BioMedical Engineering OnLine BioMed Central
Research Open Access Mathematical model describing erythrocyte sedimentation rate. Implications for blood viscosity changes in traumatic shock and crush syndrome Rovshan M Ismailov*1, Nikolai A Shevchuk2,3 and Higmat Khusanov4
Address: 1Department of Epidemiology, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA 15213, USA, 2Center for Cancer and Immunology Research, Children's Research Institute, Washington, DC, USA, 3Institute for Biomedical Sciences/Program in Molecular and Cellular Oncology, Washington, DC, USA and 4Institute of Mechanics and Seismic Stability of Structures, Academy of Science of Uzbekistan, Tashkent, Uzbekistan Email: Rovshan M Ismailov* - [email protected]; Nikolai A Shevchuk - [email protected]; Higmat Khusanov - [email protected] * Corresponding author
Published: 04 April 2005 Received: 24 January 2005 Accepted: 04 April 2005 BioMedical Engineering OnLine 2005, 4:24 doi:10.1186/1475-925X-4-24 This article is available from: http://www.biomedical-engineering-online.com/content/4/1/24 © 2005 Ismailov et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Background: The erythrocyte sedimentation rate (ESR) is a simple and inexpensive laboratory test, which is widespread in clinical practice, for assessing the inflammatory or acute response. This work addresses the theoretical and experimental investigation of sedimentation a single and multiple particles in homogeneous and heterogeneous (multiphase) medium, as it relates to their internal structure (aggregation of solid or deformed particles). Methods: The equation system has been solved numerically. To choose finite analogs of derivatives we used the schemes of directional differences. Results: (1) Our model takes into account the influence of the vessel wall on group aggregation of particles in tubes as well as the effects of rotation of particles, the constraint coefficient, and viscosity of a mixture as a function of the volume fraction. (2) This model can describe ESR as a function of the velocity of adhesion of erythrocytes; (3) Determination of the ESR is best conducted at certain time intervals, i.e. in a series of periods not exceeding 5 minutes each; (4) Differential diagnosis of various diseases by means of ESR should be performed using the aforementioned timed measurement of ESR; (5) An increase in blood viscosity during trauma results from an increase in rouleaux formation and the time-course method of ESR will be useful in patients with trauma, in particular, with traumatic shock and crush syndrome. Conclusion: The mathematical model created in this study used the most fundamental differential equations that have ever been derived to estimate ESR. It may further our understanding of its complex mechanism.
Introduction [1]. The ESR has also been found to be of clinical signifi- The erythrocyte sedimentation rate (ESR) is a simple and cance in the follow-up and prognosis of non-inflamma- inexpensive laboratory test that is widespread in clinical tory conditions, such as prostate cancer [2], coronary practice for assessing the inflammatory or acute response artery disease [3], and stroke [4]. In addition, the ESR can
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µ µ be used in the diagnosis of inflammatory conditions [4,5] = 0 (1 + 2.5 f2), (2) as well as in the prognosis of non-inflammatory condi- µ tions [6]. Some examples of recent applications of the ESR where 0 represents viscosity of the fluid, and f2 concentra- may include sickle cell disease and bacterial otitis media tion of particles. [7,8]. The ESR has been shown to be elevated in 55% of patients with otitis media [7]. Those with elevated ESR Up to now, the Einstein formula of viscosity of suspension have been shown to have a much higher risk for recur- has been the foundation for most theories describing rence [7]. In sickle cell anemia, the ESR is usually low in behavior of a suspension in a shear flow [13,14]. Most the absence of a painful crisis [8]. A low ESR is an intrinsic studies deal with precipitation of a single particle or mul- property of the sickle red blood cell rheology [9,10]. Cer- tiple diluted particles in a Newtonian viscous fluid and tainly, the ESR is only one parameter among others that a offer various corrective parameters for the Stokes formula clinician can use in the diagnosis and follow up of the [13,14]. For instance, Ozeen [13,14] deduced an approxi- above diseases. mate solution of equations for a flow of spheres that served as a basis for the formula Sedimentation of particles, in particular erythrocytes, in a Newtonian fluid (plasma), has been studied by many 3 FaVNoN=++61πµ ( (2 )), ()3 investigators based on the theory and, therefore, model of 8 Re Re interpenetrating motion of two-phase medium that take ρ µ into account aggregation of erythrocytes [11,12]. We where NRe = aV / is the Reynolds number. aimed to investigate group precipitation of particles, such as erythrocytes, in a two-phase medium (plasma), both Subsequently, some problems related to non-Newtonian theoretically and experimentally. We added the influence behavior of fluids [13,14] were also investigated. Casuell of the vessel wall on group precipitation of particles in and Schwarz applied Ericson and Rivlin's model for a slow tubes, as well as the effects of rotation of particles, the con- flow of a non-Newtonian fluid [13,14]. Applying the straint coefficient, and viscosity of a mixture as a function method of jointed expansions that uses corresponding of the volume fraction. The theory has also taken into con- non-Newtonian terms for both "internal" and "external" sideration certain experimental coefficients such as: the expansions to the formula of spherical flow, they derived coefficient of interaction between the fluid and particles, the following formula: the aggregation coefficient, the constraint coefficient of phases, the coefficient of viscosity of the mixture, and the π =++πµ 3 27 V ϕ () coefficient of rotation of a particle. The equation system FVaN61()Re , 4 2 f2 has been solved numerically. To choose finite analogs of derivatives we used the schemes of directional differences. where φ is a compound expression dependent on non- Newtonian parameters that are constant for any given Methods fluid under isothermal conditions. Lesli also investigated Stokes[13] was the first who derived an equation for non- a slow spherical flow, using the Oldroid model [13,14] for steady-state flow when he was linearizing the equation of non-Newtonian conditions. He concluded that the non- motion of a viscous incompressible fluid. In that work, Newtonian term is proportional to V3. Stokes developed a theory of resistance for a falling spher- ical body. The relationship that he derived is called Stokes' The above studies refer to the precipitation or flow of a formula: single spherical particle. However, concentrated mixtures with a large number of particles interacting during precip- F = 6πµ aV, (1) itation or in a flow are affected not only by the Stokes force shown in equations (1), (3), and (4), but also by Where µ represents viscosity of the fluid, a – the radius of other forces given in equation [14]: the sphere, V velocity of the fall, and F resistance force. Ffp()A =− ∆ , Albert Einstein investigated the disturbances caused by a 12 2 ()µµ=−0 () − particle suspended in a flow with a constant velocity gra- FfKuu12 2ρ 1 (),12 dient [13,14]. He developed a theory of resistance to shear KKfuu()µµ=− ()(| |,µµµ,a ,...), ()5 for a suspension of small spherical particles in a continu- 21 2 12, du du ous fluid medium. He proved theoretically that an Ff()mm=−ρχ0 ()(),12 increase in viscosity of a fluid carrying solid particles is 12 2 2 dt dt connected to the volume fraction of the particles via a pro- Ff()r = ρ 0χ()r ()uu− ⋅ rotu, portionality coefficient: 12 2 1 12 1
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tion of new ones that are defined by value ψ in the equa- ()rr=−ρχ0 () ⋅ tion [14]: Ff12 2 1 () uurotu12 1, µ ∂N where F ()A is a buoyancy force, F () – frictional force +∇Nu ⋅ =ψ. ()7 12 12 ∂ 2 or Stokes force resulting from the viscous force during t interaction between phases, determined by the difference In case of the absence of crushing, adhesion, and aggrega- between velocities (slippage) u1 - u2, size – a, by the quan- tion of particles or formation of new ones, i.e. when ψ = tity and shape of inclusions, as well as by the physical 0, as well as under the condition of non-compressibility of χ properties of phases; p-pressure difference, (m) – coeffi- 3 ρ ρ the material (of particles), that is ρ = constant α then α cient of constraint, 1 and 2 are densities of the first and 2 a the second phase and are determined by the theory of is a constant. Moreover, the equation of conservation of µ multiphase medium as reduced densities, K( ) – coeffi- 0α mass of the second phase[14] leads to ρρα= 3N . cient of phase interaction, µ and µ – viscosity of the first 2 2 1 2 And N, number of particles per unit volume, can be and second phase (where first and second phases corre- expressed as [14]: spond to plasma and erythrocytes respectively), f2 – con- centration of the second phase, F ()m – force related to ∂∂+∇=ρα03 ρα03 12 2 aN/ t u22 aN 08() the effect of "added masses" and caused by the accelerated motion of inclusions (particles) relative to the carrying or [14] (viscous fluid) medium, when disturbances arise at a dis- ∂N/∂t + ∇Nu = 0. (9) ()r 2 tance on the order of the size of particles; F12 – force of additional effect on particles, due to gradients in the area It should be noted that in the case of a moderate volume of average velocities in the liquid phase (the Magnus or fraction of particles in a fluid, the mixture of two incom- Zhukovskiy force). This force is a result of the difference pressible phases cannot be considered in terms of Navier between the densities of phases and the difference – Stokes' equations, even in the absence of relative rotary between pressures on the opposite sides of a flowing and radial motion [14]. This is because viscosity of sus- particle. pensions and emulsions is determined using viscosity- meters based on the measurement of a mass consumption The influence of the shape of particles, their multiplicity, of mixture Q through a tube, with diameter d and length and some other factors in the expressions for forces L, dependent on pressure ∇P [16-18]. If a Poiseuile flow µ of a Newtonian fluid is the case, then a linear connection F (), F ()m , F ()r are taken into account in coeffi- 12 12 12 between Q and ∇P holds true. It follows, then, that with cients K(µ), χ(m), χ(r). As a first approximation, the data on low concentrations, when f2 < 0.05, the resulting values of precipitation of a single particle or of a body of a corre- µ ef agree with the Einstein formula (2), but when f2 > 0.05 sponding shape can be applied here, disregarding the µ then values ef exceed values obtained from the equation direct influence of particles on each other, but taking into [14]: account the constraint of the flow of fluid around particles that is caused by the multiplicity of particles [11,14,15]. 5 µµ=+().1 f ()10 ef 022 In order to account for the Magnus effect connected to ()r Additionally, there is considerable variation among differ- F12 , it is necessary to take into account the rotation of ent authors and among different combinations of phases. particles, and in the general case, also the corresponding This variation apparently reflects the non-Newtonian kinetic parameter ω , which represents the velocity of 2 nature of concentrated viscous disperse mixtures and rotation that is independent of the field u [14]: 2 insufficiency of values ρ and µ for determining their mechanical properties. In this regard, laboratory experi- = α 3 fNa2 , ()6 ments have to be carried out for each mixture and real devices over a range of operating parameters to determine where α – coefficient accounting for the shape of parti- the loss of pressure when applying different rheological models, in particular, the model of a viscous fluid with an α π cles ( = 4 /3 for spherical particles with radius a). In effective coefficient of viscosity. addition to convection, the value of N (the number of par- ticles per unit volume) may change, due to the processes ≥ One should keep in mind that when f2 0.1, in addition of crushing, adhesion, aggregation of particles, and forma- to the shape and size of solid particles, their material may
