Coherent Hemodynamics Spectroscopy Based on a Paced Breathing Paradigm — Revisited

A novel hemodynamic model has been recently introduced, which provides analytical relationships between the changes in cerebral blood volume (CBV), cerebral blood °ow (CBF), and cerebral metabolic rate of oxygen (CMRO 2), and associated changes in the tissue concentrations of oxy-and deoxy-hemoglobin (ÁO and ÁD) measured with near-infrared spectroscopy This novel model can be applied to measurements of the amplitude and phase of induced hemodynamic oscillations as a function of the frequency of oscillation, realizing the novel technique of coherent hemodynamics spectroscopy In a previous work, we have demonstrated an in vivo application of CHS on human subjects during paced breathing [M. L. Pierro et al., Neuroimage 85, 222–233 (2014)]. In this work, we present a new analysis of the collected data during paced breathing based on a slightly revised formulation of the hemodynamic model and an e±cient ¯tting procedure. While we have initially treated all 12 model parameters as independent, we have found that, in this new implementation of CHS, the number of independent parameters is eight. In this article, we identify the eight independent model parameters and we show that our previous results are consistent with the new formulation, once the individual parameters of the earlier analysis are combined into the new set of independent parameters.


Introduction
Cerebral hemodynamics can be measured with nearinfrared spectroscopy (NIRS).4][5] The underlying physiological mechanisms of such hemodynamic changes are related to the cerebral blood °ow (CBF), cerebral blood volume (CBV) and cerebral metabolic rate of oxygen (CMRO 2 ).In addition to hemodynamic changes associated with brain activation, spontaneous hemodynamic °uctuations have also been reported.In particular, spontaneous low frequency oscillations (LFOs) (at frequencies .0.1 Hz) have been observed, re°ecting a combination of systemic cardiovascular dynamics and local metabolic and °ow regulation e®ects. 6While low frequency oscillations occur spontaneously, cerebral hemodynamic oscillations can also be induced at a speci¯c frequency.Protocols for inducing cerebral hemodynamic oscillations include paced breathing, 7 periodic head-uptilting, 8 repeated squat-stand, 9 a sit-stand maneuvers, 10 and cyclic pneumatic thigh-cu® in°ation. 11 quantitative relationship between induced hemodynamic oscillations and the underlying CBF, CBV, and CMRO 2 oscillations was recently obtained by a novel hemodynamic model introduced by Fantini. 1,12The model shows that measured total hemoglobin oscillations are directly linked to blood volume oscillations, while phase di®erences between oxy-and deoxy-hemoglobin oscillations are indicative of CBF and CMRO 2 changes.This hemodynamic model, together with measurements of hemodynamic oscillations at multiple frequencies, lead to the novel technique of coherent hemodynamics spectroscopy (CHS) that we have recently proposed 1 and demonstrated on human subjects. 2CHS allows for the determination of a set of physiological parameters related to the blood transit time in the microvasculature, microvascular blood volume, and autoregulation.We have shown previously that this method can be applied to NIRS data collected during a paced breathing paradigm, where cerebral hemodynamic oscillations were induced at multiple frequencies by varying the respiratory rate. 2 We have also shown that the data, namely the amplitude ratios of deoxy-and oxy-hemoglobin phasors, jDj=jOj, oxyand total-hemoglobin phasors, jOj=jTj, as well as the phase di®erences between them, arg(D)-arg(O), and arg(O)-arg(T), can be well ¯t with the hemodynamic model by Fantini. 1,2The resulting ¯ts yielded the values of ten physiological parameters (two parameters were set at ¯xed values).While the solutions did ¯t the hemodynamic spectra well, we recently found that the solutions of the physiological parameters were not unique 13 due to the lack of independence between some of them.Here, we revisit the paced breathing data set and apply a slightly revised formulation of the model with a reduced set of eight independent parameters.We demonstrate that the previously published solutions are consistent with the new set of independent model parameters.

Experimental methods and paced breathing protocol for coherent hemodynamics spectroscopy
The methods of obtaining frequency resolved spectra of hemodynamic oscillations (CHS) and the experimental setup for the paced breathing protocol have been described previously.

