Red blood cell dielectrophoresis in axisymmetric fields

Dielectrophoretic velocities of human red blood cells in an axisymmetric field were measured as a function of the applied voltage and the distance from the axis of symmetry. The voltage of the alternating electric field (frequency 2 MHz), applied between two concentric cylindrical metal electrodes (outer and inner radii 0.24 and 1 mm, respectively), was varied up to 19 V. Two kinds of mediums were used: (a) 90% of 2.1% glycine solution and 10% of 5.5% glucose solution and (b) 5.4% sorbitol solution. The results have shown that in both mediums the cell velocities are proportional to the square of the applied voltage and inversely proportional to the cube of the distance from the axis of symmetry, as predicted by the theory. The coefficient of proportionality (dielectrophoretic coefficient) is on the order of 10⁻²⁵ A²s⁴kg⁻¹. It depends on the donor of red blood cells and might be used for diagnostic purposes. These results will be used in future investigations of membrane adhesion, stability and fusion.     

Interaction forces between red cells agglutinated by antibody. I. Theoretical.

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutrally buoyant spheres suspended in viscous liquids undergoing a uniform shear flow has been given by Arp and Mason (1977). The method is based on the matrix formulation of hydrodynamic resistances in creeping flow by Brenner and O'Neill (1972). We describe the solution of the Brenner-O'Neill force-torque vector equation in terms of the particle and external flow field coordinates and derive expressions for the normal force acting along, and the shear force acting perpendicular to, the axis of the doublet of spheres, the latter explicitly given for the first time. The equations consist of a term comprising force and torque coefficients obtained from the matrices of the hydrodynamic resistances (functions of the distance h between sphere surfaces which have been computed), and terms comprising the orientation of the doublet axis relative to the coordinates of the external flow field and the shear stress (which can be experimentally determined). We have applied the theory to a system of doublets of sphered, hardened human red cells of group A or B antigenic type cross-linked by the corresponding antibody at a fixed interparticle distance. Working from studies of the breakup of doublets of red cells in an accelerating Poiseuille flow, given in the succeeding paper, we are able to compute the hydrodynamic force required to separate the two spheres. Previous work has shown that the theory can be applied to doublets in a variable shear, Poiseuille flow, provided the ratio of particle to tube diameter is small. In calculating the force-torque coefficients it was assumed that the cells are crosslinked by antibody with h = 20 nm.     

Dielectric Breakdown of Cell Membranes

With human and bovine red blood cells and Escherichia coli B, dielectric breakdown of cell membranes could be demonstrated using a Coulter Counter (AEG-Telefunken, Ulm, West Germany) with a hydrodynamic focusing orifice. In making measurements of the size distributions of red blood cells and bacteria versus increasing electric field strength and plotting the pulse heights versus the electric field strength, a sharp bend in the otherwise linear curve is observed due to the dielectric breakdown of the membranes. Solution of Laplace's equation for the electric field generated yields a value of about 1.6 V for the membrane potential at which dielectric breakdown occurs with modal volumes of red blood cells and bacteria. The same value is also calculated for red blood cells by applying the capacitor spring model of Crowley (1973. Biophys. J. 13:711). The corresponding electric field strength generated in the membrane at breakdown is of the order of 4 · 10⁶ V/cm and, therefore, comparable with the breakdown voltages for bilayers of most oils. The critical detector voltage for breakdown depends on the volume of the cells. The volume-dependence predicted by Laplace theory with the assumption that the potential generated across the membrane is independent of volume, could be verified experimentally. Due to dielectric breakdown the red blood cells lose hemoglobin completely. This phenomenon was used to study dielectric breakdown of red blood cells in a homogeneous electric field between two flat platinum electrodes. The electric field was applied by discharging a high voltage storage capacitor via a spark gap. The calculated value of the membrane potential generated to produce dielectric breakdown in the homogeneous field is of the same order as found by means of the Coulter Counter. This indicates that mechanical rupture of the red blood cells by the hydrodynamic forces in the orifice of the Coulter Counter could also be excluded as a hemolysing mechanism. The detector voltage (or the electric field strength in the orifice) depends on the membrane composition (or the intrinsic membrane potential) as revealed by measuring the critical voltage in E. coli B harvested from the logarithmic and stationary growth phases. The critical detector voltage increased by about 30% for a given volume on reaching the stationary growth phase.     