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also have an effect on effective viscosity and other rheo- Studies in this field usually deal with high- and low-con- logical characteristics of the mixture. Apparently, this is centration mixtures. The laws of constrained precipitation due to irregular distribution of particles, collision of par- of moderately concentrated mixtures have not been well ticles with each other, and solid walls [16-18]. investigated. The foundation of precipitation of a single particle in an infinite fluid is the Stokes law. According to The above considerations lead to an examination of group that law, the frictional force arising from the motion of ≥ precipitation of particles, when f2 0.1 in the interpene- spherical particles with diameter d and velocity V in a trating model of two- or multiphase medium. The first medium of viscosity µ is expressed by formula report of such an approach was presented in [11], where researchers used the theory of interpenetrating motion of ()M FaV= 611πµ . () two-phase mediums, taking into account the aggregation 12 of particles, in this case erythrocytes, in a fluid layer lim- The gravitational force acting on a particle depends on its ited by a free border x = L from the top and a solid border specific gravity, that is x = 0. In those studies, precipitation of particles in a fluid was considered based on the following assumptions: the ()A π Fd=−3(),ρρ g ()12 motion of particles in a fluid is taking place in the strictly 12 6 21 vertical direction under the influence of gravity; the iner- ρ ρ tial force is negligibly small; viscosity appears only during where 1; 2;g – the density of a fluid, density of the parti- interaction between particles and the fluid surrounding cle, and acceleration of gravity respectively. them; spontaneous disintegration is absent; the column of sedimentation of particles is divided into two main F ()A – buoyancy force, and, in (11) zones: clean fluid and settling particles. The influence of 12 the wall and interaction between particles were ignored. Studies [19,20] showed that the radius of the tube has an F()M – frictional force. effect on sedimentation of particles, in particular, erythro- 12 cytes. From this it follows that the group precipitation of ()A particles, such as erythrocytes, depends appreciably on Due to force F12 the particle accelerates its motion. border conditions, in this case on conditions of adhesion Aside from gravitational force, the particle is affected by and impact of particles on the vessel wall [19,20]. frictional force directed oppositely; its value increases with the increasing velocity, according to the Stokes law. In addition, in most studies the rotation of particles ()M ()r This means force F12 is canceled out by gravitational caused by force F12 was not taken into account, and ()A neither were the force of interaction between adjacent par- force F12 . Therefore, the movement proceeds with con- ()m stant velocity Vc that is determined by the equations (11) ticles or the force F12 arising from accelerated move- and (12): ment of particles. Thus, in the present work, we examine the group precipitation of particles in a liquid mixture ()ρρ−−g 2()ρρg ()r ()m V = 21 d2 = 21 a2. ()13 with forces F12 and F12 , as well determination of the c µ µ µ 18 9 constraint coefficient and of viscosity of the mixture m as functions of the volume fraction of particles. We also Sometimes one has to deal with precipitation of a large describe the effect of the vessel wall on the process of number of particles in concentrated mixtures. Formulae group precipitation of particles in a multiphase medium. for the velocity of precipitation of particles as a function of the concentration and velocity of a single particle in an Determination of the coefficient of constraint and infinite fluid can be derived using some statements of the viscosity of a mixture as a function of the volume fraction interpenetrating model [14] and the Euler equation [14]. of particles From these formulas, an equation for low concentrations Problems of constrained precipitation of solid and of particles can be easily deduced; the equation is consist- deformed particles in a fluid are the principal part of the ent with the experimental data [27]. Let us assume that in theory of two-phase flows and are of great importance for volume V there are two phases with different values of such hydrodynamic processes as: mass of exchange, sepa- specific gravity. Then particles with greater specific gravity ration by gravity, oil refining, and precipitation of eryth- start moving down canals that are being formed; this rocytes in blood [11,12,16-26]. holds true for the displacing phase, i.e. the process of mutual penetration is taking place.
