INSTITUTE OF PHYSICSPUBLISHING Phys. Med. Biol. 47 22 N249 N257 PHYSICS INMEDICINE AND BIOLOGY PII: S31-9155234248-9 NOTE Ahaemodynami model for the physiologial interpretation of in vivo measurements of the onentration and oxygen saturation of haemoglobin Sergio Fantini Bioengineering Center, Department of Eletrial Engineering and Computer Siene, Tufts University, 4 Colby Street, Medford, MA 2155, USA Reeived 26 February 22 Published 5 September 22 Online at staks.iop.org/pmb/47/n249 Abstrat We present a model that desribes the effet of physiologial parameters suh as the speed of blood flow, loal oxygen onsumption, apillary reruitment, and vasular dilation/onstrition on theonentrationandoxygen saturation of haemoglobin in tissue. This model an be used to guide the physiologial interpretation of haemodynami and oximetri data olleted in vivo with tehniques suh as optial imaging, near-infrared spetrosopy and funtional magneti resonane imaging. In addition to providing a formal desription of well-established results exerise-indued hyperemia, reperfusion hyperoxia, derease in the onentration of deoxyhaemoglobin indued by brain ativity, measurement of arterial saturation by pulse oximetry, et., this model suggests that the superposition of asynhronous ontributions from the arterial, apillary and venous haemoglobin ompartments may be at the origin of observed out-of-phase osillations of the oxyhaemoglobin and deoxyhaemoglobin onentrations in tissue. 1. Introdution Several medial imaging, diagnosti and researh tools are sensitive to the haemoglobin, in its oxygenated and/or deoxygenated forms, that is present in the blood. For example, near-infrared spetrosopy NIRS measures the optial absorption assoiated with oxyhaemoglobin and deoxyhaemoglobin, and BOLD blood oxygen level-dependent funtional magneti resonane imaging fmri is based on magneti suseptibility hanges indued by paramagneti deoxyhaemoglobin. NIRS measurements of the onentration and oxygen saturation of haemoglobin in tissue, the fmri BOLD signal and any measurement of blood oxygenation in vivo are the result of the interplay among a number of physiologial parameters suh as blood volume, blood flow and metaboli rate of oxygen. Several models have been developed for the oupling between erebral blood flow and oxygen metabolism Buxton and Frank 1997, Hyder et al 1998, for the relationship between the BOLD signal 31-9155/2/18249+9$3. 22 IOP Publishing Ltd Printed in the UK N249
N25 SFantini Tissue Probed volume: σ L. Blood vessel O 2 Figure 1. Shemati representation of our model. A blood vessel of rosssetion σ intersets for a length L the volume probed by the measurement tehnique of interest. The average blood speed within the vessel is indiated with,while Ȯ 2 represents the rate of oxygen diffusion from the blood vessel to the tissue within volume. and the erebral blood volume, blood flow and oxygen onsumption Ogawa et al 1993, Mandeville et al 1999, Hoge et al 1999, and for the dynami bioenergeti relationship between the deoxyhaemoglobin onentration measured by NIRS and musle oxygen onsumption Binzoni et al 1999. It has been shown that the interpretation of optial data an be guided by the same haemodynami modelling priniples used for the fmri BOLD signal Mayhew et al 21. In this note, we present a general haemodynami model that relates the onentration and oxygen saturation of haemoglobin in tissue to a number of physiologial parameters suh as the speed of blood flow, the loal oxygen onsumption, apillary reruitment and vasular dilation/onstrition. This model speifies the onditions that determine a higher or lower sensitivity of the onentration and saturation of haemoglobin in tissue to these physiologial parameters. The analytial relationships derived here an be used to guide the physiologial interpretation of haemoglobin-related measurements in living tissue. 