Clinical challenge

Intensive care medicine is complex, resource demanding and expensive. High-stakes decisions need to be made rapidly based on the evolving clinical state of the patient and the interpretation by clinicians of continuous physiological and ancillary data. The patient’s cardiovascular state may fluctuate over minutes with rapid transitions due to external perturbations, internal failures of the physiological sub-components, or their control. Moreover, these patients are very diverse in their physiological and disease processes. Inferring the underlying hidden physiological state of a patient from observed data is a critical problem that is encountered by clinicians across different disciplines daily. While the modality of the available data and the frequency of sampling may vary between different clinical scenarios, in almost all cases clinicians must rely on partial observations that are a peripheral reflection of the internal, hidden states that are of importance. Specifically, clinicians use observable measurements of heart rate, arterial and venous blood pressures as well as findings from the physical examination (capillary refill, extremity temperature, etc.) to estimate internal variables that are not readily observable at the bedside such as cardiac output, systemic vascular resistance, filling pressures and volumes, autonomic tone, and more. As the physiological signals are often noisy and ambiguous, creating these mental models is a notoriously hard task that requires clinical expertise. Moreover, since the patient’s state fluctuates frequently, repeat assessments by clinicians are required. The clinicians then use this mental model of the patient’s cardiovascular internal state to derive the causes of instability such as shock type (hypovoloemic, cardiac or distributive for example), and choose appropriate treatments and interventions.

Modeling background

The large amount of data available from ICUs motivated researchers from computational disciplines to develop mathematical tools which aim to support decision-making in this area. An appealing approach to the analysis of continuous physiological times series relies on mechanistic modeling, grounded in a biophysical understanding of systems. Mechanistic modeling of the cardiovascular system often requires an understanding of the cardiac cycle as a pump coupled to the arterial and venous conductance systems with associated autonomic nervous control. This understanding is derived from years of experiments analyzing human and animal cardiovascular physiology. Such models are usually based on sets of ordinary differential equations (ODEs). In a sense, they simulate the thought process of the critical-care physician at the bedside trying to estimate the underlying pathophysiology of the patient to tailor care. However, due to the complexity of even the simplest models and the number of unobserved variables and unknown parameters, such mechanistic models were mostly published in the context of well-curated “toy” datasets and in-silico simulations ([1–5]; barring a few exceptions. See also the thorough review by Chase et al [4]).

Although simplified mechanistic models of the cardiovascular system show promise, so far, to the best of our knowledge, there are none that have been developed specifically for, and tested on data collected from critically-ill patients at the bedside, nor validated in real-life settings. There are several challenges standing in the way of using mechanistic models at the ICU bedside based on real-time data; these have prevented their deployment so far, and also stood in the way of using them to gain insights into the physiology of critical illness. First, as noted above, mechanistic models require estimating multiple hidden physiological parameters and variables: inverse problems are notoriously difficult, and their solutions are often non-unique. To overcome such difficulties, prior works reduced the number of free parameters by assuming constant values for some of the parameters, often taken from the literature to describe a “typical subject” [2, 5–8]. However, physiological properties can vary considerably between subjects, and attempting to use a “one size fits all” parameter often fails to properly characterize most patients. An example where this problem is particularly acute is in pediatric critical care units, where patient weight and normative values can vary significantly [9–11]. Thus, for a model to be useful at the bedside, minimal apriori assumptions regarding the values of hidden parameters must be made. Even so, most published models are still quite complex and require estimation of many hidden variables, and thus rely on sampling of multiple invasive variables that are not readily available outside of a dedicated, specialized laboratory. Many of these models are extremely complex, as they attempt to capture cardiovascular dynamics at second or even sub-second resolution, containing multiple coupled differential equations and hidden parameters rendering them nonidentifiable. On the other hand, limiting model complexity to reduce the number of estimated parameters and variables entails a reduction in the explanatory power of such models. Of note, there are several commercially available tools for bedside clinical use that attempt to estimate hidden cardiovascular variables such as cardiac output or systemic vascular resistance that rely, at least to some extent, on simple and crude mechanistic assumptions and models (as reviewed in [12, 13] or tested in children in [14]). Some of these tools are based on analysis of the arterial pulse wave and either utilize norms derived from published population data, or, calibration based on dilution methods. These tools are mostly used in adult care and have not entered common practice in pediatric critical care, partly due to concerns regarding their accuracy stemming from some of the challenges detailed above. Moreover, the models these tools rely on, if any, are usually extremely simple and do not account for any internal regulatory mechanisms, nor do they incorporate effects over timescales longer than several seconds. Thus, there is a delicate balance as a result of this trade-off between model complexity and estimability using readily available observations.

