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Annals of Biomedical Engineering, Vol. 27, pp. 548–562, 1999 Printed in the USA. All rights reserved.

0090-6964/99/27共4兲/548/15/$15.00 Copyright © 1999 Biomedical Engineering Society

Parametric and Nonparametric Nonlinear System Identification of Lung Tissue Strip Mechanics HUICHIN YUAN,1 DAVID T. WESTWICK,1 EDWARD P. INGENITO,2 KENNETH R. LUTCHEN,1 and BE´LA SUKI1 1

Department of Biomedical Engineering, Boston University, Boston, MA, and 2Brigham and Women’s Hospital, Boston, MA (Received 3 September 1998; accepted 22 April 1999)

which displays a very long memory29 since the tail of the power law can be orders of magnitude larger than an exponential system response. The lung tissue exhibits certain complicated mechanical nonlinearities,27 namely, a strong positive lung volume dependence along the quasistatic pressure–volume curve,23 and a milder negative dynamic volume amplitude dependence,1,4 which becomes important during tidal-like breathing.17 To examine the nonlinear and linear viscoelastic properties of lung tissues during tidallike oscillatory excursions and under induced constricted conditions, several groups have applied a variety of parametric nonlinear block-structured models to pressure– flow data at the organ level.18,20,27 With tidal-like forcing, Suki et al.27 found that a Wiener model 共linear and nonlinear blocks in cascade, LN structure兲 provided a significantly better fit to the data than a Hammerstein model 共NL cascade兲. In contrast, with large amplitude forcing, Maksym and Bates20 found that the Hammerstein model was mildly superior to the Wiener model. A similar conclusion was reached for lung tissue strips.18 While parametric modeling is robust, its disadvantage compared to nonparametric approaches is that one invariably loses the generality of the identification and substantial a priori information is needed about the system. A key difficulty in applying nonparametric approaches, however, is to identify the system when it has long memory. Zhang et al.32 have demonstrated that time-domain kernel analysis such as that proposed by Korenberg,13 may not be suitable for identifying the dynamic nonlinear behavior of lung tissue, which exhibits long memory characteristics. The primary reason is that the estimated Volterra kernels become severely distorted due to truncation in memory length. As an alternative, Zhang et al.33 recently suggested to identify the system in the frequency domain. If the system is excited with a periodic input, the system response will approach a steady state in which, after a sufficient number of cycles, the effects of fading long memory on the output frequency spectrum would become negligible compared to the periodic response. With a modification of the

Abstract—Lung parenchyma is a soft biological material composed of many interacting elements such as the interstitial cells, extracellular collagen–elastin fiber network, and proteoglycan ground substance. The mechanical behavior of this delicate structure is complex showing several mild but distinct types of nonlinearities and a fractal-like long memory stress relaxation characterized by a power-law function. To characterize tissue nonlinearity in the presence of such long memory, we investigated the robustness and predictive ability of several nonlinear system identification techniques on stress–strain data obtained from lung tissue strips with various input wave forms. We found that in general, for a mildly nonlinear system with long memory, a nonparametric nonlinear system identification in the frequency domain is preferred over time-domain techniques. More importantly, if a suitable parametric nonlinear model is available that captures the long memory of the system with only a few parameters, high predictive ability with substantially increased robustness can be achieved. The results provide evidence that the first-order kernel of the stress–strain relationship is consistent with a fractal-type long memory stress relaxation and the nonlinearity can be described as a Wiener-type nonlinear structure for displacements mimicking tidal breathing. © 1999 Biomedical Engineering Society. 关S0090-6964共99兲00804-8兴

Keywords—Stress relaxation, Long memory, Fractals, Wiener model, Collagen fibers, Elastin fibers, Network.

INTRODUCTION The viscoelastic behavior of soft tissues can be characterized by the stress relaxation response to a step change in strain. The stress relaxation of lung tissue decays very slowly in time, in a power-law-like fashion.3,26 Bates et al.3 and Maksym and Bates19 suggested that power-law stress relaxation and power-law forms in the distributions of fiber characteristics 共e.g., fiber length兲 are a manifestation of the fractal-like organization of the complex extracellular 共collagen and elastin兲 matrix. From the system identification point of view, the power-law stress relaxation represents a system Address correspondence to Be´la Suki, PhD, Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215. Electronic mail: [email protected]

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Modeling of Lung Tissue Strip Mechanics

frequency-domain kernel analysis introduced by Victor and Shapley,28 Zhang et al.33 determined the frequency kernels of isolated lungs and developed a nonparametric structure test which provided strong evidence that the lung tissue behaves more like a Wiener model under tidal-like oscillatory excursions. Thus, it appears that the input wave form 共e.g., white versus color or noise versus pseudorandom兲, analysis technique 共e.g., parametric versus nonparametric兲 and the domain of analysis 共time versus frequency domains兲 can influence the estimated kernels. As a consequence, the structure and, more importantly, even perhaps the mechanistic conclusion about the system can become a function of the applied methodology. As an example, in the early 1970s, Hildebrandt and co-workers9,11 measured stress relaxation and lung impedance in excised cat lungs. Since they found a 30% discrepancy between time-domain and frequency-domain dissipation, they concluded that the difference is due to plasticity. Obviously, this is a heavy conclusion that influenced at least ten years of lung mechanics research starting from the mid-1980s. Their data, however, were not free of nonlinearities and it is very likely that nonlinearities influenced the linear analysis of their data and the above mechanistic conclusions. The purpose of this paper is to thoroughly investigate how the chosen input wave form and the analysis approach can influence the identification of a class of nonlinear systems with weak but non-negligible nonlinearities and very long memory such as the dynamic mechanical properties of lung tissues. In order to avoid contributions from airway mechanics and surfactant kinetics, we used the isolated lung parenchymal tissue strip as our model system. In particular, the stress–strain data of lung tissue strips were analyzed by: 共1兲 estimating the linear dynamics and hence the memory length of the system by deriving the impulse response directly from stress relaxation measurement; 共2兲 estimating the firstand second-order kernels in the time domain using Gaussian white-noise input and the parallel cascade algorithm proposed by Korenberg;14 共3兲 applying Victor and Shapley’s frequency-domain method28 to the steady-state stress response of the tissue to a sum-of-sinusoids strain input; 共4兲 comparing these nonparametric models to robust parametric modeling in which the data were fit with a variety of nonlinear block-structured models utilizing a priori knowledge from the study of Suki et al.27 Finally, a rigorous comparison of the models was carried out by examining their out-of-sample predictive capabilities. METHODS Design of Input Signals For a given system identification approach, it is important to choose the appropriate input signal that will