Nylon-fiber affinity selection of red blood cells and tissue culture cells on the basis of cell surface determinants

Nylon fibers coated with various lectins were used for the specific selection from mixed populations of erythrocytes or tissue culture cells with lectin receptors. Binding of human group O red blood cells to fibers treated with Ulex europaeus lectin I (H-specific) or of human group A red cells to fibers treated with Helix pomatia lectin (A-specific) was proportional to lectin concentration in the solution used to adsorb lectin to the fibers. Binding was blood group specific and increased with increasing concentrations of red cells applied to the fibers. Most adsorption of lectin to the fibers occurred within minutes; cell binding to lectin-coated fibers was almost complete within 30 min. Blood group negative Chinese hamster tissue culture cells bound non-specifically to Helix-coated fibers with a frequency of less than 10⁻⁴ input cells; the yield of viable, colony-forming cells bound to PHA-coated fibers was about 1%. Epithelial cells from cultures of amniotic fluid or fetal kidney contained 1–30% cells positive for the ABO blood group of the donor; blood group positive cells from these cultures were poorly bound to fibers coated with blood group specific lectins, though they bound readily to PHA-coated fibers, suggesting that presence of appropriate surface determinants may be necessary but not sufficient for lectin: cell binding in this system.     

Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation

Downstream from a microvascular bifurcation the distribution of blood cells in the vessel lumen is not symmetric. A diffusion process is used to model the rearrangement of red cells as blood flows between junctions in the microcirculation. A Fourier series approach is used to solve the model diffusion convection equation in slit geometry. Both flat and parabolic velocity profiles are considered. The eigenvalues, found using the Rayleigh-Ritz method, are used to find an upper bound on distance required for a symmetric red cell distribution to be obtained. The method has also been applied to cylindrical geometry and the computed symmetry recovery lengths are compared to distances between bifurcations measured in vivo. These estimates indicate that red cell distributions are frequently asymmetric in the microcirculation. Such asymmetries can have a strong effect on plasma skimming and material balance calculations.     

Red blood cell orientation in orbit C = 0.

Two modes of behavior of single human red cells in a shear field have been described. It is known that in low viscosity media and at shear rates less than 20 s-1, the cells rotate with a periodically varying angular velocity, in accord with the theory of Jeffery (1922) for oblate spheroids. In media of viscosity greater than approximately 5 mPa s and sufficiently high shear rates, the cells align themselves at a constant angle to the direction of flow with the membrane undergoing tank-tread motion. Also, in low viscosity media, as the shear rate is increased, more and more cells lie in the plane of shear, undergoing spin with their axes of symmetry aligned with the vorticity axis of the shear field in an orbit "C = 0" (Goldsmith and Marlow, 1972). We have explored this latter phenomenon using two experimental methods. First, the erythrocytes were observed in the rheoscope and their diameters measured. Forward light scattering patterns were correlated with the red cell orientation mode. Light flux variations after flow onset or stop were measured, and the characteristic times of erythrocyte orientation and disorientation were assessed. The characteristic time of erythrocyte orientation in Orbit C = 0 is proportional to the inverse of the shear rate. The corresponding coefficient of proportionality depends on the suspending medium viscosity eta o. The disorientation time tau D, after flow has been stopped, is such that the ratio tau D/eta o is independent of the initial applied shear stress. However, tau D is much shorter than one would expect if pure Brownian motion were involved. The proportion of erythrocytes in orbit C = 0 was also measured. It was found that this proportion is a function of both the shear rate and eta o. At low values of eta o, the proportion increases with increasing shear rate and then reaches a plateau. For higher values of eta o (5 to 10 mPa s), the proportion of RBC in orbit C = 0 is a decreasing function of the shear stress. A critical transition between orbit C = 0 and parallel alignment was observed at high values of eta o, when the shear stress is on the order of 1 N/m2. Finally, the effect of altering membrane viscoelastic properties (by heat or diamide treatment) was tested. The proportion of oriented cells is a steep decreasing function of red cell rigidity.     

