Diagnostic potential of blood plasma longitudinal viscosity measured using Brillouin light scattering.

Blood plasma viscosity (PV) is an established biomarker for numerous diseases. Measurement of the shear PV using conventional rheological techniques is, however, time consuming and requires significant plasma volumes. Here, we show that Brillouin light scattering (BLS) and angle-resolved spectroscopy measurements of the longitudinal PV from microliter-sized plasma volumes can serve as a proxy for the shear PV measured using conventional viscometers. This is not trivial given the distinct frequency regime probed and the longitudinal viscosity, a combination of the shear and bulk viscosity, representing a unique material property on account of the latter. We demonstrate this for plasma from healthy persons and patients suffering from different severities of COVID-19 (CoV), which has been associated with an increased shear PV. We further show that the additional information contained in the BLS-measured effective longitudinal PV and its temperature scaling can provide unique insight into the chemical constituents and physical properties of plasma that can be of diagnostic value. In particular, we find that changes in the effective longitudinal viscosity are consistent with an increased suspension concentration in CoV patient samples at elevated temperatures that is correlated with disease severity and progression. This is supported by results from rapid BLS spatial-mapping, angle-resolved BLS measurements, changes in the elastic scattering, and anomalies in the temperature scaling of the shear viscosity. Finally, we introduce a compact BLS probe to rapidly perform measurements in plastic transport tubes. Our results open a broad avenue for PV diagnostics based on the high-frequency effective longitudinal PV and show that BLS can provide a means for its implementation.

Viscoelastic phenotyping of red blood cells.

Red blood cells (RBCs) are the simplest cell types with complex dynamical and viscoelastic phenomenology. While the mechanical rigidity and the flickering noise of RBCs have been extensively investigated, an accurate determination of the constitutive equations of the relaxational kinetics is lacking. Here we measure the force relaxation of RBCs under different types of tensional and compressive extension-jump protocols by attaching an optically trapped bead to the RBC membrane. Relaxational kinetics follows linear response from 60 pN (tensional) to -20 pN (compressive) applied forces, exhibiting a triple exponential function with three well-separated timescales over four decades (0.01-100 s). While the fast timescale (τF∼0.02(1)s) corresponds to the relaxation of the membrane, the intermediate and slow timescales (τI=4(1)s; τS=70(8)s) likely arise from the cortex dynamics and the cytosol viscosity. Relaxation is highly heterogeneous across the RBC population, yet the three relaxation times are correlated, showing dynamical scaling. Finally, we find that glucose depletion and laser illumination of RBCs lead to faster triple exponential kinetics and RBC rigidification. Viscoelastic phenotyping is a promising dynamical biomarker applicable to other cell types and active systems.

Blood Rheology and Hemodynamics.

Seminars in Thrombosis and Hemostasis (STH) celebrates 50 years of publishing in 2024. To celebrate this landmark event, STH is republishing some archival material. This article represents the most highly cited paper ever published in STH. The original abstract follows.Blood is a two-phase suspension of formed elements (i.e., red blood cells [RBCs], white blood cells [WBCs], platelets) suspended in an aqueous solution of organic molecules, proteins, and salts called plasma. The apparent viscosity of blood depends on the existing shear forces (i.e., blood behaves as a non-Newtonian fluid) and is determined by hematocrit, plasma viscosity, RBC aggregation, and the mechanical properties of RBCs. RBCs are highly deformable, and this physical property significantly contributes to aiding blood flow both under bulk flow conditions and in the microcirculation. The tendency of RBCs to undergo reversible aggregation is an important determinant of apparent viscosity because the size of RBC aggregates is inversely proportional to the magnitude of shear forces; the aggregates are dispersed with increasing shear forces, then reform under low-flow or static conditions. RBC aggregation also affects the in vivo fluidity of blood, especially in the low-shear regions of the circulatory system. Blood rheology has been reported to be altered in various physiopathological processes: (1) Alterations of hematocrit significantly contribute to hemorheological variations in diseases and in certain extreme physiological conditions; (2) RBC deformability is sensitive to local and general homeostasis, with RBC deformability affected by alterations of the properties and associations of membrane skeletal proteins, the ratio of RBC membrane surface area to cell volume, cell morphology, and cytoplasmic viscosity. Such alterations may result from genetic disorders or may be induced by such factors as abnormal local tissue metabolism, oxidant stress, and activated leukocytes; and (3) RBC aggregation is mainly determined by plasma protein composition and surface properties of RBCs, with increased plasma concentrations of acute phase reactants in inflammatory disorders a common cause of increased RBC aggregation. In addition, RBC aggregation tendency can be modified by alterations of RBC surface properties because of RBC in vivo aging, oxygen-free radicals, or proteolytic enzymes. Impairment of blood fluidity may significantly affect tissue perfusion and result in functional deteriorations, especially if disease processes also disturb vascular properties.

Revisiting anemia in sickle cell disease and finding the balance with therapeutic approaches.

Chronic hemolytic anemia and intermittent acute pain episodes are the 2 hallmark characteristics of sickle cell disease (SCD). Anemia in SCD not only signals a reduction of red cell mass and oxygen delivery, but also ongoing red cell breakdown and release of cell-free hemoglobin, which together contribute to a number of pathophysiological responses and play a key role in the pathogenesis of cumulative multiorgan damage. However, although anemia is clearly associated with many detrimental outcomes, it may also have an advantage in SCD in lowering risks of potential viscosity-related complications. Until recently, clinical drug development for SCD has predominantly targeted a reduction in the frequency of vaso-occlusive crises as an endpoint, but increasingly, more attention is being directed toward addressing the contribution of chronic anemia to poor outcomes in SCD. This article aims to explore the complex pathophysiology and mechanisms of anemia in SCD, as well as the need to balance the benefits of raising hemoglobin levels with the potential risks of increasing blood viscosity, in the context of the current therapeutic landscape for anemia in SCD.

Whole blood viscosity is associated with reduced myocardial mechano-energetic efficiency in nondiabetic individuals.

INTRODUCTION: This cross-sectional study aimed to investigate the association between myocardial mechano-energetic efficiency (MEE) and whole blood viscosity (WBV) in nondiabetic adults participating in the CATAnzaro MEtabolic RIsk factors (CATAMERI) study. METHODS: 1143 participants underwent an oral glucose tolerance test and an echocardiogram for myocardial MEE per gram of left ventricular mass (MEEi) measurement. WBV was measured as: [0.12 × h] + [0.17 × (p-2.07)], where h is haematocrit and p is plasma protein levels. RESULTS: Study population includes 595 males and 548 females with a mean age of 46 ± 12 years and a mean BMI of 30.0 ± 6.2 kg/m2 . Individuals with normal glucose tolerance were 63%, while those with impaired fasting glucose, impaired glucose tolerance and or the combination of both were 14.3%, 13% and 9.7%, respectively. A univariate analysis showed that MEEi was significantly associated with sex, age, smoking, BMI, waist circumference, total cholesterol, HDL, triglycerides, fasting glucose, fasting insulin, HOMA-IR index, glucose tolerance, C-reactive protein, haematocrit, haemoglobin, plasma protein and WBV. In a multivariable regression model including variables that were significantly associated with MEEi in univariate analysis, MEEi was associated with HOMA-IR (ß = -0.144, p < .001), age (ß = -0.140, p < .001), WBV (ß = -0.129, p < .001) and glucose tolerance (ß = -0.064, p = .04). The independent association between WBV and MEEi remained statistically significant (ß = -0.122, p < .001) when antihypertensive therapy and lipid-lowering therapy were included in the model. CONCLUSION: WBV is associated with decreased myocardial MEE independently of other cardiovascular risk factors.

Decreased erythrocyte aggregation in Glenn and Fontan: univentricular circulation as a rheologic disease model.

