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Medical Physics. Ivan Tanev Ivanov. Thracian University. 2016

CHAPTER 2. RHEOLOGY AND HYDRODYNAMICS

2.1. Potential energy and forces of interaction between the particles of the substance. Molecular structure of liquids and solids. Polymers and biopolymers. Liquid crystals. Phase

transitions.

There are forces of strong interaction between the constituent particles (atoms and molecules) of any substance. These forces are usually expressed by the potential energy of interaction, Ер. Depending on the distance, r, between any pair of particles these forces cause attraction (negative Ер) or repulsion (positive Ер). At greater distances, the forces have electrostatic character - Coulombic forces of attraction or repulsion that decrease with distance, r, as 1/r2. At intermidiate distances the forces cause attraction and, after their discoverer, are called forces of Van der Waals. They originate from the interaction between permanent or induced electric dipoles and decrease with distance, r, as 1/r6. The attraction forces between similar molecules are called cohesion, while those acting between different types of molecules - adhesion. At very small distances only repulsive forces act.

Fig. 2. 1. 1. Potential energy, Ep, of the interaction between two adjacent constituent particles as a function of their displacement, r.

In physics, bodies having high density (liquids and solids) are considered as condensed matter. For condensed matter fig. 2.1.1 shows a typical curve depicting how the Ер of two adjucent constituent particles depends on the distance, r, between them. Aparently, the Ер has a single minimum (Ерo).

When the distance between the particles is ro, the potential energy of interaction has minimal value, Ерo. This position of particles is stable one (equilibrium) because each change in the displacement, r o, brings about force that restors back the equilibrium value of this displacement.

In liquid bodies, the value of Epo is closely equal to the average kinetic (thermal) energy of molecules. Therefore, the molecules of a liquid oscilate remaining at their equilibrium position for short time only, after which they jump to other equilibrium position and so on. Thus, a flow could be formed within any liquid. Even more properties originate from this molecular particularity of liquids. Liquids have no own shape and take the shape of the container in which they are placed. Except to flow, liquids readily evaporate, thereby become cooled. Diffusion within the liquids proceeds at a high rate, and gas absorbtion is very strong. The major constituent of human body is water, which is almost a perfect liquid, strongly alleviating transport processes of metabolism and gas exchange.

In solid bodies Epo is much larger than the average thermal energy of the particles. These strong interactive forces compel each particle of solids to stay at one place allowing it only to vibrate around its equilibrium position (node) that is immobile. Thus, solids do not leak, have their own shape and resist its change.

Upon the action of ouside force the solid body changes its dimentions and shape; the distance between its constituent particles also changes accordingly. The mean displacement between each pairs of adjacent particls decreases at contraction and increases at stretching. In both cases, strong forces arise between the particles acting to restore the equilibrium displacement back. These intrinsic forces are called


internal tension. Their role is to balance the ouside force and reduce the change in body shape. In crystalline solids, the nodes form a spatially regular geometric form, called a crystalline lattice.

It contains a huge number of identical elementary cells that have the same directions in space. The nodes of the crystalline lattice may containe atoms (atomic lattice), molecules (molecular lattice), ions of different sign (ion lattice), and ions of the same sign (metal lattice). The crystalline bodies have a specific melting temperature above which they pass from a solid to a liquid phase. Upon melting the distances between the constituent particles change, thereby Ep and the internal energy of the body also change and the melting body absorbs heat. This process is referred to as a phase transition of first order. Upon cooling, the reverse process, crystallization, occurs when each particle arrives at a stable position (node) and the crystalline lattice is again build.

The crystal bodies are found in two forms, as single crystals and as polycrystals. Polycrystals are build up of a multitude of adjacent, randomly oriented small crystals (crystallites). Physical properties (mechanical, optical, electrical) of monocrystalline bodies are different in various directions - anisotropism. Some monocrystalline bodies display optical birefringence.

Another example of solids is the amorphous bodies, whereat the equilibrium positions of particles are randomly distributed throughout the volume of the body. Amorphous bodies are isotropic, i.e., their properties are the same in different directions.

Upon heating the amorphous bodies pass from a solid to a viscous liquid state (melt) with a very high viscosity close to that of rubber. However, this process occurs in a wide temperature range. The mean temperature in this range is called glass transition temperature, and the process is referred to as a phase transition of second order. During this type of transition the average distance between particles does not change, and therefore, the internal energy of the body remains constant. Upon cooling of the melt a vitrification process occurs, whereat the constituent particles arrive at randomly selected equilibrium positions without building a crystal lattice. Amorphous bodies and the glass are regarded as supercooled liquids, in which the particles did no succeed to align in a regular crystal lattice due to the very high viscocity of the melt near the point of glass transition.

