Description of the nonlinear viscoelastic and thixotropic properties of human blood by the modified model of Maxwell-Gurevich-Rabinovich

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1 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Description of the nonlinear viscoelastic and thixotropic properties of huan blood by the odified odel of Maxwell-urevich-Rabinovich L. Hadjikov, N. Antonova, Ya. Ivanov Institute of Mechanics, Bulgarian Acadey of Sciences, Acad.. Bonchev street, bl.4, 3 Sofia, Bulgaria Abstract: Modeling by eans of eploying the generalized and odified Maxwell-urevich- Rabinovich (MR) odel, presented in this study, is based on our experiental data found using Low Shear 3 sinus Contraves rotational viscoeter at different regies of blood shear flow: steady and unsteady transient and sinusoidal flow. The rheological behaviour of the huan blood is characterized by pseudoplasticity, thixotropy and viscoelasticity. According to the MR odel the full strain rate at shear is a superposition of the elastic strain rate, the residual strain rate and the high elastic strain rate. The elastic odulus of shear, the initial coefficient of viscosity and the logarithic odulus of the strain rate link,,, in functional dependences and characterize the rheological behaviour of blood at the considered flow regies. Introducing nonlinear dependences of the logarithic strain odulus on shear rates for a detailed description of the observed relationships soe odifications of the MR odel are proposed. The paraeters of the proposed odels are found fro experiental data and used for blood icro structuring analysis. Key words: blood thixotropy and viscoelasticity, Maxwell-urevich-Rabinovich (MR) odel, initial viscosity coefficient, logarithic strain odulus, elastic shear odulus. Introduction Alterations of blood flow with tie is typical for thixotropic fluids and this phenoenon can be related to the influence of the inner structure on the blood flow. In the case of thixotropy when flow stopps, particles connect with each other and the process of destroying and creating new connections is well balanced, while forces of connection are weak. Higher shear rate leads to decrease of the size of the related structural units, corresponding to a less resistance to flow. When the fluid is with larger shear rates, ore acroolecules are destroyed for a unit tie. Consequently, thixotropy and viscoelasticity are related to siilar tie effects, reflecting shear rate effects. Fro this point of view theories are developed, describing the echaniss of thixotropy and viscoelasticity by eans of unified odels. The generalized odel of Maxwell- urevich-rabinovich (MR) for polyer solutions, deduced fro the thero-fluctuation theory by 66

2 Rabinovich [], has been applied in this study to odel the rheological properties of huan blood. The odel is characterized by two constants - initial viscosity coefficient and logarithic strain odulus. The observed experiental dependences for blood rheological properties at three different regies of viscoetric shear flow: steady shear, tie-dependent flow at rectangular and triangular change of shear rates and sinusoidal oscillation flow presented previously [2], are described by the MR odel. For description of the experiental relationships the MR odel is odified, including shear rate L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 dependence of its initial viscosity coefficient and taking into account the shear rate effect for any specific shear regie and range. 2. Materials and ethods 2.. Materials and experiental ethods 3-4 l blood was drawn fro a cubital vein between 8 and a.. and heparinized (5 IU/l blood ). Saples of -l were easured for up to 5 hours after blood saple preparation at 37 C keeping their native heatocrit, for shear rates ranging fro.237 to 28.5 s -. Estiations of the heatocrit of the sae saples were done, using icroheatocrit centrifuge TH2 and perforing centrifugation for 5 in at in -. Noral blood conserved with L-2 ( 3 l blood/75 l L-2 ) and a group of 22 patients (4 woen, 8 en, ean age 58,6 years) with cerebrovascular disease (CVD) were investigated. Blood saple fro a patient (feale) with chronic opioid addicts under ethadone aintenance therapy (DA) was chosen fro a group of 5 chronic opioid addicts under ethadone aintenance therapy (DA) (3 woen, 2 en, ean age ± 7.34 years). Blood was provided by the National Center for Addictions in Sofia. In addition, noral huan heparinized blood taken fro two healthy persons was centrifuged and its heatocrit was adjusted (reconstituted) within a range of 3% to 7%, i.e. to values of 3%, 4%, 5%, 6% and 7%. Rheogras of change of the shear stresses () with tie at a triangular change of shear rates (. ) were plotted by Rikadenki plotter. Hysteresis was drawn at a triangular change of shear rates t for <t<t - loading and 2t t for t <t<2t relaxation, where a). 596s and t =3s; b);. 8s and t =3s. The steady and dynaic viscosity dependence on shear rate was deterined using a rotational viscoeter with a firly set aplitude of 3 Low Shear 3 sinus (Contraves, Switzerland). All easureents were carried out by a MS / easuring syste at teperature ( T = 37, C ). The test substance was located in the gap between the rotating easuring cup and the stationary easuring bob. A Low Shear 3 sinus can shifted a rigidly ounted additional driving device (oscillation drive) fro oscillation to rotation. While easureents took place the test substance was subjected to an exactly defined frequency, thus providing forced oscillations. Apparent steady shear viscosity was deterined starting fro a shear rate of 28,5 s - and decreasing the rate in steps. Coplex dynaic viscosity was deterined by easuring the angular displaceent varying frequency fro,222 Hz up to,63 Hz at a firly set aplitude of 3. Low Shear 3 sinus allowed to register loss angle and torque M VE, whereas M V (or viscosity V respectively) and M E (or storage odulus respectively) were calculated. Sinusoidal tie-varying flow provided a basis for clear differentiation of the elastic and viscous properties of the aterial and understanding the role of viscoelasticity in a ore coplex tie-varying flow, such as pulsatile flow. 67

