From Here to Eternity: A Unified Kinetic Model for the Pathophysiology of Atherosclerotic Events
Article Outline
Abstract
Two operative pathophysiological models underlie the clinical management of ischemic heart disease: a physical model founded on the magnitude of vascular stenosis and a biochemical model founded on the inflammatory processes within the atherosclerotic plaque. Despite their complementary natures, these 2 models are implicitly competitive—the stenotic model supporting the primacy of aggressive interventional procedures and the inflammatory model supporting the primacy of conservative medical management. We unified these alternative perspectives through a kinetic model that characterizes the pathophysiology of cardiovascular events as a network of exponential transitions between the inflammatory and stenotic states. According to this model, the prevalence of the normative (nonstenotic and noninflammatory) state falls exponentially, while the prevalences of the inflammatory and stenotic states rise to a peak and then fall off exponentially. According to this model, event rate increases as a complex function of both myocardial ischemia and vascular inflammation. Although the model has yet to be prospectively validated, it provides a theoretical foundation for predicting the degree to which atherosclerotic events are due to inflammation versus stenosis and the degree to which they can thereby be prevented by treatment strategies directed at plaque stabilization or relief of ischemia.
Keywords: Atherosclerosis, Inflammation, Ischemia, Prognosis, Stenosis
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.
John von Neumann
Two operative pathophysiological models underlie the current management of ischemic heart disease: a physical model founded on the magnitude of vascular stenosis causing functional myocardial ischemia1 and a biochemical model founded on the inflammatory processes within the atherosclerotic plaque causing its anatomic disruption.2 The stenotic model leads us to perform stress tests, coronary angiograms, and revascularization procedures to restore the balance between myocardial oxygen supply and demand. The inflammatory model, on the other hand, leads us to detect the chemical mediators of risk and to prescribe pharmacological agents that mitigate their effects on the plaque (Figure 1).
Figure 1.
Putative pathophysiology and prevention of atherosclerotic events. The noninflammatory and nonstenotic state is characterized by a stable atherosclerotic plaque in the absence of a hemodynamically significant luminal stenosis (left). This normative state can transition to a vulnerable stenotic or inflammatory state (center) and thence to an irreversible morbid event (right). Depending on the processes underlying the transitions, these events might be prevented either by treating the stenosis (top) or treating the inflammation (bottom).
Each of these models has its supporters, its successes, and its shortcomings. Both comport with some aspects of the empirical evidence, but neither has been validated directly. Thus, while exercise-induced myocardial ischemia is well known to have important prognostic utility,3 the therapeutic relief of that ischemia has comparatively little prognostic benefit.4 In contrast, even though conventional risk factors are comparatively less accurate predictors of morbid cardiovascular events,5 medical therapy directed at these risk factors substantially reduces the frequency of these events.6, 7, 8 As a result, clinicians provisionally rely in varying degrees on both these models to justify their day-by-day management decisions.
Nevertheless, neither model formally acknowledges the complementary role of the other. In fact, the 2 are tacitly taken to be competitive—the stenotic model supporting the primacy of aggressive interventional procedures and the inflammatory model supporting the primacy of conservative medical management. According to Nobel Laureate Manfred Eigen, polar perspectives such as these are worse than wrong—they are “…
right, but irrelevant.”9
Consider the following scenario: John undergoes a routine medical evaluation, in the course of which an LDL cholesterol and high sensitivity C-reactive protein are observed to be elevated. He subsequently undergoes exercise-redistribution myocardial perfusion scintigraphy, which reveals reversible regional hypoperfusion. What, he asks, is his risk for a cardiovascular event, and what should be done to reduce that risk? How would things change if only the exercise test (or only the chemistries) were abnormal?