Second, the measurement frequency of physiological variables in the clinical world varies from continuous monitoring of vital signs to imaging techniques which are conducted once a day or even less. For example, in the ICU, blood pressure and arterial blood oxygen saturation are continuously monitored, while additional indirect estimation of cardiac function or volumes using tools such as echocardiograms are usually performed infrequently, every several hours to days. Models which rely on low-frequency measurements can be good for chronic conditions such as heart failure [2]. However, to estimate the state of unstable patients, mechanistic models should rely only on measurements with high temporal resolution. Of note, up until recently, [15] even these measurements that are commonly captured by the patient’s bedside monitor were not available for offline analysis or real-time ingestion for model inference.

An additional barrier to adapting cardiovascular mechanistic models to bedside use lies in the need to account for various modes of autonomic nervous control: most previously published work focused on a single, simple feedback loop, neglecting other common autonomic modulations.

Our approach: iCVS

Our motivation is translational in nature—develop an inference model that can estimate the hidden cardiovascular state, its control, and any causes for instability, in real-time and using only measurements that are readily available at the bedside for critically-ill patients. Such a model can be used both as a clinical support tool at the bedside, and to foster an understanding of the (patho-)physiology of critical illness. We derive a simple model (iCVS), which retains a high explanatory power of the cardio-vascular system including cardio-vascular control, which can capture different clinical scenarios such as bleeding or evolving hypovolemia, cardiogenic shock and distributive shock, in isolation or in combinations. iCVS takes into account the autonomic control of the cardiovascular system, which couples the dynamics of its different components, mathematically imposing coupling constraints on inferred parameters. This reduces the effective dimensionality of the parameter space over which we look for a solution to an inverse problem, allowing us to infer a larger set of unobserved parameters compared to previous models. The estimated parameters enable us to capture individual patients’ traits.

iCVS is designed to allow deployment at the bedside in the pediatric and adult intensive care unit: no prior assumptions regarding the physiological parameters of the patients are required except age and weight, and the only inputs which are required for the estimation process are arterial and venous blood pressures waveforms, which are commonly measured in the ICU. In order to gain more information about the cardiovascular state, without increasing model complexity, we introduce a novel approach that exploits both slow and fast timescales of the blood pressure trace. From the pulse contour shape of the blood pressure waveform, we extract quantities related to the peripheral resistance, stroke volume, and pulse pressure, while the main mathematical model and its inference run using the mean, smoothed, arterial and venous pressures at the longer, clinically-relevant, time scale of minutes. Thus, the main model and its inference focus on processes that evolve over minutes, and are of interest to clinicians, and further allow for a much simpler model than those that capture all the intricacies of each cardiac cycle [2]. All this is done while still capitalizing on information that can be extracted only from the arterial pressure waveform.

In what follows, we describe the iCVS model and inference system in detail and demonstrate its use and power for real-time estimation using a dataset of 10 patients admitted to a pediatric intensive care unit. While our estimation process entails inferring the full iCVS model including 15 hidden parameters, we focus our demonstration on the identification of three clinical shock states, representing a dire, hard clinical challenge, and a testable task. Shock, due to any cause is a state in which the cardiovascular system fails to deliver an adequate supply of oxygen to the tissues. There are several prototypical physiological shock states, each requiring specific interventions, without which there is a high risk for mortality or morbidity. Identifying shock and its causes is a notoriously hard task due to the nature of the observed physiological signals (demonstrated below), requiring continuous vigilance and clinical expertise. Specifically, using iCVS, we map certain hidden parameter ranges to identify (i) ongoing bleeding, (ii) distributive shock, and (iii) cardiogenic shock due to reduced cardiac contractility.

iCVS: Model overview

The goal of this work is to derive a new model of the cardiovascular system which can be deployed at the patient’s bedside, requires only routinely acquired measurements as input, and provides an inference of the hidden, internal physiological state of the patient in real-time. The model should be flexible enough to capture the different types of shock and account for the inter-subject variability which characterizes the patients in the pediatric and adult critical care unit. The full details of the model and its development are provided in the Materials and Methods section.

