Dependence of heart chamber dimensions and dynamics on chamber demands and myocardial properties

A heart chamber undergoes eccentric hypertrophy in response to a chronic elevation of stroke-displacement demand, and it undergoes concentric hypertrophy in response to a chronic elevation of systolic-pressure demand. Both of these adaptations, which occur in various combinations, involve two myocardial plastic properties, "stretch normalization" and "stress normalization". We have developed a model which predicts dimensions and dynamics of the left ventricle as functions of myocardial properties and of the loads to which the chamber is adapted. The model involves: a stress-normalization rule which describes how myocardial volume depends on average systolic pressure, cavity volumes and the responsiveness of growth to stress; a stretch-normalization rule which describes how the cavity volume of standard stretch relates to average end-diastolic and end-systolic volumes; and a pressure-volume-curve equation giving isometric pressures as functions of cavity volume and myocardial volume relative to standard-stretch cavity volume, and elastic properties including contractility. The model shows how the relations among average dimensions, dynamics and loads depend on myocardial properties, particularly contractility and the growth response to stress. These properties are the main determinants of myocardial performance. In addition to the load adaptations mentioned above, the model predicts eccentric hypertrophy incident to reduced contractility, chronic dilation incident to reduced growth response to stress, myocardial stricture incident to excessive growth response to stress, and concentric hypertrophy (similar to high-pressure adaptation) incident to deposition of inert material. It allows some refinements in the evaluation of myocardial performance and in the evaluation of the abnormal properties responsible for abnormal performance.     

Chronic beta-adrenoreceptor activation increases cardiac cavity size through chamber remodeling and not via modifications in myocardial material properties

Chronic beta-adrenoreceptor (beta-AR) activation increases left ventricular (LV) cavity size by promoting a rightward shift in LV diastolic pressure-volume (P-V) relations in association with increases in low-tensile strength myocardial (non-cross-linked) collagen concentrations. Because diastolic P-V relations are determined by chamber remodeling as well as by myocardial material properties (indexed by myocardial stiffness), both of which are associated with modifications in myocardial collagen cross-linking, we evaluated whether chamber remodeling or alterations in myocardial material properties govern beta-AR-mediated modifications in diastolic P-V relations. The effects of chronic administration of isoproterenol (Iso; 0.04 mg.kg(-1).day(-1) from 12 to 19 mo of age) to spontaneously hypertensive rats (SHRs) on LV cavity dimensions, LV diastolic P-V relations, myocardial collagen characteristics, myocardial stiffness constants [e.g., the slope of the LV diastolic stress-strain relation (k)], and LV chamber and myocardial systolic function were assessed. SHRs at 19 mo of age had normal LV diastolic P-V relations, marked myocardial fibrosis (using a pathological score), increased myocardial cross-linked (insoluble to cyanogen bromide digestion) type I and type III collagen concentrations, and enhanced myocardial k values. Iso administration to SHRs resulted in enlarged LV cavity dimensions mediated by a rightward shift in LV diastolic P-V relations, increased volume intercept of the LV diastolic P-V relation, decreased LV relative wall thickness despite a tendency to augment LV hypertrophy, and increased non-cross-linked type I and type III myocardial collagen concentrations. Iso administration resulted in reduced pump function without modification of intrinsic myocardial systolic function. However, despite increasing myocardial non-cross-linked concentrations, Iso failed to alter myocardial k in SHRs. These results suggest that beta-AR-mediated rightward shifts in LV diastolic P-V relations, which induce decreased pump function, are mediated by chamber remodeling but not by modifications in myocardial material properties.     

Mechanics of edematous lungs.

Using the parenchymal marker technique, we measured pressure (P)-volume (P-V) curves of regions with volumes of approximately 1 cm3 in the dependent caudal lobes of oleic acid-injured dog lungs, during a very slow inflation from P = 0 to P = 30 cmH2O. The regional P-V curves are strongly sigmoidal. Regional volume, as a fraction of volume at total lung capacity, remains constant at 0.4-0.5 for airway P values from 0 to approximately 20 cmH2O and then increases rapidly, but continuously, to 1 at P = approximately 25 cmH2O. A model of parenchymal mechanics was modified to include the effects of elevated surface tension and fluid in the alveolar spaces. P-V curves calculated from the model are similar to the measured P-V curves. At lower lung volumes, P increases rapidly with lung volume as the air-fluid interface penetrates the mouth of the alveolus. At a value of P = approximately 20 cmH2O, the air-fluid interface is inside the alveolus and the lung is compliant, like an air-filled lung with constant surface tension. We conclude that the properties of the P-V curve of edematous lungs, particularly the knee in the P-V curve, are the result of the mechanics of parenchyma with constant surface tension and partially fluid-filled alveoli, not the result of abrupt opening of airways or atelectatic parenchyma.     

