A method for the quantification of the pressure dependent 3D collagen configuration in the arterial adventitia.

Collagen plays an important role in the response of the arterial wall to mechanical loading and presumably has a load-bearing function preventing overdistension. Collagen configuration is important for understanding this role, in particular in mathematical models of arterial wall mechanics. In this study a new method is presented to image and quantify this configuration. Collagen in the arterial adventitia is stained with CNA35, and imaged in situ at high resolution with confocal microscopy at luminal pressures from 0 to 140mm Hg. The images are processed with a new automatic approach, utilizing techniques intended for MRI-DTI data. Collagen configuration is quantified through three parameters: the waviness, the transmural angle and the helical angle. The method is demonstrated for the case of carotid arteries of the white New Zealand rabbit. The waviness indicated a gradual straightening between 40 and 80mm Hg. The transmural angle was about zero indicating that the fibers stayed within an axial-circumferential plane at all pressures. The helical angle was characterized by a symmetrical distribution around the axial direction, indicating a double symmetrical helix. The method is the first to combine high resolution imaging with a new automatic image processing approach to quantify the 3D configuration of collagen in the adventitia as a function of pressure.

Impact of aortic grafts on arterial pressure: a computational fluid dynamics study.

OBJECTIVE: Vascular prostheses currently used in vascular surgery do not have the same mechanical properties as human arteries. This computational study analyses the mechanisms by which grafts, placed in the ascending aorta (proximal) and descending aorta (distal), affect arterial blood pressure. METHODS: A one-dimensional cardiovascular model was developed and adapted to include the graft geometry with in vitro measured mechanical properties. Pressure at the aortic root and haemodynamic parameters were computed and compared for a control, proximal and distal graft case. RESULTS: In comparison to the control case, the proximal graft increased characteristic impedance by 58% versus only 1% change for the distal graft. The proximal and distal graft increased pulse pressure by 21% and 10%, respectively. CONCLUSIONS: The mechanisms underlying pulse pressure increase are different for proximal and distal grafts. For the proximal graft, the primary reason for pulse pressure rise is augmentation of the forward wave, resulting from characteristic impedance increase. For the distal graft, the pulse pressure rise is associated with augmented wave reflections resulting from compliance mismatch. Overall, the proximal aortic graft resulted in greater haemodynamic alterations than the distal graft. Thus, it is likely that patients who receive ascending aorta grafts are more prone to systolic hypertension and therefore deserve closer blood pressure monitoring.

RC time constant of single lung equals that of both lungs together: a study in chronic thromboembolic pulmonary hypertension.

The product of resistance, R, and compliance, C (RC time), of the entire pulmonary circulation is constant. It is unknown if this constancy holds for individual lungs. We determined R and C in individual lungs in chronic thromboembolic pulmonary hypertension (CTEPH) patients where resistances differ between both lungs. Also, the contribution of the proximal pulmonary arteries (PA) to total lung compliance was assessed. Patients (n=23) were referred for the evaluation of CTEPH. Pressure was measured by right heart catheterization and flows in the main, left, and right PA by magnetic resonance imaging. Total, left, and right lung resistances were calculated as mean pressure divided by mean flow. Total, left, and right lung compliances were assessed by the pulse pressure method. Proximal compliances were derived from cross-sectional area change DeltaA and systolic-diastolic pressure difference DeltaP (DeltaA/DeltaP) in main, left, and right PA, multiplied by vessel length. The lung with the lowest blood flow was defined "low flow" (LF), the contralateral lung "high flow" (HF). Total resistance was 0.57+/-0.28 mmHg.s(-1).ml(-1), and resistances of LF and HF lungs were 1.57+/-0.2 vs. 1.00+/-0.1 mmHg.s(-1).ml(-1), respectively, P<0.0001. Total compliance was 1.22+/-1.1 ml/mmHg, and compliances of LF and HF lung were 0.47+/-0.11 and 0.62+/-0.12 ml/mmHg, respectively, P=0.01. Total RC time was 0.49+/-0.2 s, and RC times for the LF and HF lung were 0.45+/-0.2 and 0.45+/-0.1 s, respectively, not different. Proximal arterial compliance, given by the sum of main, right, and left PA compliances, was only 19% of total lung compliance. The RC time of a single lung equals that of both lungs together, and pulmonary arterial compliance comes largely from the distal vasculature.

