Model-based evaluation of the microhemodynamic effects of PEGylated HBOC molecules in the rat brain cortex: a laser speckle imaging study

: Hemoglobin-based oxygen carriers (HBOCs) were developed with the aim of substituting transfusions in emergency events. However, they exhibit adverse events, such as nitric oxide (NO) scavenging, vasoactivity, enhanced platelet aggregation, presently hampering their clinical application. The impact of two prototypical PEGylated HBOCs, Euro-PEG-Hb and PEG-HbO 2 , endowed by diﬀerent oxygen aﬃnities and hydrodynamic volumes, was assessed on the cerebrocortical parenchymal microhemodynamics, and extravasation through the blood-brain-barrier (BBB) by laser speckle contrast imaging (LSCI) method and near-infrared (NIR) imaging, respectively. By evaluating voxel-wise cerebrocortical red blood cell velocity, non-invasively for its mean kernel-wise value (v RBC ), and model-derived kernel-wise predictions for microregional tissue hematocrit, T H t , and fractional change in hematocrit-corrected vascular resistance, R’, as measures of potential adverse eﬀects (enhanced platelet aggregation and vasoactivity, respectively) we found i ) no signiﬁcant diﬀerence between tested HBOCs in the systemic and microregional parameters, and in the relative spatial dispersion of T H t , and R’ as additional measures of HBOC-related adverse eﬀects, and ii ) no extravasation through BBB by Euro-PEG-Hb. We conclude that Euro-PEG-Hb does not exhibit adverse eﬀects in the brain microcirculation that could be directly attributed to NO scavenging. mapped the spatio-temporal dynamics in cerebrocortical microperfusion by laser speckle contrast imaging before and after blood-to-HES and blood-to-PEG-HBOC exchange transfusions. We characterized the impact of the exchange transfusions in terms of model predictions derived from translating the systemic macrohemodynamic and regional microhemodynamic input parameters into (micro)regional output parameters relevant in assessing changes in intraparenchymal vascular tone and hemorheological properties (reﬂecting on platelet aggregation tendency). We have not observed signatures of enhanced vasoactivity and/or platelet aggregation, as potential adverse eﬀects developing in the parenchymal microvessels of the rat brain cortex following the application of neither our Euro-PEG-Hb molecule, nor an extensively studied HBOC molecule, PEG-HbO 2 . Further, we found no signiﬁcant diﬀerence between tested HBOCs in the systemic and microregional parameters, and in the relative spatial dispersion of T H t , and R’ as additional measures of HBOC-related adverse eﬀects. Finally, extravasation was not observed through BBB in case Euro-PEG-Hb.


Introduction
Hemoglobin-based oxygen carriers (HBOCs) have emerged as potential therapeutic agents in restoring oxygen supply [1][2][3]. Unlike hemoglobin (Hb), which under physiological conditions is confined within the red blood cells (RBCs), HBOCs are essentially Hb molecules free in plasma where they can readily interfere with nitric oxide (NO). This important modulator of vascular tone and microcirculatory fluidity of blood [4] is synthesized from L-arginine by several isoforms of NO synthase (NOS): endothelial NOS (eNOS) and neuronal NOS (nNOS), in particular. eNOS and nNOS are enzymes constitutively expressed in endothelial cells [5] and nerve terminals in the fiber region adjacent to the outer boundary of vascular smooth muscle (VSM) layer [6,7], respectively. eNOS activity is modulated by the shear of the flowing blood mechanically impacting the endothelial glycocalyx [8]. NO, as an autocrine and paracrine signal molecule, ones formed participates in numerous physiological processes, including regulation of blood pressure and blood flow, platelet aggregation, and leukocyte adhesion [4,9,10].
As a small gaseous lipophilic molecule, NO can readily diffuse from its site of production (i.e. endothelium) in all directions thus reaching the blood stream in the luminal, and the VSM cells in the abluminal direction, alike. The intravascular biological life of NO is only about 2 ms [11]. HBOCs are known to scavenge NO [12,13] at a rate 600 times higher than Hb within RBCs [14]. Hence to maintain a sufficiently relaxed vascular caliber, to inhibit platelet aggregation [15] and to prevent cell adhesion to endothelial lining [16][17][18], NO should be continuously secreted to function as a key homeostatic factor [19]. Thus HBOCs by scavenging NO could potentially upset the balance between production and elimination and thus disturb the homeostasis of the microcirculatory environment in the brain [20]. Clearly, given the extremely short intravascular lifetime, adverse effects could readily develop if the rate of NO scavenging by HBOC in blood exceeds that of production in endothelial cells [21].
As is the case with HBOCs, hemoglobin, when free in plasma, undergoes oxidation, and produces various reactive oxygen species. Moreover, it is associated with adverse vascular effects, such as increased vascular tone [22], coagulopathy [23,24] and platelet aggregation [25]. However the key role of nitrosative/oxidative [26,27] stress should be emphasized in this pathophysiological process. This may result in cellular damage, as the plasma antioxidant capacity is inadequate to counteract the oxidative cascade triggered by high levels of cell-free Hb in blood [2,28,29]. As the endothelial lining of the vasculature is in direct contact with HBOC molecules, endothelial cytotoxicity might also result [30], which can make the endothelial layer leaky, a particular concern for the blood-brain barrier (BBB). Extravasation of HBOCs through the BBB can have a dual effect: i) it can readily lead to vasoconstriction by directly intercepting the NO signaling pathway in the interstitial space between the endothelium and VSM [22,31] and ii) it can also aggravate the extravascular effects of HBOCs by triggering a cascade of inflammatory events such as leukocyte and mastocyte activation, cytokines (TNF-α, IL-4, 5 and 6) and histamine release [32,33]. Thus extravascular events could augment the microvascular disturbances. The above adverse effects may well emerge in an organ-specific manner because the rate of endothelial NO secretion varies among organs and tissues with the brain's among the highest [34][35][36].
HBOC-therapy can thus lead to decreased bioavailability of NO, which by disrupting various homeostatic systems can result in the clinical sequelae [37] with a supposedly organ-specific manifestation. HBOC-assisted interventions are primarily targeting the brain and the heart as vital organs in the interest of securing the survival of the organism. Decoration of hemoglobin with polyethylene glycol (PEG) chains is known to ameliorate some of the above adverse effects [2,22,38,39]. Our group developed Euro-PEG-Hb (a N-propionyl-maleimide-PEGylated Hb molecule) within the framework of the European Union FP6 ,Genomics and Blood Substitutes for the 21st Century Europe EuroBloodSubstitutes Consortium" [2,38] that was further characterized in this study.
Our aim was to investigate whether HBOCs, in particular Euro-PEG-Hb, have adverse effects on the cerebrocortical microhemodynamics of anesthetized rats due to either scavenging NO or extravasating through the BBB. Accordingly, first, we tested the hypothesis that PEG-HBOC molecules do increase the vascular resistance in the cerebrocortical microvessels. Second, we evaluated the potential of NO-depletion by PEG-HBOC molecules in inducing platelet and consequently RBC aggregation to an extent that would upset the spatial distribution of cerebrocortical microflow. Finally, we assessed the degree of extravasation through the BBB for Euro-PEG-Hb. Experiments targeting the first two aims were carried out under conditions of isovolemic blood-to-PEG-HBOC exchange, where laser speckle contrast imaging of the brain cortex was used to acquire the microregional RBC velocity data. A mathematical model was developed to predict changes in the key parameters of microregional perfusion such as RBC flow, tube hematocrit, blood flow, vascular resistance and wall shear stress using RBC velocity and mean arterial blood pressure as inputs of the model.

