## Abstract

Investigators have had much success solving the “hemodynamic forward problem,” i.e., predicting pressure and flow at the entrance of an arterial system given knowledge of specific arterial properties and arterial system topology. Recently, the focus has turned to solving the “hemodynamic inverse problem,” i.e., inferring mechanical properties of an arterial system from measured input pressure and flow. Conventional methods to solve the inverse problem rely on fitting to data simple models with parameters that represent specific mechanical properties. Controversies have arisen, because different models ascribe pressure and flow to different properties. However, an inherent assumption common to all model-based methods is the existence of a unique set of mechanical properties that yield a particular pressure and flow. The present work illustrates that there are, in fact, an infinite number of solutions to the hemodynamic inverse problem. Thus a measured pressure-flow pair can result from an infinite number of different arterial systems. Except for a few critical properties, conventional approaches to solve the inverse problem for specific arterial properties are futile.

- modeling
- input impedance
- effective length
- apparent compliance
- windkessel

arterial hemodynamics has historically focused on the relationship of arterial pressure and flow to the mechanical properties of arteries. The physics of blood flow is now well understood; if the input by the heart is characterized, as well as its load, then pressure and flow can be calculated with reasonable accuracy. The load of the heart can be characterized if all the mechanical properties of a given arterial system are known (i.e., vessel lengths, lumen cross-sectional areas, and compliances). The task devolves into solving a single equation describing arterial load that has many known mechanical parameters. This can be called the “forward problem.”

In recent years, investigators have focused on solving the “inverse problem,” i.e., inferring the mechanical properties of an arterial system from measured input pressure and flow. This is analogous to solving a single equation with known pressure and flow but with many unknown mechanical parameters. The inverse problem is particularly appealing to investigators who cannot measure directly specific arterial parameters but can measure input pressure and flow. Investigators have therefore attempted to solve the inverse problem by fitting reduced arterial models to measured data (6, 9, 11, 16,19, 23, 24, 34, 41). This approach reduces the number of unknowns by assuming that certain arterial parameters are zero, infinite, or simply nonexistent (cf. Ref. 37). The process of reduction eliminates the complexities arising from the particular spatial distribution of the arteries. However, different arterial system models retain different parameters (6, 19, 25, 38,49). Consequently, the choice of a particular reduced model to solve the inverse problem predetermines how measured data are interpreted. The use of reduced models to solve the inverse problem merely substitutes one problem for another.

The purpose of the present work is to test the assumption common to all methods to solve the inverse problem: information is extant in measured pressure and flow from which the mechanical properties of the arterial system can be uniquely determined. If this is not the case, then the popular endeavor to solve the inverse problem via reduced models is futile.

## BACKGROUND

### Describing the Arterial System Independent of the Heart

Arterial pressure depends on properties of the heart and the arterial system. To characterize the arterial system independent of heart properties, the concept of input impedance is employed (30). Input impedance (*Z*
_{in}) is a linear time-invariant transfer function relating pressure (P) to flow (Q), expressed in terms of frequency (ω; Fig.1)
Equation 1Because of its central importance, the steady value of input impedance is denoted peripheral resistance (*R*). It can be calculated from the ratio of average pressure (
) to average flow (
).
Equation 2Input impedance can characterize the dynamics of a system, providing that it is linear and time invariant. Nonlinearity can manifest itself in several ways, including harmonic distortion, where one input frequency yields multiple output frequencies (40). Although all arterial systems have some degree of nonlinearity, they may be treated as linear if the range of input is small enough. Although it is difficult to quantify the degree of nonlinearity (29), effects of nonlinearity have been estimated from 2% (14) to 5% (3) for normal pressures.

### Calculation of Input Impedance From Measured Pressure and Flow

The most common method to calculate input impedance requires the heart to be beating regularly. Pressure and flow are measured for a single cardiac cycle. These data sets, expressed in the time domain, are then converted to the frequency domain via discrete Fourier transform or fast Fourier transform algorithms. Input impedance is then expressed as the ratio of the harmonics of pressure to flow. For details of this calculation see Refs. 5 and 28.

To collect more information, a much more powerful technique can be applied. Rather than collecting data over a single cardiac cycle when the heart is beating regularly, data can be collected over multiple cardiac cycles when the heart is paced randomly. Random pacing increases the signal energy between multiples of heart rate. This extra information can be exploited by applying spectral analysis. It yields frequency components between the multiples of heart rate. Besides increasing the density of the data, this method allows characterization of the system at frequencies less than heart rate. Spectral analysis also provides calculation of coherence, a measure of the linearity of the pressure-flow relationship at any particular frequency. If there is no noise and the system is linear, then coherence has a value of 1. If, however, measured pressure is completely unrelated to flow, it has a value of 0. Coherence is analogous to a correlation coefficient describing the stability of the relationship of two signals. With coherence, it can be determined how well input impedance characterizes the arterial system (44).

