Study of capillary transit time distribution in coherent hemodynamics spectroscopy

A recently proposed analytical hemodynamic model [S. Fantini, NeuroImage 85, 202–221 (2014)] is able to predict the changes of oxy, deoxy, and total hemoglobin concentrations (model outputs) given arbitrary changes in blood °ow, blood volume, and rate of oxygen consumption (model inputs). One assumption of this model is that the capillary compartment is characterized by a single blood transit time. In this work, we have extended the original model by considering a distribution of capillary transit times and we have compared the outputs of both models (original and extended) for the case of sinusoidal input signals at di®erent frequencies, which realizes the new technique of coherent hemodynamics spectroscopy (CHS). For the calculations with the original model, we have used the mean value of the distribution of capillary transit times considered in the extended model. We have found that, for distributions of capillary transit times having mean values around 1 s and a standard deviation less than about 45% of the mean value, the original and extended models yield the same CHS spectra (i.e., model outputs versus frequency of oscillation) within typical experimental errors. For wider capillary transit time distributions, the two models yield di®erent CHS spectra. By assuming that Poiseuille's law is valid in the capillary compartment, we have related the distribution of capillary transit times to the distributions of capillary lengths and capillary speed of blood °ow to calculate the average capillary and venous saturations. We have found that, for standard deviations of the capillary transit time distribution that are less than about 80% of the mean value, the average capillary saturation is always larger than the venous saturation. By contrast, the average capillary saturation may be less than the venous saturation for wider distributions of the capillary transit times.


Introduction
Several hemodynamic models proposed in the literature have tried to address the complex relationship between the dynamics of tissue oxygen consumption, blood°ow, and blood volume with those of some macroscopically measured signals. These models have been particularly useful for the physiological interpretation of the blood oxygen level dependent (BOLD) signal measured in functional magnetic resonance imaging (fMRI; see for example a review by Buxton 2 ) and the optical signals measured in near-infrared spectroscopy (NIRS). [3][4][5][6][7][8][9] Some of the more sophisticated models proposed in NIRS are characterized by complex systems of di®erential equations that can be solved only numerically. 4,5,8 Recently, a new hemodynamic analytical model has been proposed by Fantini. 1 The model has been developed for both time-and frequency-domain data and can predict the absolute values or the changes of oxy, deoxy, and total hemoglobin concentrations (outputs of the model) given changes in tissue oxygen consumption, blood°o w, and blood volume (inputs of the model). In the frequency domain, phasors (i.e., complex numbers expressed in terms of modulus and phase) are associated with sinusoidally oscillating inputs and outputs. For example, in the brain the input phasors represent the normalized oscillations of cerebral blood volume, cerebral blood°ow, and cerebral metabolic rate of oxygen [cbv(!), cbf(!), and cmro 2 (!), respectively] while the output phasors represent the oscillations of oxy, deoxy, and total hemoglobin concentrations 1 [O(!), D(!), and T (!), respectively]. We named this method of inducing blood volume and°ow oscillations (and therefore oxy and deoxy hemoglobin oscillations) in target organs at di®erent frequencies as coherent hemodynamics spectroscopy (CHS). The oscillations can be induced by di®erent methods such as paced breathing, 10 leg cu®s in°ations, 11 tilt bed, etc. Since the absolute value of modulus and phase of the output phasors (which are measured) are often di±cult to control, [10][11][12] in CHS we measure the spectra of phasor ratios: D(!)/O(!) and O(!)/T (!). The hemodynamic model depends on 14 physiological parameters, which include the capillary and venous blood transit times, the baseline partial blood volumes in the three vascular compartments, and others that are described in our recent publications. 1,12 However, in CHS (which is based on measurements of phasor ratios) we have shown that the number of independent parameters reduces to six. 12 One of the assumptions of the model is that the capillary and venous transit times (two parameters used in CHS) are single-valued parameters characterizing the overall circulation in the capillary and venous compartments. In this work, we have studied the spectra obtained in CHS for the more realistic case of a distribution of capillary transit times, and we have compared the results to those of the original model. We have also investigated how the distribution of capillary transit times a®ects the average capillary and venous saturations.  (4) and (5)]; ctHb is the hemoglobin concentration in blood, and F ðcÞ is the Fåhraeus factor (ratio of capillary-to-large vessel hematocrit); the superscripts (a), (c), and (v) indicate the arterial (a), capillary (c), and venous (v) compartment values of hemoglobin saturation (S), static blood volume (CBV 0 ) and oscillatory blood volume (cbv). The oscillatory capillary blood volume [cbv ðcÞ ð!Þ] was set to zero because of the negligible dynamic dilation and recruitment of capillaries in the brain. 13,14 The transfer functions for the capillary and venous compartments represent resistor-capacitor (RC) and Gaussian low-pass¯lters, respectively 1 :