Influence of lung volume on pulmonary microvascular pressure-volume characteristics.

The pressure-volume (P-V) characteristics of the lung microcirculation are important determinants of the pattern of pulmonary perfusion and of red and white cell transit times. Using diffuse light scattering, we measured capillary P-V loops in seven excised perfused dog lobes at four lung volumes, from functional residual capacity (FRC) to total lung capacity (TLC), over a wide range of vascular transmural pressures (Ptm). At Ptm 5 cmH(2)O, specific compliance of the microvasculature was 8.6%/cmH(2)O near FRC, decreasing to 2.7%/cmH(2)O as lung volume increased to TLC. At low lung volumes, the vasculature showed signs of strain stiffening (specific compliance fell as Ptm rose), but stiffening decreased as lung volume increased and was essentially absent at TLC. The P-V loops were smooth without sharp transitions, consistent with vascular distension as the primary mode of changes in vascular volume with changes in Ptm. Hysteresis was small (0.013) at all lung volumes, suggesting that, although surface tension may set basal capillary shape, it does not strongly affect capillary compliance.     

Effect of intrathoracic pressure on pressure-volume characteristics of the lung in man.

Quasi-static pressure-volume (P-V) curves in normal seated human subjects were determined with pressure at the airway opening (Pa0) set below (negative pressure), above (positive pressure), or equal to ambient pressure. Dynamic compliance (Cdyn) during controlled continuous negative pressure breathing (CNPB) was also studied. Quasi-static P-V curves at negative pressure were decreased in slope, reflected a decrease in total lung capacity, and intersected the P-V curve obtained at ambient Pa0. At positive pressure the P-V curves showed an increase in slope and an increase in total lung capacity. During CNPB a fall in Cdyn was found. The fall in Cdyn was rapid and persisted for the duration of CNPB. Cdyn promptly returned to control levels when Pa0 was adjusted to ambient pressure.     

Factors influencing left-ventricular stiffness

The aim of the study was to investigate the relative contributions of geometrical and material factors to overall left-ventricular cavity stiffness. Left-ventricular cavity shapes were reconstructed using a computer and the variation of myocardial elastic modulus was calculated, by the finite element method, through the passive phase of diastole when rising volume coincided with rising pressure. Geometric data were obtained from biplane cineangiography, with micromanometer pressure measurements, for ten patients with left ventricular disease. Dimensional analysis was applied to the initial and derived data from which the influences of myocardial compliance, wall thickness-to-long dimension ratio, and aspect ratio (long-to-short axes) were determined. The ratio between the volume elasticity and the myocardial modulus of elasticity, the normalized stiffness ratio (NSR), is proposed as a useful index of left ventricular mechanical behaviour in diastole. The volume elasticity of the chamber is dependent not only upon the myocardium elastic modulus and the wall thickness ratio, but also on the shape of the chambe. Changes in the thickness/radius ratio of the ventricle have less effect upon its distention than those in the long dimension/radius ratio. The left ventricle becomes more spherical in shpae through diastole and hence becomes stiffer by this geometric mechanism.     