2. The model The shemati representation of the problem is illustrated in figure 1. Ablood vessel ofross setion σ,inwhihblood flows with an average speed,intersets for a length L the volume of interest. For instane, may be a voxel in fmrioroptialimaging, or the tissue region probed by NIRS, i.e. the volume that ontains most of the photon migration paths from the illumination point to the olletion optial fibre. As the blood flows within volume, the onentration of oxygen in the blood [O 2 ] may derease as a result of oxygen diffusion to tissue ells, where the derease rate is proportional to the differene between the oxygen onentrations in the plasma [O 2 ] plasma and in the tissue [O 2 ] tissue.thisdiffusion proess may be desribed by assigning to eah oxygen moleule a probability of extration per unit time k, whih depends on the permeability and surfae area of the blood vessel wall Buxton and Frank 1997. By using the speed of blood flow,the probability of oxygen extration per unit length along the blood vessel an be written as k/.asaresult, if l is the line oordinate along the blood vessel, one an write d[o 2 ] = k [O2 ] plasma [O dl 2 ] tissue = kr [O 2] 1 where r = [O 2 ] plasma [O 2 ] tissue/ [O 2 ]. By defining = kr, andassuming that and are independent of l [O 2 ] l = [O 2 ] e l 2
Haemodynami model for the onentration and saturation of haemoglobin in vivo N251 where the subsript in [O 2 ] indiates the oxygen onentration in the blood at l =, i.e. before any oxygen extration ours within volume.fromequation 2, the total rate of oxygen extration Ȯ 2 over the length L of the blood vessel is given by Ȯ 2 = [O 2 ] [ / 1 exp αȯ2 L ] σ whihshows the dependene of the oxygen extration rate on and. The average value of [O 2 ] within volume an be easily alulated as follows: [O2 ] = 1 L [O 2 ] l dl = [O 2 ] 1 e αȯ L 2. 3 L L The derivation of equation 2 isbased on the assumption that is not a funtion of l. If [O 2 ] plasma [O 2 ] tissue,whihisan approximation that has been used for brain tissue Buxton and Frank 1997, Hyder et al 1998 thespatial uniformity of implies that the ratio [O 2 ] plasma/ [O 2 ] is not a funtion of the blood oxygenation. Beause of the nonlinear oxygen equilibrium urve of haemoglobin, this assumption is not stritly orret, but Buxton and Frank found that the nonlinear orretion to equation 2 isnot signifiant Buxton and Frank 1997. However, Hyder et al, whoalso used r [O 2 ] plasma /[O 2 ], allowed for a variable diffusivity in their model for the regulation of erebral oxygen delivery Hyder et al 1998. Mayhew et al proposed that [O 2 ] tissue should not be negleted in r Mayhew et al 21, so that an inreased ellular utilization rate of oxygen an diretly inrease the probability of oxygen extration by dereasing [O 2 ] tissue and thereby inreasing the [O 2 ]gradient aross the blood vessel wall. Here, we proeed on the assumption that is uniform over the length L of the blood vessel, but we allow for temporal hanges of assoiated with hanges in [O 2 ] tissue that reflet hanges in the ellular metaboli rate of oxygen. The onentration of oxygen in the blood [O 2 ] results from two ontributions; from dissolved oxygen in the plasma [O 2 ] plasma and from oxygen bound to haemoglobin [O 2 ] haemoglobin. Beause [O 2 ] plasma normally aounts for only 1 3% of [O 2 ] Guyton and Hall 2, we will onsider [O 2 ] [O 2 ] haemoglobin,sothatthe onentration of oxyhaemoglobin in the blood [HbO 2 ] 1 4 [O 2] where the fator 1/4 aounts for the four binding sites for oxygen ateah haemoglobin moleule. The onentration of oxyhaemoglobin in the tissue volume [HbO 2 ] tissue an be written as the average onentration of oxyhaemoglobin in the blood times the blood-volume fration /. Using equation 2 and the approximate proportionality between [HbO 2 ] and [O 2 ],[HbO 2 ] tissue an be expressed as follows: [HbO 2 ] tissue = [HbO 2 ] = [HbO 2 ] L 2 L = SO 2 [HbT] L 2 σ 4 where the subsript indiates the initial value at l =, before any oxygen extration ours in volume,[hbt] = [HbO 2 ] +[Hb] is the total haemoglobin onentration in the blood we drop the subsript in [HbT] beause the total onentration of haemoglobin is not a funtion of l, and SO 2 = [HbO 2] is the oxygen saturation [HbT] of haemoglobin at l =. From equation 4 itisstraightforward to derive the following expressions for the onentrations of deoxyhaemoglobin and total haemoglobin in tissue