Materials and methods
The Regional Committee of Science and Research Ethics of the Semmelweis University approved the animal experiments.

Molecules used in the exchange transfusion experiments
PEGylated Hb derivatives were used as oxygen transport agents, while hydroxyethyl starch (HES, 6% hydroxyethyl starch 130/0.4, Voluven, in 0.9% sodium chloride injection) was used as negative reference for assessing the impact of isovolemic exchange on the cerebrocortical parenchymal microhemodynamics alone with a plasma expander not having oxygen carrying capacity.

Oxygen carrying blood substitutes (test and reference molecules)
The HBOC test and reference molecules used in this study were PEGylated hemoglobin obtained from outdated human banked blood prepared according to a protocol published earlier [38]. Specifically, we used the following two versions of PEGylated products. N-propionyl-maleimid-PEGylated Hb molecule (NPM/PEG-Hb) further referred to as Euro-PEG-Hb, developed with the support of the European Union FP6 "Genomics and Blood Substitutes for the 21st Century Europe -EuroBloodSubstitutes Consortium" was used as a test molecule. Maleimide-PEGylated hemoglobin was prepared under aerobic condition using the recipe of the Winslow group (further referred to as PEG-HbO 2 ) [40] and was used as an HBOC-reference. The final protein concentration in isotonic sodium chloride was for Euro-PEG-Hb 5.8 × 10 −4 M (5.8 g/dL) and for PEG-HbO 2 4.2 × 10 −4 M (4.2 g/dL). The conversion of PEG-Hb concentration expressed as g/dL to molar concentration was obtained using a molecular weight for PEG-Hb of 100 kDa (Hb MW = 65,000 Da plus average nPEG 5,000 Da ∼35,000 Da) as previously reported [38]. HES, a plasma expander in routine clinical use, as a negative reference molecule was administered. Due to its large molecular size (130.000 Da), it is well retained within the plasma compartment during the course of the experiment and given its osmolarity of 308 mOsm/L, it does not produce significant fluid shift in the microcirculatory beds of the body.

Animals and general anesthesia
The exchange transfusion experiments were carried out on white laboratory rats of Wistar strain (n=17, 337 ± 22 g bodyweight) under general anesthesia (Table 1).
Due to the requirements of the optical imaging of the cerebrocortical parenchymal circulation (see below), an optical window was implanted into the skull and arterial (left femoral artery) and venous lines (left femoral vein) were inserted for the exchange transfusion experiment in two surgical sessions, respectively, separated by a period of 48 hours as recovery period. In the first session, general anesthesia was induced by 4% halothane inhalation followed by an i.p. administration of ketamine hydrochloride (Calypsol -50 mg/mL; 10 mg/100 g initial dose, 3-4 mg/100 g hourly maintaining dose) and xylazine hydrochloride (0.5 mg/100 g initial dose, 0.2 mg/100 g hourly maintaining dose) during the preparation via an i.p. catheter (Braintree  Scientific, Inc., Braintree, MA 02185, USA, Type PE-10). In the second session, following a 48 hours recovery period, the experiments were carried out under general anesthesia induced by 4% halothane and maintained by urethane given in a dose of 130 mg/100 g BWt i.p.. We used halothane only for brief induction in both of these surgical sessions to avoid a halothane-induced increase of cerebrocortical NO content and pial vasodilation [34,41]. The exemplary experiment for assessing the organ-level uptake of Euro-PEG-Hb was carried out in nude mice of 20 g body weight anesthetized by an i.p. administration of ketamine hydrochloride (Calypsol -50 mg/mL; 10 mg/100 g initial dose, 3-4 mg/100 g hourly maintaining dose). For the administration of Euro-PEG-Hb, the tail vein was cannulated. In order to clear the tissues from residues of blood, the animal was sacrificed under general anesthesia augmented by an overdose of 4% halothane as the circulating blood was cleared from the animal via simultaneous large arterial hemorrhage and continuous infusion of saline via the tail vein.

Isovolemic exchange transfusion protocol
Exchange transfusion was carried out as described previously [38] (Table 1). A volume of 6.5-7 mL of arterial blood was let down via the arterial line inserted into the left femoral artery during a period of 10-12 minutes. Then, a volume of 7 mL of Euro-PEG-Hb, PEG-HbO 2 , or HES solutions, prepared in isotonic saline at concentrations of 5.8 g/dL, 4.2 g/dL, and 6.0 g/dL, respectively, were infused via the same arterial line. Using the dilution principle, percent blood-to-HBOC exchange was obtained as published earlier [38].
Accordingly, the following convention was used in the protocol timeline: −20 min (control), 0 min (beginning of exchange transfusion; post-exchange measurements are given relative to this point in time); test period was defined at 110 min. This procedure translates into a clinically relevant model of acute arterial hemorrhage [42].