Methodologies to assess blood flow in cerebral aneurysms: current state of research and perspectives.

With intracranial aneurysms disease bringing a weakened arterial wall segment to initiate, grow and potentially rupture an aneurysm, current understanding of vessel wall biology perceives the disease to follow the path of a dynamic evolution and increasingly recognizes blood flow as being one of the main stakeholders driving the process. Although currently mostly morphological information is used to decide on whether or not to treat a yet unruptured aneurysm, among other factors, knowledge of blood flow parameters may provide an advanced understanding of the mechanisms leading to further aneurismal growth and potential rupture. Flow patterns, velocities, pressure and their derived quantifications, such as shear and vorticity, are today accessible by direct measurements or can be calculated through computation. This paper reviews and puts into perspective current experimental methodologies and numerical approaches available for such purposes. In our view, the combination of current medical imaging standards, numerical simulation methods and endovascular treatment methods allow for thinking that flow conditions govern more than any other factor fate and treatment in cerebral aneurysms. Approaching aneurysms from this perspective improves understanding, and while requiring a personalized aneurysm management by flow assessment and flow correction, if indicated.

Three- and four-element Windkessel models: assessment of their fitting performance in a large cohort of healthy middle-aged individuals.

Lumped-parameter models are used to estimate the global arterial properties by fitting the model to measured (aortic) pressure and flow. Different model configurations coexist, and it is still an open question as to which model optimally reflects the arterial tree and leads to correct estimates of arterial properties. An assessment was made of the performance of (a) the three-element Windkessel model (WK3) consisting of vascular resistance R, total arterial compliance C, and characteristic impedance Zc; (b) a four-element model with an inertance element L placed in parallel with Zc (WK4-p); and (c) a four-element model with L placed in series with Zc (WK4-s). Models were fitted to data measured non-invasively in 2404 healthy subjects, aged between 35 and 55 years. It was found that model performance segregated into two groups. In a group containing 20 per cent of the dataset (characterized by low blood pressure and wave reflection) the WK4-p model outperformed the other models, with model behaviour as envisioned by its promoters. In these cases, the WK3 and WK4-s models led to increased overestimation of total arterial compliance and underestimation of characteristic impedance. However, in about 80 per cent of the cases, the WK4-p model showed a behaviour that was very similar to that of the WK3 and WK4-s models. Here, the WK4-s model yielded the best quality of fit, although model parameters reached physically impossible values for L in about 12 per cent of all cases. The debate about which lumped-parameter model is the better approximation of the arterial tree is therefore still not fully resolved.

Oscillatory shear stress and reduced compliance impair vascular functions.

Shear stress has been shown to influence endothelial cell gene expression and morphology. In particular, low and bi-directional shear stress, mimicking conditions at plaque-prone areas, down-regulates the expression of several atheroprotective genes, and up-regulates that of other genes considered as pro-inflammatory. Another mechanical situation thought to have a negative influence on vascular functions is arterial stiffness. Loss of arterial compliance occurs during ageing, in diabetic as well as in hypertensive patients. In this work we investigated the effects of these two particular hemodynamic environments (bi-directional shear stress and reduced compliance), using a recently developed perfusion system allowing to expose native arteries in vitro to complex hemodynamic environments. We were able to show that both plaque-prone shear stress and reduced compliance trigger endothelial dysfunction, but via different mechanisms. Only reduced compliance affected vascular contractility, inducing a dedifferentiation of smooth muscle cells and a consequent loss of norepinephrine sensitivity.

Anti-apoA-1 auto-antibodies increase mouse atherosclerotic plaque vulnerability, myocardial necrosis and mortality triggering TLR2 and TLR4.