BACKGROUND: In the Fontan palliation for single ventricle heart disease (SVHD), pulmonary blood flow is non-pulsatile/passive, low velocity, and low shear, making viscous power loss a critical determinant of cardiac output. The rheologic properties of blood in SVHD patients are essential for understanding and modulating their limited cardiac output and they have not been systematically studied. We hypothesize that viscosity is decreased in single ventricle circulation. METHODS: We evaluated whole blood viscosity, red blood cell (RBC) aggregation, and RBC deformability to evaluate changes in healthy children and SVHD patients. We altered suspending media to understand cellular and plasma differences contributing to rheologic differences. RESULTS: Whole blood viscosity was similar between SVHD and healthy at their native hematocrits, while viscosity was lower at equivalent hematocrits for SVHD patients. RBC deformability is increased, and RBC aggregation is decreased in SVHD patients. Suspending SVHD RBCs in healthy plasma resulted in increased RBC aggregation and suspending healthy RBCs in SVHD plasma resulted in lower RBC aggregation. CONCLUSIONS: Hematocrit corrected blood viscosity is lower in SVHD vs. healthy due to decreased RBC aggregation and higher RBC deformability, a viscous adaptation of blood in patients whose cardiac output is dependent on minimizing viscous power loss. IMPACT: Patients with single ventricle circulation have decreased red blood cell aggregation and increased red blood cell deformability, both of which result in a decrease in blood viscosity across a large shear rate range. Since the unique Fontan circulation has very low-shear and low velocity flow in the pulmonary arteries, blood viscosity plays an increased role in vascular resistance, therefore this work is the first to describe a novel mechanism to target pulmonary vascular resistance as a modifiable risk factor. This is a novel, modifiable risk factor in this patient population.

Redefining hyperviscosity in acute leukemia: Potential implications for red cell transfusions in the microvasculature.

Hyperleukocytosis is an emergency of acute leukemia leading to blood hyperviscosity, potentially resulting in life-threatening microvascular obstruction, or leukostasis. Due to the high number of red cells in the circulation, hematocrit/hemoglobin levels (Hct/Hgb) are major drivers of blood viscosity, but how Hct/Hgb mediates hyperviscosity in acute leukemia remains unknown. In vivo hemorheological studies are difficult to conduct and interpret due to issues related to visualizing and manipulating the microvasculature. To that end, a multi-vessel microfluidic device recapitulating the size-scale and geometry of the microvasculature was designed to investigate how Hct/Hgb interacts with acute leukemia to induce "in vitro" leukostasis. Using patient samples and cell lines, the degree of leukostasis was different among leukemia immunophenotypes with respect to white blood cell (WBC) count and Hct/Hgb. Among lymphoid immunophenotypes, severe anemia is protective against in vitro leukostasis and Hct/Hgb thresholds became apparent above which in vitro leukostasis significantly increased, to a greater extent with B-cell acute lymphoblastic leukemia (ALL) versus T-cell ALL. In vitro leukostasis in acute myeloid leukemia was primarily driven by WBC with little interaction with Hct/Hgb. This sets the stage for prospective clinical studies assessing how red cell transfusion may affect leukostasis risk in immunophenotypically different acute leukemia patients.

Making a virtue out of an evil: Are red blood cells from chronic mountain sickness patients eligible for transfusions?

We investigated highlanders, permanently living at an altitude of 5100 m and compared Chronic Mountain Sickness (CMS) patients with control volunteers. While we found differences in systemic parameters such as blood oxygen content, hematocrit, hemoglobin concentration, and blood viscosity, the mechanical and rheological properties of single red blood cells did not differ between the two investigated groups.

Reply to the 'Comment on "Dynamics and rheology of vesicles under confined Poiseuille flow"' by G. Coupier and T. Podgorski, Soft Matter, 2024, 20, DOI: 10.1039/D3SM01679J.

In this answer, we provide our arguments in support of the possibility to observe the single file-organization of red blood cells in microvessels and the resulting unexpectedly weak increase of blood viscosity with increasing hematocrit, the physiological relevance of which was questioned in the comment. The key element is that the equivalent diameter in 3D for the maximal hematocrit corresponding to a single file of red blood cells is about 10 µm and not 20 µm, as in 2D. In addition, the viscosity contrast (ratio between the cell internal and external viscosities) value must be chosen in our 2D simulation in a such a way that the effective viscosity (a linear combination of the internal, external and membrane viscosities) be close to that of a real RBC. Taking these two facts into account, we find a reasonable agreement between our 2D viscosity simulations data and experimental data, despite the crude 2D assumption.

Impact of anti-coagulant choice on blood elongational behavior.

Blood is a highly complex fluid with rheological properties that have a significant impact on various flow phenomena. In particular, it exhibits a non-Newtonian elongational viscosity that is comparable to polymer solutions. In this study, we investigate the effect of three different anticoagulants, namely EDTA (ethylene diamine tetraacetic acid), heparin, and citrate, on the elongational properties of both human and swine blood. We observe a unique two stage thinning process and a strong dependency of the characteristic relaxation time on the chosen anticoagulant, with the longest relaxation time and thus the highest elongational viscosity being found for the case of citrate. Our findings for the latter are consistent with the physiological values obtained from a dripping droplet of human blood without any anticoagulant. Furthermore, our study resolves the discrepancy found in the literature regarding the reported range of characteristic relaxation times, confirming that the elongational viscosity must be taken into account for a full rheological characterization of blood. These results have important implications for understanding blood flow in various physiological, pathological and technological conditions.