Clinicians betray their provisional allegiance to the stenotic or inflammatory perspective by the actions they choose in answer to these questions and by the phrases they use to justify their actions—phrases such as “the open artery hypothesis,” “risk stratification,” “supply-demand imbalance,” “the culprit lesion,” “vulnerable plaque,” “jeopardized myocardium,” “high-risk disease,” and “unstable angina.” As we shall see, a unified model that cuts through this rhetorical gloss and integrates the 2 incomplete and inconsistent perspectives within a common conceptual framework would very likely improve the accuracy of prognostic and therapeutic assessments and sharpen the rationale of future research efforts at the bench and at the bedside. Of no less importance, it would provide straightforward answers to John’s heartfelt questions.
Conventional attempts to do this rely on multivariate statistical regression. Indeed, this approach can account for the degree to which inflammatory predictors (such as C-reactive protein) and ischemic predictors (such as perfusion scintigraphy) independently contribute to overall prognosis,10 but the resultant statistical prediction rule is entirely empirical, offering no assurance that its prognostic assessments correspond with projected therapeutic benefit.11 What we need, instead, is a model that begins from a firmer theoretical foundation.
The Kinetic Model
A kinetic model fulfills this need.12, 13 Such a model is mathematically more complex than those typically encountered in the medical literature, but as we shall see, this complexity is justified by the model’s fidelity to the underlying pathophysiological principles, its correspondence with empirical observations, and its applicability to a broad range of clinically important questions.
Briefly, a kinetic model quantifies the time dependent transition from state A to state B (denoted A → B), the states being expressed in terms of absolute or relative prevalence (denoted [A] and [B]), and the time dependence being expressed in terms of a rate constant (k) or half-life (t1/2 = ln 2/k). The canonical transition of this kind is that of a monotonic exponential decay ([A] = e−kt), where [A] = 1 at t = 0, the rate of change for [A] is inversely proportional to its prevalence, and the rate constant, k, is the hazard:
Assuming that each of the state-to-state transitions leading to a cardiovascular event obeys this simple exponential law, we can construct a biologically plausible kinetic model for the process (Figure 2). According to this model, the normative state N can transition either to the inflammatory state I or to the stenotic state S. The inflammatory state, in turn, can transition to the stenotic state or to the event state E, and the stenotic state can similarly transition to the inflammatory state or to the event state.
Figure 2.
A generalized kinetic model of the pathophysiological processes outlined in Figure 1. The arrows represent potential state-to-state transitions (N = normative, I = inflammatory, S = stenotic, E = event), and the associated rate constants (k1 through k6) quantify the empirical degree to which each potential is realized (a value of 0 indicating impossibility).
The rate of each potential transition is quantified by its associated rate constant (k1 through k6), which can be interpreted in terms of the observable pathophysiological consequences summarized in Table 1. This clinical context distinguishes the model from an artificial neural network,14 to which it otherwise bears a superficial schematic resemblance, while the quantitative rate constants distinguish it from qualitative phenomenological schemas such as that championed by Braunwald.15
Table 1. Putative Pathophysiology of Atherosclerotic State Transitions
| Transition | Rate Constant | Acceleration | Deceleration |
|---|---|---|---|
| N → I | k1 | Acute coronary syndrome | Plaque stabilization |
| N → S | k2 | Stable angina | Revascularization |
| Vascular remodeling | Anti-anginal therapy | ||
| S ⇆ I | k3 and k4 | Accelerated symptoms | Clinical stability |
| I → E | k5 | Plaque rupture | Platelet inhibitors |
| Thrombosis | Reperfusion | ||
| STEMI | Thrombin inhibitors | ||
| S → E | k6 | NSTEMI | Early intervention |
| Vasospastic events | Beta blockers | ||
| Nitrates |
The mathematical representation of this kinetic model is derived in the Appendix. Its formal implementation requires the empirical enumeration of each of the rate constants from data elucidating the time course of the state-to-state transitions, and these data are currently not available in the medical literature. To illustrate the potential clinical utility of the model, then, we compiled a preliminary set of rate constants (k) and transition rates (1 − e−k) based on available observational data.16, 17, 18, 19 These parameters are enumerated in Table 2, and the resultant kinetic model is illustrated in Figure 3. According to this model, the initial prevalence of the normative state falls from unity as a simple exponential function, while the prevalence of the inflammatory and stenotic states increase from 0 (sometimes—depending on the values of the rate constants—to a single peak at a time denoted tmax). As a consequence, event rate increases as a complex curvilinear function. Two specific verifiable predictions of this particular model, then, are that the prevalence of inflammation is approximately twice that of ischemia over much of the natural history of the disease and event rate increases nonexponentially with time.