Myocardial stress equations: fiberstresses of the prolate spheroid

There are occasions in physiological research and medical practice where it is desirable to estimate the average fiberstress in a chamber wall, knowing only the pressure and dimensions. Because the contribution of a strained wall element to pressure depends on its location whereas its contribution to average stress is independent of location, an equation of this kind must involve an assumption about the stress distribution. When applied to a particular chamber, it will give an exact result only if the chamber's stress distribution is in some sense like that of the model for which the equation was derived. Since the fibers of biological chambers are continually being deposited and resorbed, they tend to exhibit similar stretches under the average conditions of the chamber. To the extent that this is so, P = (2/3) sigma v ln V0/Vc, is the best simple fiberstress equation for biological chambers. (P = transmural pressure, sigma v = volume-averaged fiberstress, V0 = volume enclosed by outside surface, Vc = cavity volume). It expresses the pressure-dimension-average-fiberstress relation of a chamber of any shape whose stresses exhibit the simplest possible distribution. One can add a term to the right side to account for the influence of stress profile complexities. That term takes the form of a moment whose value is zero at one state of distension. This "stress moment" expresses the unequal weighting of complexities on the two sides of the midwall isobar. Judging from the sarcomere length profile of the left ventricular wall, the stress moment is zero and the average fiberstress equation above is exact for average developed stress (without a second term) when cavity volume is somewhere near end-diastolic. Moreover, the departures from the relation (the effects of stress moment) are small so long as the inner and outer stresses do not differ by a factor greater than two.     

Mathematical modelling to centre low tidal volumes following acute lung injury: A study with biologically variable ventilation

Background

With biologically variable ventilation [BVV – using a computer-controller to add breath-to-breath variability to respiratory frequency (f) and tidal volume (V_T)] gas exchange and respiratory mechanics were compared using the ARDSNet low V_T algorithm (Control) versus an approach using mathematical modelling to individually optimise V_T at the point of maximal compliance change on the convex portion of the inspiratory pressure-volume (P-V) curve (Experimental).

Methods

Pigs (n = 22) received pentothal/midazolam anaesthesia, oleic acid lung injury, then inspiratory P-V curve fitting to the four-parameter logistic Venegas equation F(P) = a + b[1 + e^-(P-c)/d]^-1 where: a = volume at lower asymptote, b = the vital capacity or the total change in volume between the lower and upper asymptotes, c = pressure at the inflection point and d = index related to linear compliance. Both groups received BVV with gas exchange and respiratory mechanics measured hourly for 5 hrs. Postmortem bronchoalveolar fluid was analysed for interleukin-8 (IL-8).

Results

All P-V curves fit the Venegas equation (R² > 0.995). Control V_T averaged 7.4 ± 0.4 mL/kg as compared to Experimental 9.5 ± 1.6 mL/kg (range 6.6 – 10.8 mL/kg; p < 0.05). Variable V_Ts were within the convex portion of the P-V curve. In such circumstances, Jensen''s inequality states "if F(P) is a convex function defined on an interval (r, s), and if P is a random variable taking values in (r, s), then the average or expected value (E) of F(P); E(F(P)) > F(E(P))." In both groups the inequality applied, since F(P) defines volume in the Venegas equation and (P) pressure and the range of V_Ts varied within the convex interval for individual P-V curves. Over 5 hrs, there were no significant differences between groups in minute ventilation, airway pressure, blood gases, haemodynamics, respiratory compliance or IL-8 concentrations.

Conclusion

No difference between groups is a consequence of BVV occurring on the convex interval for individualised Venegas P-V curves in all experiments irrespective of group. Jensen''s inequality provides theoretical proof of why a variable ventilatory approach is advantageous under these circumstances. When using BVV, with V_T centred by Venegas P-V curve analysis at the point of maximal compliance change, some leeway in low V_T settings beyond ARDSNet protocols may be possible in acute lung injury. This study also shows that in this model, the standard ARDSNet algorithm assures ventilation occurs on the convex portion of the P-V curve.     

Relations among alveolar surface tension, surface area, volume, and recoil pressure

For pulmonary structure-function analysis excised rabbit lungs were fixed by vascular perfusion at six points on the pressure-volume (P-V) curve, i.e. at 40, 80, and 100% of total lung capacity (TLC) on inflation, at 80 and 40% TLC on deflation, and at 80% TLC on reinflation. Before fixation alveolar surface tensions (gamma) were measured in individual alveoli over the entire P-V loop, using an improved microdroplet method. A maximal gamma of approximately 30 mN/m was measured at TLC, which decreased during lung deflation to about 1 mN/m at 40% TLC. Surface tensions were considerably higher on the inflation limb starting from zero pressure than on the deflation limb (gamma-V hysteresis). In contrast, the corresponding alveolar surface area-volume (SA-V) relationship did not form a complete hysteresis over the entire volume range. There was a considerable difference in SA between lungs inflated to 40% TLC (1.49 +/- 0.11 m2) and lungs deflated to 40% TLC (2.19 +/- 0.21 m2), but at 80% TLC the values of SA were essentially the same regardless of the volume history. The data indicate that the gamma-SA hysteresis is only in part accountable for the P-V hysteresis and that the determinative factors of alveolar geometry change with lung volume. At low lung volumes airspace dimensions appear to be governed by an interplay between surface and tissue forces. At higher lung volumes the tissue forces become predominant.     