Assessing systemic hemodynamic parameters
The general physiological status of the animals was assessed by measuring the following parameters: core (rectal) temperature, systemic arterial blood pressure (MAP), blood pH, large arterial gas values (pO 2 , pCO 2 , BE, Sat) and large arterial tube hematocrit ( T H a ). Blood gas parameters and T H a were determined in arterial samples withdrawn from the left femoral artery with Radiometer Copenhagen ABL 77 apparatus (Copenhagen, Denmark). Core temperature, measured with a Yellow Springs Body Temperature Controller Model 73ATF (Yellow Springs, OH, USA) was maintained by controlling a heating lamp at a threshold value of the rectal temperature of 37.5 ± 0.5°C. Arterial blood pressure was measured by a Statham pressure transducer (Statham Instruments, Oxnard, CA, USA) via a catheter introduced into the left femoral artery. Blood cell flow velocity (v RBC ) was mapped within an exposed area in the parietal brain cortex by a custom made laser speckle contrast imaging (LSCI) system ( Fig. 1) [43,44] through the closed calvaria made transparent by a technique described earlier [45,46] as follows. The head was secured in a head holder. The scalp, subcutaneous connective tissue and the periosteum were removed. Hemostasis, if necessary, was carried out with low intensity bipolar electrocautery (Bipolar Coagulator, Codman & Shurtleff Inc., Randolph, MA, USA) and by BoneWax (Ethicon, San Angelo, TX, USA) applied onto the exposed cranial surface. The cranium was gently thinned by a dental drill under a stereomicroscope at 40x magnification until the internal compact lamina was reached. For the purpose of cleaning, cooling and allowing for visual control, air was blown onto the area. Within the thinned area of about 4-5 mm in diameter the epidural and pial vessels could be readily visualized. The thickness of the bone at the site of the optical measurements was about 160 micrometers as measured by video microscopy post mortem [45]. Subsequently, a metal platform with a round slit (8 mm in diameter) for optical monitoring was cemented onto the skull. Within this slit, the cured cement was gently removed by drilling along with the underlying parietal lamina of the bone until the cortical surface and pial vessels became clearly visible. The area was japanned with transparent lacquer and was covered with a plastic plate for protection during a two-day recovery period. The preparation took 3-4 hours; in the end, the peritoneal catheter was removed. This technique fully preserved the physiological conditions of the pial and intraparenchymal circulation, which were essential requirements for the testing protocols.
At the time of the measurement, the animal head was positioned in the video camera's focal plane by the metal platform. Spatial intensity data was collected by a cooled CCD camera (CoolSNAP cf High-Resolution Interline CCD camera, Roper Scientific, Inc. 3440 East Britannia Drive, Tucson, AZ 85706, U.S.A.) under laser light epi-illumination of the brain cortex using a laser diode (Hitachi Laser Diode, HL6320G, 635 nm, Thorlabs, Inc. (56 Sparta Avenue, Newton, NJ 07860, U.S.A.) equipped with a collimator lens (68 mm/1.00x, Proximity Series) and powered by an LDC201 ULN Laser Diode Controller at 6 mW (Thorlabs, Inc., 435 Route 206, Newton, NJ 07860-0366, U.S.A.). Since our aim was to test the impact of vasoconstriction and RBC aggregation in the parenchymal circulation, we set the exposure time to 40 ms in order to ensure that the sensitivity of the LSCI measurement will be extended towards the low-perfusion range as much as possible [47]. For each parenchymal perfusion map a stack of laser speckle images were acquired during 228 s at a resolution of 256 × 256 voxels corresponding to an area of 1mm 2 of which finally 4096 consecutive frames were selected for the analysis. The speckle size was approximately 20 µm computed as [48]: 1.2 · (1+M) · λ · f/# (in our case: the magnification parameter was M = 1 while f/#=13.5 cm denotes the f-stop parameter of the imaging system). The stored images were processed off-line by a MATLAB (The MathWorks, Inc., Natick, MA, U.S.A.) script of the authors for RBC velocity (v RBC ) by calculating spatial statistics as a measure of velocity-dependent blurring of the image [43]. The v RBC maps were obtained from the raw speckle intensity data by using the smallest kernel that allowed for statistical assessment of contrast changes. According to common practice its size was 5 × 5 in these analysis [49][50][51]. Accordingly, the final v RBC maps were obtained by advancing this kernel in a 1 voxel step, yielding a 252 × 252 raster corresponding to a rectangular cortical area of 1 mm 2 . Despite certain degree of inherent redundancy, the highest spatial resolution could be achieved with the aid of overlapping kernels. (Figure 1(A)).
In an in vitro validation experiment a series of LSCI measurements were made of a phantom fluid (milk containing lipid droplets as scattering particles) flowing at different linear velocities through a glass tube ( Fig. 1(B left)). The spatial statistics maps for the same segment of the glass tube calculated as in panel A are shown in a series of gray scale maps. Calculated perfusion values integrated for the imaged segment are displayed as a function of preset perfusion levels on Intensitycoded velocity maps of a scattering phantom flowing through the same segment of a glass tube is seen on the left. Calculated perfusion values integrated for the imaged segment are displayed as a function of preset perfusion levels on the right. C: RBC velocity maps obtained while perfusion pressure to the brain was lowered in a step-wise manner is shown on the left (parenchymal ROI marked framed in white). RBC velocities plotted as a function of the mean arterial blood pressure from this experiment is shown on the right (open circles) along with our data obtained earlier by a commercially available laser Doppler flowmetry instrument under the same experimental condition (closed circles). For more details, see text.
the right. Linearly fitted low-, mid-and high-perfusion ranges are shown in color. This validation curve is fully compatible to those reported in the literature [52] (calibration curve from their Fig.  3 is shown dashed). LSCI measurements were further validated in vivo in experiments capturing RBC velocity maps while the perfusion pressure to the brain was lowered in a step-wise manner by our lower-body negative pressure method [53] (Fig. 1(C left)). The mean RBC velocities were calculated for a parenchymal ROI (white box) and plotted as a function of the mean arterial blood pressure ( Fig. 1(C right)). These laser speckle flowmetry data (open circles) compare well to those we published earlier for the rat brain cortex obtained by a commercially available laser Doppler flowmetry instrument under the same experimental conditions (closed circles) [53].

Mathematical model: calculation of microregional tube hematocrit and parenchymal vascular resistance
In our experimental series parenchymal region of interests (ROIs) void of readily visible vessels of an overall size distribution of 4016 ± 1903 voxels (n=17) were selected for further quantitative analyses by our lumped microregional hemodynamic model. For the quantitative treatment of LSCI data, all vascular elements within a single 5 × 5 parenchymal kernel of the ROI are lumped into a cylinder with a given mean vascular path length (L RBC ), and total cross section area (A RBC ). This cylinder is treated as the lumped microvascular compartment within each kernel of the ROI. Based on the measurement of the mean kernel-wise RBC velocity, v RBC , our aim was to estimate the following key microhemodynamic parameters: volume flow of whole blood (Q WB ), integrated vascular caliber (r A ), microregional tube hematocrit ( T H t ), microregional vascular resistance (R) and wall shear stress (τ w ). Measured input parameters of the model were: systemic mean arterial blood pressure (MAP), large arterial hematocrit ( T H a ), and mean kernel-wise RBC velocity (v RBC ) (Fig. 2).