Materials and
Equations (4) and (5) describe the low-pass¯lter e®ect of the capillary and venous compartments, which are considered as linear time-invariant systems having cbf(!) and cmro 2 (!) as inputs, and the cerebral oxy and deoxyhemoglobin concentrations [O(!), D(!)] as outputs. We note that the transfer functions depend on two singled-value parameters, the capillary transit time (t ðcÞ ) and the venous transit time (t ðvÞ ), which determine the characteristic time constants of the capillary and venous transfer functions. The key time constants, which represent the inverse of the low-pass cuto® angular frequency, are $ t ðcÞ =e for the capillary compartment and $ ½t ðcÞ þ t ðvÞ =3:56 for the venous compartment. 1 According to this model, the average capillary saturation, hS ðcÞ i, and the venous saturation, S ðvÞ , are related to the arterial saturation, S ðaÞ , by the following relationships: where is the rate constant of oxygen di®usion from capillary blood (and from blood in the smaller arterioles) to tissue.
Because of the high-pass nature of the cerebral autoregulation process that regulates cerebral blood°o w in response to blood pressure changes, 15,16 we introduce the following relationship between cbf and cbv 1,12 : where we consider blood volume as a surrogate for blood pressure, k is the inverse of the modi¯ed Grubb's exponent, and H ðARÞ RCÀHP ð!Þ is the RC high-pass transfer function that describes cerebral autoregulation. H ðARÞ RCÀHP ð!Þ is given by the following expression: where ! ðARÞ c is the cuto® angular frequency for autoregulation.

Extended hemodynamic model: Distribution of capillary transit times
In our extension of the original model to the case of multiple capillary transit times (which we indicate here with ¼ t ðcÞ ), we introduce a distribution of capillary transit times described by a gamma probability density function (pdf) 17 hð; ; Þ: where (dimensionless) and (units of time) are constants, and ÀðÞ is the gamma function evaluated at . It can be shown that the mean, hi, and standard deviation, ðÞ, of the continuous random variable are given by hi ¼ and ðÞ ¼ ffiffiffi p .

E®ects of capillary transit time distribution on the average capillary and venous saturations
According to our hemodynamic model, the average capillary saturation for a capillary transit time is hS ðcÞ iðÞ ¼ S ðaÞ ð1 À e À Þ=ðÞ [see Eq. (6)]. A speci¯c capillary characterized by a blood transit time contributes to the saturation of a draining venule by an amount proportional to the saturation at the end of the capillary, which is S ðvÞ ðÞ ¼ S ðaÞ e À [see Eq. (7)], and to its speed of blood°ow.
In other words, due to the law of°ow conservation, the venous saturation results from a weighted average of the¯nal saturation of the con°uent capillaries, with weights given by the blood°ow velocities in each capillary. On the contrary, the average capillary saturation results from a spatial average over the capillary lengths.
In order to derive the distribution of capillary lengths and blood°ow velocities from the distribution of capillary transit times ½hð; Þ, we have assumed that Poiseuille's law is valid in the capillary compartment 8 : where R is the resistance of a vessel of length l and diameter d, and is the blood viscosity. We also assume a simple circuital equivalent relationship between capillary°ow (F ) in the capillary and pressure di®erence (ÁP ) at the extremities of the capillary 8 : where R is the resistance de¯ned by Eq. (16). Since F ¼ ðcÞ c ðcÞ , where ðcÞ is the capillary cross section (assumed to be constant) and c ðcÞ is the capillary°o w velocity, from Eqs. (16) and (17) we deduce that c ðcÞ / 1=l (assuming that ÁP is also constant). In each capillary, the transit time (), the°ow velocity ðc ðcÞ Þ, and the length (l) are related by: ¼ l=c ðcÞ . Therefore we can write that: l / ffiffiffi p and c ðcÞ / 1= ffiffiffi p . The capillary length l and the capillary°o w velocity c ðcÞ are functions of the random variable therefore they are also random variables. The pdf of c ðcÞ satis¯es: f c ðcÞ dc ðcÞ ¼ hð; ; Þd, where f c ðcÞ and dc ðcÞ are the pdf and the di®erential of c ðcÞ . Similarly the pdf of the capillary length satis¯es: f l dl ¼ hð; ; Þd where f l and dl are the pdf and the di®erential of l, respectively. Given the information provided above, we can¯nally write the expressions of the average capillary [Eq. (18) We have compared the saturations given by Eqs. (18) and (19) with those provided by the original model [Eqs. (6) and (7)], where we set the single-value capillary transit time t ðcÞ ¼ hi.