Lower inflection point and recruitment with PEEP in ventilated patients with acute respiratory failure.

The lower inflection point (LIP) on the total respiratory system pressure-volume (P-V) curve is widely used to set positive end-expiratory pressure (PEEP) in patients with acute respiratory failure (ARF) on the assumption that LIP represents alveolar recruitment. The aims of this work were to study the relationship between LIP and recruited volume (RV) and to propose a simple method to quantify the RV. In 23 patients with ARF, respiratory system P-V curves were obtained by means of both constant-flow and rapid occlusion technique at four different levels of PEEP and were superimposed on the same P-V plot. The RV was measured as the volume difference at a pressure of 20 cm H(2)O. A third measurement of the RV was done by comparing the exhaled volumes after the same distending pressure of 20 cm H(2)O was applied (equal pressure method). RV increased with PEEP (P < 0.0001); the equal pressure method compares favorably with the other methods (P = 0.0001 by correlation), although individual data cannot be superimposed. No significant difference was found when RV was compared with PEEP in the group of patients with a LIP < or =5 cm H(2)O and the group with a LIP >5 cm H(2)O (76.9 +/- 94.3 vs. 61.2 +/- 51.3, 267.7 +/- 109.9 vs. 209.6 +/- 73.9, and 428.2 +/- 216.3 vs. 375.8 +/- 145.3 ml with PEEP of 5, 10, and 15 cm H(2)O, respectively). A RV was found even when a LIP was not present. We conclude that the recruitment phenomenon is not closely related to the presence of a LIP and that a simple method can be used to measure RV.     

Effects of acute pleural effusion on respiratory system mechanics in dogs

We determined regional (Vr) and overall lung volumes in six head-up anesthetized dogs before and after the stepwise introduction of saline into the right pleural space. Functional residual capacity (FRC), as determined by He dilution, and total lung capacity (TLC) decreased by one-third and chest wall volume increased by two-thirds the saline volume added. Pressure-volume curves showed an apparent increase in lung elastic recoil and a decrease in chest wall elastic recoil with added saline, but the validity of esophageal pressure measurements in these head-up dogs is questionable. Vr was determined from the positions of intraparenchymal markers. Lower lobe TLC and FRC decreased with added saline. The decrease in upper lobe volume was less than that of lower lobe volume at FRC and was minimal at TLC. Saline increased the normal Vr gradient at FRC and created a gradient at TLC. During deflation from TLC to FRC before saline was added, the decrease in lung volume was accompanied by a shape change of the lung, with greatest distortion in the transverse (ribs to mediastinum) direction. After saline additions, deflation was associated with deformation of the lung in the cephalocaudal and transverse directions. The deformation with saline may be a result of upward displacement of the lungs into a smaller cross-sectional area of the thoracic cavity.     

Evaluation of myocardial properties from image/pressure data: chronic conditions

Since the wall/cavity ratio of a heart chamber is not a biological constant, fractional cavity-surface motion is not a valid performance index and the stresses most commonly used in the myocardial-mechanics literature are not valid expressions of pulling action or contractility. We have developed a system for analyzing and expressing left-ventricular performance and abilities which avoids these problems. It allows one to estimate the following quantities from left-ventricular image data and arterial pressures: "Fractional midwall excursion", the fractional change in a weighted average of inner- and outer-surface dimensions, which is a valid but preload-dependent expression of performance regardless of wall/cavity ratio. "Fractional midwall excursion rate", fractional midwall excursion divided by EKG-normalized ejection time, which is a relatively preload-independent expression of performance regardless of wall/cavity ratio. "Pressure safety factor", systolic pressure-making ability relative to demanded systolic pressure. "Myocardial fiberstress", the intensity of circumferential pulling force in the wall (pulling force per unit cross-sectional area). "Myocardial growth ability", the anabolic responsiveness to habitual systolic fiberstresses, expressed as the reciprocal of long-term-average systolic fiberstresses. (6) "Contractility", the stress-developing ability of the myocardium, specifically the amplitude of the developed stress-stretch function at peak activation. On the average, these quantities are related as follows: Growth ability determines average systolic fiberstresses; contractility and growth ability (or systolic stress) largely determine safety factor; safety factor largely determines fractional midwall excursion and its rate. We have developed a microcomputer program which evaluates these quantities from image/pressure data and displays them digitally and graphically.     