Table 2. Preliminary Parameters of the Kinetic Model
| State Transition | Half-Life (years) | Rate Constant (years−1) | Annual Rate (%) |
|---|---|---|---|
| N → I | 10 | k1 = 0.0693 | 6.7 |
| N → S | 30 | k2 = 0.0231 | 2.3 |
| I → S | 20 | k3 = 0.0347 | 3.4 |
| S → I | 8 | k4 = 0.0866 | 8.3 |
| I → E | 9 | k5 = 0.0770 | 7.4 |
| S → E | 27 | k6 = 0.0256 | 2.5 |
Figure 3.
A particular application of the generalized kinetic model based on the putative rate constants in Table 2. The proportional prevalence for each state (N, I, S, E) is plotted over 240 months of follow-up. The prevalence of the normative (N) state falls as a simple exponential decay. The prevalences for the intermediate stenotic (S) and inflammatory (I) states increase to a maximum (at 112 and 141 months, respectively). As a result, the prevalence of events (E) increases as a relatively flat curvilinear function over time.
The sensitivity of this model to changes in the rate constants is illustrated in Figure 4. Consistent with the empirical observations upon which the model was parameterized,18, 19 changes in event rate are more sensitive to changes in the inflammatory transition rates than to changes in the stenotic transition rates, suggesting that the former plays a greater role in the pathophysiology of morbid events than does the latter. As we shall see, these differential sensitivities have important clinical consequences.
Figure 4.
Sensitivity analyses (relative change in percent event rate on the y-axis versus relative proportional change in the rate constant on the x-axis) for individual state transitions. For this set of parameters, the curves with the greatest slope represent k1 and k5. Event rate is therefore most sensitive to relative changes in the inflammatory transitions (N → I and I → E).
Clinical Implications
Although risk and hazard are often used interchangeably, their formal distinction has important clinical relevance. Just as the physical trajectory of an object depends both on its temporal displacement (velocity) and on the rate of change of that displacement (acceleration), the prognostic trajectory of a patient depends both on the temporal threat of an adverse event (risk) and on the rate of change of that threat (hazard).13 Consequently, in the same way we change a car’s velocity by applying a suitable mechanical force (the brakes), we change a patient’s risk by applying a suitable biological force (a therapeutic intervention). Some such interventions (replacing a worn tire or relieving a stenosis) target the risk itself, while others (adjusting the tire’s pressure or the patient’s lipid levels) target its rate of change.
Nevertheless, conventional approaches to therapeutic triage20, 21 overlook these distinctions and rely on point estimates of risk alone. Even the most sophisticated of these approaches are rarely founded on plausible biological principles, but more often on obscure statistical standards such as minimization of variance or maximization of likelihood. For these reasons, the inflammatory and stenotic pathophysiological models that serve as the foundation(s) for much of current cardiovascular practice ultimately fail to meet von Neumann’s epigraphic criterion. They cannot be “…expected to work”—in fact, they do not work—because neither takes proper account of the other.
Our kinetic model bridges this divide in 2 ways. First, because it quantifies the dynamics of the state-to-state transitions, instead of the static correlations among the states, it predicts changes in risk—hazard—in addition to the level of risk predicted by a statistical model. Second, instead of relying on clinically obscure standards, such as minimization of variance, the kinetic model rests on a consistent and plausible biological foundation (the interplay between ischemia and inflammation). Consequently, its predictions will likely be better informed, richer in content, deeper in meaning, and more far-reaching in implication.