The oldest and most important example of amorphous matter is glass. It is prepared by cooling the melt of quartz sand. In order to reduce the glass transition temperature, different ingredients are added to the melt. Except its effect on the glass transition temperature, each ingredient imparts some valuable property to the glass. In such a manner various types of glass are produced. Some glasses absorb X-rays (shields for radiation protection), others are able to measure the pH of aqueous media (glass electrodes), or have a certain color and so on.

Natural and synthetic polymers (rubber, Plexiglas, Teflon, cellophane) are other important cases of amorphous bodies. Their molecules are highly elongated or branched and consist of interconnected units of a same or different type. By creating covalent bonds between the polymer molecules a new, very hard polymer material, thermoset, is produced which has a high glass transition temperature. With respect to their thermal properties, the polymers are of two types. Thermoplastics, by contrast to thermoreactive polymers, change their properties reversibly on heating, that is, they recover after cooling back to room temperature. Like crystalline bodies, some polymers have sufficient stiffness and tensile strength, while others are capable of strong reversible deformations. Polymers are easily formed into fibers and webs. They possess high resistace to electric current, moisture and chemical corrosion. Some of them are used in manufacturing artificial prostheses, cellophane membranes for artificial kidney, silicone membrane for artificial lung and for fast cover of open wounds. Some water-soluble polymers (gelatin) are used as a binding medium or a coating in the pills of drugs.

Biopolymers include bones, muscles and other human tissues, which contain biomacromolecules, primarily proteins. They have structures and properties intermediate to those of the crystalline and amorphous bodies. Visco-elasticity is their most prominent mechanical property. The aqueous solutions of biomacromolecules (polysaccharides, proteins) retain their fluidity over a wide temperature range, while they turn into the state of gel on cooling. This is a state similar to solid body whereat without removing the solvent strong cohesive bonds establish between the dissolved particles. The gel state is similar to that of the cytoplasm of cells.


Ceramics is another example of a solid body. It is prepared as a powder mix of various minerals (clay, kaolin, silica sand, etc.) and the mix is baked at high temperature (about 1500°C). The mineral particles partially melt causing their interconnection. Depending on the ratio of the initial minerals the properties of the resulting ceramics (heat and humidity resistance, porosity, resistance to corrosity, etc.) vary. Thus, ceramics with different properties are obtained such as porcelain, faience (majolica), terra cotta and more. In medicine, the ceramics are used to make a variety of prostheses.

Most substances can exist in one of the three basic physical states - solid, liquid and gaseous. Some organic compounds with moderatly elongated polar (more generally, anisotropic) molecules may also exist in the so called liquid crystal state. The latter was discovered in 1888 by the Austrian botanist Raynittser in a newly synthesized compound cholesterilbenzoat. The liquid crystal is an intermediate state between the liquid and solid states and can be obtained only in a narrow temperature range between the melting temperature of the solid crystal and a certain high temperature above which ordinary liquid results. Liquid crystals are distinguished from solid crystals in that the molecules of solids display positional and orientational order, whereas the molecules of liquid crystals display only orientational order.

Liquid crystals can flow, however, they have properties typical of the crystalline bodies - anisotropy, melting point. They are three types - nematic, smectic and cholesteric (Fig. 2.1.2). In smectic liquid crystals, the centers of gravity of the molecules can move only in one plane, while in nematic crystals they can move along the three directions. Nematic liquid crystals represent a fluid of uniformly oriented sticks, while the smektic crystals contain layers made up of sticks tightly packed in a same direction. Biomembranes contain two layers of lipid molecules (lipid bilayer, double layer), which is actually a liquid crystal of smectic type.

Fig. 2.1.2. Arrangement of the molecules of a liquid crystal of nematic type (A), smectic type (B) and cholesteric type (C).

Liquid crystals are thermotropic and lyotropic, the first alter their structure and properties above a certain temperature, the second ones above a certain concentration. The structure and properties of liquid crystals, especially cholesteric ones (e.g. their color) strongly depend on temperature, external electric fields, the presence of vapors of some substances, ect. For example, an external electric field with intensity greater than a certain limit (about 1v) reorients the molecules of liquid crystals parallel or perpendicular to the field lines. This dramatically changes the optical properties of the crystals; light absorption, optical activity, birefringence, color, etc. This change is called a Fredericks transition. This effect is used in portable monitors, televisions and light indicators. This allows the liquid crystals to be used for measuring temperature, as indicator displays, in toxicology.