The hemodynamic model
The new hemodynamic model by Fantini 1 describes sinusoidal oscillations of the cerebral concentrations of oxy-and deoxy-hemoglobin as the output of a linear time invariant system for which oscillations in CBV, CBF, and CMRO 2 act as input parameters.
The oscillatory hemoglobin concentrations and physiological parameters are represented by phasors that are indicated in bold face.The model expresses O(!), D(!), and T(!) (i.e., the phasors that describe the oscillations of oxy-, deoxy-and total hemoglobin concentrations, respectively) as a function of cbv(!), cbf(!), and cmro 2 ð!Þ (i.e.phasors that describe the oscillations of CBV, CBF, and CMRO 2 , respectively). 12We have shown previously 1,2,12,13 that taking amplitude ratios and phase di®erences between hemoglobin phasors minimizes the sensitivity to the frequency dependence of the amplitude and phase of the source of induced hemoglobin oscillations (paced breathing in this study).The phasor ratios of deoxy-to-oxy hemoglobin concentrations, and oxy-to-total hemoglobin concentrations 12 are: S ðvÞ ðhS ðcÞ i À S ðvÞ ÞF ðcÞ CBV ðcÞ 0 CBV ðvÞ 0 S ðvÞ ðhS ðcÞ i À S ðvÞ ÞF ðcÞ CBV ðcÞ S ðvÞ ðhS ðcÞ i À S ðvÞ ÞF ðcÞ CBV ðcÞ 0 CBV ðvÞ 0 where H ðcÞ RCÀLP ð!Þ and H ðvÞ GÀLP ð!Þ are the capillary and venous complex transfer function, S ðaÞ ; hS ðcÞ i; S ðvÞ , are the arterial, capillary, and venous saturations, respectively, and F ðcÞ is the Fåhraeus factor (ratio of capillary-to-large vessel hematocrit).The superscripts (a), (c), and (v) for CBV and cbv indicate partial contributions from the arterial, capillary, and venous compartments, respectively, with , where CBV 0 is the baseline blood volume concentration.We assume that the blood volumes of the arterial and venous compartments have the same frequency dependence, and we take the phase of blood volume oscillations as the phase reference.In other words, we set cbv ðaÞ ð!Þ ¼ cbv ðaÞ ð!Þ\0 and cbv ðvÞ ð!Þ ¼ cbv ðvÞ ð!Þ\0 , and the phasor ratio cbv ðaÞ ð!Þ=cbv ðvÞ ð!Þ is replaced by the real constant cbv ðaÞ =cbv ðvÞ in Eqs. ( 1) and ( 2).Furthermore, it was assumed that induced hemodynamic oscillations do not involve modulation of the cerebral metabolic rate of oxygen, therefore cmro 2 ð!Þ ¼ 0. Since there is negligible dynamic dilation and recruitment of capillaries in brain tissue, [14][15][16][17][18][19] we have set cbv ðcÞ ð!Þ ¼ 0. The following relationship between cbf and cbv 1 has been used to derive Eqs. ( 1) and ( 2): where k is the inverse of the modi¯ed Grubb exponent, and H ðAutoRegÞ RCÀHP ð!Þ is the RC high-pass transfer function with cuto® frequency !ðAutoRegÞ c that describes the e®ect of autoregulation.In Eq. ( 3), we have assumed that cbvð!Þ is representative of arterial blood pressure oscillations, so that Eq. ( 3) provides a model of cerebral autoregulation.
In general, the model parameters are S ðaÞ , the arterial saturation, , the rate constant for oxygen di®usion, t ðcÞ , blood transit time in capillaries, t ðvÞ , blood transit time in veins, CBV ðaÞ 0 , F ðcÞ CBV ðcÞ 0 , CBV ðvÞ 0 , the baseline arterial, capillary (with Fåhraeus factor, F ðcÞ , correction), and venous blood volume respectively, cbv ðaÞ , cbv ðcÞ , cbv ðvÞ , the arterial, capillary, and venous blood volume amplitudes of oscillations, !ðAutoRegÞ c , the autoregulation cuto® frequency, and k, the asymptotic °ow-to-volume amplitude ratio.Assuming S ðaÞ and to be known, the number of unknown parameters is 10.We have shown previously that CHS spectra can be ¯tted and solutions can be found for those ten unknown parameters. 2 While solutions were obtained for all parameters, the ¯ts required manual adjustments and the solutions did not necessarily correspond to the one yielding the smallest 2 value.Furthermore, we have recently shown that the solutions for these 10 parameters are not unique because some of those parameters are not independent from each other. 13peci¯cally, in addition to setting cbv ðcÞ ð!Þ ¼ 0, we have found that the model parameters can be reduced to the following set of eight independent parameters: S ðaÞ , , t ðcÞ , t ðvÞ , F ðcÞ CBV , and kCBV ðvÞ 0 =CBV 0 .After assuming a ¯xed value for the arterial saturation (98%) and setting ¼ 0:8 s À1 , the number of unknown parameters can therefore be reduced to six.Additionally, given the independence of these parameters, we have improved the inversion procedure of the ¯ts, yielding the best solution corresponding to the smallest possible 2 value.Speci¯cally, using the new description of the hemodynamic model, six independent parameters were determined by ¯tting experimental data with the model [Eqs.( 1) and ( 2)], using a built-in ¯tting procedure in MATLAB (function \lsqcurve¯t ") with the default reconstruction algorithm, which is a trust region re°ective algorithm.The reconstruction of the six unknowns was done by searching within a bounded region of the six-parameter space with the input values being the four parameters jDj=jOj, jOj=jTj, arg(D)-arg(O), and arg(O)-arg(T) (measured at multiple frequencies).Optimal sets of the six unknown parameters were found by minimizing a cost function ( 2 ), which is the sum of the residuals squared.The upper and lower limits for the parameters in the ¯tting procedure re°ect physiological ranges and are summarized in Ref. 13.We have used 54 di®erent sets of initial guesses for the six unknown parameters, which were evenly spread out throughout the range of upper and lower limits for the parameters.For each initial guess, the solution of the six parameters and the corresponding 2 value were stored at each step of the ¯tting procedure for further analysis.Eventually, the previous solutions for the original 10 model parameters 2 were combined into the new set of 6 independent parameters for comparison.