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sufficiently excite all mechanisms to be identified within the system. Thus, to examine the applicability of several identification approaches to lung tissue strip mechanics, we applied different types of input signals according to the type of mechanism to be identified 共i.e., linear and nonlinear兲 and the particular system identification technique used. These strain inputs are the following: Step Input. In theory, the direct estimation of an impulse response from the step response can be applied only to a linear system. Since for moderate strain inputs, the nonlinearities of lung tissues are mild,22,27,31 we reasoned that the impulse response as derived from stress relaxation to a small step input in strain would closely reflect the first-order Volterra kernel of the system. Therefore, by measuring the step response, we can directly assess the linear dynamics and hence the memory length of the tissue strips. Gaussian White Noise (GWN). In theory, Korenberg’s parallel cascade algorithm14 allows for time-domain kernel identification by applying a nonwhite and nonGaussian input which excites the dynamics of the system.30 However, the GWN possesses rich energy over the entire frequency bandwidth which can fully excite all nonlinear mechanisms in the system, and hence may allow for the best characterization of the nonlinearity. We applied a GWN input signal with a bandwidth of 3 Hz. This represents about 1/3 of the operating frequency range of the measuring system. Thus, if nonlinearities of up to third order are present in the system, harmonic distortions would produce energy in the output signal up to about 9 Hz and hence would allow us to identify nonlinearities of up to third order. Sparse Sum-of-Sinusoids Wave Forms. Identification of frequency kernels using Victor and Shapley’s approach28 requires a periodic signal as the strain input. Victor and Shapley28 provided a specially designed sum-ofsinusoids wave form 共VS兲 which permits direct estimation of the frequency kernels from the input and output spectrum 共see below兲. The VS signal was chosen to have a flat power spectrum and random phases. Another sparse sum-of-sinusoids wave form, called the nonsum– nondifference 共NSND兲 signal designed by Suki and Lutchen,25 was also applied to the tissue strips. The NSND wave form does not allow the calculation of the frequency kernels. Instead, by minimizing the nonlinear harmonic interactions in the output spectrum at the frequencies where input energy was specified, it provides smooth estimates of apparent transfer linear function of the system.25 If the system nonlinearity is weak, then this transfer function is a good approximation of the first-

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YUAN et al. TABLE 1. The frequency compositions and corresponding phase angles of the VS and NSND displacement inputs. VS is the Victor and Shapley wave form „Ref. 28… and the NSND is the nonsum nondifference wave form proposed by Suki and Lutchen „Ref. 25…. VS wave form Harmonic component Frequency (Hz) Phase angle (rad)

7 0.103 1.03

15 0.22 ⫺1.84

Harmonic component Frequency Phase angle (rad)

3 0.088 1.37

NSND wave form 7 19 37 0.205 0.557 1.084 ⫺0.56 ⫺0.39 0.21

order frequency kernel.27 The frequency components and their corresponding phase angles of both signals are given in Table 1. Integer-Multiple Wave Form (IM). The IM signal was the sum of the first 60 harmonics of the fundamental frequency of 0.015 Hz with a flat power spectrum and random phases. The bandwidth of the IM signal was ⬃ 1 Hz. The purpose of this wave form is to examine the out-of-sample predictive capabilities of the different models obtained from the various identification approaches. The output to the IM input predicted by these models, which were estimated from the data obtained with GWN and VS inputs, were then compared to measured output to IM inputs. Experimental Methods The force–length relation was measured in two excised lung parenchymal strips from two guinea pigs and in a calibrated steel spring. The experimental setup and protocol are similar to that used in a previous study.31 Briefly, a tissue strip in dimensions of ⬃4.5⫻4.5 ⫻10 mm was placed in a glass tissue bath 共Wilbur Scientific, Boston兲, with one end attached to a force transducer 共model 400A, Cambridge Technologies兲 and the other to a servo-controlled lever arm 共model 300H, Cambridge Technologies兲 of a displacement generator. The lever arm was driven by one of the signals described above. The input signal to the displacement generator was sent out from the digital-to-analog port of a data acquisition board 共DT2812, Data translation, Cambridge兲 then smoothed with an eight-pole low-pass filter. Using the calibrated spring, the system was properly aligned such that during sinusoidal oscillations the hysteresis area between force and displacement was minimized. The tissue strip was then preconditioned by first performing a single slow stretch to 2 g mean force followed by sinusoidal oscillations at 1 Hz for about 10 min at a mean force of 1 g. The stress relaxation of the tissue strip was first measured by applying a step of 10% in strain amplitude.

31 0.454 1.66

63 0.922 ⫺0.24

127 1.86 ⫺0.92 61 1.787 1.81

255 3.735 0.71

¯ ¯ ¯

89 2.607 0.19

97 2.842 ⫺0.44

Next, the dynamic force–length relationships were measured at 10%, and 15% peak-to-peak strain amplitudes of the length oscillations with the GWN, VS, NSND, and IM signals. These small strain amplitudes are similar to those occurring during tidal breathing.31 Both displacement and force signals were low-pass filtered at 4 and 15 Hz, then sampled at 15 and 60 Hz for the stress relaxation and the dynamic oscillatory measurements, respectively. For the oscillatory signals, the length of the sequence was 4096 points so that using a sampling rate of 60 Hz corresponded to a time period of 63 seconds. For each condition, a total of 4–8 cycles were delivered and the first cycle was discarded to avoid transients. The GWN and VS signals were also applied to the steel spring which had a similar stiffness as the tissue strips. The recorded displacement input and force output signals were normalized by the reference length and the crosssectional area of the strip to obtain strain ⑀ and stress T, respectively. Overview of System Identification Approaches Estimation of Impulse Response and Memory Length from Stress Relaxation. It has been observed for many biological tissues that the stress response to a unit step in strain is well accounted for by a power-law-type of stress relaxation:26 T 共 t 兲 ⫽ 共 At ⫺ ␤ ⫹T 0 兲 u 共 t 兲

for

t⭓0,

共1兲

where T(t) is the stress response following a step input in strain, T 0 is the stress response when t˜⬁, ␤ is the relaxation exponent, A is the relaxation amplitude, and u(t) is the unit step function. Note that for t˜0, the power-law term approaches a delta function and the system behaves as a Newtonian fluid. Our preliminary exercises showed that the change in stress, T(t)⫺T 0 , decreased linearly with time on a log–log plot. Thus, for t⬎0, the following equation: log关 T 共 t 兲 ⫺T 0 兴 ⫽⫺ ␤ log共 t 兲 ⫹log A,

