infection, where long-term management and treatment strategies are critical, adaptive observers can predict CD4+ T cell counts, viral reservoir dynamics, and drug resistance evolution [60]. This tracking procedure can be performed by the adaptive observer using patient-specific data pertinent to the HIV infection, such as viral load data and CD4+ T cell counts. Adaptive observers require a dynamical mechanistic model of the HIV infection, in which case it would be possible to estimate and monitor changes in infected cells and reservoirs [61]. These capabilities play an important role for aging HIV patients, where immunological heterogeneity and treatment complications require personalized therapeutic approaches [62]. Medical immune digital twins are pivotal for the evaluation of the long-term outcomes of purging strategies and their effect on the HIV reservoir, which are still largely fragmented [60].

3.1.2 Influenza infection

In influenza infections, adaptive observers can monitor the three stages of the infection process, which are initial viral replication, immune response activation, and either resolution of infection or further complications thereof [63, 64]. In a similar manner, the adaptive framework, coupled with a relevant influenza model and the pathogen load measurements, enables prediction of disease progression, thus allowing for optimal timing of antiviral interventions, which are particularly crucial in elderly patients where immune responses are often impaired [65]. Moreover, recent applications have demonstrated the utility of mathematical models in predicting influenza-specific CD8+ T cell responses, inflammatory responses, and optimizing treatment strategies [59, 66]. Thus, enabling the monitoring of inflammatory responses with adaptive observers could improve diagnosis and prognosis.

3.2 Antibiotic resistance

The emergence of drug-resistant pathogens represents one of the most pressing challenges in modern medicine, with traditional static models inadequately capturing the dynamic evolution of resistance mechanisms [67, 68]. Adaptive observers offer a transformative approach by enabling continuous tracking of pathogen evolution and real-time adjustment of therapeutic strategies.

In the context of antibiotic resistance, adaptive observers can monitor bacterial population dynamics, resistance gene expression, and the emergence of resistant subpopulations during treatment. This capability is essential for implementing collateral sensitivity-based cycling therapies, where the order and timing of antibiotic administration must be carefully optimized to prevent resistance development [69, 70].

Computational approaches have shown promise in developing cutting-edge methods to personalize treatment strategies and mitigate drug resistance evolution [69]. In this scenario, by iteratively updating model parameters based on patient-specific measurements, adaptive observers can predict when current therapeutic approaches may fail and recommend alternative treatment strategies before clinical deterioration occurs [71]. This proactive approach represents a significant advancement over reactive treatment modifications based on observed therapeutic failure. Modeling heterogeneous bacterial populations and predicting the evolution of resistance under different selective pressures provides clinicians with unprecedented insight into treatment optimization [69, 72, 73].

3.3 Co-infections

Co-infections, particularly viral-bacterial combinations such as influenza and Streptococcus pneumoniae, represent some of the most complex scenarios for IDT implementation. The synergistic interactions between pathogens, combined with host immune system modulation, create highly dynamic systems that require sophisticated adaptive control strategies [74, 75].

Alveolar macrophages serve as critical sentinels in respiratory co-infections, with their functional states determining infection outcomes through complex polarization mechanisms between M1 (inflammatory) and M2 (healing) phenotypes [76, 77]. Adaptive observers can track these macrophage dynamics, monitoring key cytokine profiles, such as IFN-γ and IL-6, that determine susceptibility to secondary bacterial infections [78, 79]. This capability of the adaptive framework addresses a major knowledge gap in co-infection modeling, where static representations fail to capture the spatial and temporal modulation of immune cell functions. The adaptive framework enables integration of multi-scale data, from cellular immune responses to tissue-level pathogen dynamics. For influenza-pneumococcal co-infections, adaptive observers can track the transition from viral-mediated immune suppression to bacterial invasion and systemic dissemination [63, 80]. This multi-pathogen adaptive modeling approach provides insights into the timing-dependent effects of therapeutic interventions, enabling optimization of both antiviral and antibiotic therapies.

It is important to mention that most digital twins remain in their early stages and are often far from clinical practice. Up to the author's knowledge, limited medical DT-based systems have been validated in routine clinical practice. A successful example is the artificial pancreas [81]. Heart digital twins have achieved notable advancements, such as the patient-specific cardiac digital twins for ventricular tachycardia ablation guidance [82]. However, these two examples uses are still limited to research settings as a proof-of-concept [83]. Most current MDT applications primarily perform forecasting or retrospective patient-specific simulations, which limit the possibility of a direct, prospective, head-to-head comparison between MDT models. Lastly, MDT proposal reliability and robustness are crucial. Modern MDT pipelines increasingly incorporate uncertainty quantification techniques to explicitly characterize model and data uncertainty and provide more trustworthy, clinically oriented medical digital twins [84].

In the context of adaptive observers, robustness to uncertainty is typically achieved through structural properties of the estimation law, such as input-to-state stability and Lyapunov-based adaptation mechanisms. These features allow the observer to remain stable and provide bounded estimation errors when the underlying model parameters, measurements, or disturbance inputs are uncertain bounded open sets. Additionally, adaptive observer frameworks can be combined with stochastic or set-based uncertainty representations, enabling the estimation algorithm to explicitly account for parameter uncertainties through bounded parameter estimation. Overall, these robustness properties make adaptive observers suitable for medical digital twin implementations, where data quality and physiological variability are inherent sources of uncertainty.

4 Conclusion

Medical Digital Twins hold enormous promise for personalized medicine, but their current reliance on static mechanistic models limits their effectiveness. Digital twins must continuously evolve, assimilating patient-specific data in real time to remain faithful to the biological system they represent. In this work, we propose that adaptive observers provide a natural and powerful framework to achieve this transformation. Unlike classical observers, such as the Luenberger or Kalman filter, adaptive observers jointly estimate unmeasured states and unknown parameters, enabling continuous recalibration of the computational model.

By bridging control theory with biomedical applications, adaptive observers open the door to Medical Digital Twins that are no longer static approximations but are instead dynamic, living systems. Such twins could adapt to disease progression, account for inter- and intra-patient variability, and anticipate treatment resistance. The cases of viral infections, drug-resistant pathogens, and co-infections illustrate the urgency of this capability, where evolving dynamics demand equally adaptive models.

Looking ahead, integrating adaptive observers into MDTs represents more than a technical improvement: it is a paradigm shift. It moves digital twins from being descriptive tools to prescriptive, evolving companions in clinical decision-making. The potential reward is transformative: adaptive observer-driven digital twins could become the cornerstone of a new era in precision medicine.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

JS: Conceptualization, Writing – review & editing, Writing – original draft. HN: Writing – review & editing, Conceptualization, Writing – original draft. MH-G: Writing – original draft. EH-V: Resources, Writing – original draft, Funding acquisition, Supervision, Conceptualization, Writing – review & editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This material was based on work supported by the National Science Foundation under Grant DMS-2439054.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) EH-V declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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