3 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Blood saples were provided by the National Center for Transfusion and Heatology and by the Departent of Neurology, Medical University in Sofia Theoretical ethod The generalized Maxwell-urevich-Rabinovich (MR) odel for polyers and etals, presented by Rabinovich in [] is used herein to describe the non-newtonian rheological properties of huan blood, observed in our experients. The MR odel has been deduced fro the thero-fluctuation theory. According to this odel the rate of the full deforation at shear is equal to: e () where are shear rates or full strain rate; e is the elastic strain rate; is the residual strain rate; is the high elastic strain rate and d ; (2) dt where is the shear elastic odulus and exp (3) is the initial coefficient of viscosity for the residual starin; is the logarithic odulus of the residual shear strain rate; f exp (4) 68

4 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 where is the initial viscosity coefficient for the high elastic strain; is the logarithic odulus for the high elastic shear strain rate; f is a function of the shear elastic odulus, the equilibriu shear odulus of the high elastic strain, shear stresses and strains, deterined by: f (5) 2.3. For the regie of steady shear and non-steady tie dependent viscoetric flow ( const )we consider two shear rate regions: a) <,3 s - and b) >,3 s - In the case of very low shear rates or region a) <,3 s - it follows fro our experients that the full strain rate consists of elastic and the high elastic strain rate, having the for: e (6) or d dt When =const and d f exp dt d dt the equation for the shear stress rate has the for: (7) d dt f f where, and f is the iu of f fro (5) and it is found fro experiental data. To facilitate paraeter calculation when perforing integration by eans of the finite difference ethod, we introduce functions: exp(, ) ) The initial conditions in this case are: exp( t. =, =, f (9) 69 (8)

5 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Taking into account that = const and t, the equation for takes the for: d d f exp f or it can be written in the for: d d () () with initial conditions: t. =, =, f (2) It follows fro the experiental data, observed at shear rates, 237 s - or in case b), that the high elastic strain and the full strain rate are deterined by the elastic rate and the residual strain. In this case = and f f and equation () takes the for: e (3) Then d dt d exp dt (4) or d dt, (5) where:, and exp( ), exp( ) are functions, introduced to facilitate paraeter calculation when perforing integration by eans of the finite difference ethod. 7

6 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 d d exp or it is equal to: d d (6) (7) When d, then the yield shear stress is found fro (7). d f f exp (8) exp (9) Using the MR odel we thus set forth relationship (- ) via equations (2) and (7) and relationship (-t) via equations (8) and (9). They are integrated using the ethod of the finite differences to find the odel coefficients fro experiental data Modeling of the tie dependent properties of blood Experiental results reveal that blood exhibits tie dependent rheological behavior. When,3 s we consider and integrate equations (8) and (7) with initial conditions (9), whose finite difference for is: i i i t (2) Where L. 7

