The current study is part of the innovative training network ETUDE (Encompassing Training in fUnctional Disorders across Europe; https://etude-itn.eu/), ultimately aiming to improve the understanding of mechanisms, diagnosis, treatment and stigmatization of Functional Disorders [25].

Experimental data were acquired using an experimental paradigm that is a variation of Read’s rebreathing method [26] and was previously used to investigate, e.g., medically unexplained breathlessness [27], as well as chronic fatigue and fibromyalgia [28]. Participants breathed through a mouthpiece that was connected to a Y-valve behind a visual barrier. The experimenter was located behind the barrier and could let the participant breathe either room air or air from a rebreathing bag. The rebreathing bag was initially filled with a gas mixture of 5% CO2 and 95% O2 (Carbogen, Linde). Due to rebreathing from this closed system, the inhaled CO2 concentration gradually increased leading to hypercapnia and breathlessness.

During the experiment, we recorded CO2 concentration in breathed air (capnograph, Hans Rudolph), peripheral oxygen saturation (pulse oximetry, Nonin Xpod) and respiratory flow rate (pneumotachograph, Hans Rudolph) with a sampling rate of 50Hz. For this study, we calculated single breath data for CO2 concentration. End-tidal CO2 (etCO2) was obtained by taking the maximum CO2 concentration exhaled in each breath. These single breath data were averaged over 10s intervals. Participants were instructed to rate their breathlessness on a scale from 0 (not at all) to 100 (unbearable) every 10s when an auditory cue was presented. They were informed that they would breathe air with different concentrations of CO2 and O2 that can induce either a feeling of breathlessness or no symptoms at all. However, at no point in the experiment, the actual source of breathed air was known to them. The experiment started with a baseline phase, during which participants inhaled room air for 60s. This was followed by a rebreathing phase for 150s and a subsequent recovery phase with room air for another 150s.

We recruited patients at specialized post-COVID clinics in university hospital settings who presented with post-COVID breathlessness not explained by peripheral cardiorespiratory or neurological impairments. All patients needed to provide a PCR test documenting the initial SARS-CoV-2 infection and had to be suffering from post-COVID symptoms for at least 3 months. Data collection for this rebreathing study is still ongoing, but we consider it worthwhile to inform other researchers on our modeling approach using first results. For evaluating the model, we included data from the first 5 patients (mean age ± standard deviation: 34.2 ± 13.7 years, 4 female). Healthy control participants were recruited through the intranet of the Klinikum rechts der Isar, Technical University Munich, as well as through advertisement (flyers) outside of the clinic. For this study, we included 5 healthy controls participants (mean age ± standard deviation: 35.0 ± 15.5 years, 4 female) who were matched by age and gender to the 5 patients.

On the day of the experiment, lung function tests (spirometry and diffusing capacity for CO) and a standardized neurological examination were performed to rule out any organ impairment on that very day. None of the included participants nor patients showed signs of impairment in these exams. In addition, we clinically characterized all participants using the gold standard for making DSM-5 diagnoses, i.e., the Structured Clinical Interview for DSM-5 disorders (SCID-5-CV). Furthermore, we used the patient health questionnaire (PHQ-15), a well-established tool which asks about the presence and severity of common bodily symptoms [29], and asked participants about the presence and severity of breathlessness in everyday life situations.

The study was designed in line with the Declaration of Helsinki, and the Ethics Committee of the Technical University Munich approved the study protocol prior to conduction. Informed consent was obtained from all individual participants included in the study.

The brain is not passively waiting and then reacting to sensory input but rather actively predicting sensory input based on its internal representation how certain body states are generated. Accordingly, our main assumption for mathematical modeling is that the brain holds an internal representation of how bodily states related to breathlessness are changing over time and how these changes are linked to sensorily measurable quantities such as CO2 concentration. In the following, we will refer to the bodily state reflecting the gas exchange between environment, lung and tissue as “internal respiratory state”. We assume that perception of breathlessness reflects potentially dangerous levels of this state, like perception of pain reflects damage to the body. Perception of breathlessness thus represents the brain’s estimate of a respiratory state indicating disequilibrium in gas exchange that may cause dangerous pH levels in the blood.

