Using Mathematical Models to Improve the Utility of Quantitative ICU Data

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of Quantitative ICU Data

S. Zenker, G. Clermont, and M.R. Pinsky

Introduction

Intensive care medicine is one of the areas of medicine most closely linked to applied physiology. Furthermore, it has a long tradition of being the forefront of advanced physiologic measurement technologies. The associated volume of quantitative data about a patient’s physiologic status, therapy, together with the output of off-line analy- ses, creates an information overload that profoundly reduces efficient and effective information processing. To a certain extent, this disconnection is a reason for the slow progress in utilizing such information across patients and hospital systems to improve patient care, perhaps most prominently evidenced by the failure of the physiologically valuable information provided by pulmonary artery catheterization to improve outcome in the critical care setting [1, 2]. In fact, for newer and more advanced monitoring equipment, evaluations of utility and ability to fit into proven treatment protocols is often lacking. Although the difficulty in translating the increased amount of available patient-specific information into patient benefit may in part be due to the lack of adequate therapeutic options, where clear benefit is known, actual translation of this information into practice is a primary barrier to improving patient care.

A fundamental reason for this somewhat surprising and seemingly contradictory situation of increased sophistication of monitoring and decreased efficiency of utilization of the resulting data may be the lack of focus on understanding the relationship between monitored physiological variables and the determinants of recovery from critical illness. As we have stated in the past [3], for most intensive care unit (ICU) treatments, the fundamental rationale is the restoration of perceived normal physiological status independent of understanding the process of disease or its interaction with therapies. Since the limitations of this empiric physiological approach are increasingly being recognized, novel approaches may be called for to further improve outcomes in critical care.

One approach that has gained purchase in recent years is what we call ‘functional’

monitoring [4]. The underlying principle of this approach is to obtain measurements that are more directly related to key determinants of outcome than traditional physiologic variables. For example, measures of preload do not predict responsiveness to volume loading, but measures of preload responsiveness using the exact same monitoring devices do [4, 5]. Furthermore, once functional measures are defined as the key processes to assess, novel monitoring devices percolate to the top of the potential monitoring options. For example, technologies are available to measure sublingual CO₂(PslCO₂) and tissue oxygen saturation (StO₂) that may provide more direct measures of tissue health as a key determinant of outcome, than measures of cardiac output.

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A conceptually different approach is to apply mathematical models to make better use of the available quantitative data, including functional monitoring data, by selective extraction of meaningful information from the massive amounts of available clinical data. Additionally, these approaches may eventually help to quantify and predict the relationship between measurements, patient status, and outcome, thus enabling optimization of therapy by iterative protocolized care.

Quantitative Information and How to Use it

The very intensity of intensive care monitoring results in a concentration of the highest data density present in all medical environments. Some of these data are observational and qualitative in nature, such as the nurses’ and physicians’ observations regarding patient status. However, the majority of data is quantitative, traditionally consisting of a steadily growing number of often high resolution, time- series measurements. These include data streaming of hemodynamic parameters, clinical laboratory results, ventilation related parameters, and a number of other physiologic variables, like temperature, and derived parameters such as electrocar- diogram (EKG) waveform analysis. In recent years, medical imaging technology, such as ultrasound and radiological technology, has increasingly become available at or close to the bedside in a critical care setting. These technologies, in principle, are all capable of providing quantitative information about a patient’s physiologic status.

While the amount of quantitative information has grown, there has been little change in how the critical care physician utilizes this information. Since a human decision maker is inherently limited in the amount of data he or she can process, decisions are either based on the evaluation of brief sections of the complete time series data, or on a reduced dataset. Accordingly, a large amount of information con- tained in the original high-resolution time series data is lost in such an approach.

Recent research is increasingly trying to reduce this loss of information by extract- ing additional, physiologically meaningful and easily interpretable information from the already available measurements. For example, by quantifying pulse pressure or stroke volume variation in arterial pressure measurements during positive-pressure ventilation, one may predict volume responsiveness [6 – 8].