Fig. 2. Scheme of the lumped microregional hemodynamic model developed for predicting the microhemodynamic effects of PEG-HBOC molecules in the brain cortex from RBC
velocity image data. The model requires systemic (mean arterial blood pressure and hematocrit) and microregional (RBC velocity) parameters as its inputs. In turn, the model provides predictions for a range of microregional hemodynamic parameters (RBC, plasma and whole blood flows, tube hematocrit and vascular resistance) for each and every voxel within a ROI.
As a result of the Fåhraeus effect [54,55], the tube hematocrit dynamically decreases along the arterial tree of the brain from T H a to T H t (Fig. 7, see Appendix). The relative apparent viscosity, which depends on vessel diameter and discharge hematocrit, was calculated using the in vivo viscosity law as formulated in Eq. (7) in Ref [56]. The perfusion pressure (PP) was approximated by the difference between MAP and intracranial pressure (the latter assumed to be 10 mmHg). All model predictions, except T H t , were calculated in a normalized manner relative to control as indicated by prime.
Based on these considerations, the following key model equations were derived (for their detailed derivation, along with symbols, abbreviations and definitions, see Table 5) Further where

Research Article
Vol. 11, No. 8 / 1 August 2020 / Biomedical Optics Express 4157 Finally and In our cylindrical lumped geometry, wall shear stress (as the product of the viscosity and the radial linear flow velocity gradient) was calculated in a fractional manner relative to control (τ' w = 1). The measured and calculated parameters were taken from the above equations and the derivation of the model (see Appendix) as Spatial heterogeneity of kernel-wise model predictions was assessed for the parametric maps by relative (spatial) dispersion as defined by the ratio of standard deviation and the mean.

Data acquisition, processing, and statistical analysis
Arterial blood pressure was digitized by an MP100A data acquisition system controlled by AcqKnowledge III software (BIOPAC Systems, Inc., Santa Barbara, CA, USA) at 200 Hz, at 12 bits. Statistical analysis (

Changes of model input parameters in response to isovolemic exchange transfusions
For each experiment, MAP and T H a as systemic and v RBC as regional model input parameters were measured to assess the coupled central hemodynamics/hemorheology, and peripheral hemodynamics, respectively. As to the overall temporal dynamics of the systemic model input parameters the exchange transfusion was accompanied by a marked lowering in MAP at t=0 min followed by an overshoot for HBOCs at t=60 min and return to normal levels at t=110 min. HES showed somewhat lower post-exchange MAP levels than the HBOC groups. Concerning hematocrit changes, blood-to-PEG-HBOC transfusion led to marked hemodilution, which persisted throughout the experiment with no significant differences between group means at any particular stage of the exchange protocol (Tables 2 and 3).
The exchange transfusions led to coupled changes in the systemic and regional model input parameters (Table 2). In particular, MAP and T H a decreased from its control levels to those at t=110 min in proportion to the exchange rate (Table 1). Induced by this decrease in T H a , v RBC increased from its control levels (HES: 365 ± 76.8%, PEG-HbO2: 216 ± 38.1%, Euro-PEG-Hb: 323 ± 67.4%) to those at t=110 min (HES: 481 ± 122.8%, PEG-HbO2: 401 ± 102.6%, Euro-PEG-Hb: 833 ± 561.6%). There were no statistically significant differences between control values of these parameters (with levels of significance shown in various relations in Table 3).

Spatio-temporal model predictions for microregional blood flow, tube hematocrit
and vascular resistance in the rat brain cortex before and after blood-to-PEG-HBOC exchange In a typical experiment carried out with Euro-PEG-Hb for two key parameters of microhemodynamics assessed in the parietal cortex at specific time points, the local model input parameter and kernel-wise predictions are displayed in Fig. 3. The blood-to-PEG-HBOC exchange transfusion led to hemodilution not only in the central but in the cerebrocortical microcirculation too, as seen in the significant reduction in T H t (Fig. 3, middle row panels). The relative microregional vascular resistance, R', increased in the early post-transfusion period (Fig. 3, bottom row panels) and remained slightly elevated afterwards in this exemplary case. Note that the exemplary time course of R' is within the range of R' shown in Fig. 4(C). Fig. 3. Spatio-temporal dynamics of model-predicted microhemodynamic parameters within an area of the rat brain cortex in a blood-to-Euro-PEG-Hb exchange experiment. RBC velocity (v RBC ) as input, and tissue tube hematocrit ( T H t ) and vascular resistance (R') as output parameters of the model are shown intensity-coded according to the scales on the right. Parametric maps of T H t and R' were predicted from kernel-wise v RBC and MAP data by a lumped microhemodynamic model (Fig. 2) for a parenchymal area marked as ROI on the v RBC maps.
3.3. Impact of blood-to-PEG-HBOC exchange on tissue tube hematocrit, wall shear stress, vascular resistance, and blood flow Model predictions of microhemodynamic parameters can be found in Figs. 4 and 5(A) with levels of significance shown in various relations in Table 3. Group averages of T H t during the time course of the experiment are shown in Fig. 4(A). The exchange transfusion significantly lowered T H t from its control levels hence decreasing the ratio of tissue to large arterial hematocrit (Figs. 4(A) and 5(A) and Table 3), which remained stable throughout the post-exchange period. Predictions for microregional wall shear stress, τ w ', are shown in Fig. 4(B). In addition to its slightly decreased level during the exchange, the graph shows a large inter-individual variability in all groups and times except in control where due to the relative nature of this parameter the scatter is absent in the data.
Regional blood flow (Fig. 4(D)) showed an increasing tendency in the post-exchange period, but due to the much increased inter-subject heterogeneity in the post-exchange period no difference could be demonstrated on the group level either when comparing to control or in-between groups relations ( Table 3).
As seen in Fig. 5(A), (a) drop in k indicates that regional hemodilution exceeds that in the central arterial circulation due to increased phase separation in the intracerebral arterial transit to the parenchymal microcirculation (See Fig. 7). Under the condition of this exchange Fig. 4. Model-predicted regional hemodynamic parameters following exchange transfusions. HES and various PEG-HBOC molecules were tested for their effects on regional hemodynamics in the brain cortex. Regional parameters were aggregated from the voxel-wise (microregional) values within the ROI are as follows. A: regional tube hematocrit, T H t , B: regional wall shear stress, τ w ', C: regional vascular resistance, R' and D: regional blood flow, Q WB . Exchange transfusion was carried out at 0 minute. HES: hydroxyethyl starch, EuroPEG-Hb: N-propionyl maleimide-PEGylated Hb, PEG-HbO 2 : maleimide-PEGylated oxyhemoglobin. Data are plotted as group mean ± SD. transfusion-induced hemodilution, the presence of local perturbation due to RBC aggregation or vasoreactivity were evaluated by calculating the heterogeneity of T H t (Fig. 5(B)) and R' (Fig.  5(C)), respectively given as the relative dispersion (the coefficient of variance) of voxel-wise data, RD. RD( T H t ) and RD(R') were low and did not differ with p>0.05 in any comparison among different experimental groups (Table 3). Their dynamics across the experimental protocol were also similar. Of the comparisons shown in Table 3, those for the two HBOC molecules and those for the HBOC molecules vs. HES as a negative control are of particular importance as they demonstrate that the observed markedly increased RD seen following blood-to-HES exchange was not different at 110 min from those with the HBOC molecules. This finding suggests that the HBOC molecules did not cause RBC aggregation or vasoconstrictions that would be expected to result in selectively increasing RD( T H t ) and RD(R') values, respectively.