Results
The results presented in this section were obtained by solving numerically the integrals of Eqs. (11) and (12) and Eqs. (18) and (19) Figure 1(a) shows the pdf of the capillary transit time, hð; ; Þ for ¼ 5 and ¼ 0:2 s, corresponding to hi ¼ 1 s and ðÞ % 0:45 s (i.e., 45% of hi). We used the original and the extended models to compute the four quantities of interest in CHS: jDð!Þj=jOð!Þj, Arg½Dð!Þ À Arg½Oð!Þ, jOð!Þj= jTð!Þj, Arg½Oð!Þ À Arg½Tð!Þ [see Fig. 1(b)]. In Fig. 1(b), the crosses were obtained with the original model (characterized by a single capillary transit time, which was chosen as t ðcÞ ¼ hi ¼ 1 s) and the empty circles were obtained with the extended model using the capillary time distribution of Fig. 1(a). The maximum discrepancy between the ratios of oscillation amplitudes (i.e., the ratio of phasors' modulus) calculated with the two models is about 4% (jDj=jOj) and 1% (jOj=jTj). As for the phase di®erences of phasors, the maximum discrepancies were $ 1 for both Arg(D)ÀArg(O) and Arg(O)ÀArg(T). These discrepancies are of the same order of magnitude of typical experimental errors found in the measurements of these quantities. The average capillary and venous saturation values calculated with the extended model were hS ðcÞ i mCTT % 0:66 and hS ðvÞ i mCTT % 0:50, while the capillary and venous saturations calculated with the original model were hS ðcÞ iðt ðcÞ ¼ hiÞ % 0:67 and hS ðcÞ iðt ðcÞ ¼ hiÞ % 0:44. Figure 2(a) shows the pdf of the capillary transit time, hð; ; Þ for ¼ 1:108 and ¼ 1:26 s, corresponding to hi % 1:4 s and ðÞ % 1:33 s (i.e., 95% of hiÞ. This is a wider distribution, having mean and standard deviation values that have been measured on a group of rats under resting conditions. 23 Although the distribution of transit times measured in the work of Stefanovic et al. 23 is not closely represented by a gamma distribution, here we continue considering a gamma distribution for the capillary transit times. 17 In Fig. 2(b), we show the CHS spectra obtained with the two models. The cross symbol is used for the original model whereas the empty circle is used for the extended model. The phase di®erences of phasors generated by the two models di®er by less than by the two models di®er by less than $ 13% for jDj=jOj and less than $ 10% for jOj=jTj. Note also that the spectral trends (i.e., the frequency dependence) of jDj=jOj obtained by the two models are quite di®erent. Therefore, we can state that for wider capillary transit time distributions (standard deviation approaching the mean value) the discrepancies between the two models are larger than typical errors found in the measurements of the quantities of interest in CHS.
Regarding the average capillary and venous saturation values for the original model (single capillary transit time) we obtained hS ðcÞ iðt ðcÞ ¼ hiÞ % 0:59 and S ðvÞ ðt ðcÞ ¼ hiÞ % 0:32. The extended model yielded hS ðcÞ i mCTT % 0:55 and S ðvÞ mCTT % 0:64, i.e., an average capillary saturation that is lower than the venous saturation. We note that an average capillary saturation lower than the venous saturation has been recently measured in animal models. 24 We have run calculations for di®erent values of the parameters and of the distribution function hð; ; Þ that are reported in Table 1 of the article by Jespersen and Østergaard 17 to represent measurements on rats during baseline and under various stimulations conditions. We can summarize our results by saying that for distributions of capillary transit times with ðÞ=hi . 45%, the discrepancies between the spectra generated by the two models are within the experimental errors. Furthermore, the calculations on the average capillary and venous saturations have shown that for distributions of capillary transit times having ðÞ=hi . 80% the average capillary saturation is always greater than the venous saturation. The opposite is true for wider distributions of capillary transit times.