Analysis of the Suitability of an Effective Viscosity to Represent Interactions Between Red Blood Cells in Shear Flow.

Many methods to computationally predict red blood cell damage have been introduced, and among these are Lagrangian methods that track the cells along their pathlines. Such methods typically do not explicitly include cell-cell interactions. Due to the high volume fraction of red blood cells (RBCs) in blood, these interactions could impact cell mechanics and thus the amount of damage caused by the flow. To investigate this question, cell-resolved simulations of red blood cells in shear flow were performed for multiple interacting cells, as well as for single cells in unbounded flow at an effective viscosity. Simulations run without adjusting the bulk viscosity produced larger errors unilaterally and were not considered further for comparison. We show that a periodic box containing at least 8 cells and a spherical harmonic of degree larger than 10 are necessary to produce converged higher-order statistics. The maximum difference between the single-cell and multiple-cell cases in terms of peak strain was 3.7%. To achieve this, one must use the whole blood viscosity and average over multiple cell orientations when adopting a single-cell simulation approach. The differences between the models in terms of average strain were slightly larger (maximum difference of 6.9%). However, given the accuracy of the single-cell approach in predicting the maximum strain, which is useful in hemolysis prediction, and its computational cost that is orders of magnitude less than the multiple-cell approach, one may use it as an affordable cell-resolved approach for hemolysis prediction.

Integrating UPLC-Q-TOF-MS and Network Pharmacology to Explore the Potential Mechanisms of Paeonia lactiflora Pall. in the Treatment of Blood Stasis Syndrome.

Paeonia lactiflora Pall. (PLP) is thought to promote blood circulation and remove blood stasis. This study used blood component analysis, network pharmacology, and molecular docking to predict the mechanism of PLP in the treatment of blood stasis syndrome (BSS). PLP was processed into Paeoniae Radix Alba (PRA) and Paeoniae Radix Rubra (PRR). PRA and PRR could significantly reduce whole blood viscosity (WBV) at 1/s shear rates and could increase the erythrocyte aggregation index (EAI), plasma viscosity (PV), and erythrocyte sedimentation rate (ESR) of rats with acute blood stasis. They prolonged the prothrombin time (PT), and PRR prolonged the activated partial thromboplastin time (APTT). PRA and PRR increased the thrombin time (TT) and decreased the fibrinogen (FBG) content. All the results were significant (p < 0.05). Ten components of Paeoniflorin, Albiflorin, Paeonin C, and others were identified in the plasma of rats using ultra-high-performance liquid chromatography-quadrupole time-of-flight mass spectrometry (UPLC-Q-TOF-MS). A protein-protein interaction network (PPI) analysis showed that AKT1, EGFR, SRC, MAPK14, NOS3, and KDR were key targets of PLP in the treatment of BSS, and the molecular docking results further verified this. This study indicated that PLP improves BSS in multiple ways and that the potential pharmacological mechanisms may be related to angiogenesis, vasoconstriction and relaxation, coagulation, and the migration and proliferation of vascular cells.

Influence of a Decrease in Blood Viscosity on Arterial Pressure in Normotensive and Spontaneously Hypertensive Rats.

We investigated the effect of a decrease in blood viscosity on the mean BP during isovolumic hemodilution and vasodilating activity of the endothelium in normotensive Wistar rats and spontaneously hypertensive rats (SHR). Blood viscosity was reduced by isovolumic hemodilution (replacement of 10% of circulating blood with an equal volume of plasma). Hemodilution caused the same reduction in blood viscosity by 16% in both groups of rats. In Wistar rats, a decrease in blood viscosity did not significantly change in the mean BP; no significant correlations between blood viscosity and mean BP were observed before and after hemodilution. In SHR, a decrease in blood viscosity led to a significant decrease in the mean BP by 18%. Correlations were found between the mean BP and blood viscosity in SHR before (r=0.63; p=0.028) and after (r=0.71; p=0.009) isovolumic hemodilution. In SHR, a decrease in the index of vasodilating activity of the endothelium due to a decrease in the vasodilatory response to intravenous administration of the endothelium-dependent vasodilator acetylcholine was revealed. In SHR, BP passively follows the change, in this case, the decrease in blood viscosity, which attests to impaired BP regulation in response to changes in shear stress on the vascular endothelium caused by the development of endothelial dysfunction in hypertensive animals.