Separation of cells labeled with immunospecific iron dextran microspheres using high gradient magnetic chromatography

Immunospecific magnetic microspheres, consisting of ferromagnetic iron dextran conjugated to Protein A, were used to specifically label red blood cells (RBC) for cell separation studies using high gradient magnetic chromatography ( HGMC ). When 10(7)-10(8) RBC labeled with Protein A-iron dextran microspheres were applied to a column containing 30 mg stainless steel wire placed in a 7.5 kilogauss magnetic field, 96 +/- 2% of the cells were retained in the column. These cells could be eluted by removing the magnetic field and mechanically agitating the column. The retention of labeled cells by HGMC was shown to be dependent on the applied magnetic field and the amount of wire packed into the column. HGMC in conjunction with cell labeling with immunospecific iron dextran microspheres have useful applications for the separation of specific cell types.     

On the theoretical aspects of tracer exchange in red blood cells

An analytical solution is presented for isotopic exchange between red blood cells and a finite surrounding medium where the intracellular diffusion is taken into account for two models of the red blood cell, the rectangular and the spherical. The solution is applied on experimental results from the literature with exchange of water, methanol, formamide, urea and ethylene glycol. It is seen that if the diffusion coefficient for those substances is less than 10^?7 cm²/sec, then the previous assumption about exchange between two well-stirred compartments becomes doubtful.     

Electrical Sizing of Particles in Suspensions: III. Rigid Spheroids and Red Blood Cells

The processes involved during the passage of a suspended particle through a small cylindrical orifice across which exists an electric field are investigated experimentally for an approximate prolate spheroid in the form of two tangent, rigid spheres (ragweed pollen particles) and for fresh, human red blood cells. Oscillograms of current pulses produced by both types of particles are presented and discussed in terms of particle shape and orientation and the effects of the hydrodynamic field. It is concluded that all the particles enter the orifice with their major axes aligned parallel to the orifice axis (electric field), but that during their passage some are rotated by the hydrodynamic field. Cells with their equatorial plane perpendicular to a radius of the orifice change their orientation with respect to the electric field as they are rotated, the others do not; only in the former case is there any deformation. It is shown that the bimodal or skewed size distributions can be explained on this basis, and that size (shape factor × volume) is actually a normally distributed variable (P > 95%). The average size of samples from 10 healthy adults was found to be 102.7 μ³ with a coefficient of variation of 1.8%. For a volume of 87 μ³, this corresponds to a shape factor of 1.18, an axial ratio (assuming a perfect oblate spheroid) of 0.26, and an equivalent major axis of 8.6 μ. The effect of high electric fields on red cell size distributions is mentioned.     

Improved flow cytometric method to enumerate residual cells: minimal linear detection limits for platelets,erythrocytes, and leukocytes

Increasing demand for quality control of blood products requires more sensitive methods to enumerate residual cells. Presently, the reported threshold (in cells per microliter) is 400 for red blood cells, 30-500 for platelets, and 1 for leukocytes. To examine precision and linearity in enumerating residual platelets and red blood cells, EDTA-anticoagulated blood from healthy donors was serially diluted with serum, stained in TruCount tubes using a no-lyse/no-wash procedure and a monoclonal antibody cocktail against the CD42a (FL1) and glycophorin-A (FL2) epitopes, and analyzed by flow cytometry. Leukocyte counts were determined in separate tubes. Cell preparation and analysis were performed once for 20 blood samples each and 20 times using the same specimen. Acquisition from the same tube was performed separately for platelets (threshold on FL1) and red blood cells (threshold on FL2). Multiparameter analysis was used for data evaluation. Linear results were obtained for platelets per microliter between 3,410 and 5 and for red blood cells per microliter between 54,000 and 3. For the lower cell concentrations, the coefficient of variation was 16.7% for platelets and 10.9% for red blood cells. The presented method allows the distinction between physiologically intact and ghost red blood cells. The method represents a reliable, sensitive, and accurate approach to quantify platelets and red blood cells in diluted blood. It can be applied to enumerate residual cells in plasma products and meets the increasing demand for quality control in blood components.     