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Let us determine the hydraulic radius of a canal as by S, and multiply both numerator and denominator by diameter d (in this case d corresponds to the diameter of a S single erythrocyte) and by volume V and also taking into r = .14() P account [21] that external concentration equals volume concentration. Then it follows: If the canal is cylindrical then V1 = f1V1i (19) π d2 d r ==. ()15 The flow of fluid between particles by analogy with the 44π d flow of fluid in tubes can be expressed as criterion equa- From (14) and (15) the effective diameter of the canal is tions [23]: equal to: n = m 1 () = 4S EARue , 20 de ,,()16 d P e where S – cross-section area of the canal, P – perimeter of or [23]: the canal. Multiplying both numerator and denominator by diameter d (in this case d corresponds to the diameter n ∆P 1 of a single erythrocyte) and by volume V and also taking = ARm , ()21 ρ 2 e into account [21] that external concentration equals vol- 11V de ume concentration, it follows: In the processes of precipitation when concentration of inclusions is significant and sizes of particles are small, = 4f1 () de ,,17 and in the processes of industrial filtration as well, a lam- Su inar flow is very often observed, for which m = -1, n = 1. where Su – surface of the particles; f1 – volume fraction of Then [23]: the first phase. But: ∆P Alµ = . ()22 1 3 ρ 2 ρ 2 fdN= π , 11V 1dVe 1 2 6 If one takes into account (18) and (19), the last equation (where N – number of particles per unit of volume equal can be expressed as Coseni Carman for constrained pre- 6f cipitation in a laminar flow: to 2 ), therefore (16), (17): π d3 9AVµ f 2 F = 12. ()23 4fd32 = 2 fd1 1 de . ()18 3 f2 Let's divide equation (23) by the number of particles per unit of volume and derive the resistance force of the fluid 2 acting on one particle as: (where is a coefficient of shape for spherical particles). 3 22 ∗ πρVd F = χ . ()24 For particles of irregular shape: 3 f1 (χ – resistance coefficient for precipitation of a set of = α fd1 de , particles). f2 It is known that the resistance force experienced by a par- α ( – form coefficient). ticle during precipitation in a fluid is:
We can divide the continuity equation: = χρ22 FVdccc . ()25 V1S = V1iS1 Since for particles suspended in a fluid:
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F* = Fc The difference between the coefficients of equation (30) and the coefficients calculated for small concentrations Then from (24) and (25) it follows: [22] is the factor of 0.34. If we do not ignore the fraction of velocity resulting from the proximity of two spheres, f 3 which is true for small concentrations of inclusions, then χ = 1 χ , ()26 equation (30) coincides with the Bachelor formula [22]. πβ 2 c The change in the velocity of particles depends on the con- where β – the ratio of velocity to hydraulic size; centration calculated from (29); a wide range of concen- trations is given in Figure 2. χ c – resistance coefficient for precipitation of a single par- → ticle (25) in an infinite fluid. From (24) when f1 1 it From the equation of equilibrium of forces, taking into follows: account the coefficient of constraint, it follows (13) [14]:
χ 9 − χ = c . fgρρ−+ fgfa µϕ2 () VV −= 031() π 22 122 21 f 12 when values of Reynolds numbers are small: 2 ()ρρ− g V = 21a2. ()32 c µ C 9 1 χ = , Re Based on (31) and (32) we have:
Therefore, it can be supposed that the basic mechanism is: V 1 2 = . ()33 V ϕ C χ cf χ =+c .27() Re π or taking into account (29) From the equations (26) and (27), we can derive: ϕ =+−2 2 3 f 134/,()cf2 f1 cf2 () 12/ π π 2 f22f 3 φ β =−33c + c + f , () 28( f – coefficient describing constrained precipitation). χ χ 1 Recc Re cc Assuming precipitation of a particle in suspension with where µ ρ viscosity m, density m, we can express the equilibrium equation [21] as = Vdc Rec . v 9 − fgfgfaVVρρ−+ µ2 () −= 035, () 22im 22 2 m12 According to the Stokes law, in a laminar flow we have: ρ = f ρ + f ρ . (36) π m 1 1i 2 2i χ = 3 c , Rec Allowing for (32) and V1 = 0, it follows figuring the last equation into equation (30), we can µ fV mc= 1 . ()37 derive µ 1 V2 12/ β =− + 2 () − 2 + 3 () and from (28) we can obtain cf2 c129 f1 f1 . µ 2 In accordance with our experimental data when Reynolds m =−fc 2 ()138 ffcf+−3 . () µ 1 1 1 2 numbers are small, constant C = 5. 1 When f → 1, but C = 2.5, we have the Einstein formula → 1 Equation (29) when f1 1 can be expressed as (10): β = 1 - cf2. (30) µ m =+139cf . () µ 2 1
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TheFigure relative 2 velocity of sedimentation as a function of the concentration of particles The relative velocity of sedimentation as a function of the concentration of particles. Horizontal axis: volume fraction of phase two; vertical axis: β, a change in the relative velocity.
From the calculation given in Figure 1, it follows that for- where V2 – velocity of constrained sedimentation of dis- mula (38) agrees with experimental data (up to f2 = 0.5, perse particles; when C = 2.5) obtained in a previous study [27] on the ≈ change of viscosity of suspension for a wide range of flu- Vc – velocity of sedimentation of a single particle (f2 0) ids, sizes, and materials of solid discrete particles. in a motionless fluid (V1 = 0);
The above dependence of relative velocity and viscosity of f2 – volume fraction of disperse phase f1 + f2 = 1; the mixture on volume fraction agrees with the experi- mental data [27]. Let us examine formulae (33) and (34), c – dimensionless numerical coefficient. and consider the possibility of determining an analytic calculation formula for the constraint coefficient and As is known[14], from the equilibrium equation of a two- effective viscosity, and their dependence on the concentra- phase medium, allowing for constraint motion of tion of particles. In (33) and (34) the formula for the rel- particles ative velocity of the precipitation of particles has been derived as: ∇++kkσρρ= 112()g 0
1 ()41 V 2 2 2 =−ca + c2 ()140 − f+ f 3 , () 2 1 1 9 −2 0 0 Vc µϕaVV()− +−() ρρ fg= 0 2 1 j 12 2 1
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TheFigure dependence 1 of a change in relative viscosity on the concentration of particles The dependence of a change in relative viscosity on the concentration of particles. Horizontal axis: volume fraction of solids; vertical axis: viscosity (mNsm-2)
We can derive an equilibrium equation of gravitational force, buoyancy force, and Stokes force acting on the dis- f VV= 1 . ()43 perse phase as: 2 c ϕ j