7 72 Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 / i i t i (2) For area b) or shear rates (strain rate),3 s equation (5) has the following finite difference for: i i i t, (22) i i (23) 2.5. When MR odel () is reduced to the classical Maxwell equation (24): d d (24) When the initial conditions are =, t, then the solution of (24) has the for: exp exp (25) Equation (25) is the linear case of the MR odel and it describes the relationship,, f. Shear stress-deforation relations (-) experientally found by Picart et al [4] and shown in Fig.3, are described by equation (25). The linear MR odel is used to odel stress relaxation after flow cessation to and at a rectangular change of shear rates in the area a),3 s. Its for is: pt a a exp (26) where p ; a c (27)

8 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 In area b) at,3 s, the odel for shear stress relaxation at the sae regie of shear rates, after flow cessation to, has the for: exp( pt),p (28) 2.6. Dependence of blood dynaic viscosity on shear rates is odeled by the odified MR odel (9). The logarithic strain odulus is constant. const in aterials with stable structure, opposite to aterials with varying structure. In this case equation (9) takes the for: n exp (29) To account for the shear rate effect on the fluid internal structure, we introduce function b k a (3) where, a, b, k are constants, found fro the experiental data. Experiental data found by eans of steady, tie dependent and dynaic easureents are assessed by the above presented MR theory and equations for different shear flow regies. eneral solutions t of the Maxwell-ureich-Rabinovich type equation are discussed in the case of stress relaxation (after cessation of the steady flow) and stress foration (under constant shear rate), as well as in the case of hysteresis Coputational ethods Data analysis was perfored using the software packages Jandel Scientific and Siga Plot and a rheological software specially developed for that purpose. The ethod of finite differences was used to integrate the equations odeling blood rheological properties. 3. Results 3.. Results of odeling the tie-dependent properties of blood at rectangular change of shear rates Experientally established tie dependent properties of blood at rectangular change of shear rates confir the conclusion, that blood exhibits viscoelastic behavior, which is shown in Fig.a. The 73

9 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 initial destroying and further pre-structuring of blood in rouleaux at shear rates up to.237 s - causes shear stress increase in tie, which reveals elastic behaviour of the blood saple. The process of destroying and creating new connections is not well balanced. It leads to increase of the size of the related structural units, corresponding to a bigger resistance to flow at very low shear rates. Coparison of the experiental with theoretical data obtained by using the odified MR odel (2)-(2) is shown in Fig.a. The coefficients of equation (2), found fro stress developent experiental data (Fig.a) are: 3,292 Pa, f =,728 Pa, 8, 2 Pa,, 88 Pa,, 232 and t 2,5364 s. Stress relaxation experiental data, shown in the sae figure, are fitted by equations (26) and (27). The deviation of the theoretical approxiation fro the experiental data is up to 25% at large relaxation ties, but close to % at sall relaxation ties. The coefficients of the odel (26)-(27), derived fro the experiental data are: 9, 763 Pa.s, C, 766 (deforation, at which flow has been terinated), 8, 2 Pa,, 57 Pa. Modeling of blood thixotropic properties at a rectangular change of shear rates is described in detail and presented in [3]. Fig.a. Stress growth (fro =, to =237 s - ) odeled by eq. (2) and stress relaxation - by (26)-(27) for the whole huan blood (DA), H=43%, T=37 C; experiental points - - exp, theoretical lines- - M. Fro [3] Fig.b. Stress growth (fro =, to =596 s - ) odeled by eqs. (22) - (23) and stress relaxation - by eq. (28) for the whole huan blood (DA), H=43%, T=37 C; experiental points - - exp, theoretical lines- - M. Fro [3] Fig.c. Stress growth odeled by eqs. (22)-(23) (fro =, to =.285 s - and to =.285 s - ) and stress relaxation - by eq. (28) for the whole huan blood (DA), H=43%, T=37 C; exp. points - - exp. (.285 s - ), - exp. (5.96 s - );theor. lines- - M. (.285 s - ); - M. (5.96 s - ). Fro [3] Experiental shear stress- tie dependence at rectangular change of shear rate up to,596 s - is shown in Fig.b. It ay be concluded fro the experient, that a part of RBCs are already arranged in structures or rouleaux, while another part of the are flowing as single cells. These structures for the plateau of the (-t) curve in Fig. b. When shear rates are higher, equal to, 596 s - (Fig.b),,285 s - and 5, 96 s - (Fig.c), stress developent (under constant shear rate) is odeled by equations (22)-(23). 74