To construct our mathematical model of breathlessness perception (see Fig. 2, for the equations see Appendix), we first formulated a hypothesis about the brain’s internal representation how the respiratory state will evolve. This internal representation can then be used to form predictions to optimally estimate the internal respiratory state that is not directly accessible to the brain. All our following assumptions for the construction of the model are physiologically informed. For simplification, we assume that the respiratory state can be summarized in a single variable. We further assume that the state varies only slowly from one breath to the next and is influenced by the internal CO2 concentration as well as the current activity context. Walking up a flight of stairs would amount to a high activity context as compared to standing still. Similarly, our rebreathing paradigm can amount to a high activity context. The activity context thus describes the expected influence of an activity on respiratory demands. Importantly, it can be different between individuals. We chose exhaled CO2 concentration per breath as the sensory quantity to update the respiratory state since it is experimentally accessible and can be used to approximate arterial CO2 concentration [30] that is measured by chemoreceptors. Like the internal respiratory state, the exhaled CO2 concentration is assumed to vary only slowly from one breath to the next. Thus, we hypothesize that the current respiratory state evolves from the respiratory state in the last breath and is updated by the sensory CO2 state. This process describes the brain’s internal representation of how a respiratory state is generated.

Model of breathlessness perception (a) and a visualization of the different processing steps (b). Measurement of CO2 concentration in the blood and cerebrospinal fluid (bottom, b5) is noisy and error-prone and thus needs to be combined with a prediction to obtain an estimate of the actual underlying CO2 concentration (orange, solid line in b4). Note that this internal estimate can be different from the actual CO2 concentration and will be used to update predictions about future measurements. Furthermore, the current activity context plays a role (b3). Walking up a flight of stairs leads to a high activity context, which will increase the respiratory state, while resting evokes a low activity context and a lower respiratory state. Note that while the activity context is constant throughout the simulation, its effect (shown in b3) increases and saturates after about 2 min for this participant. The respiratory state describes the current gas exchange between environment, lung and tissue cells and is not consciously accessible. The respiratory state in the last breath is used to predict the current respiratory state and can be updated by the estimated CO2 concentration as well as the activity state. How much the estimated CO2 concentration is taken into account can vary. If the sensory update is taken into account only to a very small extent, the respiratory state is mainly influenced by the prediction based on the last respiratory state and the current activity context. Thus, even though sensory measurements signal an improvement in CO2 levels (b5, in last phase with room air), the respiratory state signaling imbalances in gas exchange may show minor improvement (b2, in last phase with room air). Finally, the respiratory state needs to be translated into the perception of breathlessness (b1). Breathlessness thus reflects an internal respiratory state that signals a potentially dangerous imbalance in gas exchange

For the estimation of the expected respiratory state, the brain needs to combine the measured CO2 concentration with the internal representation described above. Since measurement of the CO2 concentration is noisy and error-prone, the brain also needs to estimate the actual CO2 concentration. For this, the brain forms a prediction based on the internal representation that the CO2 level changes slowly, but randomly, from one breath to the next. This prediction can be combined with the measured CO2 concentration to optimally estimate the actual CO2 concentration. For this estimation process, the framework of Bayes law can be used. It shows that if sensory measurement is precise, the resulting CO2 estimate will primarily rely on the sensory measurement. However, if sensory uncertainty is high, the estimate will more closely reflect the prediction based on the internal representation. As Kalman Filters are generally applied to estimate states evolving over time from noisy measurements, we used this approach to formulate the Bayesian estimation process (for the equations see Appendix). The five free parameters of this estimator, which are considered to be characteristic for each individual, can be computed from the experimental CO2 and breathlessness data from each individual participant. They are (1) the ratio of measurement uncertainty and assumed random changes of CO2 concentration, (2) a weight factor describing how much the CO2 level influences the respiratory state in every breath, (3) a parameter for the assumed activity context, and (4,5) two scaling parameters for the transformation translating the respiratory state into breathlessness perception (formulated as linear transformation comprising an offset and a gain factor).

The resulting estimated breathlessness states from the estimation model were compared to the time course of the actual breathlessness ratings from participants in the experiment. The free parameters were fitted by minimizing least-squares between actual and estimated breathlessness rating using the in-built MATLAB function lsqnonlin.

To evaluate whether the observed breathlessness ratings could also be explained by a simpler model that assumes that breathlessness is a scaled and shifted version of sensory input, we compared our model to a linear regression model of the following form:

$$b = \beta_ + \beta_ * x + \varepsilon$$

with \(b\): breathlessness, \(_\): intercept, \(_\): regression slope, x: CO2 concentration measured in the experiment and \(\varepsilon\): error term.

Furthermore, we tested whether simpler versions of our proposed model can explain breathlessness ratings equally well as the full version. Our proposed model describes the respiratory state as depending on the activity context, the respiratory state in the last breath and an estimate of the internal CO2 level. While sensory input (in this case internal CO2 level) will likely play a role to some extent in every participant, we kept this component but set up two new model variants where we (1) removed the activity context and (2) in another model removed the dependence on the respiratory state in the last breath.

Performance between the different model versions, i.e., (1) the full model, (2) without activity context and (3) without dependence on the last respiratory state and (4) the linear regression model was compared using Akaike Information Criterion (AIC) which evaluates the quality of a model fit while also taking into account the number of parameters and thus the risk of overfitting.