共2兲


provided an excellent description of the data with a very high correlation coefficient 共⬎0.99兲. Accordingly, the impulse response h 1 (t) between stress and strain of the system is the derivative of Eq. 共1兲 with respect to t:

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unit, and Q is the number of sinusoids. For a secondorder nonlinear system, the steady-state response of the system can be written as Q

h 1 共 t 兲 ⫽⫺ ␤ At

⫺ 共 ␤ ⫹1 兲

⫹C ␦ 共 t 兲

for

t⭓0,

Nonparametric Time-Domain Modeling. The crosscorrelation-based algorithms proposed by Hunter and Korenberg12 were used to fit Wiener and Hammerstein models between the input strain and output stress. Rissanen’s minimum descriptor length 共MDL兲 criterion16 was used to select the optimal combination of finite impulse response 共FIR兲 filter length and polynomial order, and to compare the resulting Wiener and Hammerstein cascades. The filter length was allowed to range from 4 to 200 samples in steps of 4 samples, with polynomial orders of 1 through 4. The optimal nonparametric Wiener and Hammerstein models were used as the first pathway in a parallel cascade model.14 The generalized eigenvector algorithm30 was used to add further Wiener systems in parallel to the initially identified Wiener and Hammerstein models. This algorithm finds the optimal path, given the data in the second-order input–residual cross correlation, to include in the model, and then uses a test based on the MDL 共Ref. 16兲 to determine whether or not the candidate pathway accounts for a statistically significant portion of the residuals. If this optimal path fails the test, no further paths need be constructed. Nonparametric Frequency-Domain Modeling. The basic idea of Victor and Shapley’s28 approach is that by using the VS input, the frequency-domain kernels can be determined directly from the input and output spectra. Using Fourier series representation, the input s(t) can be expressed as Q

兺

兺

共3兲

where C represents another parameter that cannot be estimated reliably from stress relaxation data. For t⬎0, the impulse response in Eq. 共3兲 is a simple power law and presents a very long memory. We recognize that the impulse response of a nonlinear system can in general include contributions from the diagonal elements of the higher-order Volterra kernels. However, for a system with weak nonlinearities excited with a small amplitude input, the contribution of nonlinearity to the impulse response would be negligible.

1 s共 t 兲⫽ a exp关 i 共 2 ␲ f j t⫹ ␾ j 兲兴 , 2 j⫽⫺Q j

1 r 共 t 兲 ⫽h 0 ⫹ a g 共 f 兲 exp关 i 共 2 ␲ f j t⫹ ␾ j 兲兴 2 j⫽⫺Q j 1 j

共4兲

where a j and ␾ j are the Fourier coefficient and the phase angle at frequency f j , respectively, i is the imaginary

⫹

1 4

Q

Q

兺兺

j⫽⫺Q k⫽⫺Q

a j a kg 2共 f j f k 兲

⫻exp兵 i 关 2 ␲ 共 f j ⫹ f k 兲 t⫹ ␾ j ⫹ ␾ k 兴其 ,

共5兲

where h 0 is the dc response, and g 1 and g 2 are the normalized first- and second-order frequency kernels of the system, respectively.28 The g 1 ( f ) and g 2 ( f 1 , f 2 ) are simply the Fourier transforms of the first- and secondorder Volterra kernels, respectively. For the identification of a second-order nonlinear system, the VS input requires a special set of frequencies such that the firstorder frequencies and the second-order derived frequencies 共i.e., linear combination of any two frequencies, ⫾ f j ⫾ f k 兲 are distinct. As a result, g 1 and g 2 can be directly estimated from the system response at these frequencies. It should be pointed out that if nonlinearities of order ⬎2 are present in the system, the estimated g 1 ( f ) and g 2 ( f j , f k ) could be distorted due to the contribution of the higher-order nonlinear interactions at the firstorder frequencies and the second-order derived frequencies.28 Parametric Block-Structured Modeling. The parametric modeling approach presumes an a priori model form for the system. Specifically, we assume that the lung tissue contains a linear viscoelastic subsystem L and a nonlinear zero-memory subsystem N arranged in LN or NL cascades.27 The linear viscoelastic tissue model is the power-law stress relaxation 关Eq. 共1兲兴 which has been widely used in the frequency domain by several investigators for whole lung or tissue strip.3,8,26,27,31 Previously, we found31 it necessary to include a purely viscous component R so that the linear transfer function or complex modulus G * of the tissue is described as G * 共 ␻ 兲 ⫽H ␻ ␤ ⫹ j 共 G ␻ ␤ ⫹R ␻ 兲 with

␤ ⫽1⫺

冉冊

2 H tan⫺ 1 , ␲ G

共6兲

where the parameters G and H are the tissue damping and elastance coefficients, respectively. Note that ␤ governs the frequency dependence of the real and imaginary

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YUAN et al.