N252 SFantini [Hb] tissue and [HbT] tissue = [HbO 2 ] tissue +[Hb] tissue,respetively, and for the tissue saturation StO 2 = [HbO 2 ] tissue/ [HbT] tissue : [Hb] tissue = [HbT] tissue [HbO 2 ] tissue [ = [HbT] 1 SO 2 ] 1 e αȯ L 2 L 5 [HbT] tissue = [HbT] StO 2 = SO 2 1 e αȯ L 2 6 L. 7 Equations 4 7 provide analytial relationships that relate the onentration and oxygen saturation of haemoglobin in the tissue to physiologial parameters suh as the loal rate of oxygen extration, the blood oxygenation and the speed of blood flow. During a measurement, depending on the partiular protool and the individual physiologial response, there may be variations in SO 2,[HbT], σ, or. We assume that L and remain onstant and we also ontinue to onsider the ase of a single blood vessel. The assumption of onstant L and is based on the fat that hanges in and L aused for instane by hanges in the tissue optial properties in the ase of NIRS are likely to indue smaller effets in the measurements of onentration and saturation of haemoglobin with respet to the effets of the other mentioned parameters. The hanges in [HbO 2 ] tissue,[hb] tissue,[hbt] tissue and StO 2 an be expressed as follows by differentiation: [HbO 2 ] tissue = SO 2 [HbT] σ + [HbT] [HbT] [ 2 L + σ σ αȯ2 L +1 L 2 SO 2 SO 2 +SO 2 [HbT] σ ] αȯ 2 + 8 [Hb] tissue = [HbT] [ 1 SO 2 2 L [HbT] [HbT] + σ σ L 2 SO 2 SO 2 [ L 2 ] αȯ2 L αȯ2 +1 [HbT] tissue = [HbT] [HbT] [HbT] L ] +SO 2 [HbT] σ +SO 2 [HbT] σ + σ σ 9 1
Haemodynami model for the onentration and saturation of haemoglobin in vivo StO 2 = SO 2 L 2 SO2 L SO 2 [ L 2 ] αȯ2 L L +1 +SO 2 αȯ 2 N253 +. 11 Beause the fators that multiply the parentheses ontaining the relative hanges x/x are all positive where x stands for any physiologial parameter, the sign in front of any relative hange x/x indiates the diretion of the hange plus: inrease; minus: derease in [HbO 2 ] tissue,[hb] tissue,[hbt] tissue or StO 2 assoiated with an inrease in the physiologial parameter x. The ase of multiple blood vessels within the probed volume requires a summation of theorresponding single-blood-vessel terms on the right hand side of equations 4and5to obtain [HbO 2 ] tissuem and [Hb] tissuem the supersript M indiates the multi-vessel ase. [HbT] tissuem and StO M 2 are then omputed from [HbO 2 ] tissuem and [Hb] tissuem. The more general ase where the blood vessels within volume inlude arteries, apillaries and veins an be examined by generalizing equations 4 7asfollows: [ [HbO 2 ] tissuem = SO 2 a-blood a [Hb] tissuem = [HbT] tissuem = StO M 2 = +SO 2 v-blood v +SO -blood 2 ] [HbT] { [ 1 SO2 a-blood a + 1 SO 2 -blood + } 1 SO 2 v-blood v [HbT] a + SO 2 a-blood a [HbT] + v +SO -blood 2 1 e αȯ L 2 a + L 2 -blood σ -blood -blood + v 1 e αȯ L ] 2 -blood -blood -blood σ L +SO 2 v-blood v where the supersripts a,, and v indiatethe arterial, apillary and venous haemoglobin ompartments, respetively, and we have onsidered that the oxygen extration ours only in the apillaries i.e. = for arteries and veins so that we have dropped the subsript in SO 2 a-blood and SO2 v-blood,andwehave set [HbT] to be the same for all blood vessels for simpliity. The fat that the small-vessel haematorit is typially lower than the large-vessel haematorit an be aommodated in the model byusing different values for [HbT] a-blood, [HbT] -blood and [HbT] v-blood rather than fatorize a ommon value for [HbT] as done in equations 12 14. However, the regional differene between the haematorit in small and large vessels is hard to estimate in vivo. Data in the literature indiate that the small-vessel to large-vessel haematorit is on the order of.8.9 Grubb et al 1974. 12 13 14 15
