page_title: Hemorheology
section_title: Maxwell model
hierarchical_section_title: Hemorheology / Blood viscoelasticity / Maxwell model
caption_attribution_description: English: Simple cubic model showing elastic and viscous deformation components
context_page_description: Hemorheology, also spelled haemorheology, or blood rheology, is the study of flow properties of blood and its elements of plasma and cells. Proper tissue perfusion can occur only when blood's rheological properties are within certain levels. Alterations of these properties play significant roles in disease processes. Blood viscosity is determined by plasma viscosity, hematocrit and mechanical properties of red blood cells. Red blood cells have unique mechanical behavior, which can be discussed under the terms erythrocyte deformability and erythrocyte aggregation. Because of that, blood behaves as a non-Newtonian fluid. As such, the viscosity of blood varies with shear rate. Blood becomes less viscous at high shear rates like those experienced with increased flow such as during exercise or in peak-systole. Therefore, blood is a shear-thinning fluid. Contrarily, blood viscosity increases when shear rate goes down with increased vessel diameters or with low flow, such as downstream from an obstruction or in diastole. Blood viscosity also increases with increases in red cell aggregability.
context_section_description: Maxwell Model concerns Maxwell fluids or Maxwell material. The material in Maxwell Model is a fluid which means it respects continuity properties for conservative equations :  Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids. Maxwell model is made to estimate local conservative values of viscoelasticity by a global measure in the integral volume of the model to be transposed to different flow situations. Blood is a complex material where different cells like red blood cells are discontinuous in plasma. Their size and shape are irregular too because they are not perfect spheres. Complicating moreover blood volume shape, red cells are not identically distributed in a blood sample volume because they migrate with velocity gradients in direction to the highest speed areas calling the famous representation of the Fåhræus–Lindqvist effect, aggregate or separate in sheath or plug flows described by Thurston. Typically, the Maxwell Model described below is uniformly considering the material (uniform blue color) as a perfect distributed particles fluid everywhere in the volume (in blue) but Thurston reveals that packs of red cells, plugs, are more present in the high speed region, if y is the height direction in the Maxwell model figure, (y~H) and there is a free cells layer in the lower speed area (y~0) what means the plasma fluid phase that deforms under Maxwell Model is strained following inner linings that completely escape from the analytical model by Maxwell.
  In theory, a fluid in a Maxwell Model behaves exactly similarly in any other flow geometry like pipes, rotating cells or in rest state. But in practice, blood properties vary with the geometry and blood has shown being an inadequate material to be studied as a fluid in common sense. So Maxwell Model gives trends that have to be completed in real situation followed by Thurston model  in a vessel regarding distribution of cells in sheath and plug flows.
  If a small cubical volume of blood is considered, with forces being acted upon it by the heart pumping and shear forces from boundaries.  The change in shape of the cube will have 2 components:
  Elastic deformation which is recoverable and is stored in the structure of the blood.
  Slippage which is associated with a continuous input of viscous energy.
  When the force is removed, the cube would recover partially.  The elastic deformation is reversed but the slippage is not.  This explains why the elastic portion is only noticeable in unsteady flow.  In steady flow, the slippage will continue to increase and the measurements of non time varying force will neglect the contributions of the elasticity.
  Figure 1 can be used to calculate the following parameters necessary for the evaluation of blood when a force is exerted. 
  Shear Stress:  
  Shear Strain:  
  Shear Rate:  
  A sinusoidal time varying flow is used to simulate the pulsation of a heart. A viscoelastic material subjected to a time varying flow will result in a phase variation between  and  represented by . If , the material is a purely elastic because the stress and strain are in phase, so that the response of one caused by the other is immediate. If  = 90°, the material is a purely viscous because strain lags behind stress by 90 degrees. A viscoelastic material will be somewhere in between 0 and 90 degrees.
  The sinusoidal time variation is proportional to . Therefore, the size and phase relation between the stress, strain, and shear rate are described using this relationship and a radian frequency,  were  is the frequency in Hertz.
  Shear Stress:  
  Shear Strain:  
  Shear Rate:  
  The components of the complex shear stress can be written as:
  Where  is the viscous stress and  is the elastic stress.
  The complex coefficient of viscosity  can be found by taking the ratio of the complex shear stress and the complex shear rate:
  Similarly, the complex dynamic modulus G can be obtained by taking the ratio of the complex shear stress to the complex shear strain.
  Relating the equations to common viscoelastic terms we get
caption: Figure 1 - Displacement due to Elastic and Viscous Effects
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