Testing on clearance and extravasation of labeled Euro-PEG-Hb by whole-body and organ-level NIR fluorescence imaging
In an exemplary experiment, tissue uptake of Euro-PEG-Hb molecules was determined by nearinfrared (NIR) fluorescence imaging following our recently published approach [57] in vivo and in blood-cleared excised organs of a nude mouse of 20 g BWt ex vivo. A batch of Euro-PEG-Hb molecules have been labelled by IRDye800CW; a dye with peak fluorescent emission in the NIR range. A period of 90 minutes was allowed for incubation at room temperature. An amount of 100 µL containing the labeled substance was injected via the tail vein under general anesthesia . Heterogeneity in regional hemodynamics following exchange transfusions. The impact of exchange transfusion by HES and various HBOC molecules in terms of RBC aggregation and vasoreactivity were evaluated by assessing the regional spatial heterogeneities in microregional tube hematocrit (B) and vascular resistance (C), respectively. Regional to large arterial hematocrit ratio (A) indicates that regional hemodilution exceeds that in the central arterial circulation. Note that RD( T H t ) and RD(R') for the HBOC and HES molecules do not differ, which is taken as the lack of adverse effects manifested in RBC aggregation or vasoactivity. Data are plotted as group mean ± SD. and 27 hours and 30 minutes were allowed for maximal molecular uptake in the organs of the animal. Imaging the emission of labeled molecules was done in the NIR energy range at a spatial resolution of 170 µm by the LI-COR Pearl NIR Optical Imager (LI-COR Biosciences GmbH, Bad Homburg, Germany). Figure 6 shows in vivo high-resolution whole-body image of the circulating and extravasated HBOC molecules. Following sacrifice, the organs were cleared from blood by infusion of normal saline via the abdominal aorta. The organs were subsequently removed and the extent of the molecular retention was determined by capturing NIR fluorescence intensities from the exposed organ surfaces. Euro-PEG-Hb apparently showed long enough half time, so that through the time of our test period (110 min post-exchange) the level of this test molecule in the plasma remained well maintained (Fig. 6). Also note, that its clearing from the brain was faster as compared to that from suturae and neck muscles as became apparent by 240 min (Fig. 6(D)). Removing blood from tissues Fig. 6(F) demonstrated that this difference was indeed due to extravasation in the suturae and neck muscles. Images taken of removed organs Fig. 6(G, H) evidenced that Euro-PEG-Hb did not escape into the brain, as brain did not exhibit fluorescence intensities, when compared to the heavily loaded kidney Fig. 6(H) (disregarding the slight specular reflection due to the nearby intensively fluorescing kidney).

Discussion
In this study, we developed a noninvasive optical imaging platform combined with mathematical modeling for predicting the effects of blood substitutes in the microcirculation of the rat brain cortex: a key information for a vital organ in blood substitute research and applications. Using an isovolemic blood-to-PEG-HBOC exchange model with HES as negative reference, we mapped the spatio-temporal dynamics in cerebrocortical microperfusion by laser speckle contrast imaging before and after blood-to-HES and blood-to-PEG-HBOC exchange transfusions. We characterized the impact of the exchange transfusions in terms of model predictions derived from translating the systemic macrohemodynamic and regional microhemodynamic input parameters into (micro)regional output parameters relevant in assessing changes in intraparenchymal vascular tone and hemorheological properties (reflecting on platelet aggregation tendency). We have not observed signatures of enhanced vasoactivity and/or platelet aggregation, as potential adverse effects developing in the parenchymal microvessels of the rat brain cortex following the application of neither our Euro-PEG-Hb molecule, nor an extensively studied HBOC molecule, PEG-HbO 2 . Further, we found no significant difference between tested HBOCs in the systemic and microregional parameters, and in the relative spatial dispersion of T H t , and R' as additional measures of HBOC-related adverse effects. Finally, extravasation was not observed through BBB in case Euro-PEG-Hb.
Owing to the long history of blood substitute research, a variety of approaches have been applied to study their vascular effects, especially their vasoactivity [3,20]. Among them, direct observation, visualization and assessment of individual vessels or parenchymal vascular networks for their reactions were prevalent [58]. Several studies on pial arterioles using intravital microscopy were reported on assessing the effect of HBOCs [59]. As to imaging, positron emission tomography (PET) was applied in liposome encapsulated hemoglobin investigations to provide information about the body-and organ-level distribution of blood substitutes and about their effect on cerebral blood flow, cerebral blood volume and oxygen delivery [60,61]. Due to the limited spatial resolution of PET, no information on microhemodynamics could be obtained. Data are therefore in need on the spatio-temporal impact of HBOCs on parenchymal microhemodynamics in particular in vital organs, like the brain, in order to critically evaluate their potential adverse effects preferably in a pre-clinical setting with relevance in future clinical applications [62]. This study is perhaps the first attempt in this direction making use of high-resolution perfusion imaging empowered by mathematical modeling.

Blood levels of Euro-PEG-Hb and PEG-HbO 2 molecules are stable throughout the post-exchange period
Isovolemic exchange reduces the arterial hematocrit in two phases: first during the phase of acute hemorrhage due to autoinfusion, followed by hemodilution due to isovolemic exchange [38]. We chose the 110 min post-exchange time point for testing, because by then the post-exchange perturbation in MAP (Table 2) was over and because both systemic and regional tube hematocrits ( Table 2 and Fig. 4(A), respectively) reached a steady level of hemodilution already at 20 min post-exchange. We anticipated that in our experimental model HBOC levels in blood would be sustained throughout the experiment as for these two PEG-Hb molecules we earlier reported a less than 1% renal excretion rate over a period of 150 minutes [38]. In fact in this study we demonstrated that through the post-exchange period (inclusive of the test time point of t=110 min) the level of our test molecule in the plasma remained well maintained (Fig. 6). More recently using a quantitative analysis of the temporal profile of blood NIR fluorescence data obtained in a mouse top-load model we showed that the blood level of the wild type HBOC molecule (i.e. Euro-PEG-Hb) indeed remained unchanged for 2 hours post-injection [57].