### Forward Problem: Predicting Impedance Given Mechanical Properties

#### Motivation for solving the forward problem.

The theory of blood flow in an artery was well developed by the mid-1950s (29, 32). The next logical step was to use this theory as the basis to describe blood flow in an entire arterial system. Of particular interest was how the mechanical properties of the arterial tree determine the pressure-flow relationship measured at its entrance. That is, given measured arterial mechanical properties, the goal was to predict input pressure and flow (or input impedance). This will be termed the “forward problem” (Fig.2
*A*). Solution to the forward problem promised to explain pressure pulse morphology and indicate which arterial properties determine ventricular afterload.

#### Conventional methods of solving the forward problem.

After several attempts to use simple models to predict pressure and flow, the basis for the solution arose in the process of elucidating the physical basis of ballistocardiography (33). Originally, an electrical analog model was constructed in which current was made analogous to flow and voltage was made analogous to pressure. This model was improved and parameter values were updated as more information became available (22, 31). These developments culminated in a model presented by Westerhof et al. (47,48) that had realistic input impedance. It was not the last version of this evolving model, but it has formed the basis for several new species (4, 42). Describing the large arteries, these models used resistances or modified windkessels to represent the smaller arterial beds. Although it is no longer necessary to implement this model as an electrical analog, the structure, parameter values, and basic equations have continued to be used with great success (4, 17, 39-42). Much of our understanding of arterial system dynamics can be attributed to one or another version of this model. Because many of the questions originally motivating its development have been resolved, recently there has been a fundamental shift in the type of questions being posed.

### Inverse Problem: Inferring Mechanical Properties From Pressure and Flow

#### Motivation to solve the inverse problem.

Now that the forward problem has been, for the most part, solved, the “inverse problem” has become the dominant challenge. That is, given input pressure and flow (or input impedance) of an actual arterial system, the new goal is to determine arterial mechanical properties (Fig. 2
*B*). Because there are innumerable arteries, the focus has been limited to determining the properties of the larger arteries. Of particular interest in recent years is ascribing changes in input impedance, pressure, or flow to changes in mechanical properties. Thus the effect of disease states or pharmacological agents is to be attributed to their direct action on particular mechanical properties. By comparing Fig. 2
*A* with Fig. 2
*B*, it becomes clear that solving the inverse problem is considerably more challenging than solving the forward problem: much less information is available. In this case, there is only one known value (*Z _{in}
*) and numerous unknown values (i.e., numerous arterial compliances, lumen areas, and lengths).

#### Early attempts to solve the inverse problem.

Two historical approaches were originally used to relate input pressure and flow to arterial mechanical properties. In one approach, a model known as the windkessel was developed that described the arterial system as a compliant chamber that stores blood. Arterial pressure and flow or, alternatively, input impedance was related to total arterial compliance and total peripheral resistance. In the other approach, the arterial system was described by the infinitely long tube model. Input impedance was related to local arterial compliance and cross-sectional area of this vessel (30).

Once flow could be adequately measured in vivo, it became clear that neither of the historical models adequately described the data (30, 47, 49). The windkessel, neglecting pulse wave propagation and reflection, fails to adequately describe input impedance at higher frequencies, where propagation and reflection play a prominent role. The infinitely long tube, lacking a term describing total arterial compliance, fails to describe input impedance at low frequencies, where total arterial compliance plays a critical role. These two simple models, both lacking specific information about the spatial topology of the arterial system, were rejected as bases from which to solve the inverse problem.

#### Conventional methods to solve the inverse problem.

On careful scrutiny, conventional methods to solve the inverse problem are revealed to be variations of a single approach. First, a model of the arterial system with unknown parameters is assumed. Then the model is fit to pressure and flow or, equivalently, to input impedance. The values of the parameters represent the retrieved information.

With a putative physiology-based model, the quality of the information retrieved from data is assumed to be high if estimated error of each of the parameter estimates is small. If the model is too simple, then it cannot fit the data well, and there is large error in the parameter values. For this reason, the windkessel and the infinitely long tube have been rejected outright as models from which to retrieve information. However, if there are too many parameters, this will also cause large estimation error, since the data could be described adequately by multiple sets of parameter values. For this reason, a fully distributed arterial system model is rejected. The challenge faced by investigators relying on curve fitting has been to achieve the optimal balance of the number of unknown parameters and the goodness of fit (1).