The diastatic pressure-volume relationship is not the same as the end-diastolic pressure-volume relationship

The end-diastolic pressure-volume (P-V) relationship (EDPVR) is routinely used to determine the passive left ventricular (LV) stiffness, although the diastatic P-V relationship (D-PVR) has also been measured. Based on the physiological difference between diastasis (the LV and atrium are relaxed and static) and end diastole (LV volume increased by atrial systole and the atrium is contracted), we hypothesized that, although both D-PVR and EDPVR include LV chamber stiffness information, they are two different, distinguishable P-V relations. Cardiac catheterization determined LV pressures, and conductance volumes in 31 subjects were analyzed. Physiological, beat-to-beat variation of the diastatic and end-diastolic P-V points were fit by linear and exponential functions to generate the D-PVR and EDPVR. The extrapolated exponential D-PVR underestimated LVEDP in 82% of the heart beats (P < 0.001). The extrapolated EDPVR overestimated pressure at diastasis in 84% of the heart beats (P < 0.001). If each subject's diastatic and end-diastolic P-V data were combined to form a continuous data set to be fit by one exponential relation, the goodness of fit was always worse than if the diastatic and end-diastolic data were grouped separately and fit by two distinct exponential relations. Diastatic chamber stiffness was less than EDPVR stiffness (defined by the slope of P-V relation) for all 31 subjects (0.16 +/- 0.11 vs. 0.24 +/- 0.15 mmHg/ml, P < 0.001). We conclude that the D-PVR and EDPVR are distinguishable. Because it is not coupled to a contracted atrium, the D-PVR conveys passive LV stiffness better than the EDPVR. Additional studies that fully elucidate the physiology and biology of diastasis in health and disease are in progress.     

An alternative method for analysing pressure-volume curves produced with the pressure chamber

Abstract. An equation is derived relating tissue water potential to relative water content. This equation may be used to lit a single curve to a set of data such as the standard pressure volume measurements made with a pressure chamber. From such a single curve fitting operation, estimates of the parameters involved may be found, and this allows calculation of such quantities as bulk modulus of elasticity, osmotic potential at full turgor and at the turgor loss point, pressure potential, and the weight of symplastic water. The method of analysis has several advantages, which are illustrated using pressure chamber data obtained from leaves of Lombardy poplar, Populus nigra L. 'italica'.     

Position of the embryos of American mink in the fetal chamber at different stages of development]

The spacial position of American mink embryos is characterized by regular changes and is associated with the development and formation of provisory embryonic organs and the uterus. After the implantation the longitudinal axis of the embryo's body lies perpendicularly towards the long axis of the uterus horn. From the end of the 22nd day till birth the embryo moves along the antimesometral side of the fetal chamber by rotation counter clockwise relative to the point of attachment of the alantois stalk. On the 20th day prior to delivery the embryo's body bent as a coil takes a vertical position its fore-part is disposed in the yolk sac cavity, and the hinder part is in the exocoelom. During 17 days before birth the embryo "rolls out" from the yolk sac cavity and occupies the low position in the longitudinal posture of the body. During the following 6 days the prefetus moves towards the opposite side wall of the fetal chamber, takes the upper position and keeps a longitudinal position till the end of the embryonic life.     

Sigmoidal equation for lung and chest wall volume-pressure curves in acute respiratory failure.

To assess incidence and magnitude of the "lower inflection point" of the chest wall, the sigmoidal equation was used in 36 consecutive patients intubated and mechanically ventilated with acute lung injury (ALI). They were 21 primary and 5 secondary ALI, 6 unilateral pneumonia, and 4 cardiogenic pulmonary edema. The lower inflection point was estimated as the point of maximal compliance increase. The low constant flow inflation method and esophageal pressure were used to partition the volume-pressure curves into their chest wall and lung components on zero end-expiratory pressure. The sigmoidal equation had an excellent fit with coefficients of determination >0.90 in all instances. The point of maximal compliance increase of the chest wall ranged from 0 to 8.3 cmH2O (median 1 cmH2O) with no difference between ALI groups. The chest wall significantly contributed to the lower inflection point of the respiratory system in eight patients only. The occurrence of a significant contribution of the chest wall to the lower inflection point of the respiratory system is lower than anticipated. The sigmoidal equation is able to determine precisely the point of the maximal compliance increase of lung and chest wall.     