Euro-PEG-Hb is retained within the microvasculature of the brain cortex
The hemoglobin tetramer dissociates into dimers depending on the oxygen fractional saturation and protein concentration [2,63]. Given the negative effects associated with the presence of dimers in plasma, research focused on developing hemoglobin tetramers that either did not dissociate into dimers or, if they did so, they did not extravasate because their size was artificially increased [2]. Thus the key aspect in PEG-Hb design strategy is to increase overall molecular volume to an extent sufficient in preventing or minimizing extravasation [2,38,39]. The efficacy of PEGylation in this regard depends on the extent the overall hydrodynamic volume exceeds the effective pore size of the capillaries in the actual microvascular bed. The issue is rather complex though as PEGylation itself interferes with the molecular structure of Hb and thus tends to enhance its dimerization less for Euro-PEG-Hb than for PEG-HbO 2 [63]. It has been well documented, that extravasation strongly augments adverse vasoactivity due to NO scavenging, intercepting the NO pathway and over-oxidizing the perivascular tissue; all depending on the molecular volume of HBOCs [39,64].
In principle, the tight junctions of the endothelial lining of the BBB would prevent molecular passage typically larger than the size of blood gases like O 2 , CO 2 . Earlier studies indeed showed no leakage of HBOC products across the BBB [65]. Most recently, using a sensitive immunohistochemical approach, MP4OX (prepared with a protocol analogous to PEG-HbO 2 ) has been shown to cross the BBB under conditions identical to ours, namely following a 50% isovolemic exchange, albeit in amounts with no significant difference when compared to sham [66]. Our NIR imaging data confirmed these findings in that no sign of extravasation of Euro-PEG-Hb was found in the brain (Fig. 6). This suggests that the BBB effectively prevents the extravasation of Euro-PEG-Hb molecules which restricts their access to the smooth muscle layer of the cerebrocortical vasculature. Hence, vasoactivity as one of the adverse effects targeted in this study should not be expected resulting from extravascular NO scavenging or intercepting the NO pathway due to HBOC leakage into the brain interstitial space. Nonetheless, intravascular scavenging of NO should be considered as a viable alternative in the remaining discussion, as it can potentially lead to vasoreactivity and RBC aggregation.

Effect of exchange transfusion on whole blood relative apparent viscosity and microhemodynamics
The luminal surface of the endothelial cells is lined with the glycocalyx dynamically attracting plasma proteins from the blood stream thus forming the much thicker endothelial surface layer (ESL) [67]. The presence of ESL is known to have various hemodynamic and rheological consequences. Given the dynamic interaction between plasma proteins and ESL, viscous resistance to flow during hemodilution has been shown to depend on the composition of plasma for the following reasons [68]. In fact, ESL is degraded by dilution by artificial plasma replacement has a similar effect to its enzymatic removal [69]. This leads to increased internal vessel diameter, which in turn increases tube hematocrit by enhancing multi-file pattern of RBC flow [67,70]. This results in a hematocrit-dependent increase in whole blood viscosity with a peak close to a vessel diameter of ∼10 µm [70]. In addition, ESL transmits the force by the shear of the flowing blood via the coupled extra (i.e. glycocalyx) and intracellular matrix to intracellular anchoring sites, where the transmitted mechanical force modulates eNOS activity and thus the amount of NO released by the endothelium [8]. The released NO thus depends on viscosity, too [71]. In particular, extreme hemodilution with elevated plasma viscosity increases perivascular concentration of NO and microvascular perfusion relative to that seen under control conditions [72]. Most likely due to the nonlinear nature of the mechanical coupling between the glycocalyx and the intracellular matrix, the mitigating effect of decreased blood viscosity on NO release is not as potent but nevertheless is present [71]. While the apparent viscosities of the test and reference solutions were not measured in this study, their value are either known or can be approximated based on data in the literature. The apparent viscosity of rat plasma at 37°C was reported 1.6 cP [73], that of HES as 1.74 cP [74], and that of PEG-HbO 2 dissolved in Ringer's lactate in a concentration of 4.2 g/dL at 37°C as 2.5 cP [40]. Considering the similar structures of the two HBOC molecules, the viscosity of Euro-PEG-Hb should be in proportion of its concentration (5.8 g/dL) to that of PEG-HbO 2 (4.2 g/dL) thus yielding 3.5 cP. Hence the apparent viscosity of HES is the closest to that of plasma, while those of the two HBOC solutions are higher. Nevertheless, they are much lower than that of whole blood, which is also shear rate-dependent due its non-Newtonian properties [73]. Since the estimated rheological parameters reflect upon the aforementioned diluent viscosities, too, (especially under the conditions of isovolemic exchange transfusion of around 50%) the impact of altered relative apparent viscosity of whole blood on our model estimates, such as relative wall shear stress should be evaluated.
To what extent whole blood apparent viscosity could be influenced by diluent viscosity was reported earlier [75] in an experimental setting decreasing the arterial hematocrit from 40% to 30% by diluting with 10% molecular weight dextran solution, a scenario comparable to ours. The application of these data to the actual rates of exchange in our study (E in Table 1) yields the following average relative apparent viscosities of whole blood in the experimental groups as <η HES > 0.7, <η PEG−HbO2 > 0.8, and <η Euro−PEG−Hb > 0.9. These differences in η due to marked SD of E (Table 1) did not get reflected in the Eq. (5)-based relative wall shear stress values when derived using the above adjusted relative apparent viscosities. We hypothesized that the apparent rheological behavior mainly depends shear-rate under physiological conditions and diluent viscosity is of little significance. Indeed, no statistically significant difference was revealed in relative apparent viscosity estimates (Table 4.) further substantiating the validity of calculating R' based on Eq. (4). Therefore we obtained model estimates for relative vascular resistances (normalized to control as in case of R') based on Eq. (4) and assessed the effect of test molecules at the group level (as it is presented in Table 3). The ultimate effect of decreased whole blood viscosity on regional vascular resistance emerges from the combined and opposing effect of a decreased viscous resistance to flow and an increased regional wall shear stress (Fig. 4(B)), the latter increasing the shear-induced endothelial NO release thus decreasing R' (Fig. 4(C)). Indeed, a decrease in whole blood viscosity was associated with an increase in mean RBC velocity (Table 2) and wall shear stress concomitant with a decrease in vascular resistance (Fig. 4(B, C)); these latter findings were present only in the means with no significant difference between groups ( Table 3). Given that R' at 110 min did not significantly differ between the test and reference groups, the supposed vasoconstrictive effect of the HBOC molecules either was not seen in the brain, or completely ameliorated by the enhanced eNOS activity by the increased wall shear stress.