parts of G * ( ␻ ). However, G, H, and ␤ are not independent, but related via the second part of Eq. 共6兲. The real part of G * is the storage or elastic modulus, and the imaginary part is the loss modulus which is related to energy dissipation during cyclic deformation. The model not including R, is called the constant phase model since the phase angle of the modulus is independent of frequency. A mathematical framework of Eqs. 共1兲 and 共6兲 and its molecular basis has been offered by Suki et al.26 Additionally, if the system is linear, one can show that the exponent ␤ is equal to the stress relaxation exponent in Eq. 共1兲 and the coefficient R of the Newtonian term is equal to C in Eq. 共3兲.26 The nonlinear tissue block is zero memory, and its input–output relationship is described by either a secondor a third-order polynomial with a 1 ⫽1, a 2 , and a 3 corresponding to the first- second-, and third-order coefficients, respectively. Several studies have reported that the order of the nonlinearities present in the lung is ⬍4.18,27,33

input can be predicted. To examine the predictive ability of the models, the estimated parameters from each model fitting were used to predict the response of the tissue strips to the IM input signal. Using convolution, the prediction of the response of the nonparametric timedomain models and the parametric block-structured models is straightforward. However, the same calculation involves some complications in the nonparametric frequency-domain model identified from the VS input since it requires the knowledge of the frequency kernels over all frequencies in the bandwidth of interest. The Victor and Shapley’s approach28 provides the first- and second-order kernel values only at the first-order and second-order derived frequencies, respectively, that is, the kernel values are sparse in the frequency domain. Therefore, to predict the system response to an independent broadband input, we interpolated the frequency kernels to the desired frequencies using low-order polynomials. RESULTS

Data Analysis System Identification. The linear dynamics and memory length of the tissue strip was estimated from the stress relaxation data. The tissue strip properties were identified with the following three different identification approaches. 共1兲 Using the parallel cascade algorithm of Korenberg,14 the nonparametric time-domain 共NPT兲 kernels were estimated from fitting the GWN data with a linear model 共NPT-1兲 and second- and third-order Wiener 共NPT-W2 and NPT-W3, respectively兲 and Hammerstein 共NPT-H2 and NPT-H3, respectively兲 models. 共2兲 For the nonparametric frequency-domain 共NPF兲 approach, the frequency kernels of a linear model 共NPF-1兲 and a second-order nonlinear model 共NPF-2兲 were identified from VS data measured from both the tissue strips and the spring. 共3兲 The dynamic stress–strain data obtained from the GWN, VS, and NSND inputs were fit to the following parametric 共P兲 block-structured models: Linear model 共P-1兲, second- and third-order Wiener models 共P-W2 and P-W3, respectively兲 and Hammerstein models 共P-H2 and P-H3, respectively兲. Prior to model identification, the linear trends were removed from the oscillatory stress–strain data so that the stress signal was zero mean. The parameters of the nonparametric time-domain and the parametric blockstructured models were estimated by minimizing the normalized sum-of-squares error 共NSSE兲. In the case of the parametric models the NSSE was minimized using a global optimization algorithm.5 Model Validations. Once the system is identified with one of the above approaches, its response to any type of

Estimation of Impulse Response from Stress Relaxation The stress relaxation (T⫺T 0 ) measured in a lung tissue strip and the corresponding impulse response h 1 (t) derived using Eq. 共3兲 are shown in Figs. 1共A兲 and 1共B兲, respectively. Following a small ringing for t⬍1 s, which is due to the apparatus, the tissue strip displayed a slowly decaying nearly perfect power-law stress relaxation behavior as apparent from the linear decrease of stress on a log–log plot. This is consistent with previous findings in the literature.3,26 The value of the exponent ␤, which governs the slope of decay in stress, was very small (0.06⫾0.002) indicating an extremely long memory of the lung tissues. Nonparametric Time-Domain Modeling The optimal Wiener system had a memory length of 80 samples 共1.33 s兲, followed by a third-order polynomial. This system predicted 99.93% of the output energy in the data. When a Hammerstein structure was used, the optimal model was found to have a third-order polynomial, followed by a filter with a memory length of 148 samples 共2.47 s兲. This model achieved accuracies of 99.95% in the sample. Furthermore, it resulted in a lower MDL cost than the optimal Wiener model, despite the increased number of parameters. After starting with either of the above models, the generalized eigenvector algorithm30 was unable to construct any additional pathways which accounted for statistically significant fractions of the residuals. The linear block of the Hammerstein model, shown in Fig. 2共A兲, was less noisy than that of the Wiener model 共not shown兲 and displayed some ringing. However, even in


FIGURE 1. „A… The stress relaxation „ T ⴚ T 0 … or stress response to a strain step input as a function of time „closed symbols…. Solid line is the linear regression in the log–log domain according to Eq. „2…. The parameters A and ␤ are the elastic modulus and slope or relaxation exponent in Eq. „1…. „B…: The impulse response h 1 „ t … „ t >0… derived from the stress relaxation in Eq. „3…. The symbol „䊊… at t ⴝ0 represents an undefined value of C in Eq. „3….

the Hammerstein model, the first-order kernel decayed rapidly below the noise level, making it impossible to recognize that the kernel follows a power function and hence estimate the exponent ␤. The second-order Volterra kernel, h(t 1 ,t 2 ), shown in Fig. 2共B兲, was zero except for the diagonal elements h(t,t). The diagonal elements h(t,t), being proportional to h 1 (t), were very small and comparable to estimation noise.

Nonparametric Frequency-Domain Modeling To evaluate the ability of this method to identify and justify tissue nonlinearities, we compared the mean values and their standard deviations of the first- and secondorder frequency kernels of the calibrated steel spring 共Fig. 3兲 to those of the tissue strip 共Fig. 4兲. For the spring, the real part of g 1 ( f ) or storage modulus is largely invariant with frequency. The imaginary part of g 1 ( f ), which corresponds to dissipative properties, is three orders of magnitude smaller than the elastic modulus and statistically not significantly different from zero. Furthermore, the average values of the second-order frequency kernels g 2 are very small and most of them are

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FIGURE 2. The first- „A… and second-order „B… Volterra kernels estimated from the nonparametric time-domain approach. The inset in B shows the diagonal elements of the second-order kernel.

statistically not different from zero. Thus, the spring appears to be a perfectly elastic 共zero imaginary part兲 and linear 共zero second-order kernel兲 system. In contrast to the spring, the characteristics of the estimated g 1 and g 2 for the tissue strip show that the real part of g 1 increased steadily 共linearly with the logarithm of frequency兲 and the imaginary part of g 1 is only about ten times smaller than the elastic modulus 共Fig. 4兲. Thus, while the real part of the tissue was only ⬃2 times higher than that of the spring, the imaginary part or loss modulus of the tissue was more than two orders of magnitude higher than that of the spring indicating the presence of important dissipation in the tissue. The results obtained for the tissue strip are consistent with previous findings in the literature.21,22,31 The g 2 estimated from the tissue strip has much larger magnitudes with small standard deviations 共SDs兲 compared to the spring in Fig. 3 which had large SDs centered around small values. This indicates the presence of important nonlinearities in the tissue. Interestingly, the real part of g 2 also appears to exhibit a definitive structure: slices along both frequency axes are similar to the real part of the first-order kernel. In fact, we fit g 1 to the linear model of Eq. 共6兲,

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YUAN et al.

