Post-COVID breathlessness: a mathematical model of respiratory processing in the brain

Persistent breathlessness is estimated to affect more than 25% of patients after COVID-19 [1]. While some patients present with impaired lung function and carbon monoxide diffusing capacity [2], others have neither measurable pulmonary [3, 4] nor cardiac impairments [5] despite profound breathlessness. In general, there is only a moderately strong relationship between peripheral organ dysfunction and patients’ breathlessness, and a considerable number of patients lack any measurable organic symptom correlate [1]. Recently, concepts based on the processing of respiratory information in the brain have been developed that describe how persistent breathlessness that is not sufficiently explained by organ dysfunction could manifest [6‐10]. These concepts highlight that perception of symptoms occurs in the brain, even if the initial cause resides in body periphery, and that symptoms can be just as authentic and disabling when peripheral organs are intact, but information relayed from sensors to the brain is misprocessed. Therefore, investigating how bodily signals are processed in the brain should be an integral part of the search for possible disease mechanisms in addition to the examination of peripheral organ impairments.

The environment around us is constantly changing. To keep the body in homeostasis, the brain must monitor all relevant processes in the body and adjust them as soon as they exceed setpoints such as a certain core temperature, pH or glucose level [11]. In the case of breathing, different receptors signal information about lung mechanics, cardiac function, carbon dioxide (CO₂) concentration and pH levels in the blood. The brain needs to measure and integrate these signals to obtain information about the current respiratory state, i.e., the gas exchange between the environment, lungs, and tissue cells [12, 13]. This involves two problems: (1) Sensory information coming from receptors is inherently noisy which makes sensing of a signal prone to errors. (2) Sensory input always follows an event and consequently is delayed. Therefore, reactive control of bodily states will often be too slow leading to over- or undershooting the desired setpoint, which could, e.g., in the case of pH levels, be life-threatening.

This implies that in many scenarios, reactive control mechanisms will not be sufficient. Conversely, it is crucial that the brain predicts deviations from setpoints in advance and adjusts breathing in anticipation of actual changes, e.g., in pH. To predict future changes of bodily states, the brain needs to form an internal representation that describes how such bodily states evolve over time (see Fig. 1). This internal representation is often called an internal model. It needs to be dynamically adapted based on newly available information. For example, changes such as increased lung ventilation due to training, or decreased lung function due to disease (as in Fig. 1b) need to be incorporated. This means that the internal representation is built from past experiences. Based on these, predictions can be developed (see Fig. 1) to handle noisy measurements and obtain an optimal estimate of the underlying body state [9, 14, 15]. This is comparable to driving on a familiar road at night: even if visibility is poor, our knowledge of how the road is developing improves our perception and makes driving easier. In a similar way, the brain obtains an optimal estimate of the actual underlying body state from the combination of sensory input and prediction. The relative contributions of noisy measurements and predictions (see Fig. 1c) are determined by their relative precision. If sensory input is very noisy and imprecise (like when driving at night and vision is poor), more reliance will be put on predictions (our knowledge of the road), and the resulting estimate is shifted toward these. Thus, predictions will dominate the estimate of the body state (see Fig. 1c). In contrast, if sensory input is precise (driving during the day and good vision), the brain’s estimate of the body state will more closely reflect the actual sensory input. The brain’s best educated estimate about the underlying body state is thus a combination of predictions based on internal representations and sensory input. This is described by Bayes’ Law, a statistical framework that can explain different perceptual phenomena [16, 17] and is often used to model perception [15]. It is important to highlight that the brain’s estimate about body states is not necessarily consciously perceivable and that probably a further step is necessary that translates this estimate into conscious perception.

While internal representations are crucial to correctly interpret the noisy information around us and to deal with bodily perturbations in an adequate and timely manner, visual illusions demonstrate that predictions based on internal representations can also misdirect perception. Visual illusions (such as the checkerboard illusion [18]) are often caused by strong predictions that bias our perception, leading to a discordance between the perceived reality and the physical reality. Similar to perception of stimuli arising outside the body, internal representations can bias perception of stimuli arising inside the body [19]. If the internal representation about the processes causing the respiratory state is defect, measurements will be incorrectly interpreted, and breathlessness could arise even if the sensory input does not signal any abnormalities—just as an optical illusion is perceived and becomes one’s own reality despite not corresponding to physical reality. Importantly, even though objective knowledge about the actual physical reality is present, it usually does not ‘correct’ perception.

At present it is unclear how and where these internal representations are implemented in the brain, although there is some evidence that the insula [20, 21] and cerebellum [22‐24] are involved in updating and maintaining internal representations. Here, mathematical models can provide relevant insights by revealing constraints to which the physiological mechanisms must be subjected. Such models implement a quantitative description of assumed internal representations and estimation processes of bodily states. In our model, we assume an internal respiratory state that describes gas exchange between environment, lung and tissue cells. The current internal representation evolves from that of the last breath via updating from sensory measurements of CO₂ concentration in the blood and cerebrospinal fluid. The perceived breathlessness reflects the brain’s estimate of this respiratory state signaling a normal versus potentially dangerous disequilibrium in gas exchange.