Influence of storage and buffer composition on the mechanical behavior of flowing red blood cells.

On-chip study of blood flow has emerged as a powerful tool to assess the contribution of each component of blood to its overall function. Blood has indeed many functions, from gas and nutrient transport to immune response and thermal regulation. Red blood cells play a central role therein, in particular through their specific mechanical properties, which directly influence pressure regulation, oxygen perfusion, or platelet and white cell segregation toward endothelial walls. As the bloom of in-vitro studies has led to the apparition of various storage and sample preparation protocols, we address the question of the robustness of the results involving cell mechanical behavior against this diversity. The effects of three conservation media (EDTA, citrate, and glucose-albumin-sodium-phosphate) and storage time on the red blood cell mechanical behavior are assessed under different flow conditions: cell deformability by ektacytometry, shape recovery of cells flowing out of a microfluidic constriction, and cell-flipping dynamics under shear flow. The impact of buffer solutions (phosphate-buffered saline and density-matched suspension using iodixanol/Optiprep) are also studied by investigating individual cell-flipping dynamics, relative viscosity of cell suspensions, and cell structuration under Poiseuille flow. Our results reveal that storing blood samples up to 7 days after withdrawal and suspending them in adequate density-matched buffer solutions has, in most experiments, a moderate effect on the overall mechanical response, with a possible rapid evolution in the first 3 days after sample collection.

In silico modeling of patient-specific blood rheology in type 2 diabetes mellitus.

Increased blood viscosity in type 2 diabetes mellitus (T2DM) is a risk factor for the development of insulin resistance and diabetes-related vascular complications; however, individuals with T2DM exhibit heterogeneous hemorheological properties, including cell deformation and aggregation. Using a multiscale red blood cell (RBC) model with key parameters derived from patient-specific data, we present a computational study of the rheological properties of blood from individual patients with T2DM. Specifically, one key model parameter, which determines the shear stiffness of the RBC membrane (µ) is informed by the high-shear-rate blood viscosity of patients with T2DM. At the same time, the other, which contributes to the strength of the RBC aggregation interaction (D0), is derived from the low-shear-rate blood viscosity of patients with T2DM. The T2DM RBC suspensions are simulated at different shear rates, and the predicted blood viscosity is compared with clinical laboratory-measured data. The results show that the blood viscosity obtained from clinical laboratories and computational simulations are in agreement at both low and high shear rates. These quantitative simulation results demonstrate that the patient-specific model has truly learned the rheological behavior of T2DM blood by unifying the mechanical and aggregation factors of the RBCs, which provides an effective way to extract quantitative predictions of the rheological properties of the blood of individual patients with T2DM.

Association of haemolysis markers, blood viscosity and microcirculation function with organ damage in sickle cell disease in sub-Saharan Africa (the BIOCADRE study).

Sickle cell anaemia (SCA) is a monogenic disease with a highly variable clinical course. We aimed to investigate associations between microvascular function, haemolysis markers, blood viscosity and various types of SCA-related organ damage in a multicentric sub-Saharan African cohort of patients with SCA. In a cross-sectional study, we selected seven groups of adult patients with SS phenotype in Dakar and Bamako based on the following complications: leg ulcer, priapism, osteonecrosis, retinopathy, high tricuspid regurgitant jet velocity (TRV), macro-albuminuria or none. Clinical assessment, echocardiography, peripheral arterial tonometry, laboratory tests and blood viscosity measurement were performed. We explored statistical associations between the biological parameters and the six studied complications. Among 235 patients, 58 had high TRV, 46 osteonecrosis, 43 priapism, 33 leg ulcers, 31 retinopathy and 22 macroalbuminuria, whereas 36 had none of these complications. Multiple correspondence analysis revealed no cluster of complications. Lactate dehydrogenase levels were associated with high TRV, and blood viscosity was associated with retinopathy and the absence of macroalbuminuria. Despite extensive phenotyping of patients, no specific pattern of SCA-related complications was identified. New biomarkers are needed to predict SCA clinical expression to adapt patient management, especially in Africa, where healthcare resources are scarce.