Auto-antibodies to apolipoprotein A-1 (anti-apoA-1 IgG) were shown to promote inflammation and atherogenesis, possibly through innate immune receptors signalling. Here, we aimed at investigating the role of Toll-like receptors (TLR) 2 and 4 on anti-apoA-1 IgG-induced atherosclerotic plaque vulnerability, myocardial necrosis and mortality in mice. Adult male apolipoprotein E knockout (ApoE)-/- (n=72), TLR2-/-ApoE-/- (n=36) and TLR4-/-Apo-/- (n=28) mice were intravenously injected with 50 µg/mouse of endotoxin-free polyclonal anti-apoA-1 IgG or control isotype IgG (CTL IgG) every two weeks for 16 weeks. Atherosclerotic plaque size and vulnerability were assessed by histology. Myocardial ischaemia and necrosis, respectively, were determined by electrocardiographic (ECG) changes assessed by telemetry and serum troponin I (cTnI) measurements. Impact on survival was assessed by Kaplan-Meier analyses. In ApoE-/- mice, anti-apoA-1 IgG passive immunisation enhanced histological features of atherosclerotic plaque vulnerability (increase in neutrophil and MMP-9 and reduction in collagen content), induced a substantial cTnI elevation (p=0.001), and increased mortality rate by 23 % (LogRank, p=0.04) when compared to CTL IgG. On a subgroup of ApoE-/- mice equipped with telemetry (n=4), a significant ST-segment depression was noted in anti-apoA-1 IgG-treated mice when compared to CTL IgG recipients (p< 0.001), and an acute ST-segment elevation myocardial infarction preceding mouse death was observed in one case. The deleterious effects of anti-apoA-1 IgG on atherosclerotic plaque vulnerability, myocardial necrosis and death were partially reversed in TLR2-/-ApoE-/- and TLR4-/-ApoE-/- backgrounds. In conclusion, anti-apoA-1 auto-antibodies seem to be active mediators of atherosclerotic plaque vulnerability, myocardial necrosis, and mortality in mice through TLR2- and TLR4-mediated pathways.

Cytosolic-free calcium in smooth-muscle and endothelial cells in an intact arterial wall from rat mesenteric artery in vitro.

The regulation of cytosolic-free calcium concentration of smooth-muscle and endothelial cells was mainly studied on cultured cells where the cross talk between these two coupled cell types is lost. In the present study, the cytosolic-free calcium concentration in the endothelial and the smooth-muscle cells was examined in an intact arterial wall in vitro. Strips of the main branch of rat mesenteric artery were used. Cytosolic-free calcium concentration [Ca2+]i was estimated by determining the fluorescence ratio of the two calcium probes, Fluo-4 and Fura red. The emitted fluorescence of both probes was measured with a confocal microscope. We showed that potassium and phenylephrine, which increase the cytosolic -free calcium concentration of the smooth-muscle cells, also indirectly influence the calcium concentration in the endothelial cells. By simultaneously determining [Ca2+]i in the endothelial and the smooth-muscle cells of an arterial strip, we observed that when calcium increases in the endothelial cells in response to acetylcholine, it slightly decreases in the smooth-muscle cells. We conclude that the regulation of [Ca2+]i in the arterial endothelial cell, depends according to the stimuli either upon the endothelial cells themselves, or upon the coupled smooth-muscle cells.

Determinants of pulse pressure.

We have searched to define the major arterial parameters that determine aortic systolic (Ps) and diastolic (Pd) pressure in the dog. Measured aortic flows were used as input to the 2-element windkessel model of the arterial system, with peripheral resistance calculated as mean pressure divided by mean flow and total arterial compliance calculated from the decay time in diastole. The windkessel model yielded an aortic pressure wave from which we obtained the predicted systolic (Ps,wk) and diastolic (Pd,wk) pressures. These predicted pressures were compared with the measured systolic and diastolic pressures. The measurements and calculations were performed for 7 dogs under control conditions during aortic occlusion at 4 locations (the trifurcation, between the trifurcation and diaphragm, the diaphragm, and the proximal descending thoracic aorta) and during occlusion of both carotid arteries. Under all conditions studied, the predicted systolic and diastolic pressures matched the experimental ones very well: Ps,wk=(1.000+/-0.0055) Ps with r=0.958 and Pd,wk=(1.024+/-0.0035) Pd with r=0.995. Linear regression for pulse pressure (PP) resulted in PPwk=(0.99+/-0.016) PP with r=0.911. We found the accuracy of prediction equally good under control conditions and in the presence of aortic or carotid artery occlusion. Multiple regression between pulse pressure and arterial resistance and total arterial compliance yielded a poor regression constant (R2=0.19), suggesting that the 2 arterial parameters alone cannot explain pulse pressure and that flow is an important determinant as well. We conclude that for a given ejection pattern (aortic flow), 2 arterial parameters, total arterial resistance and total arterial compliance, are sufficient to accurately describe systolic and diastolic aortic pressure.