The esters of cholesterol form cholesteric liquid crystals containing parallel planes. The molecules in each plain are arranged as in the nematic crystals. The neighboring planes are displaced by a certain angle which depends on the temperature and the external electric field. When the temperature approachs a


certain value (37°C) the angle will change, resulting in a change in the color of the crystal. This is used for accurate indication of the skin temperature in patients.

The solid phase of the liquid crystal is called gel state. The transition between gel state and liquid crystal state is an example of a first order phase transition. High temperatures also induce a change in the conformation of the macromolecules of biopolymers. In this case, the thermally induced change in the conformation results in abolition of biological activity called denaturation.

2.2. Deformation of solid bodies. Types of deformation and deformation curve. Hooke's law for elastic deformation. Elastic, viscous and viscoelastic deformations of solids

and biological tissues.

Under the action of external forces (mechanical, electrical, etc.) the solid bodies as well as human tissues sustain deformations. As a physical term the deformation (or strain) of a body means any change in its shape and dimensions. During its deformation the solid body generates an intrinsic resistive force which counteracts the ouside deforming force. During their deformation the solid bodies demonstrate their passive mechanical properties - elasticity, hardness, strength and others. In traumatology and orthopedics, in sport medicine and occupational medicine it is necessary to apprehend the passive mechanical properties of biological tissues and substituting prostheses. The deformations and passive mechanical properties of solids and tissues are object of study for the physical discipline rheology (from Greek, rheo – a flow, logos - science).

The main types of deformations in solids are tensile deformation, compression, bending, twisting and shear (angular deformation) - fig. 2.2.1.

Tensile strain: the force acts to pull the body apart;Compressive strain: the force squeezes the body;Shear stress: the force causes one part of the body to slide on another part.

Fig. 2.2.1. Main types of deformations.

Let's have a longitudinal tensile deformation of a body and Lo denotes the initial length of the body and L indicates the final length of the deformed body. Then L = L-Lo is called absolute longitudinal deformation and = L / Lo is the relative longitudinal

deformation (Fig. 2.2.2). Upon imposing an outside force to the body, the constituent particles of the body are forcibly

displaced from their equilibrium positions. Hence, the distance between these particles changes giving rise to internal force that resists the external force F. With increasing the deformation the internal force increases and, at a given deformation it balances the external force. At this equilibrium state the internal force is denoted as internal stress . This equilibrium state establishes within a very short time. During the longitudinal tensile deformation the arised internal stress is = F/So, where F is the deforming force, and So is the initial cross-section of the body (Fig. 2.2.2).

Each small deformation ( << 1) is always elastic (reversible). This means that after removal of


the deforming force, the body recovers back its initial size and shape. Elasticity of a material is its ability to restore back its original shape after tensile deformation when the outside stress or load is removed. The elastic deformations obey the Hook's law: = /E. Here, E is the modulus of elasticity (Young's modulus, elastic modulus). The modulus of elasticity is equal to that stress, , producing deformation L = Lo. The modulus of elasticity is a measure of the resistance the bodies show during the initial step of their deformation. The reciprocal value of E, 1/Е = α is refered to as elasticity coefficient, or elasticity.

Bodies that have a small modulus of elasticity (rubber, tendons, joints) are referred to as flexible (deformable) ones. Bodies that have a high modulus of elasticity (steel, bone) are referred to as rigid bodies or bodies with great strength.

Fig. 2.2.2. Changes in the dimensions of a body during tensile strain.

Besides the longitudinal size, L, the transversal size, d, and the volume, V, of the body also change during the tensile deformation. The relative transverse deformation is defined by the formula = d/do. The ratio / = is called Poisson's coefficient. It is less than 0.5 for all

materials. The relative change in the volume is given by the expression V /Vo = (1 – 2). The other types of deformations and their relevant parameters are defined similarly.

Fig. 2. 2. 3. Stress-strain curve for tensile deformation of solids.

In the state of equilibrium deformation, there is a correlation between the magnitude of the applied force, F, respectively the internal stress, , and the arised deformation, . This

correlation is usually determined experimentally and expressed graphically by the so called deformation curve, also known as stress-strain curve. For most solids (crystals, some amorphous bodies), the deformation curve for tensile deformation has the form shown in Fig. 2.2.3. Several regions are well distinguished on this curve; linear region, perfect plasticity region, strain hardening region, necking and fracture region.