An assay for dielectrophoresis: Applications to electromagnetically induced membrane adhesion and fusion

A new assay for dielectrophoresis, consisting of a platinum wire placed along the axis of symmetry of a hollow metal cylinder, was constructed. The main advantage of this device is that because of the simple axisymmetrical electric field the experimental data can be easily interpreted. Preliminary experimental data for adhesion and fusion of red blood cells are reported. The most interesting observation was that fusion of cell membranes can be induced without applying DC electrical pulses, merely because of the dielectrophoretic effect. A possible explanation of this phenomenon is suggested.     

Measurements of potential differences in human subjects induced by motion in a superconducting magnetic field.

We have attempted to measure the electromotive forces (emfs) induced in human beings moving at a constant speed in a highly dense magnetic field. Experiments were initially conducted on a set of models, and then directly on human subjects. The models consisted of single circular loops of Tygon tubing (I.D., 0.635 cm; O.D., 0.9525 cm) filled with normal saline solution, with circumferences of 20, 40, 60, 80, and 100 cm. The models were connected to an amplifier via silver/silver-chloride electrodes. Each saline loop was mounted on a movable platform, with the plane of the loop perpendicular to the platform's axis; the platform was enabled to move at known constant speeds into and out of the bore of a 1.89-T magnet. The human subjects were then substituted for the saline loops, with the long axis parallel to the direction of motion, and with standard EKG electrodes placed at 180 degrees successively on the ankle, calf, lower thigh, upper thigh, chest, and head. In all cases, for human subjects and models, the peak induced voltage was directly proportional to the speed of movement and the square of the circumference of the bounded cross-sectional areas. Thus, for the saline loops, the correlation coefficient between induced voltage and circumference was .998, and for human subjects, .947. Under the loose assumption that for equal circumferences the bounded areas in human subjects were equal to those in the circular loops, the induced emfs in human subjects were consistently about 13% greater than those in the loops. At a mean speed of 1.18 m/s, the chest had a peak induced voltage of 260 mV, while the voltage at the ankle had a peak of 19.8 mV. The experimental data were used to estimate the corresponding induced-current density at the pericardium, 17 mA/m2. We conclude for a human subject moving at constant speed along the body's long axis into a magnetic field that Faraday's law is closely followed for various cross-sections of the body. Further, in those cases in which the magnetic field and its gradient are not well-established, one can use saline-filled loops to estimate approximate values of voltages induced in human subjects.     

Asymmetric flows of spherical particles in a cylindrical tube

To study the rheological behavior of blood cells in various flow patterns through narrow vessels, we analyzed numerically the motion of blood cells arranged in one row or two rows in tube flow, at low Reynolds numbers. The particles are assumed to be identical rigid spheres placed periodically along the vessel axis at off-axis positions with equal spacings. The flow field of the suspending fluid in a circular cylindrical tube is analyzed by a finite element method applied to the Stokes equations, and the motion of each particle is simultaneously determined by a force-free and torque-free condition. In both cases of single- and two-file arrangements of the particles, their longitudinal and angular velocities are largely affected by the radial position and the axial spacing between neighboring particles. The apparent viscosity of the asymmetric flows is higher than that of the symmetric flow where particles are located on the tube centerline, and this is more pronounced when particles are placed farther from the tube centerline and when the axial distance between neighboring particles is reduced.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

Shear‐induced diffusion of red blood cells measured with dynamic light scattering‐optical coherence tomography