Differences in the mechanical properties of the rat carotid artery in vivo, in situ, and in vitro.

The elastic properties of carotid arteries of spontaneously hypertensive rats (SHR) and normotensive controls (Wistar-Kyoto rats [WKY]) were examined in vivo, in situ, and in vitro. The changes of internal diameter were measured with a high-resolution A-mode echo-tracking device simultaneously with the intra-arterial pressure at the carotid. The internal diameter at mean arterial blood pressure (MBP) was substantially smaller in vitro than in vivo in SHR (-33.8%) and WKY (-48.3%). The arterial distensibility was lower in vitro in all arteries compared with in vivo conditions (SHR, -30.1%; WKY, -60.4%; at MBP) despite a reduced incremental elastic modulus in vitro (SHR, -56.9%; WKY, -45.1%; at MBP). However, the in vitro and in vivo measurements show consistent elastic behavior of the carotid arteries between both strains of rats. Carotid arteries from WKY were also examined in situ. Although no significant reduction in internal diameter could be observed in situ, distensibility was dramatically decreased (-87% at MBP). These results emphasize the importance of considering the original vascular geometry when determining elastic properties of arteries. We conclude that experimental conditions are likely to be a critical determinant for the assessment of the mechanical properties of conduit vessels.

Contributions of vascular tone and structure to elastic properties of a medium-sized artery.

Isobaric compliance and distensibility of the radial artery were recently reported to be normal or slightly increased in untreated hypertensive patients. However, these findings provide no information on the intrinsic mechanical properties of the wall material. To address this question, we determined intima-media wall thickness, wall-to-lumen ratio, and incremental elastic modulus in the radial artery of 25 untreated hypertensive patients with blood pressure of 150 +/- 14/103 +/- 6 mm Hg (mean +/- SD) and 25 matched control subjects with blood pressure of 118 +/- 9/79 +/- 6 mm Hg. High-resolution echotracking for assessment of internal diameter and intima-media wall thickness was combined with measurements of blood flow velocity by Doppler and blood pressure by photoplethysmography. In addition, isobaric compliance and distensibility and incremental elastic modulus were measured at peak diameter during reactive hyperemia after a 5-minute brachial occlusion. No significant difference was found between the two groups for isobaric compliance or distensibility at baseline or during hyperemia. However, incremental elastic modulus at 100 mg Hg tended to be lower in hypertensive patients than control subjects (1.9 +/- 1.1 versus 2.5 +/- 1.2 mm Hg x 10(4), P = .1) in resting conditions. Hypertensive patients and control subjects had similar internal diameters (2.47 +/- 0.32 versus 2.41 +/- 0.35 microm), but intima-media wall thickness and wall-to-lumen ratio were significantly increased in hypertensive patients compared with control subjects (0.268 +/- 0.032 versus 0.236 +/- 0.025 mm -P < or = .01- and 0.220 +/- 0.038 versus 0.195 +/- 0.028 -P < or = .05-, respectively). Peak hyperemic blood flow response (hypertensive patients versus control subjects: 349% versus 360% increase from baseline) and reactive hyperemic dilation (7.2% versus 7.9%) were similar in amplitude and duration in the two groups. These results suggest that wall thickening is an adaptive process that reduces wall tension in hypertensive patients while preserving a normal mechanical behavior of the radial artery. This is most likely accomplished by modification of the incremental elastic modulus of wall components rather than by a change in vascular tone.

Flow-diameter phase shift. A potential indicator of conduit artery function.

This study assesses (1) the relation of the very-low-frequency vasomotion (< 0.02 Hz) of the radial artery of young healthy volunteers to regional blood flow and (2) its distribution in the upper extremities. Radial artery diameters from comparable sites were measured on contralateral extremities in 18 young healthy volunteers by an echo tracking system simultaneously with blood flow velocity determined by continuous wave Doppler and blood pressure acquired by photoplethysmography in the middle finger. A synchronous global pattern of vasomotion was detected on contralateral radial arteries, suggesting the presence of either a centrally located pacemaker or a humoral system. Modulation of sympathovagal balance in 8 subjects did not significantly alter either the frequency or amplitude of the very-low-frequency vasomotor waves. Matching patterns of diameter and flow oscillations of the very-low-frequency type recorded at the same site were obtained in 10 strictly nonsmoking volunteers for given periods of time. A consistent phase lag was observed between flow and diameter signals. Flow always preceded the diameter fluctuations by a mean (+/- SEM) course of 20.8 +/- 1.56 seconds. Although the physiological basis for oscillatory behavior remains for the moment highly speculative, these results suggest that the very-low-frequency vasomotion pattern in this conduit vessel might be a flow- or shear stress-dependent phenomenon. Shear stress changes at the endothelium modulate vascular tone through the release of vasodilators. The noninvasive assessment of the diameter-flow relation may thus offer a new way of addressing vascular wall function in medium-sized and large arteries in subjects with cardiovascular risk factors.