Semi-automated red blood cell core detection in blood micro-flow.

In microcirculation, red blood cells (RBCs) tend to migrate toward the centre of the vessel leaving a region of a cell depleted layer or cell-free layer (CFL) at the vessel wall and a core of RBCs at the centre. This heterogenous distribution of cells has an effect on the blood apparent viscosity and the exchanges of gases and nutrients between the RBCs and the vessel. Understanding the formation of the CFL and obtaining accurate measurement of it is paramount for furthering development of devices such as drug administration. This paper presents a general semi-automatic method to quantify the thickness of the CFL for different channel geometries and image quality. It enables the use of a method based on intensity, a method using the gradient of the intensity, or a method based on spatiotemporal variation. The main features are reported, the performance is demonstrated on experimentally obtained image sets and accuracy is validated using synthetic images with known CFL thickness. A pure automatic detection is limited by the most visually correct using the spatiotemporal method, however proposed thresholding through automatic detection allows for quality controls through manual adjustments. With a semi-automatic approach RBC core variability between 3 % to 8 % was found when the test user was tasked with repeating the analysis of the same set. The presented method provides, users without programming ability with a user-friendly interface that can extract CFL automatically with quality control through manual adjustments.

Erythrocyte deformability profile evaluated by laser diffractometry in patients with multiple myeloma: Re-examination of our cases.

BACKGROUND: Multiple myeloma is a complex pathology which represents about 10 % of all hematological neoplasms. It can often present changes in the hemorheological profile and, in relation to this last topic, our aim is to evaluate the hemorheological profile in a group of multiple myeloma patients, with reference to erythrocyte deformability. METHODS: We have examined the profile of the erythrocyte deformability in multiple myeloma enrolling 29 patients; this profile, expressed as elongation index at several shear stress, has been obtained using the diffractometric method. RESULTS: By comparing normal controls and MM patients, a significant decrease in erythrocyte deformability, especially at low shear stresses, but we did not observe any significant differences about this profile subdividing the whole group of MM patients according to the degree of bone marrow plasma cell infiltration, to the red blood cell distribution width and to the serum values of LDH. CONCLUSIONS: In this paper we have taken in consideration all the hypothesis for a possible explanation of the behaviour of this a reduced erythrocyte deformability in multiple myeloma. Erythrocyte deformability interferes with the physiological release of oxygen to tissues, with several clinical implications.

Effects of sarpogrelate on blood viscosity.

The objective of the present study was to evaluate the effects and safety of sarpogrelate hydrochloride (sarpogrelate) in patients with elevated blood viscosity (BV), after 12 and 24 weeks of twice (BID) or thrice (TID) daily administrations of sarpogrelate (100 mg). The participants received oral sarpogrelate administration for 24 weeks and visited the hospital every 12 ± 2 week for blood viscosity measurements at shear rates of 5 and 300 s-1. The BV measured at shear rate of 5 s-1 in male patients decreased significantly from 18.91 cP at the baseline to 16.3 cP after 24 weeks of sarpogrelate administration (13.6 % drop, p < 0.001). The BV measured at 5 s-1 in female decreased more significantly from 17.5 cP at the baseline to 13.4 cP after 24 weeks of sarpogrelate administration (23.0 % drop, p < 0.001). In summary, sarpogrelate may be considered as a possible therapeutic option for improving BV in patients with elevated blood viscosity. In particular, the reduction of the low-shear BV with the help of a viscosity-reducing drug such as sarpogrelate may be considered as a potentially new pharmacological tool for microvascular disease.

A theoretical model for the Fåhræus effect in medium-large microvessels.

This paper presents a mathematical model capable to reproduce a celebrated phenomenon in blood microcirculation known as Fåhræus effect, since its discovery by Robin Fåhræus (1929). This consists in a decaying of the relative hematocrit in small vessels as the vessel diameter decreases. The key point of the model is a formula, direct consequence of the basic principles of fluid dynamics, that links the relative hematocrit to the reservoir hematocrit and the vessel diameter, which confirms the observed behavior. To test the model we selected, among the few experiments carried on since then, those performed by Barbee and Cokelet (1971). The agreement is remarkable. An extended comparison is also carried out with a celebrated empirical formula proposed by Pries et al. (1992) to describe the same phenomenon.