Current mathematical study of cardiovascular variabilities follows three main approaches. First, attempts have been made to construct simple, generally novel quantitative indices to classify broad physiological or pathophysiological conditions. Ratios of low to high frequency spectral power have been used to gauge ‘sympathovagal balance’ (Malliani *et al.* 1994), Mayer wave frequency power has been used to classify congestive heart failure patients (Teich *et al.* 2000) or predict death after heart attacks (Bigger *et al.* 1992*a*, *b*), and the slope of very low frequency blood pressure oscillations has been used as a correlate for cardiovascular mortality (Makikallio *et al.* 1999). When successful, this work produces useful classification tools for clinical populations. However, to produce physiologically insightful models of these variabilities, a set of equations to explain the relations between them is necessary. These more complete explanations form the latter two approaches. One approach uses principles of applied mathematics and physics to describe cardiovascular rhythms via complicated, generally non-linear models, for example, non-linear delay differential equations (Cavalcanti & Belardinelli, 1996; Ottesen, 1997), difference equations (de Boer *et al.* 1987), non-linear oscillators (Abbiw-Jackson & Langford, 1998), regularization theory (Seydnejad & Kitney, 2001) and non-linear fluid dynamics (Olufsen, 1999; Olufsen *et al.* 2000). Since these models are based on simple physical principles, they can explain the physical origins of cardiovascular oscillations and could lead to unique insight. However, current application of these mathematical approaches produces models that are quite difficult to evaluate statistically and provide at best a theoretical framework for possible mechanisms. The other approach uses application of linear time series analysis and systems identification to study the connection between these variabilities (Pagani *et al.* 1986; Cerutti *et al.* 1996; Chon *et al.* 1997; Mullen *et al.* 1997). In many cases, these are misapplied to construct generally improperly validated linear stationary models of cardiovascular variables such as heart rate, blood pressure, lung volume and sympathetic nerve activity. Moreover, although easily interpreted, these models have been limited largely to single-input relations for mechanistic inferences (Baselli *et al.* 1988; Parati *et al.* 1995; Kocsis *et al.* 1999) and routinely ignore differences across multiple subjects. Only these latter two approaches will be considered here; although simple indices may have clinical utility and may even lead to more complete formulation of potential models, an explanation of the relation between physiological variables requires a set of equations that specify possible inter-relations.

#### Model criteria

Any model applied to cardiovascular variabilities must meet certain criteria. Of course, what is paramount is whether the model makes an accurate prediction. Generally, this accuracy must be more than a minimal, statistically significant relation, since it should capture a large portion of the true relation between variables (Box *et al.* 1994). The validity of the model should be assessed by simulations and goodness of fit statistics, whereas reliability of the posited parameters can be assessed easily from their confidence intervals; it is best if the model is simple, having few parameters and postulating simple quantitative relations (Box *et al.* 1994). The idealizations and assumptions should have a natural physical interpretation; only then will the model will be useful for further modelling and experimentation (Koopmans, 1995). Moreover, the equation terms should represent physiologically meaningful variables expected to relate to one another and interindividual variability should be expressed and explained, since humans are not phenotypically homogeneous. Lastly, it is useful if the model can be compared with alternative models of the same phenomena (Kaplan *et al.* 1991).

It should be noted that there is generally a tradeoff between accuracy and simplicity in applied science (Koopmans, 1995). A model that provides highly accurate predictions is usually neither simple nor physically interpretable. However, simplicity and interpretability are the more important criteria for mathematical models of human physiology. There is little economic, social or military need to have highly accurate predictions for the timing of the next heartbeat, the amplitude of the next blood pressure wave or the size of the next lung volume. The need is for explanatory, generalizable models of cardiovascular function rather than purely predictive machines of cardiovascular variability.

#### Models deriving from principles of applied mathematics and physics

Generally, although theoretically possible, statistical validation or analysis is rarely attempted for models deriving from principles of applied mathematics and physics because of the complexity in the transfer between input and output and the often large number of implicit parameters. However, despite inadequate empirical validation, these models can provide a rich source of physical intuitions due to their dependence upon known physiological or physical principles. Most are constructed from principles of hydrodynamics and hydrostatics.

Most of these models, realized as differential or delay differential equations, lump the entire cardiovascular system into a small number of variables for simulation and analysis. A routine starting point for many of these models is the Windkessel model of cardiovascular fluid flow that consists of resistors, a capacitor and a non-linear function. For example, Cavalcanti & Belardinelli (1996) used this to provide an oscillator that can be tuned to produce spectra mimicking typical Mayer waves. An idealized model of the Windkessel equations was also used by de Boer *et al.* (1985) to produce time series, power spectra, and cross-spectra that were comparable to those observed in human subjects. Further, Abbiw-Jackson & Langford (1998) modified the Windkessel system by incorporating two pumps, one for each side of the heart, and adding the baroreflex via a non-linear transfer from pressure to heart rate. This model displayed persistent oscillations that could be abolished by a decrease in the modelled baroreflex gain or an increase in venous volume, presumably reflecting the effect of ageing.