Discussion
In this work, we have investigated the e®ect of a capillary transit time distribution on the spectra obtained with the novel technique of CHS and on the computed values of average capillary and venous saturations. We have de¯ned new formulas that incorporate the capillary transit time distribution in the calculation of oxy and deoxy hemoglobin oscillations [Eqs. (11) and (12)]. The CHS spectra derived with this new extended model have been compared with those obtained with the original model based on a single capillary transit time. We have found that whenever distributions are relatively \narrow" (i.e., distributions having mean value and standard deviation such that ðÞ=hi 45%) the spectra obtained with the two models yield the same values to within experimental errors. For wider distributions, the two models yield increasingly di®erent spectra.
We have also investigated the question of how the choice of baseline parameters a®ect the CHS spectra (data not shown). In particular from our preliminary data, we have found that changing , (up to 20% of the value assumed in this study) and the baseline blood volumes (up to three times of the values assumed in this study), has negligible e®ect on the spectra compared with the e®ect of the distribution of capillary transit times. In fact, we recall that CHS spectra are based on amplitude ratios and phase di®erences of oxy-and deoxy (or oxy and total) hemoglobin oscillations rather than on the absolute values of the oscillating hemoglobin species. Therefore, we argue that the choice of the baseline parameters has a lesser impact on the CHS spectra than on the computed amplitude and phase of the individual oscillations of the hemoglobin species.
We have also tested our model for the calculation of the oxygen extraction factor [de¯ned as OEF ¼ ðS ðaÞ À S ðvÞ Þ=S ðaÞ ] in the two di®erent situations addressed also by Jespersen and Østergaard 17 : (a) for a single transit time (taken to be the mean value of the distribution of transit times considered in case b); (b) for a distribution of transit times. What we have found is that, for the baseline parameters and distribution of transit times of Fig. 1 For the calculations of the average capillary and venous saturations we have derived the capillary blood°ow velocity and capillary length distributions from the distribution of capillary transit times. For this purpose we have used Poiseuille's law for the resistance of a blood vessel and a simple circuital equivalent relationship between the pressure difference at the extremities of the blood vessel and the blood°ow in the vessel. Calculations of the average capillary and venous saturations have shown that for capillary transit time distributions characterized by ðÞ=hi . 80%, the average capillary saturation is always greater than the average venous saturation, but the opposite is true for wider distributions.
For the results of this work we have assumed no change in oxygen metabolic rate. This is only an approximation, since if (the net rate of oxygen transfer from blood to tissue) does not change (as we have assumed), the cerebral metabolic rate of oxygen oscillates in phase with cerebral blood°ow. More precisely, the relationship between cerebral blood°ow oscillations (cbf) and cerebral metabolic rate of oxygen oscillations (cmro 2 Þ for ¯xed is 25 : cmro 2 ¼ cbf ð1 À S ðvÞ hS ðcÞ iÞ. We note that in this case the changes in oxygen consumption are only induced by°ow oscillations (and associated oscillations in the oxygen concentration in capillary blood) and their average value over a period is zero (which is not the case during a period of brain activation that is associated with an increased cerebral metabolic demand). Given the fact that, in the model equations, cbf and cmro 2 only appear combined in their di®erence [cbf ð!Þ À cmro 2 ð!Þ], the above observations lead to the conclusion that an in-phase contribution from cmro 2 ð!Þ has an e®ect that is equivalent to a reduction in cbf ð!Þ.
Given the autoregulation high pass relationship between volume and°ow oscillations [Eq. (8)], we can show that a reduction of cbf(!) amplitude will not a®ect the CHS spectra as long as equal arterial and venous volume oscillations are considered. 12 In addition to a single capillary transit time, the original model also considers a single venous transit time, thus assuming that microvascular hemodynamics can be represented by e®ective transit times instead of the actual distribution of blood transit times. We have run some additional calculations to study the dependence of CHS spectra on the distribution of venous blood transit times by also using a gamma distribution with ht ðvÞ i ¼ 3 s and ðt ðvÞ Þ ¼ 1:22 s. We have found that the distribution of venous transit times has a more negligible e®ect on the calculated CHS spectra than the distribution of capillary transit times.

Conclusions and Future Directions
In this work, we have compared the spectra calculated in CHS by using two models: (a) our original model which considers e®ective, single-valued capillary and venous transit times to describe the entire microvasculature; (b) an extended model which considers a distribution of capillary transit times for the calculation of CHS spectra. We have found that our original model can reproduce the spectra obtained with the extended model (to within typical experimental errors) for distributions having standard deviation [ðÞ] and mean value (hi) verifying ðÞ=hi . 45%. Future work will further quantify the di®erence between the original and extended models in terms of the values of the six physiological parameters that are obtained by¯tting the measured CHS spectra. 12 More speci¯cally, we will generate CHS spectra with the extended model, assign random errors to the data points, and run the inverse procedure based on a¯t with our original model (considering only a single capillary transit time). In this fashion, we will quantify the errors on the recovered physiological parameters associated with the approximation of distributions of capillary and venous blood transit times with single, e®ective values.
Another future direction of our work will be concerned with the robustness of our results when di®erent distribution functions are chosen. In fact, the gamma distribution used here and previously proposed 17 does not always closely match experimentally measured capillary transit time distributions. 23 Finally, the fact that in our calculations we have neglected changes in the metabolic rate of oxygen [cmro 2 ð!Þ ¼ 0] means that this study is applicable to cases where the high-pass¯lter relationship be-tween°ow and volume is the relevant one, and the e®ects of oxygen consumption changes on the CHS spectra can be neglected, as we have argued in the discussion. We plan on further exploring whether cases in which changes in metabolic rate of oxygen may not be neglected (such as functional brain studies) require additional considerations and determine new results.