Mikroskopische Beiträge zum Druck-Volumen-Schema des Zellwasserhaushaltes anhand von Einzelzellen und Zellfäden

Summary Microscopic contributions to the pressure-volume-diagram of the cell-water relations as demonstrated with single cells and cell filaments.The diagram described byRichter (1978), which is based on the relationship between osmotic potential and cell water volume, is used for the evaluation of microscopic measurements on algae in osmotically active solutions. Pressure-volume curves forEremosphaera viridis andSpirogyra spp. are very similar to those obtained from organs of higher plants in experiments with a pressure chamber or a thermocouple device. Tissue counterpressure and cell wall water have no influence on data from experiments on single cells with osmotic methods, which therefore give probably less biased values than tissues or organs. Differences in the elasticity of the cell wall, the resistance of the chromatophore against volume changes of the protoplast, and the relative volumes of protoplasm and vacuole show up very distinctly. -values for the elasticity of the cell wall were calculated for the cells analysed. The usefulness of the Richter diagram could fully be confirmed for single cells.

     

A model of muscular control of upper airway geometry

Some disorders of the upper airway in humans are marked by decreased cross-sectional area and increased airway wall compliance. Based on our observations from studies performed in the isolated upper airway of dogs, we hypothesized that the size, and perhaps the geometry, of the airway was altered by changes in the relative activation levels of various muscle pairs. This could be accomplished either by altering the intensity of the neuromuscular input, or by activating muscle pairs which have different geometric orientation to the airway. We developed an analytic relationship to allow us to vary the stimulus level driving any one of six muscle pairs, each with a different anatomic orientation, to evaluate the relationship between those parameters and upper airway volume. With data generated from bilateral electrical stimulation of upper airway muscles, we described a shape factor which allowed us to predict the maximum force produced at optimal length. These findings were applied to a length/tension curve common to striated muscle to allow us to examine the muscle behavior at lengths other than optimal. The position of each muscle was described in spherical coordinates relative to an elastic cylinder, which represented the isolated, sealed upper airway. These coordinates defined the direction in which the force generated by each muscle pair would be applied. Three compliance constants determined the change in airway dimensions produced by the muscle force. This system and its variables were used to calculate the change in volume of the sealed upper airway chamber resulting from muscle contraction.(ABSTRACT TRUNCATED AT 250 WORDS)     

Pulmonary vascular compliance and viscoelasticity

When dog lung lobes were perfused at constant arterial inflow rate, occlusion of the venous outflow (VO) produced a rapid jump in venous pressure (Pv) followed by a slower rise in both arterial pressure (Pa) and Pv. During the slow rise Pa(t) and Pv(t) tended to converge and become concave upward as the volume of blood in the lungs increased. We compared the dynamic vascular volume vs. pressure curves obtained after VO with the static volume vs. pressure curves obtained by dye dilution. The slope of the static curve (the static compliance, Cst) was always larger than the slope of the dynamic curve (the dynamic compliance, Cdyn). In addition, the Cdyn decreased with increasing blood flow rate. When venous occlusion (VO) was followed after a short time interval by arterial occlusion (AO) such that the lobe was isovolumic, both Pa and Pv fell with time to a level that was below either pressure at the instant of AO. In an attempt to explain these observations a compartmental model was constructed in which the hemodynamic resistance and vascular compliance were volume dependent and the vessel walls were viscoelastic. These features of the model could account for the convergence and upward concavity of the Pa and Pv curves after VO and the pressure relaxation in the isovolumic state after AO, respectively. According to the model analysis, the difference between Cst and Cdyn and the flow dependence of Cdyn are due to wall viscosity and volume dependence of compliance, respectively. Model analysis also suggested ways of evaluating changes in the viscoelasticity of the lobar vascular bed. Hypoxic vasoconstriction that increased total vascular resistance also decreased Cst and Cdyn and appeared to increase the vessel wall viscosity.