In the present work, we test the plausibility of this hypothesized mechanism by evaluating whether our model can describe the relationships between individual breathlessness ratings and CO₂ levels measured in a rebreathing experiment.

By writing down our proposed mechanism as a quantitative mathematical model, we render our theory about processing of respiratory information in the brain testable. We hypothesize that breathlessness ratings from a very heterogeneous sample including healthy participants and patients with post-COVID can be simulated by a model that assumes the same underlying processes in all individuals and that differences in breathlessness are explained by different weighting and thus influence of these processes on the final percept.

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.

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).

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:with \(b\): breathlessness, \({\beta }_{0}\)_: intercept, \({\beta }_{1}\): 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.

Table 1 displays the clinical characteristics of all included patients and healthy control participants. These characteristics were not part of the modeling procedure, nor were they considered for statistical analyses to evaluate differences between patients and healthy control participants. Table 2 shows diagnoses of all participants as obtained with the clinical interview for DSM-5 disorders (SCID-5-CV).

Although all participants inhaled air with the same CO₂ concentrations, both at baseline and at the beginning of the rebreathing phase, i.e., received a similar sensory stimulus, breathlessness ratings varied considerably between participants (see Figs. 3 and 4). This was true both for the maximum perceived breathlessness and for the development of breathlessness over time. Differences in breathlessness patterns could be observed between the two groups; however, there were also substantial differences between individual patients, as well as between individual healthy participants. While some participants recovered rapidly after the rebreathing phase, i.e., breathlessness decreased back to low ratings, others remained breathless even when they were breathing room air (compare e.g., P1 to P5 in Fig. 4). Despite these very different patterns, our model showed good performance in its capability to replicate the observed time course of individual breathlessness. Using only CO₂ concentration as input, it did not simply mirror this input but was also capable of describing breathlessness ratings that were uncoupled from the actual sensory input. This was for example the case in P1, where breathlessness increased throughout the experiment and stayed high, even though CO₂ concentration had decreased back to baseline.

Table 3 shows Akaike Information Criterion (AIC) for variants of the proposed model as well as a linear regression model. The lower the AIC, the better the model performance. The diverse breathlessness patterns observed in the experiment were poorly explained by a linear regression model which assumes that breathlessness is a scaled and shifted version of the sensory input, i.e., CO₂ concentration. In none of the participants, it performed better than the full proposed model or variants of it. Similarly, simpler version of our proposed model (1) without activity context and (2) without dependence on the respiratory state in the last breath predicted breathlessness ratings in general less well than the full model. A model variant without the activity context only led to slightly better predictions in 2 out of 10 participants. Similarly, the model variant without dependence on the respiratory state in the last breath only improved model prediction slightly in 2 out of 10 participants. However, in most participants, our full model showed a decisive improvement in model performance when compared to variants of it or the linear regression model.

The fact that our mathematical model could simulate our experimental data does not necessarily mean that it is the only possible model. It is also still greatly simplified. For example, it is unlikely that CO₂ concentration is the only sensory input used to update the respiratory state. Breathing also evokes, e.g., proprioceptive signals that provide information about lung mechanics such as the breathing frequency. In addition, for sudden changes in breathlessness perception that are decoupled from changes in CO₂ concentration (see e.g., P3 & H3), our current model shows poor performance. Here threshold effects could be implemented in future versions of the model to allow simulations of such patterns. Furthermore, the sample size in this study only allowed to show that in general our model can predict different breathlessness patterns but did not allow for analysis of group differences, neither for model parameters nor for experimental data. Despite these limitations, we present our model at this stage of development because it could already describe experimental data very well, especially in view of the small set of parameters needed to describe a complex behavior.

Our model enables to test hypotheses about the processing of (post-COVID) breathlessness in the brain. While our hypothesis of how respiratory signals are processed in the brain is so far supported by results, further experimental tests are required to validate, and potentially refine it. Especially in post-COVID patients such as P1, an independent test of respiratory chemoreception could help to answer the question, whether sensory damage, e.g., to the carotid bodies or to central chemoreception [47], may have played a role in maintaining an inadequate internal representation of respiratory state. Another obvious consequence of the hypothesis would be that relief from breathlessness should be possible by readjusting the internal representation so that it adequately reflects a healthy state. One may assume that this already occurs during pulmonary rehabilitation programs, although not explicitly addressed [48]. Here, our model could provide a means to monitor which parameters are improved by rehabilitation. Finally, a possible method of providing improved sensory input is biofeedback, which has recently been suggested for post-COVID treatment of dysregulation of the autonomic system [49]. For example, monitoring the blood oxygenation level or, via transcutaneous CO₂ monitoring, even the CO₂ level, could show patients that their respiratory state is normal despite feeling breathless. Such a cognitive input might have a small effect but could help in gradually readjusting the internal representation.