FIGURE 3. The meanⴞSD „standard deviation… of the first- and second-order frequency kernels g 1 „ f … and g 2 „ f 1 , f 2 …, respectively, of a calibrated linear spring. The top panels show the real and imaginary parts of g 1 „ f …. The bars denote ⴞ1 SD. The bottom panels show the real and imaginary parts of g 2 „ f 1 , f 2 … as a function of the two frequencies. The open symbols above the mesh denote 1 SD above the data value. The mean and SD values were obtained at each frequency by averaging ⬃15 data segments with an overlap percentage of 50%.

and found mean values of ␤ (0.052⫾0.001) similar to that estimated from the stress relaxation (0.060⫾0.002) indicating a good agreement between the frequencydomain data and the stress relaxation measurements. Parametric Block-Structured Modeling The optimal parameters estimated from fitting the GWN and VS data with various parametric blockstructured models having different structure and degree of nonlinearity are summarized in Table 2. The values of ␤ inferred from all linear and nonlinear models were ⬃0.055 varying within 5%, which are consistent with

that estimated from the stress relaxation and from the real part of g 1 of the nonparametric frequency kernels. For both GWN and VS inputs, the errors NSSE between model fit and data for both the nonlinear P-W2 and P-H2 models are reduced significantly by ⬃90% from that of the linear model which has only one less parameter. However, the fitting errors of the third-order nonlinear models 共P-W3 and P-H3兲 were reduced by only 1% for the VS data, and by ⬃10% for the GWN data as compared to the corresponding second-order nonlinear models. This suggests that a third-order nonlinearity was present in the system, since with a large number of data


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FIGURE 4. The meanⴞSD of the first- and second-order frequency kernels g 1 „ f … and g 2 „ f 1 , f 2 …, respectively, of a tissue strip „see the legend of Fig. 3….

points or degrees of freedom, the F test16 showed that 10% decrease in error for the GWN data was significant enough for inclusion of an additional parameter. Since the GWN input is far richer than the VS input, it is not surprising that the VS input was not able to resolve the third-order nonlinearity. The values of damping and elastance coefficients 共G and H, respectively兲 were independent of the characteristics of input and the type of models. This, again, is evidence that the dynamic stress–strain behaviors for strain amplitudes used in this study are mostly dominated by the linear viscoelastic subsystem. Importantly, the polynomial coefficient a 2 varied within 5% for the different type of models and inputs, which suggests that the second-order nonlinearity was robustly estimated from

both the VS and GWN data. For the same order of nonlinearity, the errors of the Wiener model were slightly smaller 共1%–3%兲 than that of the Hammerstein model. Finally, we found significant discrepancies between the values of a 3 estimated from the GWN and VS inputs, which again confirms that the VS input may not be suitable for identification of third-order nonlinearities. Therefore, we conclude that a third-order tissue nonlinearity affected the GWN data, which is in agreement with the nonparametric time-domain identification results. We also fit our block-structured models to the stress–strain data obtained using the NSND signal as strain input. The parameters estimated from the NSND inputs are nearly identical to those obtained from the VS inputs 共Table 3兲.

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YUAN et al. TABLE 2. The optimal parameters obtained from the parametric block-structural models in a tissue trip. Units: G, H: N/cm2; R: N s/cm2; a 2 : „N/cm2…ⴚ1; a 3 : „N/cm2…ⴚ2. P-1: parametric linear model; P-W2: parametric second-order Wiener model; P-W3: parametric third-order Wiener model; P-H2: parametric second-order Hammerstein model; and P-H3: parametric third-order Hammerstein model. VS is the Victor and Shapley wave form „Ref. 28… and GWN is Gaussian white noise. Input

Model

G

H

R (10⫺4 )

a2

VS

P-1 P-W2 P-W3 P-H2 P-H3

0.0323 0.0323 0.0324 0.0324 0.0324

0.3696 0.3697 0.3705 0.3696 0.3703

7.071 7.061 7.079 7.075 7.086

4.844 4.822 1.865 1.862

P-1 P-W2 P-W3 P-H2 P-H3

0.0356 0.0336 0.0339 0.0336 0.0339

0.3573 0.3600 0.3631 0.3602 0.3628

0.9898 3.0160 3.0100 3.0990 3.1130

4.611 4.606 1.856 1.862

GWN

a3

⫺9.87 ⫺1.30

⫺37.16 ⫺5.27

NSSE (%) 0.3610 0.0256 0.0253 0.0269 0.0266 0.360 0.0428 0.0384 0.0428 0.0396

the errors for all models were below 1%. The linear models, i.e., NPT-1, NPF-1, and P-1, provided comparable prediction errors. For each type of model, inclusion of nonlinearity always provided better prediction by reducing NSSE by at least 50% from the corresponding linear model. Regarding the nonparametric time-domain nonlinear models, the Hammerstein model resulted in 20% smaller prediction error compared to the Wiener model of the same order. For both the Hammerstein and Wiener models, the NSSE of the third-order nonlinear models was smaller only by 4% than that of the secondorder models. The largest reduction in prediction error occurred for the parametric second-order Wiener model 共P-W2兲 which reduced the NSSE by ⬃90% from that of the linear model. The NSSE values of the P-W3 and P-H3 models were smaller than those of the P-W2 and P-H3 by only ⬃10%. However, the F test showed that the reduction of the NSSE was significant. The P-W2 model provided slightly better prediction than that of the P-H2 共the NSSE was 10% smaller兲, which is also consistent with the fitting results in Table 2 that P-W2 fit the data slightly better than P-H2. Interestingly, the NSSE of the P-W2 model with only four parameters is over 65% and 40% smaller than that of the NPT-H2 with over 140