9 − From the equations (40) and (43), the constraint coeffi- ρρ0 fg−+0 fg µϕ a2 f() V − V = 0, 2 21212 j 21 2 cient can be determined as: ()42 f += ϕ = 1 . ()44 fV11 fV 2 2 0. j 1 2 ()− 2 + 3 2 − cff1 1 1 cf2 The parameters correspond to the parameters in the study [14]. The second equation (42) describes a rest condition From the analysis of the experimental data for small vol- of the mixture; that is, the motion of dispersed particles in ume fractions (f2 < 0.005), it follows that there are three one direction accompanies the motion of fluid in the types of functions [14] for the determination of a opposite direction. constraint coefficient corresponding to different types flu- ids in which spherical particles are precipitating: From the equation (42) for precipitation of a single parti- cle (f ≈ 0) in (φ = 0) in the at-rest mixture, we can derive f 0 j ϕ =− 1 ,..k ≈÷13 18() 45 the Stokes formula, however the velocity of constrained f 1 1 ()− 2 precipitation is: 1 kf12
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f f µ ϕ = 1 , k ≈÷56() 46 µ =− 11 ,..k ≈÷13 18 f − 2 M 1 1 1 kf22 − 2 ()1 kf12 f µ f µ = 11 , k ≈÷56() 50 ϕ = 1 ,.k ≈÷006() 47 M − 2 f + 3 1 kf22 1 kf32 f µ µ = 1 1 ,.k ≈÷006 Comparing equations (44) – (47), we can conclude that M 1+ kf 3 formula (44) deduced from [28] is similar to formula 32 We can summarize here: equation (45) has been derived (46). from the consideration of the cell model for uniform dis- tribution of particles; equation (46) deals with the ran- / 2 13 dom allocation of particles; and equation (47) is the case When cff2 ()11− + 3 ≈ and c ≈ 5 ÷ 6; that is, 1 1 of significant formation of clumps and alignment of par- under conditions (44), a change of constraint coefficient ticles in chains [14]. Therefore, equations (50) allow one is described, depending on the volume fraction of ran- to determine viscosity of a mixture for all three types of domly dispersed particles. As described previously [14], distribution of particles. Thus, the formulae for calcula- when the concentration of the disperse phase in a mixture tion of the constraint coefficient and viscosity of a mixture increases, numerical values of volume and shear viscosity as functions of the volume fraction of the disperse phase start to differ from each other, and the motion of the mix- have been fully described. ture cannot be described by Navier – Stokes equations [14]. Forces in a two-phase medium that affect group precipitation, in particular, precipitation of erythrocytes
When concentrations (f2 < 0.005) and values of viscosity in blood agree with the Einstein formula, and when concentrations As follows from the above, there can be four basic inter- determined in the laboratory are high, then the viscosities acting forces when particles are precipitating in a mixture: considerably exceed the values derived from the formula for moderate and concentrated viscous disperse mixtures 1. The buoyancy force occurring due to the difference of non-Newtonian nature. In addition, the rheological between densities of phases; parameters are insufficient for the determination of the mechanical properties of those moving mixtures. 2. The frictional force or Stokes force; it results from inter- action between the two phases; In particular, one of the models of viscous fluid is a model taking into account the effective coefficient of viscosity, as 3. The force of "added masses" arising from either acceler- an additional rheology parameter. Moreover, it is ated or constrained motion of particles; assumed that the effective coefficient of viscosity (viscos- ity of a mixture) reflects not only a shape and size of solid 4. The force of additional effect on particles which is due particles, but also their material and irregularity of their to gradients in the area of average velocities in the liquid center of gravity. phase (the Magnus or Zhukovski force).
Following the approach described in[14] for precipitation These forces should be taken into consideration when cre- µ of particles in a fluid with viscosity m, we can derive the ating a mathematical model of group precipitation in a following equality multiphase medium.
V µ The first force, buoyancy force, is defined by formula: 2 = f 1 . ()48 1 µ Vcm FfP()A =− ∆ . ()51 Using equation (48) and allowing for equation (43) [28], 12 2 it follows that where f2 – volume fraction of the second phase (particles).
µ φ µ ∆ m = f 1 (49) P – gradient of pressure, or the difference of densities of phases. We can also substitute equation (44) into equation (49), and from the equations (45)–(47) and (49) we can derive The second force, frictional force, or Stokes force, is [28] defined by the difference of velocities (slippage) |V1 - V2|,
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the size, quantity and shape of inclusions, as well as phys- f1V1 + f2V2 = 0 (60) ical properties of the phases. From [11] we can write down the following equation for the frictional force (5), (6): Which means that in each point, the volume flow of the precipitating solid phase accompanies upward motion of 1 the fluid. This phenomenon has been considered in the ()µ 3 = αµf2 ψ ()()− ()previous paragraph. The following is the formulae for FN12 1 fVV212. 52 N forces of the additional effect on particles, caused by gra- µ φ Instead of viscosity of plasma 1 (f2) we can use the vis- dients in the area of average velocities in the liquid phase: cosity of the mixture determined in formula (50) rr=−ρχ0 () − × Ff12 2 1 () VVrotV12 1 ()61 f χ = 1 k ≈÷13.. 