Mechanistic models of the cardiovascular system are developed from first principles, tie together changes in blood pressures and volumes, and usually use the latter as the dependent dynamical variables in the set of ODEs [5]. We derive the model in terms of intravascular pressures since arterial and venous blood pressures are directly measurable (in contrast to blood volume). Briefly, our proposed model contains a one-chamber heart that acts like a pump [5], an arterial component and a venous component, and further assumes the peripheral organs can be lumped as a linear resistor (see Fig 1). In addition and importantly, the model contains three regulatory units which are described below. The two dynamical equations which describe the time-dependent evolution of the intravascular pressures are written as follows (the full derivation can be found in Materials and methods): (1) (2) Here P_a and P_v are the mean arterial and venous pressures respectively, Hr is the heart rate, R is the systemic vascular resistance and P_p is the pulse pressure. In addition, C_a and C_v are the arterial and venous compliances respectively, ΔV_v0 is the difference between minimal and maximal unstressed venous volume, I_ex is the intravascular volume change, and S_tot serves as a general marker for the activation of the autonomic nervous system (see below).

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Fig 1. A schematic representation of the cardiovascular model.

The model is a serial circuit and contains one heart chamber which acts like a pump, organs, and arterial and venous compartments. Heart rate, contractility, vascular resistance and unstressed venous volume are affected by the baroreflex (S_b) which is determined by the arterial pressure, and by S which is an independent component that does not depend on the current physiological state. In addition, intra-vascular volume can change through I_ex and the vascular resistance is also modulated by M_SVR. Created with BioRender.com.

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Our proposed model contains fifteen hidden parameters (see Table 1), and five time-dependent observables: mean arterial pressure, mean venous pressure, heart rate, peripheral resistance, and pulse pressure (Table 2). We later show how the mechanistic model can be used to estimate the hidden parameters based on the observables, and identify clinical states based on the observed data (Fig 2).

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Fig 2. Overview of the estimation process.

A. Arterial and venous pressure waveforms are recorded via arterial and central venous catheters that are connected to pressure transducers. Specifically, in this paper waveforms are sampled from critically-ill children hospitalized in an intensive care unit and recorded at 125Hz. B. The raw data is analyzed and five measurements are extracted—the observables. C. The observables are used as an input for the full model (iCVS) estimation and using constrained nonlinear optimization we obtain the set of parameter values that best fit observables. Created with BioRender.com.

https://doi.org/10.1371/journal.pcbi.1010835.g002

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Table 1. List of estimated parameters.

https://doi.org/10.1371/journal.pcbi.1010835.t001

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Table 2. List of measurements.

https://doi.org/10.1371/journal.pcbi.1010835.t002

iCVS can simulate different shock states.

The interplay between the fifteen parameters of the model can generate different clinical scenarios. In this work we focus on three different shock states which are described by three hidden parameters. Briefly, shock is a failure of the cardiovascular system to supply adequate blood and oxygen to the body tissues. In order to describe the hypovolemic shock state, we use I_ex (intra-vascular volume change), which is negative in case of intravascular fluid or blood loss. A distributive shock state is equivalent to a negative M_SVR (non-autonomic vascular resistance modulation). In the next section, we also discuss the cardiogenic shock state which is caused by a reduction in K (heart contractility).

Fig 3 presents examples of different shock conditions of artificial neonate patients which are generated by the mechanistic model. The internal parameters of the subjects are set and two of them (I_ex and M_SVR) are presented (Fig 3A, 3B, 3I, 3J, 3Q and 3R). The observable parameters (Fig 3C–3G, 3K–3O and 3S–3W) are simulated according to the iCVS model (see Material and methods). The baroreflex, which depends on the current arterial pressure and affects the observables is also presented (Fig 3H, 3P and 3X).

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Fig 3. Simulations of the iCVS model for different shock conditions.