Mathematical modeling of microhemodynamics based on high-resolution perfusion imaging
Recent large-scale 3D models of hemodynamics and oxygenation of the brain cortex has been developed utilizing the power of two-photon microscopy in obtaining high-resolution structural data of the intracortical microvascular network in an in vivo setting [76][77][78]. Albeit at a magnitude lower spatial but at higher temporal resolution, the LSCI method used in our study, could effectively provide dynamic data (i.e. v RBC ) needed for hemodynamic modeling in a superficial 2D raster of voxels positioned over a parenchymal area within the brain cortex [43,78] ( Fig. 1). To supplement the missing topological data in our model, we relied on the size distribution of microvessel segments described earlier for the rat brain cortex [79]. Solutions with the most degree of freedom requiring the least computational effort were sought for. Accordingly, voxels were assumed independent and their embedded microvasculature were treated in a lumped manner as a "black box", more precisely as a cylindrical vascular segment. The consequence of not having specific information on vascular caliber and length in 3D space is that vascular resistance and all the parameters derived from it (e.g. shear stress via vessel radius) can only be calculated relative to a control level. Our approach in this regard was similar to the one we followed earlier when modeling hemodynamics and oxygenation in the brain cortex using functional near-infrared spectroscopy [80]. In order to avoid the use of dimensioned constants, which can be difficult to estimate, and to reduce the number of model parameters, the equations estimating hemodynamic and rheological variables were written in a unitless (non-dimensional) form, by representing dimensioned variables in relative terms, defined as the ratio of perturbed and baseline values denoted by prime. Our lumped microhemodynamic model accounts for the fundamental hemodynamic and rheological relationships adapted to the lumped vascular system under study. When estimating microcirculatory parameters, we rely on measured systemic (MAP, T H a ) and regional (v RBC ) parameters as model inputs (Table 2). In yielding model outputs, we build on the following relationships. To estimate R', the Hagen-Poiseuille equation and the in vivo viscosity law of Pries et al [56]. are considered. The Fåhraeus effect describes the progressive drop in tube hematocrit relative to discharge hematocrit ( T H and D H, respectively) [54,[81][82][83] is the basis of gaining access to T H t . As the blood undergoes separation of its phases, T H/ D H decreases along the bifurcating arterial tree (Fig. 7). In the arterial tree, this results in a gradually contracting distribution space for RBCs, which in turn increases RBC velocity over that of plasma [83,84]. (Note, that in the venous tree the phenomenon repeats itself in a reversed manner.) Indeed, earlier Eke demonstrated that T H t could be determined by measuring microregional velocities for RBC and plasma derived from their respective transport functions through microareas of the brain cortex by high-resolution video imaging [85]. In this study, however our access was limited to v RBC , the parameter reflecting upon the microregional transport for the RBC phase only. Thus we needed to rely on k= T H/ D H in obtaining T H t =k˙DH, where D H was treated as constant throughout the circulation [86], since conservation of mass apply in this case. Because the two phases are well mixed in the large arteries, we can readily approximate D H by T H a and thus further yield estimates for k= T H t / T H a and subsequently for T H t as a function of vessel radius [See Eqs. (11), (16), (17)].
Under control conditions, our key model predictions, such as for k and T H t , are in good agreement with their physiological values reported in the literature under conditions of isovolemic hemodilution by 6% hetastarch in saline for the hemispheres of the rat brain [87]. At the level of overall HES group averages of T H a of 40% and 25% as corresponding levels of MAP (Table  2) in this study, our estimates for T H t of 31% and 18% (Fig. 4(A)) compare well to values obtained by these authors. As to the effect of isovolemic exchange transfusion, we found v RBC increased ( Table 2). Similar findings were reported in an exchange-transfusion study for arterioles and venules in a golden Syrian hamster skinfold chamber model under conditions when blood viscosity was maintained sufficiently high [72].

Euro-PEG-Hb and PEG-HbO 2 molecules do not exhibit marked adverse effects on microhemodynamics in the brain cortex
The aim of this investigation was to assess the impact of two hemoglobin-derivatives on NO availability, which is expected to increase vasoactivity and induce aggregation of RBCs. If the available NO is reduced by quenching, both effects can be captured in the model predictions for microregional vascular resistance and hematocrit, respectively. As seen in Table 3, we found no difference in R' over that seen with HES in neither relations within the HBOC groups or in between the two HBOC groups, which precludes the presence of critical NO levels in the microcirculation of the brain cortex resulting from scavenging by the PEG-Hb molecules. RBC/platelet aggregation in blood is a complex phenomenon [88][89][90]. At normal levels, NO inhibits platelet aggregation and adhesion to the endothelium. Decreased NO bioavailability by HBOC-related quenching of NO can weaken this important homeostatic mechanism. Under normal conditions, RBC internal viscosity is very low, unlike when with platelets they form aggregates under which condition their contribution to the viscosity of blood becomes disproportionately large [91]. This would actually shift the Fåhraeus-Lindqvist relationship between apparent viscosity and tube diameter with its point of inversion around capillary dimensions to larger diameters [91] with a consequence of shutting down perfusion in the range of microvessels with diameters below the actual point of inversion. Due to an inherent link between the Fåhraeus-Lindqvist and Fåhraeus phenomena, this scenario must manifest in disturbed v RBC , T H t , and k. This was not observed in our study because mean levels of v RBC (Table 2), T H t , and k ( Fig. 4(A) and Fig. 5(A), Table 3) beyond post-exchange 20 min were well maintained. Also, in the same post-exchange period the unchanged heterogeneity of T H t , and R' indicate that the NO quenching by HBOC molecules resident in plasma was below the rate sufficient for triggering RBC aggregation or vasoreactivity ( Fig. 5(B,C)).
In summary, the followings can be considered as likely reasons for the demonstrated lack of adverse effects of Euro-PEG-Hb and PEG-HbO 2 molecules in the brain cortex. Because the HBOC molecules do not penetrate the BBB, extravascular NO scavenging and intercepting the NO signaling pathway (as known strong triggers of adversely increased vasoactivity) are missing. Following blood-to-HBOC exchange, the actual regional intravascular level of NO is a result of balance between NO production by eNOS and nNOS activities [92], elimination in RBCs [11], diffusion through the extravascular space [5], and scavenging by the HBOC molecule free in plasma [12,13]. In spite of the extremely short intravascular lifetime of NO, adverse effects do not develop in the brain because its NO content is the highest among the organs [34] suggesting high eNOS and nNOS activities. Therefore, even though extreme hemodilution is needed to deliver therapeutic quantities of HBOC in the circulation, the former cannot decrease NO to an extent that would result in considerable impairment of cerebrocortical hemodynamics and rheological behavior. Notwithstanding the additional increase by NO-scavenging effect, our findings attest that such hemodilution does not reach the critical level needed to induce adverse vasoconstriction, RBC and platelet aggregation.