N254 SFantini 3. Disussion The relationships derived here provide indiations on the relative ontributions of SO 2, [HbT], σ, and to the average onentration and saturation of haemoglobin in tissue, and on the onditions that affet suh ontributions. The quantitative preditions of this model, however, should be used with aution and their appliability evaluated for eah partiular ase. In fat, in addition to the assumptions already mentioned uniform and over the length of the blood vessel, time-independent and L our model simplifies the treatment of the partial ontributions within the volume by assuming that all volume elements within ontribute equally to the measurementof the haemoglobin-related parameters see equation 3. This may not always be the ase, for example, beause of the non-uniform density of the photon migration paths within in NIRS. In this ase, a quantitative treatment of the problem would require the introdution of spatially dependent weight funtions Graber et al 1993oramore rigorous treatment of photon transport in tissue Arridge and Hebden 1997. Furthermore, the size of individual blood vessels also plays a role in determining their relative ontribution to the optial measurements Liu et al 1995. When the size of is smaller, as may be the ase for optial imaging and fmri voxels, this problem is minimized but one has to onsider that SO 2 -blood forsmall portions of apillaries may depend on and,thus introduing an additional variable inthe model. Nevertheless, our modelprovidessimple analytial relationships of general appliability, at least for qualitative analyses, whih would not be available with more omplex, quantitative models. Our model is appliable to the desription of equilibrium states or hanges indued by transitions from one equilibrium state to another. However, this model is not restrited to stationary onditions but it is also appliable to dynamial proesses, provided that the variations assoiated with these proesses our on a time sale that is longer than the time needed to reah equilibrium. Examples of suh proesses are the pressure-indued volume osillations of the arterial and venous ompartments, whih our on a time sale of seonds, muh longer than the essentially instantaneous pressure volume equilibrium proess determined by the inompressibility of blood. In the ase of hanges assoiated with the blood flow, with the ellular metaboli rate of oxygen, or with a vasular bolus of oxyhaemoglobin or deoxyhaemoglobin, the time needed to reah equilibrium is determined by the vasular transit time arossvolume.thistime depends on the size of and on the speed of blood flow. An in vivo study on rhesus monkeys has found a erebral vasular mean transit time in the range 2 6 s Grubb et al 1974, whih an be onsidered as an upper limit for the vasular transit time through the volume of interest mm 3 to m 3 onsidered here. These equilibrium requirements should be onsidered when using thismodel for quantitative analyses. However, qualitative analyses of the relative ontributions to [HbO 2 ] tissue,[hb] tissue,[hbt] tissue and StO 2 from the physiologial parameters onsidered here an be performed on a more general basis. A number ofresults reported in the literature find a formal desription in the model presented here. For example, equations 1 and14 aount for the observed inrease in [HbT] tissue as a result of the dilation of blood vessels i.e. positive σ in equation 1 and apillary reruitment i.e. inrease in in equation 14 during musle exerise Quaresima et al 1995. Equation 11 desribes how an inrease in the speed of blood flow positive may exatly ompensate a moderate inrease in the oxygen onsumption positive walking exerise Quaresima et al 1995, underompensate a signifiant inrease in oxygen onsumption running exerise Quaresima et al 1995 or overompensate the loal oxygen demand reperfusion hyperoxia Smith et al 199, Hampson and Piantadosi 1988. In fat, the term with has a positive sign in equation 11sothat an inrease