Each column presents a simulation of one artificial patient with a given shock state: A-H—hypovolemic shock state, I-P—distributive shock state and Q-X—combined hypovolemic and distributive shock state. The hidden parameters are fixed, and the observables are simulated accordingly (see Methods). Two of the hidden parameters are presented for each patient: A, I, Q Time dependent intra vascular volume change (I_ex). In A, Q liquid withdrawal occurs over a period of 15 minutes. B, J, R Time course of M_SVR. In J, R a reduction in M_SVR occurs over a period of 15 minutes. C-H, K-P, S-X—the resulting observables: C, K, S Arterial pressure (P_a), D, L, T Venous pressure (P_v), E, M, U Peripheral resistance multiplied by arterial compliance (RC), F, N, V Heart rate (Hr), G, O, W Pulse pressure (P_p). H, P, X The time dependent magnitude of the baro-reflex (S_b).

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Panels A-H in Fig 3 present a simulation of a patient in a hypovolemic shock state, as can happen with ongoing bleeding. The intravascular volume change (I_ex) is negative (Fig 3A), representing a reduction of intravascular volume over time, while the non-autonomic SVR modulation (M_SVR) remains constant (Fig 3B). As expected, we see that the arterial and venous pressures decrease (Fig 3C and 3D). The heart rate and the vascular resistance reflexively rise (Fig 3E and 3F) due to baroreflex activation which acts to maintain blood pressure and compensates for the reduction in the intravascular volume (panel H). Note, that despite the baro-reflex activation that increases the contractility, the pulse pressure (Fig 3G) decreases due to the change in venous pressure and return (see also Eq 22 in Material and methods).

Panels I-P in Fig 3 present a simulation of a patient with a distributive shock state. The parameter I_ex is zero (Fig 3I), reflecting constant intravascular volume, while M_SVR decreases (Fig 3J), reflecting a reduction in the peripheral resistance which is not caused by the autonomic nervous system. It is possible to see in Fig 3N that the heart rate increases (due to the baroreflex activation, see panel P), the pulse pressure increases as well (due to the baroreflex and rise in venous pressure), and the peripheral resistance decreases due to the reduction in M_SVR (Fig 3M).

Panels Q-X in Fig 3 present a simulation of a patient with a combined hypovolemic and distributive shock state. The reduction in the arterial blood pressure (Panel S in Fig 3) is more pronounced than in either shock state separately.

Of note, in addition to simulating shock states, iCVS can emulate common clinical behaviors, such as a fight-or-flight response that is elicited by pain or a noxious stimulus, mediated by autonomic tone modulation (see S1 Fig).

Estimation pipeline of the cardiovascular parameters from bedside-acquired blood pressure waveforms

The estimation pipeline is schematically illustrated in Fig 2 and is described below and in Material and Methods. Briefly, the first step includes the initial processing of the raw data and extraction of five observables. In the next step, the observables serve as an input to an optimization function which is based on iCVS and we estimate the hidden parameters and the full model.

Extracting the observables from the raw data.

As a first step, we use waveforms of venous and arterial blood pressure tracings, acquired at a high frequency that allow extraction of the following five observables: mean arterial pressure, mean venous pressure, heart rate, peripheral resistance, and pulse pressure (see Material and methods). We then use both slow and high-frequency features of the data (Fig 4). From the slow time scale we calculate the mean arterial and venous pressures. From the fast timescale (the scale of a single heartbeat), we extract: heart rate (timing of consecutive beats), peripheral vascular resistance (up to a constant, as detailed in Material and methods), and pulse pressure. These quantities, which cannot be derived without the waveforms recorded at high-frequency (a sampling rate which is much higher than the heart rate and can capture the shape of a single beat), are then smoothed and used for model inference at the slower timescale of minutes. The peripheral resistance is extracted using a non-calibrated pulse contour analysis method relying on assumptions of a simplified Windkessel model and rectangular ejection pattern during systole that has been described in detail elsewhere [16, 17].

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Fig 4. Extraction of slow and fast time scales features from venous and arterial blood pressure recordings.

A,B. An example of the raw data, sampled at 125 Hz frequency by the bedside patient monitor. A. Arterial blood pressure, B. Venous blood pressure. Note how these real-life data are noisy, fluctuating, and riddled with artifacts related to patient movement and care. C-G. Features that are extracted from the raw data. C,D. Mean arterial and venous pressures (slow time scale features), E-G. Heart rate, peripheral resistance and pulse pressure (fast time scale features).