#### Fundamental assumption.

The conventional methods to solve the inverse problem have a common fundamental assumption: that the information about specific arterial properties is contained in measured pressure and flow or, equivalently, in calculated input impedance. That is, all methods are based on the implicit premise that there is a unique solution to the hemodynamic inverse problem.

## THEORY

The mechanical properties of arteries, i.e., lengths, areas, and compliances, determine input impedance of a vascular bed. These three mechanical properties, along with properties of blood (density and viscosity), determine the hemodynamic properties of the arteries, i.e., capacitance, inertance, and resistance. In small vessels, capacitance effects become negligible and inertance and resistance effects dominate. In large vessels, capacitance and inertance effects dominate and resistance becomes negligible (28, 47).

The transmission line equations provide a link between hemodynamic properties, i.e., resistance, inertance, and capacitance, and the mechanical properties of a vessel and the blood it contains, i.e., length (*l*), cross-sectional area (*A*), volume compliance per unit length (C′), viscosity (μ), and density (ρ) (29, 30). The transmission description is derived from the Navier-Stokes equations and a condition of continuity. The solution is simplified by assuming that second-order terms of the Navier-Stokes equations are negligible and by neglecting the nonlinear terms. Following a solution for a thick-walled constrained compliant vessel, the three related qualities, i.e., resistance (*R*′), inertance (*L*′), and capacitance (*C*′; all normalized by length), can be expressed in terms of lumen cross-sectional area, blood density, and blood viscosity (28, 29,32)
Equation 3a
Equation 3b
Equation 3cExpressed in this way, a vessel's capacitance (a hemodynamic property) is mathematically equivalent to its compliance (a mechanical property). The parameters *M*′_{10} and ε′_{10} are frequency dependent and, ultimately, functions of α
Equation 4awhere α is the Womersley parameter
Equation 4bThese three hemodynamic properties, in turn, affect two phenomenological parameters: characteristic impedance,*Z*
_{0}, and pulse wave velocity (or phase velocity),*c*
_{ph}
Equation 5a
Equation 5bInput impedance of a vessel depends on properties of the vessel itself and properties of the vessel's termination. The load at the end of a vessel (*Z*
_{l}) is a frequency-dependent quality that can represent any number of vessels in series and in parallel. The vessel-load combination illustrated in Fig.3 can thus describe the hemodynamics at the entrance of any vessel, providing that the pressure-flow relationship is sufficiently linear. The equation describing input impedance of a tube terminated by a load *Z*
_{l} is derived elsewhere (37).
Equation 6For large vessels, resistance is negligible. This can be illustrated by taking the limit of the resistance as *A* ≫ 1 in *Eq. 3a
*
Equation 7In this case, *Z*
_{0} and*c*
_{ph} degenerate into real values, depending on ρ, *A*, and C′
Equation 8a
Equation 8bWith the tools developed thus far, the question can be addressed as to whether there are unique solutions to the hemodynamic inverse problem. Suppose that there are two different arteries, each terminated by loads with the same value, *Z*
_{l} (Fig. 3). Let the first artery have parameter values *A*, *C*′, and*l* and the second artery have values *Â*,Ĉ′, and *l̂*. These mechanical properties yield phenomenological properties *c*
_{ph},*Z*
_{0}, *Z*
_{in} and*ĉ*
_{ph}, *Ẑ*
_{0},*Ẑ*
_{in}, respectively. Let the second artery be a factor β larger than the first and a factor 1/β less compliant, so that
Equation 9
Substituting *Â*, Ĉ′, and*l̂* into *Eq. 8* illustrates that
Equation 10
Substituting *Eq. 10
* into *Eq. 6
*illustrates that the input impedances of these two arteries are identical for all positive values of β
Equation 11Thus a particular artery can be large and stiff or small and compliant, and there would be no detectable difference in the measured input impedance.

The implications are that any single vessel in an arterial tree can be replaced with a larger or smaller vessel with a different compliance without altering the ratio of pressure to flow at the entrance of an arterial tree. By extension, any particular branch of an arterial tree can have an infinite combination of parameters that will yield the same input impedance at the tree's entrance. This is illustrated in Fig.4, where three very different architectures are represented, all with identical input impedance.