18() 53 m 1 1 In addition, coagulation or aggregation of multiple parti- ()− 2 1 kf21 , cles can take place, which is described by these equations (7), (8), (9) [14]: f χ = 1 , k ≈÷56() 54 m 1− kf 2 dN r 22 +=NdivV ψ ,62() dt f χ = 1 ,.k ≈÷006() 55 Where N – number of units, ψ – total velocity of aggrega- m + 3 1 kf32 tion [14]. Expression (53) has been derived while applying the cell ψ = KN2, (63) model for uniform distribution of particles, expression (54) deals with irregular distribution of particles, and where k – aggregation coefficient. expression (55) is the case of significant formation of aggregations and alignment of particles in chains. There- Based on the above, the mathematical model for precipi- fore, from the equations (54) and (55), one can deter- tation of particles in a two-phase medium can be mine the viscosity of the mixture for all three types of formulated. distribution of particles in a fluid. Instead of function ψ ψ ()= −25. (f2) from [11] we can use ff22 . The mathematical model for group precipitation in a two- phase medium (mixture) in a flat tube The third force: the force is related to the action of "added Let us consider precipitation of particles in a fluid on the masses," i.e. this force arises from either constrained or basis of the hydrodynamic theory. Experiments on precip- accelerated motion of particles: itation of erythrocytes in plasma carried out in cylindrical tubes of various diameters have shown that the radius of the tube significantly affects the process of precipitation ()m dV dV Ffm =−ρχ0 12 − ()56 [29]. 12 2 2 dt dt where constraint coefficient χ(m) from formulae (44), It follows then that group precipitation of particles such as (45) and (46) can be defined as [14]: erythrocytes, depends on border conditions, in this case the conditions of adhesion. The small radius of a tube f defines the area of velocities of flow of the liquid phase χ = 1 k ≈÷13.. 18() 57 m 1 1 during precipitation of a particle. Therefore, particles − 2 ()1 kf21 , experience a differential of pressure from the surrounding fluid. That is why, in this study, the fluid (carrying phase) f is considered a viscous phase. χ = 1 , k ≈÷56() 58 m 1− kf 2 22 Let us consider precipitation of a particle taking place between vertical parallel walls. We will place X axis f χ = 1 ..k ≈÷006() 59 between the walls pointing upward, Y axis also parallel m + 3 1 kf32 and equidistant from the walls but horizontal, and Z axis is horizontal and perpendicular to the walls. Let us Horizontal movement of particles under the influence of assume that the horizontal length of the tube is 2 h, and gravitational force can be explained, if one considers the its height is L. We will also assume that the motion is not flow of precipitating particles. In a homogeneous mixture taking place along Z axis. This leaves us with two-dimen- with a zero average flow rate [24],
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sional movement of particles. Group precipitation of the or for specific cases particles in the mixture can be described mathematically as the motion of two-phase interpenetrating mediums µ µ = f11 ≈÷ m ,..k1 13 18() 71 according to the theory of Rakhmatullin [21]. − 12/ ()1 kf12 It follows that equations for the two-phase flow in a Car- tesian space coordinate system can be expressed as [30]: f µ µ = 11 , k ≈÷56() 72 m − 1 ()1 kf22 ∂ ∂ ∂ G ∂ ∂ ∂ ∂ ∂ ∂ () ρµdU1 =− P + U1 − 2 + µV1 + V1 + µρW11+ U ++x 1 f111f 2 divV111 f f11 FX11, dt ∂xx∂ ∂x 3 ∂y ∂x ∂yz ∂ ∂x ∂z 12
dV ∂P ∂ ∂V 2 G ∂ ∂W ∂V ∂ ∂V ∂V ()y ρ 1 = − f + f µµ 2 1 − divV + f 11+ + f µρ 11+ ++FY, ()64 1 111∂ ∂ ∂ 111∂ ∂ ∂ ∂ 11 ∂ ∂ 12 11 µ dt xy y 3 z y z x y x µ = f11 ≈÷ () dW ∂P ∂ ∂W 2 G ∂ ∂U ∂W ∂ ∂V ∂W ()z ,.k 006 73 ρ 1 =−f + f µµ2 1 − divV + f 11+ + f µρ 11+ ++FZ. m 1 1 1 ∂ ∂ 11 ∂ 111 ∂ ∂ ∂ ∂ 11 ∂ ∂ 12 11 + dt zz z 3 x z x y z y ()1 kf32 for the second phase [30]: µ Where 1 – viscosity of plasma. dU ∂P ()x We can write down the continuity equation as ρρ2 =−f −+FX, 2 dt 2 ∂x 12 22 ∂ () ∂ρρ∂ u ρρdV2 =− P −+y () 111+ = 074, () 2 f2 FY12 22, 65 ∂ ∂ dt ∂y t x ∂ () ρ ddW2 =− P −+z ρ 2 f2 FZ22, ∂ρρ∂ u dt ∂y 12 222+ = 075, () ∂t ∂x Continuity equations for each phase are expressed as [30]: The equation of numerical concentration then becomes: ρρ∂ ∂ ρ∂ ρ d 1111111U V W + + + = 0, ∂N ∂Nu dt ∂x ∂y ∂z + 2 = ψ , ()76 ()66 ∂t ∂x dρρ∂ U ∂ ρV ∂ ρW 222222+ + + 2 = 0. dt ∂x ∂y ∂z where N – number of aggregates, ψ – total velocity of aggregate formation. G ∂U ∂V ∂W divV = 11+ + 1, ()67 Three forces determine the force of interaction between 1 ∂ ∂ ∂ x y z phases. and the equation of numerical concentration is then [14] ()µ () () =++mr () FF12 12 F 12 F 12 . 77 ∂N ∂NU + 2 = ψ ()68 The frictional force can be derived from formula (52) ∂t ∂x 13/ Assuming that the components of velocity V and W are ()µ f FN= αµ 2 ψ ()fuu()− , ()78 negligibly small compared to component U, and that the 12 m N 212 horizontal length of the canal is negligibly small com- pared to the height, that is, V <<> U, W <<> U and L Ŭ 2h, which takes into account the number of aggregates N. we can then estimate some terms and simplify the system Equation (78) also allows an investigator to take into of equations ; instead of U , U we will use u and u account the changes in the number of particles per unit of 1 2 1 2 volume, the average distance between them, the radius of
du ∂P 4 ∂ ∂u ∂ ∂u particles, volume of the disperse phase, etc. These geomet- ρµµρ1 =−f + ()(),f 1 + f 1 −+Fg 1 dt 11∂xx3 ∂ mm∂xx∂ 1 ∂x 12 1 rical parameters have a significant effect on the kinematics ()69 ddu ∂P ρ 2 =−f ++Fgρ , of the aggregation processes. 2 dt 2122∂x µ From [11] we can derive the expression for the total veloc- where m – effective coefficient of viscosity of the mixture that is derived from the formula: ity of aggregate formation φ = -KN2. (79) µ f m = 1 ()70 µ 12/ 1 2 ()− 2 + 3 − cff1 1 2 cf2