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Demonstrating iCVS estimation of cardiovascular shock states on real-world data from critically-ill patients

In this section, we illustrate the estimation process for a sample of 10 critically-ill children, hospitalized in an intensive care unit (ICU) at a large, tertiary children’s hospital. These children were admitted to the ICU either for routine observation post-operatively (for the control samples) and were stable hemodynamically, or presented with various shock states, for example, due to massive post-operative bleeding, cardiac dysfunction, infections or other causes, see Table 3 for more information. The dataset which we used to test the model’s performance is detailed in [15]. It contains physiological waveforms recorded from critically-ill children at sampling rates of 125Hz for all invasively-recorded pressures; however, any high-frequency recording that contains the full arterial blood pressure waveform is adequate for the model’s purposes. In addition, the times of all medical interventions, including drug consumption and fluid administration are recorded. Note how these real-life data are noisy, fluctuating and riddled with artifacts related to patient movement and care (see Fig 4A and 4B).

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Table 3. Patients information.

For each patient, age, weight, cause of admission to the intensive care unit and the cardio-vascular state as determined by an expert clinician are given.

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As noted above, iCVS is able to deal with patients of varying weights from neonates (a couple of kilograms) to adult-sized patients, as we demonstrate below. The hemodynamic data for all patients in this ICU are collected as routine practice [15, 17]. The diagnostic labels for shock (either hypovolemic/ hemorrhagic, distributive, cardiogenic, or a combination of the above) were derived from a prospective and retrospective review of at least two expert clinicians. These labels were also corroborated using objective measurements from the medical record, when available. For example, from quantification of blood loss postoperatively, assessment of cardiac function or systemic vascular resistance using clinical examination, drugs administered, and ancillary diagnostic tests.

The challenge of estimating the causes for cardiovascular instability and shock from real-world data can be appreciated by examining Figs 5–8. Examination of panels A-E in each figure, which depict the five observables, shows that patients’ data are “messy” with marked fluctuations and trends arising from multiple physiological sources (due to the shock cause and others), unlike the clean simulated data shown before. Moreover, it is evident that simple rules linking changes in these observables to the shock cause may be misleading.

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Fig 5. Demonstration of iCVS model inference results for a neonate patient (body weight < 5 Kg), in a hypovolemic shock state.

iCVS results for patient number 1 (see Table 3) are presented. A-E.—The observables: mean arterial pressure (A), mean venous pressure (B), heart rate (C.), RC—the peripheral resistance multiplied by the arterial compliance (D) and pulse pressure (E). F-H. Gray circles—optimal parameters which are achieved in each segment of 300 sec. Black line—a smoothed version of the estimated parameters (see Material and methods). F. Relative intra vascular volume change (). G. Non autonomic vascular resistance (M_SVR). H. Maximal relative contractility (). Note that in this patient which suffers from a hypovolemic shock state is identified as having negative by the model.

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Fig 6. iCVS model inference results for an adolescent patient (body weight ∼ 60 Kg), in a distributive shock state.

iCVS results for patient number 5 (see Table 3) are presented. A-E. The observables: Mean arterial pressure (A), mean venous pressure (B), heart rate (C), RC—the peripheral resistance multiplied by the arterial compliance (D) and pulse pressure (E). F-H. Gray circles—optimal parameters which are achieved in each segment of 300 sec. Black line—a smoothed version of the estimated parameters (see Material and methods). F. Relative intra vascular volume change (). G. Non autonomic vascular resistance (M_SVR). H. Maximal relative contractility (). Note that this patient, suffering from a distributive shock state, is correctly identified as having negative M_SVR by the model.

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Fig 7. iCVS model inference results for a neonate patient (body weight < 5 Kg), in a combined hypovolemic, cardiogenic and distributive shock state.

iCVS results for patient number 4 (see Table 3) are presented. Dots mark chest opening. A-E.—The observables: mean arterial pressure (A), mean venous pressure (B), heart rate (C), RC—the peripheral resistance multiplied by the arterial compliance (D) and pulse pressure (E). F-H. Gray circles—optimal parameters which are achieved in each segment of 300 sec. Black line—a smoothed version of the estimated parameters (see Material and methods). F. Relative intra vascular volume change (). G. Non autonomic vascular resistance (M_SVR). H. Maximal relative contractility (). Note that in this patient becomes non-negative and the contractility increases after the chest opening procedure, compatible with an improving in cardiac function and bleeding cessation as a result of the chest opening procedure. We note that in panel H a single point with an extremely large value is not plotted, at 9.66 minutes after chest opening ( mmHg). This point correlates with a short transient increase in the venous and arterial pressures, possibly resulting from a bolus of a vasoactive drug given to the patient at the time.