Not only does a kinetic model produce clinically realistic predictions when parameterized by rate constants derived from retrospective empirical data,22, 23, 24, 25 but it continues to do so even when pushed to extremes (Figure 5). Thus, a 10-fold increase in the inflammatory rate constants—mimicking an acute coronary syndrome—produces a 70-fold increase in 6-month event rate (from 0.14% to 9.2%), which is in close agreement with a statistical prediction model based on 22,645 patients in the Global Registry of Acute Coronary Events (GRACE).22
Figure 5.
Simulation of an acute coronary syndrome by a 10-fold increase in the value of the inflammatory rate constants (k1 and k5). Note the marked increase in event rate (E) compared with Figure 3, and the peak in the inflammatory state (I) at tmax = 11 months.
A recent study of cholesterol-induced atherogenesis provides additional indirect confirmation of these predictions. Just as predicted by our kinetic model (and in contrast to a statistical model), the expression of inflammatory mediators in ApoE-deficient mice increased to a peak similar to that in Figure 5 and decreased thereafter, even as the atherosclerotic plaque continued to enlarge.23 These dynamics parallel the time course of morbid events following coronary stenting.24 Similarly, an increase in k6 or a decrease in k3 is sufficient to mirror the otherwise paradoxical associations between plaque growth and inflammation observed in some patients.25
The prognostic implications of alternative therapeutic strategies might thereby be modeled in terms of quantitative differences in the state-to-state transition rates—surrogate markers of disease progression26—rather than ad hoc qualitative differences in the underlying pathophysiological processes.15 According to the current model, for example, a halving of k2 (a proxy for relief of ischemia) reduces 5-year event rate by only 6%, while a halving of k1 (a proxy for anti-inflammatory therapy) reduces event rate by 36%. Halving both rate constants reduces event rate (additively) by 44%. These reductions are consistent with a number of clinical trials showing greater improvement in outcome by lipid lowering6 than by revascularization.27 If such inferences were prospectively verified, this model would allow one to predict the incremental benefits of treatment strategies directed at stabilization of the atherosclerotic plaque versus restoration of myocardial blood flow. Here, then, is the way to provide rational answers to John’s questions about risk and benefit.
Kinetic models have additional relevance to the design and interpretation of therapeutic trials. Suppose you are performing a clinical trial to assess the effect of some therapeutic intervention on some biochemical marker of inflammation. Your entry criteria call for enrollment of subjects exhibiting no evidence of ischemia or inflammation over the preceding year. Without knowing it, then, you just happen to be conducting this trial during the period t < tmax (when, as in Figure 5, our model predicts inflammation to be on the increase). As a result of this inopportune timing, you will observe a paradoxical increase in your inflammatory marker even if the intervention actually reduces the rate of inflammation. Similarly, if you happen to conduct the trial during the period t > tmax (when the model predicts inflammation to be on the decline), you will now observe a decrease in the inflammatory marker even if the intervention has no effect. These errors can be avoided by evaluating the rate constants themselves (the hazards) rather than the frequencies of the outcomes (the risks).
Currently, these rate constants quantify only the average hazards among a group of patients. Although this is sufficient for strategic planning, it is the specific hazard in a particular member of the group that is needed for patient management. Fortunately, the model can be adapted to provide such patient-specific estimates. Rate constants, after all, are not really constant. Just as chemical rate constants vary with temperature, atherosclerotic rate constants vary with the magnitude of inflammation and ischemia—the greater these magnitudes, the greater the rates. Once these relations are defined, the inflammatory and stenotic markers observed in an individual patient can be used to generate a set of patient-specific rate constants, and these, in turn, can be used to create a patient-specific kinetic model from which we can derive patient-specific predictions of outcome. Modest changes in one or another rate constant, for example, readily account for observed differences in outcome between men and women.28
Even so, clinical atherosclerosis is not restricted to a single lesion or to a single vascular region. Instead, it comprises a heterogeneous spectrum of lesions undergoing continual transitions between the stenotic and inflammatory states. One of the strengths of our model is its ability to capture the complexity of this process. In chemical kinetics, the outcome depends on the slowest of the component transitions (the so-called rate-limiting step). In clinical kinetics, just the opposite is the case, the observed outcome being dependent on the fastest set of transitions (reflective of the most dynamic lesion). The lesions manifesting these accelerated transitions are likely to be the very same ones characterized by clinicians as anatomically “culprit” or biochemically “vulnerable.”