Conclusions
The combination of high-resolution optical imaging of brain parenchymal circulation and mathematical modeling of its key parameters empowered us in demonstrating that HBOCs, in particular Euro-PEG-Hb the novel molecule we had developed earlier, have no adverse effects on the brain's microcirculation despite their potential in interfering with the NO-homeostasis. We conclude, that the well-known NO scavenging effect of HBOCs must have been overridden by the rate of endothelial NO secretion in the microcirculatory beds of the brain cortex. We argue that these findings may well help refining the picture on the safety of HBOCs as blood substitutes in clinical applications.
where t RBC is the mean transit time of red blood cells (RBCs) through their volume of distribution, V RBC .
The ROI of the LSCI measurement was selected void of visible vessels from this size and up. Under these conditions the majority of microvessels within the imaging voxels becomes of capillary size with an actual distribution of microvessel diameters within the 2-11 µm range as reported by Weiss [79]. The morphological data of Weiss also enables us to make model predictions for tissue tube hematocrit (see below) in absolute terms [79].
As each RBC travels the ROI along its path, L RBC , at its velocity, v RBC , Eq. (6) can be written as where L RBC is the mean vascular path length for RBCs, and A RBC is the total vascular cross section area for the traversing RBCs. Given that RBCs and plasma traverse the microcirculation in the form of plug and bolus flow, respectively, their corresponding mean vascular path lengths, L RBC = L plasma = L, total cross section areas are the same, A RBC = A plasma = A, and thus Eq. (7) can be taken further to where r A is the total cross section area-based vascular radius, T H t =V RBC /V WB is the tube hematocrit within the ROI.
The whole blood perfusion (Q WB ) in the same vascular compartment is related to RBC perfusion via the discharge hematocrit, D H t =Q RBC /Q WB , as Due to the conservation of mass throughout the cerebral circulation the discharge hematocrit remains at its input level ( D H a ) across subsequent generations of arterial branching ( D H a = D H t ) as schematically illustrated in Fig. 7 (left panel). In contrast, the tube hematocrit as a result of phase separation along the arterial tree (known as the Fåhraeus effect [54,55]) decreases towards the periphery reaching its minimal value in the capillaries (right panel). In the large arterial circulation, phase separation is minimal as red blood cells flow well-mixed in plasma. Thus in our model we regard T H a = D H a D H t .
The Fåhraeus effect within the cerebral arterial circulation can be characterized by the ratio of tube and discharge hematocrit, denoted as k. As mentioned above, due to the mass of conservation, the discharge hematocrit is constant throughout the circulation [86], which in our model allows defining k for the lumped tissue vascular segment as the ratio of tube hematocrit in the cerebrocortical tissue with respect to that measured in the femoral artery Substituting D H t from Eq. (11) and then Q RBC from Eq. (8), we obtain According to the Hagen-Poiseuille equation, the overall microvascular resistance depends on the viscosity of blood (η) and r A . Since R can also be derived from Ohm's law with the aid Fig. 7. Relations of discharge and tube hematocrits in the cerebral circulation. A dichotomically branched arterial tree was generated by a fractal model for the purpose of this schematics. Discharge (left) and tube hematocrits (right) were plotted in gray scale code (far left) for subsequent segments of the arterial tree from its input in the large arterial circulation (lower index a) all the way to its output at the tissue (capillary) level (lower index t). Note the apparent homogeneity in the distribution of the plotted parameters. Due to the lack of pase separation, discharge ( D H) and tube ( T H) hematocrits are equal at the input of the system. Owing to the conservation of mass, D H is uniform within the system, hence D H a = D H t . As a result of dynamic separation of phases along the arterial tree, T H becomes progressively smaller than D H (Fåhraeus effect [54,55]) reaching its minimal value at the tissue level within the ROI ( T H t ).
of previously calculated Q WB , the two expressions can be combined. For the lumped vascular segment of the ROI both Ohm's hydrodynamic law and the Hagen-Poiseuille equation can be applied to write the circulatory resistance, R, as The perfusion pressure (PP) is taken as the difference between the measured mean arterial pressure (MAP) and the intracranial pressure (ICP, assumed to be 10 mm Hg). The in vivo relative apparent viscosity of blood, η, can be assessed by the in vivo viscosity law [Eq. (7) in Ref [94]. ] The constant terms of this equation are: C = (0.8 + e (−0.075·D) ) −1 + 1 1 + 10 −11 · D 12 + 1 1 + 10 −11 · D 12 , and η * 0.45 =6 · e (−0.085·D) + 3.2 − 2.44 · e (−0.06·D 0.645 ) .
Here, D is the lumped vessel diameter (i.e. 2r A ) and η* 0.45 is the viscosity of blood at arterial discharge hematocrit of 0.45. We can approximate the hematocrit ratio as a function of vessel radius similarly to Eq. (2) of Pries et al [94]. as k = T H t T H a = T H a + (1 + T H a ) · (1 + 1.7 · e (−0.415·D) − 0.6 · e (−0.011·D) ).
After rearrangement, the unknown T H t can be expressed as a function of measured arterial tube hematocrit T H t = T H 2 a + T H a · (1 + T H a ) · (1 + 1.7 · e (−0.415·D) − 0.6 · e (−0.011·D) ). (18) and the unknown lumped vessel radius (D=2r A ) by expressing r A from Eq. (13) The mean microvascular path length is considered unaltered during the experiment, so the unknowns can be determined in a normalized manner from test and control values noted as prime Having Eq. (17) and Eq. (14) substituted into Eq. (21) the ultimate expression for the unknown r' A is obtained with terms either known or measured. The equation has to be solved numerically since an analytical solution does not exist. r lumen = r lumen (T) r lumen (C) = tube Hct tissue (T) · v RBC (T) · η(T) · Hct arterial (C) · PP(C) tube Hct tissue (C) · v RBC (C) · η(C) · Hct arterial (T) · PP(T)