Haemodynami model for the onentration and saturation of haemoglobin in vivo N255 in the blood speed ause an inrease in StO 2, while the term with has a negative sign so that an inrease in the oxygen onsumption auses a derease in StO 2. Furthermore, equation 11 indiates that the relative inrease in should math the relative inrease in to exatly ompensate its effet on StO 2. An important point of equation 1 isthatahange in the speed of blood flow, by itself, does not modify [HbT] tissue,whihisonly affeted by the partial volume of blood in the tissue whih isproportional to σ andby the total haemoglobin onentration in the blood see also, equation 6. By ontrast, if, the speed of blood flow does have an effet on the onentrations of oxyhaemoglobin and deoxyhaemoglobin in tissue if =, the terms that multiply / in equations 8and9beome zero. As an be seen in equations 8 and9, the ontributions from to [HbO 2 ] tissue and [Hb] tissue have opposite signs with respet to the orresponding ontributions from. An inrease in i.e., a positive indues an inrease in [HbO 2 ] tissue and a derease in [Hb] tissue. This aounts for the fat that the inrease in the erebral [HbO 2 ] tissue and the derease in [Hb] tissue observed during brain ativation result from a greater inrease in blood flow with respet to the inrease in the oxygen onsumption indued by neuronal ativation Foxand Raihle 1986, illringer and Chane 1997. Studies on animal models have shown that erebral blood flow inreases are mostly determined by inreased flow veloity rather than by apillary reruitment Berezki et al 1993, so that is the relevant parameter to desribe blood flow in this ase. Equations 8 and9 only predit in-phase same sign or 18 out-of-phase opposite signs hanges in [HbO 2 ] tissue and [Hb] tissue.inpartiular, in-phase hanges are assoiated with modifiations to the total haemoglobin onentration in the blood [HbT] orblood partial volume σ, while 18 out-of-phase hanges are assoiated with modifiations to the initial blood oxygen saturation SO 2, oxygen utilization rate αȯ2 or speed of blood flow. While either in-phase Wolf et al 1997 or 18 out-of-phase Elwell et al 1994 osillations in [HbO 2 ] tissue and [Hb] tissue are typially reorded, intermediate phase shifts have also been observed Taga et al 2. These intermediate phase shifts an result from the superposition of out-of-phase ontributions from different blood vessels, for instane arteries, apillaries and veins, as onsidered in equations 12and13. For example, if the partial blood volumes assoiated with arteries and veins, a and v,respetively, osillate with an arbitrary phase differene, then [HbO 2 ] tissuem and [Hb] tissuem may also osillate with an arbitrary phase differene beause of the different relative ontributions of a v and to [HbO 2 ] tissuem and [Hb] tissuem see equations 12 and13. In fat, the arterial/venous partial volume ontribution ratio is SO 2 a-blood/ SO2 v-blood in equation 12, while it is 1 SO 2 a-blood/ 1 SO2 v-blood in equation 13. Equations 12 14 also illustrate how the volume osillations of the arterial and venous ompartments an be used to measure the arterial saturation pulse oximetry Mendelson 1992 and the venous saturation spiroximetry Franeshini et al 22, respetively. In fat, by assuming that a osillates at the heartbeat frequeny and that v osillates at the respiratory frequeny, equations 12 and14 showthat the ratio of the amplitudes of the [HbO 2 ] tissuem and [HbT] tissuem osillations at the heart rate and respiratory frequeny are equal to SO 2 a-blood the arterial saturation or SaO 2 andso 2 v-blood the venous saturation or SvO2, respetively. 4. Conlusion We have presented a haemodynami model that an be used to guide the physiologial interpretation of oxyhaemoglobin and deoxyhaemoglobin onentration measurements in
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