10 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 The higher shear rate leads to decrease of the size of the related structural units, corresponding to a less resistance to flow. The last two (-t) rheogras at shear rates of,285 s - and 5,96 s - in Fig.c reveal the well-known stress overshoot behaviour of the huan blood. It is seen that the MR odel describes very well the experientally observed tie-dependent properties. Stress relaxation described by the MR odel is shown in the sae figures. It is seen in Fig.a and Fig.b, that stress relaxation after flow cessation has not been copleted for the tie of the experient perforance (about s). Experiental results for tie dependent properties of blood and their odeling with the proposed MR odel (22)-(23) at shear rates of,285 s - and 5,96 s - are shown in Fig. c. Stress overshoot is observed after which blood is flowing. Following the experiental observations we conclude that blood exhibits elastic and plastic (residual) deforation. The MR odel in this case is based on the proposal, that the full strain rate consists of the elastic strain rate e and the residual strain rate. Coparison shows that the MR odel - equations (22) and (23), adequately describes this flow echanis. It is seen in the sae figure that the stress relaxation after cessation of flow (Fig. c), odeled by (28) at this shear rate, describes adequately the process. In this case when flow is terinated, particles connect to each other and the process of destroying and creating new connections is well balanced. Paraeters of odel (22)-(23) calculated fro the experiental data and coefficients, describing stress developent at shear rates,596 s -,,285 s - and 5,96 s - - Fig. b and Fig.c, are given in Table a. In the case of stress relaxation (after the cessation of a steady flow starting fro shear rates.596 s -,.285 s - and 5.96 s - to s - ) equation (28) describes the observed phenoena. Coparison of experiental points and theoretical lines are shown in Fig. b and Fig. c. The coefficients of odel (28) for the whole huan blood are given in Table b. Table a. Nuerical values of the paraeters of the equations (22)-(23), odeling stress developent at rectangular change of shear rates fro to,596 s -,.285 s - and 5.96 s - for the whole huan blood (DA), H=43%, T=37 C. Shear rates Paraeters of eqs (22) and (23),[Pa.s],[Pa.s] ) ( t,[ s].596 s -,6736 3,92 2,8,725 2,285 s -,5779 3,292 8,84,3 5,96 s -, ,292 5,292 8,39438,5 Table b. Nuerical values of the paraeters of equations (28), odeling stress relaxation after cessation of the rectangular flow, starting fro different shear rates -,596 s -,.285 s - and 5.96 s -, for the whole huan blood (DA), H=43%, T=37 C Paraeters of eq (28) Shear rates o,[pa] o / o,[s - ].596 s -,9854,3422,285 s - 8,84, ,96 s - 57,83,9438 The progressive breakdown and foration of rouleaux under shear rates is proposed to be one of the ajor causes of blood thixotropic behavior. The increase in shear rate leads to a transient reduction in size and nuber of these associated structural units, corresponding to a saller resistance against flow. At 75

11 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 higher shear rates,285 s - and 5, 96 s - RBC aggregates can not for larger structures and fast flowing is observed. In those cases shear stresses reach a full relaxation and only elastic deforation is revealed. This behavior is odeled by equation (28). The proposed MR odel does not describe the observed stress overshoot in this for. For that purpose, the odel can be copleted including the energy of activation of RBC aggregates due to the change of their structure Results of the odeling blood tie-dependent properties at triangular change of shear rates Experientally found hysteresis for noral conserved huan blood (3 l blood/75 l L-2), H=38%, T=37 C at triangular change of shear rates t 2t t for <t<t - loading and for t <t<2t discharge is shown in Fig.2. The for of the hysteresis loops depends on the applied shear rates and on the duration of shear Fig.2: a). 596s and t =3s; b);. 8s and t =3s. The observed experiental relationships (-) under triangular change of shear rates a) and b) are odeled by equation (9). The MR odel has been odified taking into account shear rate effect on structuring applied through the logarithic odulus of the residual strain rate at shear. This odulus has the for of relation (3), while at relaxation that of eq. (3): 2 3 b a c (3) The nuerical values of the odel equation paraeters found fro the experients, are shown in Table 2. Fig. 2. Blood hysteresis loops at triangular change of shear rates, odeled by the odified MR odel (9) a). 596s and t =3s; b);. 8s and t =3s. for noral huan conserved blood (3 l blood/75 l L-2), H=38%, T=37 C, LS3 Contraves 76