Model Performance and Predictability A model can only be effectively evaluated by examining its ability to predict the response to an independent input. One example is shown in Fig. 5共A兲 where we compare the predicted output of the block-structured linear P-1 and the nonlinear P-W3 models to actual measured output when the input was an IM signal. The linear model provided a good prediction of the stress output and the output of the P-W3 model appears to be slightly better only around the peak values. However, Fig. 5共B兲 shows that the P-W3 model always produced much smaller errors between model predicted values and measured data than the linear model, especially more so around the peak stresses. The superiority of the P-W3 model is even more evident from the corresponding frequency spectra 共Fig. 6兲. By definition, the linear model can only predict a smooth linear response of the system. However, we see that the nonlinear model matched almost all harmonic distortion and cross-talk interactions resulting from tissue nonlinearities. The NSSE between model predicted values and measured data are summarized in Table 4. For the parametric block structured models, the parameters estimated from the GWN were used for model prediction. First, note that

TABLE 3. Parameters estimated from the parametric block-structural models in a tissue strip. For units and definitions, see Table 2. Input VS

NSND

Model

G

H

R (10⫺4 )

a2

NSSE (%)

P-1 P-W2 P-H2

0.0387 0.0387 0.0388

0.4096 0.4096 0.4094

4.72 4.63 4.61

12.44 5.32

1.2638 0.0686 0.0724

P-1 P-W2 P-H2

0.0368 0.0367 0.0368

0.4089 0.4090 0.4089

5.08 5.32 5.52

12.13 5.20

1.2200 0.0544 0.0615


557

TABLE 4. Model performance of different models in predicting system response to an IM input. Model Nonparametric NPT-1 NPT-W2 NPT-W3 NPT-H2 NPT-H3 NPF-1 NPF-2

NSSE (%) a

0.3486 0.1726 0.1667 0.1362 0.1307 0.3800 0.11

Parametric P-1 P-W2 P-W3 P-H2 P-H3

0.3610 0.0452 0.0422 0.0499 0.0451

a

FIGURE 5. „A… The time-domain measured and predicted stress from the parametric block-structured linear „P-1… and second-order Wiener „P-W2… models. „B… The error, that is the difference between measured and model predicted stress „ T d and T m , respectively…, as a function of time.

parameters and the NPF-2 model with over 40 parameters. DISCUSSION The primary goal of nonparametric nonlinear system identification is to provide the first- and higher-order kernels of a system without presumption about the structure of the system.15 Once the kernels are determined, the structure of the kernels may provide inference on the relationship between the linear and nonlinear elements within the system, and hence perhaps on the underlying

Nonparametric time (NPT) domain models: -1: linear model; -W2: second-order Wiener model; -W3: third-order Wiener model; -H2: second-order Hammerstein model; -H3: third-order Hammerstein model. Nonparametric frequency (NPF) domain models: -1: linear model; -2: second-order nonlinear model.

nonlinear mechanisms. For example, if the slices parallel to the axis of a second-order time-domain kernel were proportional to the linear impulse response, the system could be a Wiener model, whereas one would infer the possibility of a Hammerstein model if the second-order time-domain kernel had nonzero elements only along the diagonal. In this study, we investigated and compared the robustness and predictive power of a variety of nonlinear system identification techniques in lung tissue strips using strain amplitudes relevant for tidal breathing. We found that for a mildly nonlinear system with a long memory, a nonparametric nonlinear system identification in the frequency domain is preferred over the widely used nonparametric time-domain techniques which confirms the conclusion of Zhang et al.33 obtained in isolated lung lobes. Additionally, if a parametric nonlinear model that captures the long memory of the system with a small number of parameters is also available, then the robustness of the model identified with the VS signal can be substantially increased without compromising its predictive ability. Below, we discuss the advantages and drawbacks of each identification technique with regard to robustness and predictive ability in the presence of long memory. Experimental Identification of the First-Order Kernel and Memory Length

FIGURE 6. The frequency spectrum of the measured stress data and the spectra of the stress predicted by using the parametric block-structured linear „P-1… and third-order Wiener „P-W3… models.

One simple method to obtain an apparent linear impulse response of a system is to estimate it from the unit step response 共e.g., stress relaxation in a tissue strip兲. In principle, this method can only be applied to a linear

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system. Since the step response of a nonlinear system includes contributions from the higher-order Volterra kernels, the impulse response derived from the stress relaxation may be a biased estimate of the first-order kernel. However, the lung tissue is mildly nonlinear and hence the effects of tissue nonlinearity on the stress relaxation are expected to be much smaller than those of the linear dynamics. We tested this by halving the amplitude of the step input in strain. This had virtually no effect on the results. Additionally, the value of the relaxation exponent ␤ obtained from the stress relaxation measurements is close to the ␤ values derived from all other nonlinear models we have studied. Therefore, we believe that the apparent impulse response derived from stress relaxation is a reasonable estimate of the first-order Volterra kernel and hence the linear mechanical properties of lung tissue at the level of the tissue strip. Accordingly, the amplitude of the first-order time-domain Volterra kernel of the tissue strip is a decaying power law 共Fig. 1兲 with an exponent 1⫹ ␤ (⬃1.06) giving rise to a long memory of the system. Unfortunately, the stress relaxation data cannot be used to estimate higher-order kernels. Nonparametric Time-Domain Kernel Identification In theory, the time-domain kernel analysis for nonlinear system identification requires only that the system be time invariant and the functional series converge.24 The identification of time-domain kernels generally involves a large number of parameters, which increases dramatically with not only the order of nonlinearity but also with the memory length of the system. This is particularly problematic in the case of the lung tissue since, due to the long memory, the identification requires large data sets and many parameters. Long recordings are either not available in biological experiments or can lead to an extreme computational burden to identify the nonlinear system. Zhang et al.32 have demonstrated that using Korenberg’s fast orthogonal algorithm,13 the truncation of memory length or inappropriate assignment of nonlinearity order can severely distort the identified kernels. It is worth noting that in their study, the system was identified using a flow input which is analogous to strain rate rather than strain, which was used in our study. As a consequence, the memory problem was more severe in their study since their system had an extremely long memory length that did not decay to zero.32 In this study, the impulse response we identified directly from experimental data approached the noise level in seconds. For the time-domain kernel identification approach, we chose the parallel cascade method proposed by Westwick and Kearney30 which works to decrease the number of parameters. Note that a parallel cascade expansion does not require any assumption about the sys-