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ϕ = −25. At t = 0 u1 = 0, u2 = 0, (89) From expression (78) when ()ff22 , and based on the formula from [31] we can obtain 3 (N0 is the number of particles in 1 mm , a – the radius of a particle) 13/ ()µ = αµ f2 − FNm fu21() u 2, ()80 at t>=====000 and at XLu ,, u P P 12 1 2 amm N === at X000 u12, u ()90 at y=± h u =0 here α – coefficient of shape for the particles in question. 1 This problem can be solved numerically. We will use the following initial data: ()m Instead of the force of "added masses" F12 we can use =⋅−−4 = = =22 = formula (56), namely a5 10,.,.,.,./ f10 0 55 f 20 0 45µ 0 1 5 mNcm g 9 81 cm s 0 = 3 03== ρ1 10. 3310820191gcm/,ρ2 ./, gcm h ., cm () ==9 ()mm () du du Lcm10 , α for sphericall particles Ff=−ρχ0 12, ()81 2 12 22 dt dt Analysis of the experimental and numerical results and where χ(m) – coefficient of constraint, which is derived their comparison from one of the following formulae Figure 3 shows the vertical distribution of concentrations of the disperse phase at various time intervals. More pre- () χ m = f1 ≈÷ cisely, it shows the concentration at various points along ,..k1 13 18() 82 − 12/ X axis at various intervals of time. In figures 3, different ()1 kf12 curves correspond to the distribution of concentration in the tube after t = 5 min, t = 25 min, t = 60 min from the ()m f χ = 1 , k ≈÷56() 83 beginning of the process. It was noted that the change of − 1 1 kf22 concentration of the disperse phase along the height of the tube at various intervals of time, corresponds to the () χ m = f1 ≈÷ decrease of volume fraction of the disperse phase in the ,..k1 13 18() 84 − 12/ upper part and its increase in the lower part of the tube. ()1 kf12 Moreover, aggregation of particles reinforces the precipita- The Magnus or Zhukovski force is defined by the tion of erythrocytes. expression As one can see, when aggregation coefficient κ is 10-5 the Ff()r =−ρχ0 ()r () uurotu. ()85 effect is observed, if we compare the curves in figure 3. It 12 21 12 1 should be noted that the wall layer has a considerable Therefore, the closed equation system to solve the prob- effect on precipitation of erythrocytes. Since particles lem has been obtained: bounce off the walls, the aggregation of particles in the central part of the tube is enhanced. As a result, a big du ∂P 4 ∂ ∂u ∂ ∂u ()µ ()mmr () ρµµ1 =−f + ()()f 1 + f 1 −−FF−+Fgρ , aggregate of particles precipitates faster than a single par- 1 dt 11∂xx3 ∂ mm∂xy∂ 1 ∂x 12 12 12 1 ()86 du ∂P ()µ ()mr () ticle. This effect was described in the theoretical studies, as ρρ2 =−f ++++FFF g. 2 dt 2 ∂x 12 12 12 2 well as shown experimentally. In order to determine the Using equations (74) and (75) we can obtain influence of the vessel wall, we placed the blood of a patient in capillaries of various diameters. During precip- ∂ itation, erythrocytes descended 18 mm after one hour. ()fu+ fu = 087. () ∂x 11 2 2 R Since the precipitating particles cannot flow through the When = 015. they descended 11 mm, but when h bottom of the tube, the following holds true: R = 002. by 6 mm. f1u1 + f2u2 = 0. (88) h
Equation (88) gives us the rest condition of the mixture. That means that the influence of the vessel wall on precip- Let us use the force of interaction between phases given in itation of erythrocytes is considerable in capillary tubes of [11] to derive the closed equation system. First, it is neces- lesser diameter. Also, according to figure 3, the curves of sary to formulate initial and border conditions. precipitation are different at various time intervals.
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CurvesFigure of3 the change of concentration of the disperse phase along the height of the tube at various moments of time Curves of the change of concentration of the disperse phase along the height of the tube at various moments of time. Curves 1 and 2 when t = 60 min, 3 and 4 when t = 25 min, 5 and 6 when t = 5 min. Curves 1, 3 and 5 correspond to precipitation with- out border conditions, and 2, 4 and 6 with border conditions. Aggregation coefficient K = 10-5 for all curves. Horizontal axis: concentration of the second phase (f2); vertical axis: distance (millimeters)
Figure 4 shows the curve of precipitation of particles as a described by means of a two-velocity interpenetrating, function of time, for various values of aggregation coeffi- two-phase (or multiphase) model. cient K, both in the theoretical (curves 1–3, 5, 6) and experimental studies (curve 4). According to Figure 4, the- In general, ESR has a complex mechanism where, on the oretical curves are consistent with the experimental data. one hand, an increase in the concentration of erythrocytes leads to a decrease in β, the change in the relative velocity Discussion of sedimentation (figure 2). On the other hand, such an This work addresses the theoretical and experimental increase may result in an increase in the quantity of investigation of single and group precipitation of particles rouleaux. Our mathematical model takes into account the in homogeneous and heterogeneous (multiphase) influence of the vessel wall on group precipitation of par- medium, as it relates to their internal structure (coagula- ticles in tubes, and also viscosity of the mixture, con- tion or aggregation of solid and/or deformed particles). straint, and rotation of particles. This model can also Such processes cannot be described by means of a single describe ESR, using the coefficient of the velocity of eryth- velocity model of a non-Newtonian fluid. Precipitation of rocyte precipitation. particles in a viscous homogeneous relaxing medium and the motion of flow of the fluid are directed oppositely, As mentioned earlier, ESR measurement remains the due to the gravitational force. These processes can be method of choice in evaluating different clinical condi- tions [5]. In our view, the ESR could also be useful in
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ofFigureCurves aggregation of4 the changeare different in the borders of separation between fluid and mixture as a function of time, when values of total velocity Curves of the change in the borders of separation between fluid and mixture as a function of time, when values of total velocity of aggregation are different. k : 1 - k = 5·10-7; 2 - k = 1·10-6; 3 - k = 5·10-7; 4-experimental curve corresponds to rheumatism; 5 - k = 1·10-5; 6 - k = 5·10-5. Dotted curves match the results of equations (86), with the border conditions (89) and (90). Hori- zontal axis: time (minutes); vertical axis: the height of the tube, millimeters.