12 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Experientally deterined hysteresis loops of noral conserved huan blood under triangular change of shear rates and the odel results, found using the MR odel odification (9), are shown in Fig.2. The odel describes very well the observed loading and discharge curves and the hysteresis loop, except for the special experiental data. The coparison shows that equation (9) can be used to calculate the hysteresis loops. Table 2. Nuerical values of the paraeters of the odel equation (9) for hysteresis loop for noral huan blood conserved by L-2 (3 l blood / 75 l L-2), H = 38 %, T = 37 C. Models o o, Pa.s Paraeters of eq. (9), Pa a k b C M l (9),(3).596 s - 793,38 2,4393-2,67,5 28,954 M d (9),(3).596 s - 793,38,3 228, ,55-955,42 M l (9),(3),8 s - 82,93,2286,96 -,9 M d (9),(3),8 s - 82,93,589 -,428 -,38,2556 To verify odel (25), the experiental results found by Picart et al. [4] were used to describe the shear stress-deforation (product of tie and shear rate) relationship for a huan blood saple, H=7%, under different shear rates - -3 s -, s -, -2. s - and s -, LS4, T=25 C. The coparison is shown in Fig.3. It is seen in the figure that the relationship shear stresses-strain ( ) is very well described by the linear solution of the MR odel (24)-(25). The nuerical values of the paraeters of equations (24) and (25) calculated fro experiental data are shown in Table 3. Table 3. Nuerical values of the paraeters of odels (24)-(25) for huan blood, T=25 C, LS 4, obtained by Picart et al [4], [s - ] o o, [Pa.s] Paraeters of odel eqs. (24)-(25) o, [Pa], [Pa] ,7578 2, ,75 2,484 22, ,32 27, ,37 8,398 39,8 77

13 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Fig.3. Modeling the relationship - by eans of eq. (25) for a huan blood saple, H=7%, under different shear rates - -3 s -, s -, -2. s - and s -, LS4, T=25 C (fro Picart et al.[4]). Experiental data: exp ( -3 s - ); - exp2 (3. -3 s - ); - exp3 ( -2 s - ); - exp4 (3. -2 s - ); Theoretical lines: - - M ( -3 s - ); - M2 (3. -3 s - ); - M3 ( -2 s - ); - exp4 (3. -2 s - ); - M4 (3. -2 s - ) Results of odeling the rheological properties of blood under steady shear The apparent dependence of viscosity on shear rates under steady shear flow ( ) is odeled by equation (32): exp, (32) Shear rate effect on the structure foration is presented by eq. (3) b k a, where, a, b, k are constants found fro the experiental data and given in Table. 4. The relationships ( ) under steady flow is odeled by equation (9). 78