tem structure since any system which has a finite Volterra series representation, also has a parallel Wiener cascade representation. If the number of identified pathways in the parallel cascade method is small compared to the memory length, then the total number of parameters to be identified is significantly smaller than in a completely general nonlinear system identification scheme. In our case, the algorithm30 identified only a single pathway. Using a Hammerstein structure resulted in a slightly improved first-order kernel allowing for the estimation of longer memory length, which is consistent with the findings of Maksym.20,21 This nonlinear model provided a fairly good fit and reasonable predictive ability. The first-order kernel 关Fig. 2共A兲兴 is qualitatively similar to the impulse response estimated from the stress relaxation 关Fig. 1共B兲兴. Most of the useful information lies in the t⫽0 point which is due to the fluid-like behavior of the system for t˜0 and hence the kernel value would approximate the resistance parameter R in Eq. 共6兲. Unfortunately, the estimated linear kernel for t⬎0 would not allow us to conclude that the impulse response follows a power law such as that derived from the stress relaxation 共Fig. 1兲. The fact that the estimated impulse response is not smooth could be caused by truncated memory length, incorrect assumption about system structure, and/or order of nonlinearity as well as the noise level in the measurement. Thus, we conclude that timedomain nonlinear system identification methods should not be the choice to identify weakly nonlinear systems with long memory.

Nonparametric Frequency-Domain Identification An alternative approach for identification of a nonlinear system with a long memory is the frequency-domain kernel analysis as suggested by Victor and Shapley28 and adopted recently by Zhang et al.33 Following several periods of periodic excitation, the system approaches a steady state, and the effects of transients due to long memory become negligible compared to the steady-state response. Indeed, this hypothesis is supported by the fact that the response of the tissue to a single sinusoidal input contains appreciable energy only at the fundamental and second-order harmonic. The validity of the estimated first-order frequency kernel is confirmed by the fact that the values of ␤ obtained from fitting the constant phase model 关Eq. 共6兲兴 to the first-order frequency kernel and directly derived from stress relaxation are practically the same. With regard to the second-order kernel, we note that in a linear spring the second-order kernel is essentially zero. In contrast, the second-order kernel of the tissue strip is different from zero with coefficients of variations ⬍12%. Additionally, the kernel also displays a


definite structure with projections to both frequency axes similar to the first-order kernel. This suggests a Wienertype structure. The frequency-domain technique also has disadvantages. The kernel values for a second-order system can be accurately estimated only at the primary and secondary derived frequencies for the first- and second-order kernels, respectively. To obtain broadband frequency kernels, one has to interpolate between kernel values assuming that the frequency kernels are continuous and smooth. The first-order kernel values at the desired frequencies can be estimated to a reasonable accuracy due to the smoothness of the spectrum. However, interpolation of the second-order kernel values in two dimensions is more difficult because of the sparse frequencies at which the kernel values can be obtained from the data. Thus, errors in the numerical interpolation of the twodimensional kernels can directly affect the predictive performance of the model. Although the nonparametric frequency-domain second-order nonlinear model 共NPF-2兲 gave a good prediction 共Table 4兲, most likely the interpolation errors caused larger prediction errors in the NPF-2 model with 36 parameters than the parametric Wiener model P-W2 with only four parameters but with smooth second-order kernels. Additionally, for a higherorder nonlinear system, the system response to different linear combinations of the input frequencies would share the same output frequency. As a consequence, the estimated low-order kernels would be distorted. Nevertheless, it appears that our frequency kernels g 1 and g 2 estimated from the tissue strips were not biased by the presence of any third-order nonlinearity that must exist because the VS input did not appear to excite the system enough to make a significant contribution to the output. Techniques have been provided for minimizing the frequency overlap problem for interaction of higher-order nonlinearities.28 However, they involve more computations and experimentally they may not be feasible. In conclusion, the frequency kernel analysis overcomes the long memory issue that appears in the time-domain analysis. It can be easily extended to the nonwhite input spectrum33 and hence serves as a good candidate for nonparametric nonlinear system identification for weakly nonlinear biological systems with long memory behavior. Parametric Block-Structured Modeling In contrast to the nonparametric modeling approaches, the parametric modeling generally requires a priori knowledge about the system, which in turn often leads to significant reduction in the number of parameters. The parametric block-structured models account for the viscoelastic properties of tissues with tissue damping G, tissue elastance H, and a Newtonian viscosity parameter R, and incorporates tissue nonlinearity with second- or

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third-order zero-memory polynomials. It is astonishing that the parametric block-structured nonlinear models, which have only four and five parameters for the secondand third-order nonlinearity, respectively, provided much better model fits and predictive power than any of the nonparametric nonlinear models, all of which have at least ten times as many parameters. We believe that the reasons are that 共1兲 the parametric model explicitly contains a slowly decaying power-law type of stress relaxation, and hence embraces the long memory nature of lung tissues and 共2兲 both the first- and second-order kernels of the model are necessarily smooth. The striking error reduction from the linear model to the nonlinear models 共⬃90%兲 indicates the presence of important tissue nonlinearity, which is consistent with previous findings.27 Also, it appears that though marginally, the Wiener model is better than the Hammerstein model, and it provides the best balance between quality of model fit and number of free parameters. This is consistent with the studies of Suki et al.27 in ventilated dogs and of Zhang et al.33 using frequency-domain analysis in isolated lung lobes. It is important to point out that the parametric block-structured models allow for robust estimation of the properties incorporated whereas the kernels identified in either the time or frequency domain are much more susceptible to noise.32 On the other hand, if the mechanism built into the parametric models are incorrect, the estimated model parameters will be biased and hence the analysis leads to false characterization of the system properties. The most important criterion of the frequency composition of the VS input is that all cross-talk interactions should be separated. This, however, requires a very sparse frequency composition with the first harmonic well separated from the fundamental frequency. In our VS signal, the first frequency containing energy was the seventh harmonic of the fundamental frequency 共Table 1兲. As a consequence, the required data length in one cycle is relatively long. In this study, we also found that the parametric block-structured model performed the best. Since the parameters were estimated using an iterative global optimization,5 fitting the parametric models may not need the sparse VS. Recently, in a study by Suki et al.,27 a similar Wiener-type model was fit to pressure–flow data obtained in dog lungs in situ. The input flow signal was the nonsum nondifference signal25 which has fewer constraints than those of the VS. The NSND frequency components are chosen to be not integer multiples of each other and only the input frequencies cannot be obtained as a linear combination of each other. As a consequence, the NSND signal requires much shorter data length per cycle if the goal is to cover the same frequency bandwidth. In our NSND signal, the third harmonic of the fundamental already contains energy 共Table 1兲. Thus, measurement time is more than 2.5