Making better use of already available measurements certainly is desirable, both with respect to avoiding the possible risk to the patient introduced by new, possibly invasive monitoring equipment, and to limiting the already extreme resource utilization in intensive care medicine. The close relationship of such approaches to a model-based approach to intensive care data analysis will be discussed below. How- ever, to make meaningful therapeutic decisions, it is insufficient to simply measure physiologic states. The pathway to successful therapy based on physiological monitoring is to understand their relationship to recovery from disease.

For the traditionally measurable physiologic variables, like blood pressure and pulse rate, this relationship can be very complicated, and our understanding of it remains limited. A possible solution to this dilemma may be to monitor functional parameters that are more directly related to key determinants of response to therapy and outcome [9].

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Functional Physiologic Monitoring

In the attempt to develop monitoring techniques that would provide more directly meaningful information about tissue health, and could thus potentially serve as therapeutic target variables the improvement of which would hopefully lead to an improvement in outcome. Most prominent of these in recent years have been the measurement of tissue CO₂and StO₂.

The relationship of tissue CO₂levels with tissue perfusion has been known for several decades [10]. Only recently, however, have practical devices been developed that allow for the non-invasive bedside measurement of tissue CO₂ levels through the sublingual mucosal surface as PslCO₂. This technique has been shown to closely correlate with both local and systemic blood flow in various relevant scenarios [11 – 14]. The key feature of this technology that may justify its classification as

‘functional monitoring’ is that the readings may directly reflect microcirculatory blood flow, which is the key determinant of tissue health, as opposed to monitoring systemic blood flow, which may be misleadingly high through functionally irrelevant shunting mechanisms despite insufficient tissue perfusion at the microcirculation level in critical disease, resulting in potentially high clinical utility both for diagnosis and guiding of therapy [15 – 18]. Additionally, such monitoring may have potential in the pre-hospital and battlefield settings, since it is non-invasive and may have dynamic response characteristics in the detection of functionally relevant hemodynamic fluctuations comparable to more invasive measurements [19].

Another non-invasive technique that may have functional capabilities is the spec- troscopic measurement of StO₂. Here, near infrared spectroscopy is used to quantify the amount of oxygenated and deoxygenated hemoglobin in the tissue (typically muscle) microcirculation [20]. This method, in combination with non-invasively applicable perturbations, such as temporary arterial or venous occlusion, may yield information about functionally relevant disturbances of both oxygen utilization and supply, with potential applications to critical illness [21 – 24]. There are also indica- tions of its ability to identify the severity of disease and predict outcome in both laboratory and clinical settings [25 – 30]

While the technologies discussed above show promise both in obtaining a more easily interpretable estimate of the patient’s physiologic status and may help to predict outcome, their value as therapeutic targets remains to be demonstrated, and will possibly be subject to similar limitations as the traditionally available physiologic measurements. While clearly providing additional insight, it seems unlikely that the complexity of physiological interactions that determine outcome in critical illness will be reflected by single, albeit functional measurement, closely enough to justify guiding therapy based on these alone. However, one can address these limitations directly by using mathematical modeling that embraces the inherent complexity of the interactions between various traditional and functional variables, disease state, and response to therapy.

The role of mathematical models has mostly been restricted to aiding data interpretation by deriving parameters from raw measurements that are more easily interpretable by the physician. However, mathematical models may be even more powerful when applied in other ways that have this far only rarely been successfully imple- mented in clinical practice. Recent developments in both affordable computing power and analytical as well as numerical methods may make these approaches feasible in the foreseeable future, as described below.