Although these physical approaches produced seemingly useful models, most of those assessing cardiovascular variability suffer serious limitations. All have lacked quantitative or statistical validation; the implicit assumption seems to be that if it produces physiological-looking waveforms, the model must be physiological. It is also unclear why diastolic or systolic pressure are usually considered; for these type models, they are merely arbitrary local maxima and minima on the continuous analogue pressure waveform that should be in the model. Generally, no noise is input to these models so that the entire trajectories of the waveforms produced by it are completely predictable. The above models are also open to the criticism that lumping cardiovascular variables, while simplifying the model, can invalidate the underlying physical assumptions. These limitations can be avoided. For example, although not directly examining slow wave oscillations, Olufsen and colleagues (Olufsen, 1999; Olufsen *et al.* 2000) successfully applied a full set of hydrodynamic equations to predict fluid and pressure waveforms in the entire arterial tree. Even though extensive validation is not physically possible, they did provide some quantitative validation from excellent fits to MRI flow data. Therefore the strength of these approaches, the application of physical principles, can be realized, but in most applications to cardiovascular variability this strength has actually proved illusory.

#### Time and frequency domain models

Spectral analysis has yielded important insights, including delimiting frequency responses of the cardiovascular system and potential causal relations among cardiovascular and respiratory variables. Spectral analysis can be accomplished directly by non-parametric fast Fourier transform (FFT) analysis and windowing in the frequency domain (Priestley, 1994) or indirectly by autoregressive multi-parameter modelling of the time series (Barbieri *et al.* 1997). Rather than reproducing the power spectrum, simplified autoregressive models have the potential of quantifying essential delays and gains by which blood pressure, cardiac interval, sympathetic activity and lung volume affect each other. Moreover, non-stationary variants of these autoregressive models have the capacity to treat trends in the data and can be extended to certain classes of non-linear models. However, the spectral models used to date have either only assessed a single input-single output relation or have been too complex (i.e. over-parameterized) for direct validation.

In general, these models posit a linear and stationary relation between independent and dependent variables. Throughout this literature it is generally further assumed that a linear relation holds between the directly measured variables. However, it is possible that suitable, non-linear transforms of the input data should be used; non-linear effects are amply documented in cardiovascular control systems, for example, saturation and threshold elements (Eckberg, 1980), respiratory gating (Seals *et al.* 1993), hysteresis (Rudas *et al.* 1999), sinoatrial node transduction (Michaels *et al.* 1987), CO_{2} effects on arterial diameter (Ursino & Lodi, 1998) and mechanical saturation of aortic diameter (Wesseling *et al.* 1993). Nonetheless, this approach has provided some information on the characteristics of cardiovascular oscillations. For example, it is clear that cardiac interval spectral power is inversely proportional to respiratory frequency (Hirsch & Bishop, 1981; Brown *et al.* 1993). Moreover, cross-spectral derived phase relations indicate that cardiac interval is in phase with arterial pressure in supine humans, but lags pressure in upright humans (Saul *et al.* 1991; Taylor & Eckberg, 1996). A close coherence between arterial pressure Mayer waves and both cardiac interval and sympathetic nervous oscillations is usually shown; however, cross-spectral phase between arterial pressure and sympathetic activity at the Mayer wave frequency provides no firm support for either a predominant feedback or feedforward mechanism (Taylor *et al.* 1998*b*). Thus, simple spectral models provide the characteristics of the oscillations, but do not provide insight as to whence they derive.

In contrast, autoregressive linear stationary time series modelling imposes causality upon the model and hence the power spectrum estimate. In addition, the model selection is, in principle, an objective procedure. This provides the advantage of a methodology which can identify the feedforward and feedback links between variabilities. One example is the autoregressive model proposed by Nakata *et al.* (1998) to explore the links among heart rate, systolic pressure and sympathetic activity at the Mayer wave frequency. This novel approach suggested that sympathetic oscillations predict ≈70 % of the power in systolic Mayer waves, an effect partly abolished by α-receptor blockade. Ironically, the novelty of this approach prevents direct comparison to previous work, all of which used cross-spectral and coherence analysis generally at fixed temporal frequencies. Moreover, confidence intervals for each subject's relative power contributions were not provided and interindividual and intergroup variability in the relative contributions were not assessed. Thus, the estimates could intersect with zero and vary widely from subject to subject, limiting the usefulness of this model. This has been the case with other similar autoregressive models examining syncope (Di Virgilio *et al.* 1997; Mainardi *et al.* 1997) and interactions between lung volume and heart rate (Mullen *et al.* 1997).