14 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 Fig. 4. Shear stress as a function of shear rates (9) and apparent viscosity as a function of shear rates (32) for noral huan blood saples with different heatocrits, respectively: experiental points (exp) and theoretical odel lines ( or 2) for: - exp ; exp (H=3%); - ; (H=3%); - exp ; exp (H=4%); - ; (H=4%); - 2 ; 2 (H=4%); - exp ; exp (H=46%); - ; (H=46%); - 2 ; 2 (H=46%); - exp ; exp (H=5%); - ; (H=5%); - 2 ; 2 (H=5%); - exp ; exp (H=6%); - ; (H=6%); - 2 ; 2 (H=6%); - exp ; exp (H=7%); - ; (H=7%); 2 ; 2 (H=7%). Table 4. Nuerical values of the paraeters of the odified Maxwell-urevitch-Rabinovich odel (9) for the whole noral huan blood with different heatocrits under steady shear at 37 C, LS 3 Contraves Modified Maxwell-urevitch-Rabinovich odel (9) Paraeters Heatocrit H, % , [Pa.s] ,83 272, 572,3,[Pa] 2,24 3,692,4327 5,58 27,8 35,25 b,359.97,8784 2,3 6,86 -,979 a 3, ,53 5,9 2,2 -,6886 6,974 k,75,6,4,3575,3862, Results of the odeling blood viscoelastic properties by eans of the odified MR odel The experiental dependence of the viscous part of the coplex viscosity of huan blood of a patient with CVD (H=49%, T=37 C) on the oscillatory shear rate, and the theoretical curves found using odel equations (33) and (34), are shown in Fig.5. Dynaic viscosity - oscillatory shear rate dependence 79

15 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 of blood is described by the linear MR odel (33) and by the odified MR odel (34). The odified odel describes very well changes of the dynaic viscosity, especially within the range 2 s - 2 s -. It is seen in the sae figure that the experiental data can be presented as separate linear parts with different paraeters a and n, using the linear odel (33). The nuerical values of the paraeters of eqs. (33) and (34), derived fro the experiental oscillatory shear rate dependence of the viscous part of the coplex viscosity of blood of a patient with CVD are shown in Table 5. Linear case - Model, equation (33): lg a nlg (33) Nonlinear case - Model 2, equation (34): n. exp (34) Fig.5. The oscillatory experiental shear rate dependence of viscous part of the coplex viscosity of huan blood of a patient with CVD - exp :H= 49%, T=37 C, LS3 sinus and theoretical lines - M ; - M2. found by odeling the dynaic viscosity of blood eploying the MR odel (33) and (34) respectively. Table 5. Nuerical values of the coefficients of eqs. (33) and (34) of the experiental oscillatory shear rate dependence of the viscous part of the coplex viscosity of the huan blood of a patient with CVD, shown in Fig.5. Paraeters of the eqs.(33) and (34) Model : 26,7 49, 3 4,68, H=49% A o o,[pa.s, [Pa] N Model 2,92 373,54 ]. Model :,267 3, 46,3. 2,6. Model : 3, 26, 7 73,97. 8,66 8

16 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., Discussion The experiental relationships provided by the steady, tie dependent and dynaic rheological easureents were odeled by the discussed MR theory and by equations for different shear flow regies, whereas the latter were experientally realized.. For the regie of steady shear and non-steady tie dependent viscoetric flow ( const ) we considered two shear rate areas: a) <,3 s - and b) >,3 s - In case a) stress developent revealed elastoplastic behaviour, while in case b) viscoelastic one. Stress relaxation in case a) was not copleted, while in case b) shear stresses attained full relaxation and elastic deforation was revealed, only. The observed experiental relationships (-) under triangular change of shear rates a) and b) were odeled by equation (9), which is an exponential general solution t of the Maxwell-ureich-Rabinovich type equation in the case of hysteresis. The MR odel was odified taking into account the shear rates effect on the structuring applied via the logarithic odulus of the rate of residual strain at shear. Blood steady flow experiental relationships ( ) were odeled by the sae equation (9). The corresponding ( ) relationships for different heatocrits were odeled by equation (32). Increase of the initial coefficient of viscosity for the residual strain and the logarithic odulus of the rate of residual strain at shear with increasing the heatocrit, presented in Table 4, was observed. Dynaic viscosity - oscillatory shear rate dependence for blood was described by the linear MR odel (33) and the odified MR odel (34). The odified odel describes very well changes of the dynaic viscosity, divided in three separate linear parts, especially within the range 2 s - 2 s -.The elastic odulus of shear ; the initial coefficient of viscosity and the logarithic odulus of the velocity of the deforation provide a functional relation between,,, and characterize the rheological behaviour of blood at the considered flow regies. For accurate description of the observed relationships soe odifications of the MR odel were proposed, introducing nonlinear dependences of the logarithic deforation odulus on shear rates. The paraeters of the proposed odels were found fro experiental data. Very siilar olecular or icroscopic processes associated with viscoelasticity and thixotropy provide the basis of a unifying description of these diverse rheological properties. enerally these properties are connected with the effects of the fluid inner structure on the rheological behaviour. The change of blood flow in tie is typical for viscoelastic and thixotropic fluids and this phenoenon is related to the influence of the inner structure on blood flow. In the case of thixotropy when flow is stopped, particles connect with each other and the process of destroying and creating new connections is well balanced. Connection forces are weak. The higher shear rate leads to decrease of the size of the related structural units, corresponding to a less resistance to flow. Siilarly, the acroolecules of the viscoelastic fluid for a network of teporal links, and the destruction and creation of links is balanced. 8