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Fig 8. Illustration of the estimation process for a neonate patient (body weight < 5 Kg) control patient, post cardiac surgery.

iCVS results for patient number 9 (see Table 3) are presented. A-E. The observables: mean arterial pressure (A), mean venous pressure (B), heart rate (C), RC—the peripheral resistance multiplied by the arterial compliance (D) and pulse pressure (E). F-H. Gray circles—optimal parameters which are achieved in each segment of 300 sec. Black line—a smoothed version of the estimated parameters (see Material and methods). F. Relative intravascular volume change (). G. Non autonomic vascular resistance (M_SVR). H. Maximal relative contractility ().

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Panels F-H in Figs 5–8 present the results of model estimation for three hidden parameters that are crucial to elucidate the causes for cardiovascular shock. A hypovolemic state is described by a negative (the intra-vascular volume change divided by the difference between the maximal and minimal unstressed venous volume ΔV_vo). This magnitude does not depend on the patient weight or blood volume, has units of , and thus allows comparison between patients with different sizes. The vasodilatory state is represented by M_SVR (non-autonomic vascular resistance) which is expected to be negative in a distributive shock state. The cardiac function is described by the Maximal relative contractility: , where K_max is the maximal contractility that the ventricle can generate (see Eq 21), C_ven is the compliance of the ventricle, and C_a is the arterial compliance. For constant arterial and ventricular compliances, is proportional to the maximal heart contractility. is expected to decrease in case of cardiogenic shock or cardiac dysfunction.

Each gray circle in panels F-H in Figs 5–8 corresponds to the optimal parameters which are obtained in a given time segment. The black line in each panel is a moving average of the results of different segments (see Material and methods).

Fig 5 illustrates model inference for a sample patient with post-operative bleeding. This is a neonate that presented with significant intravascular volume loss after cardiac surgery. Panels A-E show decrease in the arterial and venous blood pressures, while SVR increases and pulse pressure decreases, as expected. Of note is that for an unknown reason, the heart rate of this patient does not reflexively increase, as one would expect for a bleeding patient. Panel F shows that even without the expected heart rate increase, our model correctly identifies the negative intravascular volume change (Fig 5F). The maximal relative contractility seems stable or even increases (for comparison see the changes in for a patient with cardiac dysfunction in Fig 7).

Fig 6 illustrates the model inference results for an adolescent patient which was clinically identified as having distributive shock, specifically a vasoplegic state seen sometimes after solid organ transplantation. In the time window shown here, the mean venous and arterial pressures increase, as also the heart rate and the pulse pressure. Yet, the mean arterial pressure is low and the heart rate is high, supporting the clinical observation of a shock condition. We can further see that the peripheral resistance decreases over time. Our model indeed correctly identifies the cause for shock: M_SVR which reflects the non-autonomic component of the vascular resistance is negative, in agreement with the clinically detected distributive shock. Moreover, the estimation process yields roughly zero , meaning that no intra-vascular volume change is detected, as well as demonstrating stable maximal relative contractility.

Fig 7 presents the estimation process for an unstable patient with a combined hemorrhagic, cardiogenic, and distributive shock. This was a neonate with massive bleeding post cardiac surgery, combined with left ventricular dysfunction and inadequate systemic vascular resistance. The patient was extremely unstable despite multiple interventions and underwent a chest opening procedure in the ICU (grey arrows in Fig 7) which resulted in marked improvement in cardiac function and bleeding cessation. We see that arterial blood pressure increases after the chest opening. In addition, the venous pressure decreases—compatible with an improvement in the heart function (Fig 7A and 7B). Note how the model correctly identifies the marked improvement in the maximal contractility and immediately after the procedure.

Finally, a neonate control patient that was admitted after thoracic surgery and had an uneventful post-operative course is presented in (Fig 8). As expected, the estimated intravascular volume change () and the non-autonomic SVR modulation (M_SVR) are not negative, and there are no marked fluctuations in the maximal relative contractility ().