The inverse Womersley problem for pulsatile flow in straight rigid tubes.

In this study a numerical solution for the problem of pulsating flow in rigid tubes is described. The method applies to the case of known flow rate waveform, as opposed to Womersley solution where the pressure gradient was the known quantity. The solution provides the pressure gradient and wall shear stress waveforms as well as the instantaneous velocity profiles. Results show that the method can be used to study the blood flow characteristics in large arteries.

Computer simulation of arterial flow with applications to arterial and aortic stenoses.

A computer model for simulating pressure and flow propagation in the human arterial system is developed. The model is based on the one-dimensional flow equations and includes nonlinearities arising from geometry and material properties. Fifty-five arterial segments, representing the various major arteries, are combined to form the model of the arterial system. Particular attention is paid to the development of peripheral pressure and flow pulses under normal flow conditions and under conditions of arterial and aortic stenoses. Results show that the presence of severe arterial stenoses significantly affects the nature of the distal pressure and flow pulses. Aortic stenoses also have a profound effect on central and peripheral pressure pulse formation. Comparison with the published experimental data suggests that the model is capable of simulating arterial flow under normal flow conditions as well as conditions of stenotic obstructions in a satisfactory manner.

Theoretical study of dynamics of arterial wall remodeling in response to changes in blood pressure.

The dynamics of arterial wall remodeling was studied on the basis of a phenomenological mathematical model. Sustained hypertension was simulated by a step increase in blood pressure. Remodeling rate equations were postulated for the evolution of the geometrical dimensions that characterize the zero stress state of the artery. The driving stimuli are the deviations of the extreme values of the circumferential stretch ratios and the average stress from their values at the normotensive state. Arterial wall was considered to be a thick-walled tube made of nonlinear elastic incompressible material. Results showed that thickness increases montonically with time whereas the opening angle exhibits a biphasic pattern. Geometric characteristics reach asymptotically a new homeostatic steady state, in which the stress and strain distribution is practically identical with the distribution under normotensive conditions. The model predictions are in good agreement with published experimental findings.

Predicting systolic and diastolic aortic blood pressure and stroke volume in the intact sheep.

We developed a mathematical model describing the interaction between the heart and the arterial system. The model was constructed and tested on basis of invasive hemodynamic data in six sheep. Data from a first group of three animals (49 cardiac cycles) were used to assess a template time-varying elastance curve for the left ventricle, while the baseline steady-state data of a second group of three animals were used to assess reference cardiac and arterial parameters in sheep. The model is fully characterized by nine parameters, which were converted into 6 dimensionless numbers using the Buckingham pi theorem. The model was then used to generate LV pressure and volume and aortic pressure and flow for 86 conditions obtained by varying parameters 50 to 200% of their reference value. Systolic (SBP) and diastolic (DBP) blood pressure and stroke volume (SV) were determined from these model-generated curves and multiple linear regression analysis yielded the following expressions: SBP = Pisovolumic [0.638 - 0.0773 Emax C + 0.0507 RC/T] (r2 = 0.89); DBP = Pisovolumic [0.438-0.0712 Emax C + 0.0655RC/T] (r2 = 0.88) and SV = LVEDV [1.265-1.040 LVEDV/(LVEDV - Vd) + 0.125 Emax C-0.0777RC/T] (r2 = 0.93) with Pisovolumic = Emax (LVEDV - Vd), Emax and Vd being the slope and intercept of the end-systolic pressure-volume relation, R and C the total peripheral resistance and compliance, LVEDV the left ventricular end-diastolic volume, and T the cardiac cycle length. These expressions were validated using data from the second group of three animals obtained during vena cava occlusion at baseline and during administration of dobutamine (61 cycles). The correlation between measured and predicted values was 0.98, 0.97 and 0.92 for SBP, DBP and SV, respectively. Compared to the measured values, SBP and DBP were, on average, underestimated by 5 and 6mmHg, respectively, and SV overestimated by 1.4 ml. We conclude that the derived expressions for blood pressure and stroke volume remain valid in the intact sheep for various hemodynamic conditions, and, taking into account their dimensionless form, may hold in other species and in humans.