Quantitative measurements of intravascular microscopic dynamics, such as absolute blood flow velocity, shear stress and the diffusion coefficient of red blood cells (RBCs), are fundamental in understanding the blood flow behavior within the microcirculation, and for understanding why diffuse correlation spectroscopy (DCS) measurements of blood flow are dominantly sensitive to the diffusive motion of RBCs. Dynamic light scattering‐optical coherence tomography (DLS‐OCT) takes the advantages of using DLS to measure particle flow and diffusion within an OCT resolution‐constrained three‐dimensional volume, enabling the simultaneous measurements of absolute RBC velocity and diffusion coefficient with high spatial resolution. In this work, we applied DLS‐OCT to measure both RBC velocity and the shear‐induced diffusion coefficient within penetrating venules of the somatosensory cortex of anesthetized mice. Blood flow laminar profile measurements indicate a blunted laminar flow profile and the degree of blunting decreases with increasing vessel diameter. The measured shear‐induced diffusion coefficient was proportional to the flow shear rate with a magnitude of ~0.1 to 0.5 × 10^?6 mm². These results provide important experimental support for the recent theoretical explanation for why DCS is dominantly sensitive to RBC diffusive motion.      

Determination of aciclovir and ganciclovir in human plasma by liquid chromatography-spectrofluorimetric detection and stability studies in blood samples

A sensitive HPLC method has been developed for the assay of aciclovir and ganciclovir in human plasma, by HPLC coupled with spectrofluorimetric detection. Plasma (1000 microl), with 9-ethyl-guanine added as internal standard, is submitted to protein precipitation with trichloroacetic acid solution 20%. The supernatant, evaporated to dryness at 37 degrees C, is reconstituted in 100 microl of a solution of sodium heptanosulfonate 0.4% adjusted with acetic acid to pH 2.60 and a 30 microl volume is then injected onto a Nucleosil 100-5 microm C18 column. Aciclovir and ganciclovir are analysed by spectrofluorimetric detection set at 260 nm (excitation) and 380 nm (emission) using a gradient elution program with solvents constituted of acetonitrile and a solution of sodium heptanosulfonate 0.4% adjusted to pH 2.60. The calibration curves are linear between 0.1 and 10 microg/ml. The mean absolute recovery of aciclovir and ganciclovir are 99.2+/-2.5 and 100.3+/-2.5%, respectively. The method is precise (with mean inter-day C.V.s within 1.0-1.6% for aciclovir and 1.2-3.5% for ganciclovir), and accurate (range of inter-day deviations -1.6 to +1.6% for aciclovir and -0.4 to -1.4% for ganciclovir). The method has been applied in stability studies of ganciclovir in patients' blood samples, demonstrating its good stability in plasma at -20 degrees C and at room temperature. The distribution of ganciclovir and aciclovir in plasma and red blood cells was also investigated in vitro in spiking experiments with whole blood, which showed an initial drop of ganciclovir and aciclovir levels in plasma (about -25%) due to the cellular uptake of aciclovir and ganciclovir by red blood cells. The method has been validated and is currently applied in a clinical study assessing the ganciclovir plasma concentration variability after administration of valganciclovir in a population of solid organ transplant patients.     

Plug Effect of Erythrocytes in Capillary Blood Vessels

As an idealized problem of the motion of blood in small capillary blood vessels, the low Reynolds number flow of plasma (a newtonian fluid) in a circular cylindrical tube involving a series of circular disks is studied. It is assumed in this study that the suspended disks are equally spaced along the axis of the tube, and that their centers remain on the axis of the tube and that their faces are perpendicular to the tube axis. The inertial force of the fluid due to the convective acceleration is neglected on the basis of the smallness of the Reynolds number. The solution of the problem is derived for a quasi-steady flow involving infinitesimally thin disks. The numerical calculation is carried out for a set of different combinations of the interdisk distance and the ratio of the disk radius to the tube radius. The ratio of the velocity of the disk to the average velocity of the fluid is calculated. The different rates of transport of red blood cells and of plasma in capillary blood vessels are discussed. The average pressure gradient along the axis of the tube is computed, and the dependence of the effective viscosity of the blood on the hematocrit and the diameter of the capillary vessel is discussed.     