In Fig 9 we present results summarizing the model’s inference for the 10 patients. Panel A depicts the ability of our model to correctly identify intravascular volume loss: the patients marked in blue are those that had documented post-operative bleeding. Note that for these patients mean change in blood volume is more negative than their counterparts. Conversely, panel B depicts the mean non-autonomic component of systemic vascular resistance modulation. As noted above, a negative value indicates a maladaptive response with vasodilatation, as one can see in distributive shock states (for example sepsis or vasoplegia). Since the parameter K_max is not comparable between different patients, this parameter is not presented in Fig 9. The patients marked in red were identified by expert clinicians as having abnormally low SVR. Indeed, overall, these patients were identified by our model as having lower values of M_SVR. Additionally, the patients marked in pink were treated by vasoconstrictors, drugs that directly affect vascular smooth muscle to increase systemic vascular resistance, an indirect indication extracted from the electronic health record of at least a somewhat distributive state.

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Fig 9. Estimated and M_SVR for ten subjects.

For each subject, the model is applied for a time window of 25 minutes (see Methods). The analysis is done over segments of 300 seconds and the average values of (panel A) and M_SVR (panel B) are presented. A. The average value is presented for all sample patients. Blue—patients which are documented with post-operative bleeding, black—no post operative bleeding is documented (average value of the hypovolemic group is −0.016 min⁻¹, average value of the non-hypovolemic group is 0.0003 min⁻¹, P_value = 0.026). B. The average M_SVR value of all sample patients is presented, ordered differently than in A. As there is no objective criteria for a distributive state, unlike blood loss, we resort to identifying such patients by either expert opinion or indirect evidence from the electronic medical record. In red—patients who were identified by clinical experts as suffering from distributive shock states, in pink—patients who were treated by vaso-constrictors (drugs that directly affect vascualr smooth muscle to increase systemic vascular resistance), in black—patients who were not documented with distributive shock states (average value of the distributive group (red) is −0.43, average value of the non-distributive group is −0.07, P_value = 0.077). Error bars in A and B represent standard error of the mean.

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As an additional verification of model estimation performance on real data, we also compare between the measured observables: , and and reconstructed values for the observables which are obtained using model inference. The observables are reconstructed well by the iCVS model, with a Pearson correlation of 0.95–0.99 (see supplementary information). Furthermore, as is elaborated in the supplementary to this article, we demonstrate the robustness of our findings to the estimation algorithm.

Ethics statement

The study was approved with waiver of informed consent by the Research Ethics Board at The Hospital for Sick Children, approval number 1000048904.

Mechanistic model

The goal of this section is to detail iCVS: a mechanistic model of the cardio-vascular system that links measurable variables to clinically relevant physiological parameters that can not be measured directly. We use a lumped parameter model with several compartments: the circulatory system is represented as a closed loop with two compliant vessels (the arterial and venous compartments) and one pure resistance vessel (the organs) [5, 23]. The heart contains one chamber that acts as a pump. The venous-arterial flow I_H is generated by the heart and is given by the product of the heart rate Hr and the stroke volume SV: (3)

The arteriolar and capillary I_C flows are modeled as a linear resistor: (4) where P_a and P_v are the arterial and venous mean pressures, respectively, and R is the peripheral resistance. The evolution of arterial and venous volumes (V_a and V_v respectively) is given by the following differential equations: (5) (6) where I_ex is the intra-vascular volume change—either loss of volume due to bleeding or extra-vascular capillary leak or conversely intravenous fluid administration.

Eqs 5–6 relate between blood volumes dynamics and other hidden cardio-vascular variables. The challenge however is that blood volumes cannot be directly measured in patients. We therefore reformulate the mechanistic model in terms of blood pressures (which are measurable); this allows us to infer hidden cardio-vascular parameters based on observable data.

The pressure in the venous and arterial compartments is derived from the vessel’s volume and compliance as follows: (7) Where α ∈ {a, v} represents either the venous or arterial components, respectively, C_α the compliance, and the unstressed volume.

For the arterial compartment, we assume constant unstressed volume [5]. The venous unstressed volume is modulated by an autonomic activation S_tot (see below): (8)

Assuming constant C_a [17], differentiating Eq 7 and rearranging, we obtain: (9)

Substitute Eqs 3, 4 and 9 in Eqs 5 and 6 we get: (10) (11) where we assume that the unstressed arterial volume is constant [5] and thus , and also use the approximation [17].