17 L. Hadjikov et al. / Series on Bioechanics, Vo.23, No., 27 When we aintained larger fluid shear rates, ore acroolecules were destroyed for a unit tie. Consequently, thixotropy and viscoelasticity were related to siilar tie effects. In this sense, appropriate theories were developed, describing the echaniss of thixotropy and viscoelasticity - see. Thurston.[5], P. Riha [6] and D. Queada [7]. The presented MR theory and the proposed equations describe in detail the rheological behaviour and characteristics of blood at different regies of viscoetric shear flow. 6. Conclusion The odified odel of A.L.Rabinovich or Maxwell-urevich-Rabinovich (MR) odel was chosen to describe the experientally observed rheological properties of blood under steady and nonsteady flow conditions. The MR odel enables one to investigate changes of the icrostructure of blood undergoing deforation through blood paraeters found fro experiental data. By eans of the Low Shear 3 sinus oscillation rheoeter, shear stresses, viscosity and storage odulus were deterined, varying frequency and shear rates as ruling paraeters. Rheogras of change of viscosity ( ), respectively shear stresses () after sudden step increase of shear rates (. ) fro to.237 s - ;.596 s - ;.285 s - and 5.96 s - as well as under triangular change of shear rates, are plotted. Model equations were proposed through MR odel odifications, introducing nonlinear dependences of the logarithic deforation odulus on shear rates. eneral solutions of the Maxwell-urevich- Rabinovich type equation were discussed in the case of stress foration (under constant shear rate) and stress relaxation (after the cessation of the steady flow) at a rectangular and triangular change of shear rates and under steady and oscillation sinusoidal flow. The rheological paraeters obtained fro the odified odel of Rabinovich describe the icroechanical rheological properties of huan blood. The proposed equations can be successfully used to coplete the syste of equations for blood under different flow regies. Acknowledgeents The authors thank Mag. Physics I. Ivanov for the assistance, Assoc. Prof. Dr. I.Velcheva fro the Departent of Neurology, Sofia Medical University, and the colleagues fro the National Center for Transfusion and Heatology in Sofia for the blood saples provided. This work was done within a project TH 32/23 with the National Fund for Scientific Investigations to the Ministry of Education and Science of Bulgaria. References [] Rabinovich A.L., 97. Introduction in the echanics of the reinforced polyers, Mir, Moscow (in Russian) [2] Antonova N, 998. Blood viscoelastic and thixotropic properties, J. Theoretical and Applied Mechanics, Year XXVIII, No.3, [3] Hadjikov L., Antonova N., Ivanov Ya., Ivanov I. Modeling of the tie-dependent properties of blood by the odified odel of Maxwell-urevich-Rabinovich. In: Proceedings of the 2 nd Eurosuer School on Biorheology & Syposiu on Microechanobiology of Cells, Tissues and Systes, Varna, Bulgaria, 7-2 Septeber, 26, Avangard Publisher, Sofia, 27, [4] Picart C, Piau J-M, alliard H, Carpentier P, 998, J. Rheology, 42(), -2 [5] Thurston.B.,979. Rheological Paraeters for the Viscosity, Viscoelasticity and Thixotropy of Blood. Biorheology, 6, [6] Riha P.,982, The unified description of viscoelastic and thixotropic properties of huan blood, Rheol. Acta 2, [7] Queada D., Droz R.,983. Blood Viscoelasticity and Thixotropy fro Stress Foration and Relaxation Measureents: a Unified Model. Biorheology, 2,

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