## DISCUSSION

This exercise illustrates that there are an infinite number of solutions to the hemodynamic inverse problem. That is, any number of different arterial system architectures can yield the same input impedance, entrance pressure, and flow. This has a direct impact on the ability to solve the hemodynamic inverse problem. Given a particular input impedance or input pressure and flow, it is impossible to determine the lengths, areas, or compliances of individual arteries. As a corollary, it is impossible to determine effective reflection sites. This work extends the work of Campbell et al. (10), who illustrated that there is an infinite combination of arterial length and terminal load that will yield the same input impedance.

### Arterial System Modeling Run Amok

The need for a simple model that has few parameters and fits the data well has led to a new era of cardiovascular modeling. Simplistic models that failed to solve the forward problem have resurfaced. Their very simplicity, once perceived as a liability, is now often perceived as an asset. Although the large number of models reported in the literature prevents an exhaustive summary, three categories of models can be identified.

The first is the family of windkessel models, based on the classical windkessel. Most notably, Westerhof (47) added an element representing the characteristic impedance, resulting in the three-element windkessel (49). The three-element windkessel has been extraordinarily popular, yet it does not address the observed deviation from the exponential pressure predicted in diastole. To account for oscillations in the diastolic pressure curve, Goldwyn and Watt (19) presented a five-element windkessel consisting of a “proximal compliance” and a “distal compliance” separated by an inertance (16). To capture the viscoelastic nature of arteries, Canty et al. (11) made the compliance of the windkessel viscoelastic. Given enough time, it is conceivable that every possible combination of inertance, compliance, and resistance will eventually be reported (50).

The second is the family of tube models based on the infinitely long tube, bringing pulse propagation and reflection into the picture. To add reflection in an early attempt, Taylor (43) terminated a uniform tube with a resistance. This led to unrealistic peaks in the frequency domain. To describe the changes in arterial properties in the longitudinal direction, the tube was given elastic tapering (45) and both geometric and elastic tapering (46). However, more popular has been the T tube consisting of two tubes in parallel, one representing the system supplying the head and upper limbs and the other supplying the trunk and the lower limbs (34). Burattini and Campbell (7) added complexity by terminating the T tube with three-element windkessels. Liu et al. (25) added even more complexity by making the tubes viscoelastic and terminating the tubes with four-element windkessels. The recent combination of tubes with lumped models has considerably expanded modeling possibilities.

The third family of models departs from the assumption of linearity found in the first two families. Although arterial vessels are known to have pressure-dependent characteristics, investigators initially avoided models that required numerical evaluation. They have since begun to face this challenge. For instance, Liu et al. (24) made the compliance of the classical windkessel an exponential function of pressure. Later, Burattini et al. (9) used a different nonlinear function to add nonlinearity to the compliance of the three-element windkessel. This was followed by Li et al. (23), who made the compliance of the three-element windkessel an exponential function of pressure. The process of adding nonlinearity to lumped models was suggested first by Otto Frank in 1899 (38).

Because of the implications of the present theory, it is clear that no model can retrieve particular mechanical properties of arteries from measured input impedance, pressure, and flow alone. Individual vessel size, stiffness, and spatial distribution are irretrievable. Thus attempts to fit reduced models to data in order to interpret the data are suspect. However, use of reduced models to characterize the data is completely compatible with the present work. That is, if a simple model is to be used to mimic an arterial bed or to describe the data with a limited number of parameters, then any particular model that fits the data sufficiently is acceptable.

### Effective Length of the Arterial System

There have been several attempts to solve very limited aspects of the inverse problem. One remarkably enduring approach is to estimate the effective length of the arterial system (8, 12). In this case, a tube with length *l* is assumed to be terminated by a particular load, *Z*
_{l}. This model is then fit to input impedance (or, equivalently, pressure and flow) to determine the value for *l*. Campbell et al. (10) challenged this approach when they showed that there is an infinite combination of *l* and *Z*
_{l} that yield the same measured input impedance. Therefore, the estimated length (*l*) depends on the load (*Z*
_{l}) assumed. In this case, the information about the site of reflection is simply not contained within the measured data. Recently, there have been attempts to overcome this troublesome challenge by constraining the assumed load (6). However, constraining the load does not obviate the limitation presented in the present work. Given a particular arterial load, there are still an infinite number of parameters describing the tube itself that yield the same input impedance.

### Resistance in the Large Arteries

The theoretical development assumed that resistance was negligible in the large arteries. Indeed, in the human systemic arterial system, the large arteries contribute only a small percentage of total peripheral resistance. When there is significant resistance in a particular artery, *Eq. 8* does not apply, and there are not an infinite combination of compliances and sizes that result in the same value of input impedance. Therefore, the basic limitation to the inverse problem only applies here to the larger arteries. Vascular beds consisting of smaller vessels may have more information contained in the measured input pressure and flow.