One evident problem of autoregressive models is the potentially large number of parameters. Since the model selection procedure defines lags iteratively from zero, a model that includes a coefficient with lag *t* typically includes all coefficients with lags less than *t* (Box *et al.* 1994; Ljung, 1999) As a result, not only are the parameters for an adequate fit reached quickly within a small lag range, but also the number of parameters are inflated if all lags less than *t* are included. It is not uncommon for standard autoregressive models of 4 Hz cardiovascular time series to include in the order of 40–100 parameters (Barbieri *et al.* 1997). Although effectively increasing the variance that the model can predict (Penm & Terrell, 1982; Duong, 1984), it excludes interpretation of the model's significance to the physiology. That is, the existence of 40 different time relations between heart rate and arterial pressure changes is difficult to reconcile with accepted concepts of haemodynamic control. In addition, even more parameters may be introduced because autoregressive models allow for arbitrary feedback interactions between independent and dependent variables. For example, a 1000 time point series of lung volume, heart rate, blood pressure and sympathetic activity will exhaust the degrees of freedom with time lags representing only 1.2 % of the entire data length, if allowed full interactions (Box *et al.* 1994). One major reason for this lack of parsimony is the dominant use of standard scalar or vector autoregressive models rather than the more parsimonious transfer function models introduced by Box and Jenkins over 30 years ago (Box *et al.* 1994), perhaps due to the relative unavailability of software. Thus, the standard approach generally leads to many more estimated parameters and obscures the relation between input and output. Not only are longer (and, perhaps, more meaningful) lags not considered, but the parameters can be capricious. For example, current blood pressure could be linked causally to future lung volumes. Nonetheless, a large number of closely spaced, commensurately sized parameters are common in linear autoregressive models of cardiovascular variabilities (Di Virgilio *et al.* 1997; Korhonen, 1997; Mainardi *et al.* 1997). This suggests that currently proposed models have poor predictive reliability and limited physiological generalizability and that models with few, sparse parameters might improve our understanding of cardiovascular oscillations.

Although simplified autoregressive models can potentially quantify essential delays and gains that define interactions among variabilities, they generally assume that the cardiovascular rhythms are stationary. This may mean that they only partially model the data due to the presence of significant non-linearities and non-stationarities (Hayano *et al.* 1993; Jasson *et al.* 1997; Badra *et al.* 2001; Mangin *et al.* 2001). The attempt to fit non-linear data to a linear model can result in model parameters that depend largely on the distribution of the input and output, not on the actual causal relation between them. As a result, the goodness of fit would be highly variable. Such effects have been documented in frequency domain estimates of baroreflex gain (Badra *et al.* 2001). Therefore, it would seem that non-stationary variants are the most advantageous autoregressive models for cardiovascular time series. They have the capacity to treat trends in the data and can be extended to certain classes of non-linear models. Although such autoregressive modelling techniques have been used (Chon *et al.* 1996; Di Virgilio *et al.* 1997), they generally have had limited validity since they usually fit only small sections of the data with the same parameters. This effectively represents applying sequential stationary models, thereby limiting their predictive accuracy.

#### A simple model

Although linear models have limitations, a conservative starting point for understanding cardiovascular variabilities is to force a linear model that is as simple as possible. If at least a partial linear relation exists between the input and output variables, then positing the simplest possible relation will probably reveal it. The simplest of linear relations are easily related to physical principles of fluid dynamics. For example, a simple model expressing arterial pressure Mayer waves as a linear combination of heart rate and sympathetic activity can be derived from Poiseuille's law. This model can be limited to two weighted inputs, each with a single time lead, allowing direct assessment of the time relations and the relative contributions of heart rate and sympathetic activity to Mayer wave amplitude. In addition, the simple model facilitates characterization of both trait (e.g. intersubject) and state (e.g. high sympathetic outflow) differences in Mayer wave genesis. An application of this model was able to account for approximately half the variance in Mayer wave amplitude (Myers *et al.* 2001). Moreover, it showed that sympathetic activity contributes to Mayer wave amplitude most when sympathetic activity is high, but that heart rate was a much more potent contributor, regardless of state differences (see Fig. 6). This simple model suggests that Mayer waves can be generated by mechanisms other than sympathetically mediated vasoconstriction, such as autochthonous vascular smooth muscle contractions (Siegel *et al.* 1976). Moreover, this demonstrates that the simplest possible bivariate linear autoregressive model can lead to testable hypotheses to guide further experimental work.

#### Other mathematical explanations and characterizations

The bulk of this review concerned modelling with regression, providing fits to physiologically observed quantities to predict their values. However, there are attempts to characterize time series with intrinsic scaling relations, fractal analysis, or series entropy. Under many conditions heart rate and blood pressure variability have a relatively large DC component, even when the signal is detrended. Some investigators have examined this and related measures (de Boer *et al.* 1985; Yeragani *et al.* 1993; Cerutti *et al.* 1996; Ivanov *et al.* 1999; Heneghan & McDarby, 2000; Mateo & Laguna, 2000; Teich *et al.* 2000), often with an interest to use these relations to assess pathophysiological conditions (Turcott & Teich, 1996; Makikallio *et al.* 1999; Porta *et al.* 2000, 2001). However, these applications can be highly problematical for both practical and theoretical reasons. Simple spectral power may provide predictors as good as (Bigger *et al.* 1993) or even better than (Teich *et al.* 2000) more complicated analyses. Moreover, slope estimates over very slow frequencies are, in fact, unstable, and pure white noise switching with Brownian motion at relatively long intervals can generate similar results (Pilgram & Kaplan, 1999). Thus, many of these metrics may simply be epiphenomena of non-stationarity.