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times less in the NSND signal than in the VS signal which may be advantageous in certain biological applications. Despite these differences between the VS and NSND inputs, the parameters estimated using these inputs are nearly identical 共Table 3兲. Thus, from the practical viewpoint, using the NSND signal combined with an appropriate parametric block-structured model provides the optimal approach to characterize the macroscopic rheological properties of lung tissues. Implications for Potential Mechanisms Regardless of approach 共except for the nonparametric time-domain technique兲, we found that the mechanical properties of the lung tissue are consistent with a stress relaxation behavior that follows a slowly decaying power law. Even though power-law stress relaxation or equivalent impedance for soft tissues has been reported,3,8,10,18,26,27,31–33 our analysis is the first to unequivocally confirm it using a nonparametric and nonlinear approach unbiased by a priori assumptions and nonlinearities. While the basic mechanism giving rise to this peculiar viscoelastic property of lung tissue remains controversial, several mechanisms have been proposed.6,22,26 Suki et al.26 proposed that the power-law stress relaxation behavior is a consequence of the ‘‘reptation’’ motion of the collagen–elastin fibers in the extracellular matrix whereby the fibers undergo a series of highly constrained ‘‘worm-like’’ displacements under the influence of external stresses. Also, identifying the connective tissue network as the dominant factor of the macroscopic mechanical behavior of lung tissues, Mijailovic et al.22 proposed a stick-and-slip motion between fibers during load transfer as the primary source of the dissipative properties of tissue. An alternative mechanism was proposed by Fredberg et al.6 in which tissue dissipative properties are a consequence of the cross-bridge cycling rates within the contractile cells. However, the latter somewhat contradicts our recent findings that the mechanical properties of lung tissue strips do not depend on the metabolic state of interstitial cells.31 Predictions of the other two models qualitatively agree with the observed tissue linear properties, but neither of them could provide a satisfactory explanation for all dynamic characteristics of tissue mechanics. The appearance of harmonic distortion in the output 共Fig. 6兲, and the fact that including nonlinearities reduces the error by as much as 90%, clearly indicate the presence of at least second-order tissue dynamic nonlinearities. One of the goals of this study was to identify the nonparametric kernels of the lung tissue, which in turn can suggest structure, and hence provide more insight into the underlying mechanisms. While the time-domain techniques failed to reveal the functional structure of the lung tissue with mild nonlinearity and long memory at

the organ level32 as well as at the tissue strip level in this study, the block-structured modeling exercise indicated that the Wiener model was slightly better than the Hammerstein with regard to model fit and predictive ability. This is consistent with the whole lung studies.27,33 However, for whole lungs we found27 that the Wiener model was far superior than the Hammerstein model, i.e., the Wiener models provided at least 30% smaller NSSE in ventilated open chest dogs. Similarly, using a new and more robust structure test in the frequency domain, Zhang et al.,33 in isolated lung lobes, found strong evidence that the system behaves as a Wiener structure. One possibility for the small difference between Wiener and Hammerstein models in this study is the lack of surfactant in the tissue strip. The surfactant decreases the surface tension in the alveolar sacs and the alveolar mouth which can have an effect on the three-dimensional geometrical configuration of the alveolar structure. The lack of this phenomenon in the tissue strip may decrease the strength of any nonlinearity which in turn may abolish the difference between these two system structures. With regard to structure, to our knowledge, no mechanistic model that would be based on first principles, has been proposed for the Wiener structure. Recently, Bates2 derived a nonlinear model of lung tissue rheology. The analysis of Bates2 is based on how the fibers in the tissue modeled as rigid rods, rearrange following a step input in strain. The total stress response of the model is nearly separable in instantaneous stress and time-dependent relaxation. Since the instantaneous stress response is a nonlinear function of strain amplitude, Bates’ model provides a possible mechanistic basis of Fung’s quasilinear viscoelastic theory.7 Furthermore, due to this separability in instantaneous stress and relaxation, the model is structurally equivalent to the Hammerstein model whose time-domain off-diagonal elements are zero. Bates argues that this is a consequence of the fact that elastic recoil and stress relaxation arise from two different and noninteracting mechanisms: the first related to how fibers rearrange following a step in strain and the second being a consequence of how the fibers diffuse back to a random configuration. It is important to realize, however, that the step response of a nonlinear system contains information only about the diagonal elements of the higher-order kernels. Thus, it is possible that the tissue strip behaves more as a Hammerstein structure when probed with a step input. Our study shows that the offdiagonal kernels cannot be completely negligible. We speculate that when the tissue is excited with a complex strain signal which probes the off-diagonals, additional mechanisms 共e.g., fiber flexibility, interaction between fibers, or fibers and ground substance兲 may also contribute to the stress response and the tissue behaves more like a Wiener structure.


SUMMARY We have examined the applicability of several system identification approaches to identify the mechanical properties of lung parenchymal tissues. The time-domain kernel analysis is suitable to identify the nonlinearity and perhaps the general structure but not the detailed linear dynamics of the system most likely due to the fractal type long memory. The frequency-domain kernel identification can avoid the memory problem and appears to be a successful alternative approach for nonparametric identification. However, it is limited in identifying higher-order nonlinearities. Parametric nonlinear models which capture the long memory of the system with only a few parameters, can provide robust parameter estimation with high predictive ability. The implications are that our modeling exercise provides strong evidence unbiased by a priori assumptions and nonlinearities, that the linear dynamics follow a fractal-type long memory stress relaxation. Finally, from the practical point of view, the block-structured model combined with the NSND input signal may be the most efficient approach for characterizing the macroscopic rheological properties of parenchymal tissue. Further studies are warranted to characterize the microscopic origins of these kernels. ACKNOWLEDGMENTS This study was supported by NSF and NIH.

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