### Requirement for Additional Measurements

The results of this work indicate that additional measurements, other than input pressure and flow, are required to determine mechanical properties of specific arteries. For instance, measurement of pressure or flow at two locations can yield pulse wave velocity (2, 18). Pulse wave velocity is, in turn, related to cross-sectional area and compliance of the vessel between the two points of measurement (*Eq. 8b
*). Compliance of an individual artery can also be determined from measured pressure and lumen area at a particular location (21). These methods have been used with great success for determining mechanical properties of individual vessels in animal preparations.

Recent technological advances have made it feasible to collect this additional required data from humans noninvasively or by minimally invasive techniques. Angiography can noninvasively determine vessel size (20, 27) and, thus, constrain the number of possible solutions to the inverse problem illustrated in Fig. 4. Transcranial Doppler ultrasonography, for instance, can measure flow in a variety of previously inaccessible locations (13). With new measurement techniques, some of the reasons originally motivating the solution of the hemodynamic inverse problem from input pressure and flow alone have become moot. Although it is becoming easier to measure hemodynamic variables at various locations in an arterial tree and, thus, to determine mechanical properties of individual vessels, determining mechanical properties of the arterial system as a whole would require an enormous number of measurements. The question of what information can be gleaned from pressure and flow measured at a particular location will only become more important with the development of new measurement techniques.

### Information Content in Pressure and Flow

The conventional approach to solve the inverse problem is to fit a model to experimentally measured input pressure and flow. There is an inherent limit to this approach to analyzing data: information can be retrieved or destroyed but not created ex nihilo by clever assumption. The fully distributed model, the classical windkessel, and the infinitely long tube have physiologically interpretable parameters (37). However, the fully distributed model has been rejected as a vehicle to solve the inverse problem, because it has too many parameters. The classical windkessel and the infinitely long tube models have been rejected, because they fit the data poorly and lack the basic properties under study. The challenge has been to find the ideal model that simultaneously *1*) has physiologically interpretable parameters, *2*) has a limited number of unknown parameters, *3*) fits the data exactly, and *4*) has all the properties of interest.

### Limited Solutions to the Hemodynamic Inverse Problem

There is hope, however, of solving limited aspects of the inverse problem from measured input pressure and flow alone. Because*Z*
_{0} = *Ẑ*
_{0} and C′*l* = Ĉ′*l*′, the limitation to solving the inverse problem described above does not apply to estimation of total compliance (C′*l*) or characteristic impedance (*Z*
_{0}) of an individual artery. Thus estimation of characteristic impedance, *Z*
_{0}(15, 26), or total arterial compliance, C_{tot} = ΣC′*l *(36), is still theoretically possible. In Fig. 4, all the models have the same*Z*
_{0} and C_{tot}. Table1 summarizes recognized unique solutions to limited inverse problems on the basis of input impedance. These parameters have in common the property that they are not involved in the transformations in Fig. 4 (*R*) or are insensitive to it (*Z*
_{0} and C_{tot}).

Two particular arterial system properties, total arterial compliance and characteristic impedance, deserve special attention. Estimation of total arterial compliance must assume the windkessel model (36), and estimation of characteristic impedance must assume the infinitely long tube model (2). These two historical models *1*) have physiologically interpretable parameters, *2*) have a limited number of unknown parameters, and *3*) fit the data exactly at very low and very high frequencies. However, they do not have all the properties of interest. Most notably, the windkessel, with the assumption of infinite pulse wave velocity, does not have any pulse transit delay. The infinitely long tube lacks pulse wave reflection. However, the windkessel can be used to evaluate the effect of pulse transit delay on input impedance, stroke work, and pulse pressure, and the infinitely long tube can be used to evaluate the effect of reflection (35). It is perhaps fitting that the two fundamental models of the arterial system, once rejected for their simplicity, can play a vital role in solving aspects of the inverse problem.

## Acknowledgments

The authors thank the members of the Center for Cerebrovascular Research and the University of California, San Francisco, Arteriovenous Malformation Study Project for support.

## Footnotes

Portions of this work were supported by National Institutes of Health Grants K24 NS-02091 and T32 GM-08440.

Address for reprint requests and other correspondence: C. M. Quick, Center for Cerebrovascular Research, University of California, San Francisco, 1001 Potrero Ave., Rm. 3C-38, San Francisco, CA 94110 (E-mail: quickc{at}anesthesia.ucsf.edu).

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- Copyright © 2001 the American Physiological Society