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Mathematical Modeling as an Aid to Data Interpretation

The simplest mathematical model routinely used in the daily practice of intensive care medicine may be the electric circuit analog that gives rise to the oversimplified concept of total peripheral resistance from analysis of pressure and flow data. Using Ohm’s Law, the circulation is presumed to behave like an electrical circuit where resistance can be defined as the driving pressure (arterial pressure either absolute or minus central venous pressure) divided by flow (cardiac output). A more complex example of model based data interpretation is ‘pulse contour analysis’ used to estimate cardiac output [31]. In this approach, a simplified mathematical representation of the vascular system coupled to the heart is used to derive changes in cardiac output from changes in the invasively or non-invasively measured arterial pressure waveform assuming a defined or estimated central arterial stiffness or elastance. A large number of specific incarnations of this basic idea have been proposed, with some implementations now having reached the maturity and stability to be marke- ted as out-of -the-box solutions for less invasive monitoring of cardiac output in the ICU such as Pulsion’s PiCCO, LiDCO’s LiDCOplus, and Edward’s FloTrac devices.

Unlike vascular resistance, the mathematical constructs required to derive cardiac output estimates from the arterial pulse pressure are too complex to routinely perform manually at the bedside. As a final example, we mention techniques to assess heart rate variability, which itself allows unique insights into autonomic control of cardiovascular function.

The robustness of the derived parameters is a function of the implicit assumptions made in constructing the mathematical model, as well as the specific mathematical techniques used. Many of the techniques that estimate cardiac output from arterial pressure pulse or heart rate variability from the cardiac event time series analyze the data by assuming that the underlying process can be described using a simple mathematical model characteristic for the particular analysis technique. For example, the frequency domain quantification of heart rate variability is often based on the Fourier transformation of the R-R interval signal by modeling the temporal variability as a sum of (possibly phase shifted) sinusoids of varying frequencies. The underlying assumption of signal stationarity is often violated in practical applications. More recent techniques like wavelet transforms may achieve similar results under weaker assumptions, leading to potentially more robust results [32, 33]. Thus, one may not assume that, just because a device reports a derived parameter com- mon to another device, its accuracy and robustness will be similar without specific knowledge of the technical details of the analysis.

Mathematical Modeling: What can we Learn from Physics and Engineering?

While the application of mathematical models to data interpretation in critical care has already proven its usefulness, the potential of mathematical modeling is markedly underutilized as compared to the current practice in most of the natural sciences and engineering. Several centuries ago, the level of quantitative understanding of physical phenomena was in many ways similar to our current understanding of disease processes, in that it was largely qualitative. While serving as a basis for building mental models of reality, the predictive power of such qualitative models, which can be considered their ultimate test of validity, was naturally limited. Only

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with the advent of mathematical physics, ushered in by the work of Isaac Newton, did true testable prediction become feasible. Since then, the interaction between theoretical physics (building mathematical models of physical phenomena) and experimental physics has been extraordinarily fruitful. This benefit is three fold. First, mathematical models can help to understand processes underlying observed phenomena, and help to verify whether a hypothetical mechanism can indeed explain the observed behavior of a system. Second, mathematical models (‘theories’) can predict previously unobserved phenomena, which can then be verified in experiments, thus deepening our understanding of the underlying laws of nature and improving our assessment of a given model’s validity. And third, once validated for certain scenarios, mathematical models can also be used to perform virtual experiments that would be too costly or outright impossible to perform in reality. Vali- dated mathematical models have made many of today’s technological achievements possible by informing both the design process of technological artifacts and being utilized in real time control of technological processes.

The most desirable applications of mathematical modeling for the practice of medicine would probably be:

a) aiding the understanding of complex physiologic mechanisms;

b) performance of virtual experiments that would be costly, unethical, or impossible in reality; and,

c) the prediction of future system behavior, and, closely related, the control of physiologic processes (optimization of therapy).

While application ‘a’ has met with numerous successes in the past, substantial diffi- culties still have to be overcome before applications listed in ‘b’ and ‘c’ approach bedside applicability in the critical care setting.