In this context, a consortium of physician-scientists recently called for the development of “[a] quantitative method for cumulative risk assessment of vulnerable patients…that may include variables based on plaque, blood, and myocardial vulnerability.”29, 30 Our kinetic model serves this purpose. The inflammatory axis of the model comprises aspects of both “vulnerable plaque” and “vulnerable blood,” while the stenotic axis is an indirect representation of “vulnerable myocardium.”
The key advantage of this model, then, is its ability to unify alternative conceptual perspectives (just as Maxwell’s equations unified the electric and magnetic fields) from which we can deduce empirically testable predictions (just as Maxwell deduced the constancy of the speed of light from his equations). The general form of this model can be applied to similar problems in other domains: the development of heart failure as the event state resulting from hypertension or ischemia as the intermediary states, to name but one example.
Two limitations of the current model deserve comment. First, the model fails to consider a variety of other potentially relevant genetic and environmental factors, such as insulin resistance,31 which might serve as additional intermediate transitional states, or that some of these transitions (N → I and N → S) might be reversible. Although a more extensive model can be constructed to incorporate these additional considerations, it comes at the cost of increased computational intensity.12 Second, we cannot emphasize too strongly that the transition rates employed here are intended only for pedagogical purposes and must be determined prospectively before clinical application using quantitative methods of measurement analogous to those in chemical kinetics.32 We hope that our preliminary exposition of this kinetic model will encourage others to perform these empirical determinations. If some of the resultant transitions exhibit time dependencies more complex than that associated with a monotonic exponential decay (as do some chemical kinetic processes), the model will have to be modified to incorporate these higher order dynamics.
In summary, just as one cannot properly assess the dynamic processes resulting in blood flow without simultaneous consideration of pressure and resistance, one cannot assess the dynamic processes resulting in atherosclerotic events—the corporal flow “…from here to eternity” in the words of Kipling—without simultaneous consideration of the physics of anatomic stenosis and the chemistry of plaque instability.33, 34 The kinetic model described herein performs that function. Although not yet prospectively validated, the model is formally scientific in design. In unifying 2 otherwise disparate pathophysiological mechanisms, it uses accepted biological principles and mathematical techniques, makes a number of specific verifiable predictions of palpable clinical and economic importance, and is supported by the available body of indirect empirical evidence. As a result, the model serves to reveal to interventional cardiologists the limits of their myopic focus on the lumen and to medical cardiologists the limits of their similarly myopic focus on the plaque.
Von Neumann, we think, would approve.
APPENDIX
Assuming that each of the state-to-state transitions in Figure 2 obeys a simple exponential law, we can construct a kinetic model for the process in which the set of transitions is represented as a simultaneous series of linear differential equations (hazard functions):
The rate constants can be evaluated empirically from published follow-up studies or prospective investigations using conventional methods derived from chemical kinetics.31 Thus, if inflammation or stenosis alone is a sufficient cause of atherosclerotic events, the rate constants representing the pure form of that model will have values > 0, while those representing the alternative model will have values equal to 0. Counterfactual transitional elements are thereby automatically pruned from the model. Standard errors, confidence intervals, and significance tests for these rate constants can be calculated using conventional statistical techniques.
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PII: S0002-9343(06)00567-5
doi:10.1016/j.amjmed.2006.04.021
© 2007 Elsevier Inc. All rights reserved.