patients with the traumatic shock and traumatic crush traumatic shock [36]. The analysis of the certain rheolog- syndrome. In general, conditions such as hemorrhage, ical changes caused by the experimental traumatic shock trauma, and burns may result in blood viscosity changes was conducted by Tatarishvili et al [35]. In total, 40 adult [32]. Erythrocyte aggregation caused by trauma, burns, male laboratory rats were used, of which 20 were used for and hemorrhage was described in detail by Knisely [33]. experiments and 20 served as controls [35]. Kennon's method was used to produce traumatic shock. Erythrocyte In the 21st century, at the time of high-speed transporta- aggregability was measured using the "Georgian tion and motor-vehicle accidents, critical states, such as technique"[37]. Traumatic shock resulted in significant traumatic shock, have become more widespread. Trau- changes in all investigated blood rheological parameters matic shock results in generalized vasoconstriction, and (erythrocyte aggregability, deformation, and systemic therefore, this condition may result in microcirculatory hematocrit). The experimental group of rats showed an disorders that have been associated with increased blood increase in erythrocyte aggregability, which was more viscosity [34,35]. Traumatic shock, due to increased blood than 2 times higher compared to control animals. How- viscosity, may further decrease the cardiac output [34]. ever, the hematocrit and erythrocyte deformability decreased in experimental rats compared to control ani- Theoretically, any trauma may result in extensive intravas- mals [37]. Despite the small sample size, these results cular erythrocyte aggregation, irrespective of the presence were statistically significant [37]. Tatarishvili et al. con- or absence of traumatic shock [36]. However, generalized cluded that "the blood rheological disorders in the micro- intravascular blood slowing and stasis, as well as erythro- circulation related in the present experiments should be cyte aggregation, is much more severe in the presence of considered as a significant factor determining the severity
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of the damage to the tissues during the development of decrease in blood viscosity [41,42]. The damage of parti- the traumatic shock" [37]. cles at different erythrocyte adhesiveness within rouleaux has been studied in the previous research [47]. From those As mentioned earlier, the ESR measurement might also experiments it follows that the damage is proportional to become a useful diagnostic test in patients with crush syn- yield velocity [46]. On the other hand, an increased ESR drome. Disasters, such as earthquakes, are a key source of in patients with traumatic shock and crush syndrome is victims with crush syndrome. Earthquakes in Armenia in due to an increased number of rouleaux, however, as it 1988 [38] and in Japan in 1995 have caused multiple has been shown earlier, the quantity of rouleaux is largely cases of traumatic crush syndrome. [39] Victims of such dependent on the quantity of erythrocytes [47]. Going earthquakes have been trapped for prolonged periods. back to the beginning of the discussion, those rats that The other possibility for the crush injury and crush syn- had lower survival (second group) had, according to our drome is, for example, the severe and transient pressure hypothesis, a larger quantity of rouleaux formation, due that occurs when a limb is run over by a heavy vehicle. to an initially larger quantity of erythrocytes [41,42]. Finally, crush syndrome may also affect patients with stroke or drug overdose [40]. Such patients could crush a Conclusion part of the body with their own weight while uncon- The mathematical model created in our study accounts for scious. The lower limbs and, less frequently, upper limbs the influence of the vessel wall on group precipitation of are predominant sites of injury in victims with traumatic particles in tubes, viscosity of the mixture, constraint and crush syndrome. Traumatic crush syndrome caused by a rotation of particles. The determined viscosity of a mixture crush injury to the torso has also been described [39]. and constraint of particles depend on volume fraction. In Rheological changes caused by traumatic crush syndrome theoretical studies, the difference of curves of precipita- have been studied in rats that were exposed to 6 hours of tion, during the initial and later intervals of time, has been compression of soft tissues on the thigh. Narcotized rats established. with crush syndrome were shown to have increased blood viscosity. In these animals, total peripheral vascular The theoretical curves of precipitation of particles at vari- resistance was shown to correlate well with changes in ous values of aggregation coefficient K show that an blood viscosity at different shear rates. [41]. Rats' blood increase of K increases the velocity of precipitation. Like- viscosity was predominantly determined by erythrocyte wise, the presence of the tube wall enhances precipitation aggregation at low shear. [41] These rats were also divided of particles. The Magnus force and the force connected to into two groups: low resistant and high resistant to shock. acceleration of particles in a relatively liquid phase do not Low resistant animals exhibited high hematocrit and have a significant effect on the precipitation of particles. plasma viscosity and decreased deformability of erythro- From our experimental and theoretical data, we can con- cytes. On the contrary, blood viscosity increased, inde- clude that the behavior of the curve of velocity of erythro- pendent of shear rates (10 – 300 sec -1). Severe cytes depends on the timing of observations (interval of hemoconcentration and blood hyperviscosity developed time is 5 min). Time-course measurements of erythrocyte in all low resistant animals. High resistant rats developed precipitation more accurately reflect the state of the sufficient hemodynamics and oxygen supply to tissues. human body than the known determination of velocity of On the other hand, high resistant rats exhibited less signif- erythrocytes. icant hemoconcentration and an increase in blood viscos- ity [42]. Determination of the ESR should be conducted at certain intervals of time, i.e. in a series of periods not exceeding 5 According to our current research, increased ESR in minutes each. Differential diagnosis of various pathologi- patients with traumatic shock and crush syndrome is due cal conditions when ESR is applied can be conducted to an increased number of rouleaux. Our understanding using the aforementioned timed method of determining of changes in the ESR in traumatic shock or in crush syn- ESR. Different pathological states, when this method is drome could be summarized as follows: crush syndrome used, would have different initial blood viscosity parame- or traumatic shock (due to hemorrhage) may result in an ters. We have shown that an increase in blood viscosity increase in peripheral resistance[34,43-45]. Increased during trauma results from an increase in rouleaux forma- peripheral resistance may result in a decrease in cardiac tion. We have shown that the time-course method of ESR output, which subsequently leads to a decrease in blood will be relevant in patients with trauma, in particular, with flow velocity and, thus, a decrease in yield velocity and traumatic shock and crush syndrome. Therefore the shear stress[46] A decrease in blood flow velocity, due to number of rouleaux is dependent on the initial quantity trauma, may result in a decrease of erythrocyte adhesive- of erythrocytes and is increasing sharply with the decreas- ness within rouleaux [46], since even a relatively high ing yield velocity [47] which may result from increased yield velocity (300 m-1) did not result in a significant
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