Mathematical Modeling can help to Understand Mechanisms

Perhaps the most developed of the above mentioned applications of mathematical models to medicine is the application to the understanding of physiological mechanisms. While the observation of a spontaneous variability in heart rate is quite old, and its identifiable spectral components were quickly associated with functional components of the autonomic nervous system, some uncertainty remained with respect to the exact mechanisms by which the high frequency component of heart rate variability is linked to the parasympathetic branch of the autonomic nervous system, while the low frequency component is linked to both sympathetic and parasympathetic activity [34, 35]. The development of mathematical models has helped to understand how cardiopulmonary interactions as well as resonance mechanisms in the relevant physiological control loops may explain the observed phenomena [36]. With application to intensive care, we have recently been able to show how the clinical observation of a relationship between various clinical indicators of sepsis severity and outcome and the low-frequency power of heart rate variability may at least partly be explicable as a saturation phenomenon on the basis of the non-linear characteristics of the baroreflex feedback loop [37], leading to a physiologically accurate correlate of the previously claimed ‘sympathetic failure’ in a situation of severely elevated sympathetic activity.

Specifically, there are two interacting mechanisms that may contribute to the weakening of low frequency power in sepsis: The reduction of gain in the effector branch of baroreflex through vasodilatory effects of inflammation, which has been

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shown theoretically to contribute to reduced low frequency power [36], and saturation of the sigmoidal non-linearity in the central component of the baroreflex feedback loop in the hemodynamically stressed condition of sepsis.

Mathematical Modeling may help to Perform otherwise ‘Impossible’

Experiments in silico

Once a valid mathematical model of the physiologic processes of interest has been developed and validated in a defined environment, it can be utilized to explore system responses under a wide range of conditions inexpensively and without doing harm to study subjects, whether they are humans or animals. These investigations may take the form of simulated laboratory experiments, where a mathematical model incorporating both animal physiology and experimental setup may allow to evaluate effects of variations of both properties of the animals and parameters of the experimental protocol, or even of simulated clinical trials for the virtual evaluation of therapeutic strategies [38].

To illustrate the use of virtual experiments to supplement real life experimental work, we explored the effects of the inevitable inter-animal variability in physiological response on outcome in an animal model of hemorrhage [39]. Figure 1 shows the calibration results of a mathematical model designed to replicate the animal study of hemorrhagic shock in silico for a survivor and a non-survivor ani- mal, top panels showing actual measurements for each animal, bottom panels simulation results after manual parameter adjustment. The mathematical model describes both the animal’s physiology and the experimental procedure, which consisted of repeated bleeding episodes triggered by the animal’s recovery to a blood pressure of 40 mmHg, and resuscitation triggered by its final cardiovascular decompensation. Potentially, a complete experiment can be simulated from one set of initial conditions, allowing one to theoretically explore effects of alterations in physiological parameters and initial conditions (fitness of the host), treatments and their temporal relationship to the physiological response to the insult and outcome.

After we verified our ability to simulate the actual experimental physiological data in a qualitative fashion, we strove to assess the model behavior under differ- ing conditions, starting with virtual experiments that are simpler than the real-life protocol. Figure 2 shows an example where outcome (blood pressure after approx.

17 hours) is shown as a function of the bleeding rate and the total volume bled in a single constant-rate bleeding expressed as a fraction of total blood volume when using the parameter set from the survivor pig in Figure 1 as a starting point. The survivable total blood loss shows a threshold behavior, with bleeding rate affecting this threshold only at low bleeding rates when slower compensatory mechanisms like inter-compartmental volume shifts start to matter. Figure 3 shows the dependency of outcome under a single bleed of fixed volume and rate on the tissue sensitivity to hypotension, as expressed as the midpoint of a sigmoidal damage rate/

pressure relationship, and on the strength of the sympathetic response, as expressed by a scaling factor on the peripheral effector sites. As can be seen, qualitative behavior in this scenario is as expected, with higher sensitivity to hypoper- fusion, and lower sympathetic response range being associated with a lower sur- vivability.

Subsequently, this model was used to simulate the real-life experimental protocol.

The same plot of dependency of survival on hypotension sensitivity and baroreflex

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Fig. 1. Example calibration results for mathematical model of hemorrhagic shock experimental protocol and physiology in a pig model. Top panels show actual experimental data (arterial blood pressure and heart rate, as well as beginning and end of controlled bleeding episodes) for a surviving and non-surviving animal. Bottom panels show simulation results after parameter adjustment. Note that heart rate does not drop in the simulation when the non-surviving animal dies (blood pressure falls and stays low, in agree- ment with the experiment) since cardiac rythmogenesis is represented in a very simplified fashion in the current model.