Nonlinear separation of forward and backward running waves in elastic conduits.

A new method for the separation of forward and backward running waves in elastic conduits, with possible extension to the arterial system, has been developed. The mathematical model is based on the one-dimensional flow equations which allow the treatment of non-periodic or transient pressure and flow pulses. The method is fully nonlinear, i.e. no linearizing assumptions are made. The method includes the effects of convective acceleration and pressure-dependent vessel compliance. A first approximation for the fluid friction at the wall is also included. The application of the method requires the knowledge of the elastic properties, the instantaneous pressure and flow, as well as the instantaneous spatial derivatives of pressure and flow. Analysis of simulated data shows good results and suggests that the proposed method, unlike previous quasi-nonlinear and frequency domain methods, can be applied to strongly nonlinear and/or nonperiodic flows. The method predicts that if a linear analysis is applied to a nonlinear system errors arise.

Modeling of the wave transmission properties of large arteries using nonlinear elastic tubes.

We propose a new, simple way of constructing elastic tubes which can be used to model the nonlinear elastic properties of large arteries. The tube models are constructed from a silicon elastomer (Sylgard 184, Dow Corning), which exhibits a nonlinear behavior with increased stiffness at high strains. Tests conducted on different tube models showed that, with the proper choice of geometric parameters, the elastic properties, in terms of area-pressure relation and compliance, can be similar to that of real arteries.

Residual strain effects on the stress field in a thick wall finite element model of the human carotid bifurcation.

A three-dimensional finite element model of the carotid artery bifurcation was constructed in order to determine the stress field and assess the modification of the stress field when residual strain is taken into account. Residual strain in the carotid bifurcation was characterized by experimental observations. According to these observations, a geometrical model of the carotid artery was constructed to exhibit a state free of strain. Appropriate boundary conditions were applied to yield the correct geometry in the unloaded state, and physiological levels of pressure and axial stretching were applied. The model took into account the varying thickness of the arterial wall along the bifurcation. For modeling purposes, the material was considered to be hyperelastic, incompressible, homogenous and isotropic. For comparison, a similar model of the carotid artery which does not include the effects of residual strain was also created. The results demonstrate that in the model of the carotid artery bifurcation with residual strain, the distribution of maximum principal stress along the inner wall and the circumferential stress throughout the wall is much more uniform than in the model without residual strain. The ratio between the stress at the inner and the outer walls is highest at the lateral wall of the carotid sinus; this is the same location known to be a site of low and oscillatory fluid wall shear stress, and the principal location of early intimal thickening. These results suggest that the localization of atherosclerosis in the carotid artery may be due to local variations in both fluid wall shear stress and solid wall stress.

Effects of friction and nonlinearities on the separation of arterial waves into their forward and backward components.

In this paper we examine the importance of fluid friction and nonlinearities due to the area-pressure relationship and to the convective acceleration on the separation of arterial pressure and flow waves into their forward and backward components. Experiments were run in straight uniform nonlinearly elastic tubes. Different degrees of fluid friction and nonlinearities, covering the physiological range, have been tested. We predicted the forward and backward running pressure components using two wave separation methods: the classical linear method (Westerhof et al., Cardiovasc. Res, 6,648-656, 1972) and the first order correction (FOC) method (Pythoud et al., Trans ASME J. Biomech. Engng, in press) which takes nonlinearities and fluid friction into account. We found that the two methods yield somewhat different predictions. The differences tend to increase with the degree of fluid friction and nonlinearities and are typically of the order of 4-8%. We further compared the transmission ratio of forward and backward waves predicted by both methods. The transmission ratio was found to be overestimated by 10% by the classical linear method. The nonlinear method gave more accurate estimates, consistent with theory. We conclude that, for in vivo applications, the classical linear method should be the method of choice because it is simpler to use and the erros involved (4-8%) are comparable to measurement erros.