Simulations of the model

In Fig 3 and S1 Fig simulations of the iCVS model are presented. The hidden parameters are fixed and the observables are simulated according to Eqs 10–16 and 22. The following values for the hidden parameters are used: P_set = 50 mmHg, K_b = 0.1838mmHg⁻¹, Hr_max = 180 Bpm, Hr_min = 100 Bpm, , , , , , , ΔV_v0 = 65 ml. In Fig 3 S = 0.2. The initial conditions of the mean arterial and venous pressure are set to 48 mmHg and 4 mmHg respectively.

Extracting the observable variables from the raw data

The raw data which is used in this work is a continuous (sampled at 125 Hz) measurement of the venous and arterial pressure waveforms. The data analysis is performed as follows:

1. Using the algorithm developed by [16, 17], we extract the following measurements from the raw data (see Table 2): (mean arterial pressure), (mean venous pressure) (heart rate), (pulse pressure) and , which is an approximation for the peripheral resistance multiplied by the arterial compliance and scaled by a constant (α_RC): .
2. Outliers are found and replaced—Outliers are defined as points which are smaller than the Lower threshold value—three local scaled median absolute deviation below the the local median within a sliding window of 20 sec, or higher than the Higher threshold value—three local scaled median absolute deviation above the the local median within a sliding window of 20 sec. The outliers are replaced by the following: Lower threshold value for elements smaller than the lower threshold value, and upper threshold value for elements larger than the upper threshold value.
3. A linear interpolation is performed in order to obtain a data-set in 10 Hz frequency.
4. A low pass filter is applied and the frequency components above Hz are removed.
5. The data of each subject is divided into segments of 300 sec. Only Valid segments are used for further analysis. A valid segment should satisfy the following criteria: , , , , .

Sample subjects analysis

For each subject in Fig 9, the estimated parameters of 12 segments of 5 minutes are averaged. Each segment is 300 seconds long, and the segments are partly overlapped—each segment starts 100 seconds after the previous one. Segments which are not valid are not analyzed.

For the analysis in Fig 9, for each subject the first recorded 25 minutes time window that satisfies the follows is analyzed: the subject did not receive intra-venous fluids bolus during the time window, 15 minutes before and 15 minutes after.

In Figs 5, 6, 7 and 8, the above described 25 minutes time windows of patients 1,5,4,9 are presented, respectively.

The estimation procedure

Definition of the cost function.

Based on the iCVS model, we define a cost function which consists of a set of terms that link between the observables (Table 2) and the hidden parameters (Table 1). We first define the following expressions: , , (relative contractility), , , , and .

The cost function is defined as follows: (23) where: (24) (see Eq 14), (25) (see 15), (26) (see Eq 22), (27) (see Eq 11), (28) (see Eqs 10 and 15).

R_modulation(t) is defined in Eq 16, and S_b(t), S(t), M_SVR(t) are defined below: (29) (30) (31)

The parameters {N_H, N_R, N_P, N_I, N_a} in Eq 23 normalize the different terms such that they have no units, and are adjusted in each time interval of the analysis: , ,, , and N_I is the upper bound of . The parameters {w_H, w_R, w_P, w_I, w_a} adjust the relative contribution of each term to the cost function. Throughout this work we use the following values: w_H = 1, w_R = 0.2, w_P = 0.2, w_I = 0.2, w_a = 10. The values of {w_H, w_R, w_P, w_I, w_a} are chosen such that the contributions of the terms , and to the cost function have the same order of magnitude, and the contributions of and (the terms that contain derivatives) are larger. The optimal parameters are searched within a certain physiological range (see Table 4).

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Table 4. Range of hidden parameters.

Minimal and maximal values of P_set are the 10th and 90th percentile of the mean arterial pressure respectively, adapted from [11]. Bounds of minimal heart rate are the 5th percentile and 25th percentile (adapted from [10]). Bounds of maximal heart rate are the 75th percentile and 95th percentile (adapted from [10]). In addition, and are the values of minimal and maximal arterial compliance, respectively. See [24].

https://doi.org/10.1371/journal.pcbi.1010835.t004