In vitro evaluation and study of the survival of erythrocytes preserved in a saline-adenine-glucose-mannitol medium for 35 days

The saline-adenine-glucose-mannitol (SAGM) solution for resuspension of red cells was evaluated on 30 blood units tested over 42 days and compared to 5 red cell concentrates collected on the conventional CPD medium. Total and extra-cellular hemoglobin, potassium, pH, ATP and DPG concentrations, osmotic fragility, schizocyte formation, and red cell antigenicity were studied through the storage period. Chromium survival studies of autologous donated red cells were performed in 10 donors. Red cell concentrates resuspended in SAGM solution showed at the 35th day of conservation at 4 degrees C, a mean storage hemolysis of only 0.66%, an ATP concentration of 67% of the initial value, a schizocyte proportion of less than 1.5%, a mean 24 hour posttransfusion viability of 88.33% and a mean red cell T 1/2 survival of 25 days 10 hours. No alteration of common blood group antigens could be found after storage of red cells for 42 days.     

Staining Tissue of the Central Nervous System with Luxol Fast Blue and Neutral Red

The tissue is fixed in 10% neutral saline formalin for 1 day to 3 wk depending on the size of the block, dehydrated and embedded in paraffin. The sections are stained at 57° C for 2 hr, then at 22° C for 30 min, in a 0.0125% solution of Luxol fast blue in 95% alcohol acidified by 0.1% acetic acid. They are differentiated in a solution consisting of: Li₂CO₃, 5.0 gm; LiOH-H₂O, 0.01 gm; and distilled water, 1 liter at 0-1° C, followed by 70% alcohol, and then treated with 0.2% NaHSO₃. They are soaked 1 min in an acetic acid-sodium acetate buffer 0.1 N, pH 5.6, then stained with 0.03% buffered aqueous neutral red. Sections are washed in distilled water, 1 sec, then treated with the following solution: CuSO₄·5H₂O, 0.5 gm; CrK(SO₄)₂·12H₂O, 0.5 gm; 10% acetic acid, 3 ml; and distilled water, 250 ml. Dehydration, clearing and covering complete the process. Myelin sheaths are stained bright blue; meninges and the adventitia of blood vessels are blue; red blood cells are green. Nissl material is stained brilliant red; axon hillocks, axis cylinders, ependyma, nuclei and some cytoplasm of neuroglia, media and endothelium of blood vessels are pink.     

Electrical hemolysis of human and bovine red blood cells

Summary The external electric field strength required for electrical hemolysis of human red blood cells depends sensitively on the composition of the external medium. In isotonic NaCl und KCl solutions the onset of electrical hemolysis is observed at 4 kV per cm and 50% hemolysis at 6 kV per cm, whereas increasing concentrations of phosphate, sulphate, sucrose, inulin and EDTA shift the onset and the 50% hemolysis-value to higher field strengths. The most pronounced effect is observed for inulin and EDTA. In the presence of these substances the threshold value of the electric field strength is shifted to 14 kV per cm. This is in contrast to the dielectric breakdown voltage of human red blood cells which is unaltered by these substances and was measured to be 1 V corresponding in the electrolytical discharge chamber to an external electric field strength of 2 to 3 kV per cm. On the other hand, dielectric breakdown of bovine red blood cell membranes occurs in NaCl solution at 4 to 5 kV per cm and is coupled directly with hemoglobin release. The electrical hemolysis of cells of this species is unaffected by the above substances with exception of inulin. Inulin suppressed the electrical hemolysis up to 15 kV per cm. The data can be explained by the assumption that the reflection coefficients of the membranes of these two species to bivalent anions and uncharged molecules are field-dependent to a different extent. This explanation implies that electrical hemolysis is a secondary process of osmotic nature induced by the reversible permeability change of the membrane (dielectric breakdown) in response to an electric field. This view is supported by the observation that the mean volumes of ghost cells obtained by electrical hemolysis can be changed by changing the external phosphate concentration during hemolysis and resealing, or by subjecting the cells to a transient osmotic stress immediately after the electrical hemolysis step. An interesting finding is that the breakdown voltage, although constant throughout each normally distributed ghost size distribution, increases with increasing mean volume of the ghost populations.