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Fig. 2. Simulated dependency of outcome represented by arterial blood pressure on the total bleeding rate (x-axis) and the relative bleeding volume (y-axis) of a single constant rate bleeding episode when using the survivor parameter set from Figure 1. Each rectangle represents one simulated experiment, tone encodes blood pressure after 60,000 seconds of simulated experimental time, with lighter tones corresponding to higher blood pressures.

response range depicted in Figure 3 for the simple single constant rate bleeding is shown in Figure 4 for the real-life experimental protocol. As can be seen, a non- trivial dependency of outcome on variations in parameters is observed. While definitive interpretation of these simulation results awaits further experimental and theoretical validation, a key insight gained using this only qualitatively calibrated model was that the use of this particular experimental protocol, which bases its decisions on the response of the animal, may introduce a non-trivial, and for some regions of states and parameters somewhat paradoxical, dependency of outcome on the properties of the individual animal. More fit animals led to good responsiveness that may in fact lead to longer shock duration and thus worse outcome (Fig. 4).

These findings are relevant to the trauma literature wherein previously healthy trauma victims often present with compensated shock but carry a worse prognosis because of delayed resuscitation owing to the false sense of security that mentation and a normal blood pressure give the acute health care providers. In this setting, the modeling effort helped to understand some of the dynamic properties of the experimental protocol in its interaction with the animals’ physiology, and, thereby, aided the ongoing improvement of the real life laboratory experiments and clinical practice.

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Fig. 3. Simulated dependency of outcome on the midpoint of the sigmoidal rate of tissue injury/arterial pressure relationship, and thus effectively tissue sensitivity to hypotension (x-axis) and the baroreflex effector response range (y-axis) of a single constant rate bleeding episode when using the survivor parameter set from Figure 1 as reference. Each rectangle represents one simulated experiment, tone encodes blood pressure after 60,000 seconds of simulated experimental time, with lighter tones corresponding to higher blood pressures.

Mathematical Modeling for Prediction and Selection of Therapeutic Strategies The application of mathematical models to predict future developments in the individual patient and evaluate and optimize available therapeutic strategies with respect to outcome is the area of application with the largest potential benefits, since it could potentially support an approach to therapeutic decision making that is, at the same time, more individualized and more quantitative than the current para- digm of evidence-based medicine. Unfortunately, this area of application is also the most challenging and least developed. Current day implementations are mostly limited to small physiologic subsystems, oftentimes based on essentially empirical line- arized representations of the underlying physiology.

An example of the application of control theory to therapy is the closed-loop blood glucose control with infusion pumps, where the dysfunctional physiological control loop is replaced by a system consisting of a glucose sensor, a computer that makes the insulin dosing decisions, and a pump that injects insulin [40]. Other examples of model-based control include the closed-loop control of blood pressure and opioid dosing during surgery [41, 42].

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Fig. 4. Simulated dependency of outcome on the midpoint of the sigmoidal rate of tissue injury/arterial pressure relationship, and thus effectively tissue sensitivity to hypotension (x-axis) and the baroreflex response range (y-axis) of the full experimental protocol, when using the survivor parameter set from figure 1 as reference. Each rectangle represents one simulated experiment, tone encodes blood pressure after 60000 seconds of simulated experimental time, with lighter tones corresponding to higher blood pressures (here, a higher number of 120 × 120 = 14400 simulations was run to better resolve the structure at the boundaries of survival/death).

While this type of application can increase efficiency by reducing work load on care personnel while at the same time minimizing the possibility of human error, it seems doubtful whether the achievable increase in control precision over the level an experienced ICU nurse can achieve will create more than an incremental improvement in outcome, although that incremental improvement may be large.