Results
The measured spectra from all eleven subjects of jDj=jOj, jOj=jTj, arg(D)-arg(O), and arg(O)-arg (T) over the paced breathing frequencies are shown in Fig. 1.The solid black lines correspond to the ¯tted spectra with the minimum 2 value.The shaded gray areas correspond to all other solutions of the six parameters with a slightly greater 2 value than the smallest one, so that the ¯ts fall within one standard deviation of the data points.Since multiple solutions for each subject were obtained by the ¯tting routine, Fig. 2 shows the average solutions for each of the six parameters, with error bars corresponding to the standard deviation.Since we have analyzed this data set previously with ten unknowns, 2 we have converted the 10 previous parameters into the newly de¯ned six parameters.The converted solutions from our previous results are shown in Fig. 2 by the grey dots.It is noteworthy that the converted solutions for all but one subject (#5) fall within three standard deviations of the six parameter solutions.The 10 parameter solutions, while not unique, can therefore be converted into unique solutions that are consistent with the new data analysis.The only exception is subject #5, where, however, the ¯t of the data was not as good as the other subjects, especially in the amplitude ratios.
Each of the six parameters of the hemodynamic model in°uences the shape of the CHS spectra.We have summarized this dependence previously for the 10 parameter solutions. 2In Table 1 we report the updated summary in terms of the new set of six parameters.Table 1 is a visual summary of the key spectral features and speci¯es how they are a®ected by each one of the model parameters within the reported physiological ranges.Large arrows indicate a stronger dependence than that associated with the small arrows.We observe that some of the e®ects reported in Table 1 do not hold in general but give a guidance and reference for interpreting measured coherent hemodynamics spectra.

Discussion
We have analyzed previously collected CHS data during a paced breathing paradigm.We have previously shown that the CHS spectra can be analyzed by the novel hemodynamic model introduced by Fantini. 1,2The initial description of the model was treating each one of 10 ten model parameters as independent. 2In particular, baseline concentrations of blood volume in the three compartments as well as the blood volume oscillations in each compartment were assumed to be independent parameters.We have recently demonstrated that this assumption was not correct, but that, instead, the number of independent parameters can be reduced to six, only giving access to ratios of baseline blood volume and oscillations between microvascular compartments.Also, the inverse of the Grubb's exponent, k, is not an independent variable, but rather linked to the venous-to-total blood volume ratio.
Converting the previously obtained results 2 into the new set of six independent parameters showed that the previous results are consistent with the ¯tting results obtained with the new description (Fig. 2).However, it shall be pointed out that the previous results (gray dots in Fig. 2) were obtained by ¯tting the data to the hemodynamic model considering also the contributions from the capillary volume oscillations, cbv ðcÞ ð!Þ, which have been set to zero in this new formulation.This di®erence could explain remaining discrepancies between the solutions of the six parameters.

Conclusion
The introduction of quantitative photon migration methods and hemodynamic models have played a key role in the e®ort of translating NIRS measurements into relevant physiological, functional, or diagnostic parameters.The major requirement for such models to be e®ective is to achieve a good compromise between (1) their ability to describe the complex structure and biology of tissues and (2) a su±ciently simpli¯ed treatment that limits the number of free parameters.In this paper, we have reported the analysis of coherent hemodynamics spectra collected in a paced breathing protocol.This analysis was based on a new hemodynamic model that, while treating the complexity of the cerebral microvasculature, 1 results in a limited number (eight) of independent physiological parameters.This number is smaller than the set of 12 parameters considered in a previous implementation. 2he 8 independent parameters are the arterial saturation, rate of oxygen transfer from blood to tissue, blood transit time in the capillary and venous compartments, relative capillary-to-venous blood volume ratio, relative arterial-to-venous blood volume changes in response to a hemodynamic challenge, cuto® frequency for cerebral autoregulation, and the product of the high-frequency °ow-tovolume amplitude ratio times the venous-to-total blood volume ratio.These model parameters provide a relevant description of the cerebral vasculature, its blood °ow, and its dynamic response to physiological or functional perturbations, so that they can be important for a range of functional and diagnostic studies.

Fig. 1 .
Fig.1.Measured CHS spectra of all eleven subjects (symbols), and best ¯ts obtained with the hemodynamic model (lines), as well as all solutions within one standard deviations of the experimental data (shaded area) for all eleven subjects.For each subject, the right panel shows the amplitude ratio spectra ratio spectra (jOj=jTj and jDj=jOj), and the left panel shows the phase di®erence spectra (Arg(O)-Arg(T) and Arg(D)-Arg(O)).