A more comprehensive approach would seem necessary to obtain further improvements in outcome over unassisted human decision making. Such an approach would need to exploit the ability of mathematical models to incorporate arbitrary amounts of quantitative data as well as, in principle, an unlimited number of complex physiologic interactions into its current estimate of patient state and its prediction of future developments, and the effects of therapeutic options. Specifi- cally, this would involve making a best estimate of model parameters and states based on all available data, to then use the dynamic mathematical model to propa-

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gate the estimated system state into the future, and predict effects of available therapeutic interventions. Realizing such an interactive approach would markedly improve the usefulness of continuous, high precision measurements of physiologic variables in the ICU environment. Regrettably, some fundamental obstacles will have to be overcome first before such an approach is realized.

The most important obstacle is the lack of validated, quantitatively correct models describing sufficiently large parts of the relevant physiology of critically ill patients. This, however, does not represent a fundamental problem since it seems reasonable to assume that the human body as an, albeit extremely complex, physical system, should be amenable to mathematical description. The increasing amount of mechanistic insight into the body’s functioning generated by the basic sciences can serve to define the structural framework of relevant mathematical models. The complex, hierarchical structure of this organism, which spans several orders of magni- tude in spatial scales alone, may necessitate an at least partially stochastic description.

This insight leads to the next fundamental obstacle. When applying mathematical models to physiologic processes, it has traditionally been attempted to estimate a single set of parameters and/or system states of the model from available experimental data that represent a ‘best fit’, which, when using a least squares approach, corresponds to a maximum likelihood estimate. Unfortunately, there is often more than one solution equally compatible with the actual observations when using mathematical models complex enough to represent significant parts of our knowledge of the structure of the underlying physiologic processes (a phenomenon termed ‘ill- posedness of the inverse problem’). Although this plurality defines real life as well, it does complicate decision support algorithms that presumably focus on single point goals. Still, this ‘ill-posedness’ is a reflection of clinical reality. We should, therefore, develop ways to quantify these multiple solutions. At the very least, such solutions would allow us to get a quantitative handle on the present amount of uncertainty about a patient’s status based on our currently available information on physiology, bedside monitoring, and known treatments. To achieve this goal, we have recently proposed a methodology based on a Bayesian approach to probabilis- tic reasoning that derives density estimates in parameter and state space from the available observations and an underlying deterministic or partially stochastic model of pathophysiology [43]. Using this approach the patient’s status is described by a multidimensional and potentially multi-modal probability distribution, with con- centrations of probability density corresponding to possible differential diagnoses.

Subject to the availability of valid models, this approach would not only allow one to quantify the current uncertainty about patient status, but would also enable one to explore which additional observations (diagnostic interventions) could most effectively contribute to decreasing uncertainty about patient status. Additionally, propagation of state densities would allow for prediction and evaluation of therapeutic options while fully reflecting the uncertainty present, thus giving the physician an idea about the quality of the data he or she is basing their decisions on, which, in terms of patient safety, appears as an absolute prerequisite. This methodology could, in principle, be validated in controlled clinical trials, allowing for an approach to medical decision making that would satisfy the criteria of evidence- based medicine, while taking into account all known information about the individual patient at the same time.

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Conclusion

Functional physiological monitoring is a promising approach to making more directly interpretable measurements. The application of mathematical models to the critical care setting will allow one to both more accurately assess the patient’s physiologic status based on all available quantitative and qualitative information and infer individually optimal therapeutic strategies. This approach has the potential to become a powerful tool for improving outcomes and optimizing resource utilization in the critical care setting. However, most of its possibilities are still far from being realized, and only proof-of-concept examples have reached the maturity to allow beside application. Turning the conceptual framework outlined above into a useable aid for day-to-day decision making in the ICU of the future will require coordinated and highly interdisciplinary efforts from many scientific disciplines, including, but not limited to, medicine, physics, biology, and mathematics. We believe, however, that the eventual results, both in terms of improvements of care and in the deepening of our understanding of pathophysiologic processes relevant to critical illness, may well justify the investment.

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