The initial linear portion OA of the curve is the elastic region where Hook's law is valid. The slope, tg (), of this portion is equal to the modulus of elasticity, E. The smaller the slope, the more elastic and flexible is the material. Conversely, the stiff (rigid) materials exhibit greater slope of the linear portion. Such materials have high resistance to deformation. Point A is called proportionality limit.


The AB region also corresponds to a fully reversible, elastic deformation without any residual deformation. However, this deformation is larger than that calculated by the Hook's law. Provided the deforming force at the point B is reduced to zero the deformed body will restore its size back to its initial value in the point O. The point B is called limit of elasticity and the point of yielding as well. The B is called yield strength.

The deformation, described by the region OB, is equilibrational, occurs rapidly, and does not depend on the time.

A new type of partially reversible deformation, called plastic deformation, takes places in the BC zone of plasticity or yielding. For example, if the deforming force at the point C is reduced to zero the deformed body will shrink along the line CC (fig. 2.2.4). The OC segment is called residual strain (deformation) while the СС segment is the elastic deformation. Plasticity is the ability of a material to undergo some degree of permanent deformation without failure. Plastic deformation will take place only when the elastic range has been exceeded, beyond the point B.

At its final stage the deformation in the plasticity zone is also an equilibrational one, however, its approach to the final equilibrium value consumes a substantial time. This time-depending step of plastic deformation is designated cold (plastic or viscous) creep (flow). The plastic creep in solids occurs after the application of a supercritical force that begins slowly but steadily to extend the body until the strain reachs its equilibrium value. It is due to re-arrangement of molecules and partial eradication of internal structural defects in the crystal lattice of deformed body.

Fig. 2. 2. 4. Stress-strain curve for tensile deformation of solids.

Cold creep phenomenon is typical for crystal bodies that have micro defects in their crystal lattice. Any deviation from the ideal structure of the crystal lattice is called a micro defect. For example, when a node lacks a particle this defect is called vacation. The most common defects are due to the impurity particles which may be incorporated either in the

nodes of the crystalline lattice (substitution) or between the nodes (internode position). In addition, most crystals have mosaic (granular) structure because they are composed of crystallites - small blocks of monocrystals. Neighboring crystallites have slightly shifted crystal lattices, therefore, a transition layer is formed between them. This layer contains an entire line or plane of deformation called dislocation.

Plastic deformation is accompanied by local raptures and formation of micro cracks within the overcritically deformed solid body. During the crack formation, the micro defects of crystal lattice start to move, disrupting a number of bonds between the molecules and creating new bonds. This process of remodeling the intermolecular bonds eradicates most of the cracks and is called self-healing of the highly deformed crystal. However, this self-healing is slow and time consuming process. For example, under the influence of external force the dislocations shift to another place. In a perfect crystal without impurities the displacement of dislocations ends to the outer surface and the body becomes more hardened. In real crystals with impurities, the dislocation moves until reaching an impurity atom where several new dislocations will be created. This produces new local ruptures and crackings, which reduce the strength and stiffness of the body. However, if the concentration of impurity atoms is higher than a certain limit, they impede the movement of dislocations and mechanical properties improve. This explains the positive effect of such preliminary processing of metals as dopping (introduction of trace concentrations of dopant atoms), hardening (creating granular structure of the crystal by heat treament), tempering, anealing and others.


In the CD region of strain hardening, elastic deformation reappers and a new modulus of elastic deformation can be defined. This effect is due to the pre-arrangement of the molecules which leads to the so called strain hardening of the body.

Necking, in engineering and material sciences, is a significant decrease in the local cross-sectional area of highly deformed body during the last stage of tensile deformation. It is due to the relatively large amount of stress applied disproportionately in a small region of the deformed body. Necking precedes the final rupture of deformed body beyond the point D (fig. 2.2.4). The so called breaking point, D, indicates the ultimate tensile stress, D, over which the local structural defects can no longer be cured and the body tears apart. In some materials the final rupture is not preceded by a cold creep and the obtainable final extension is very small. Such materials are classified as fragile (brittle).

During the elastic deformation of a body a portion of reversible, elastic energy is stockpiled in the deformed body. The area of the triangle OAA' (fig. 2.2.4) is called modulus of resilience and quanitatively equals the elastic energy stored. This energy returns back when the body is allowed to recover its original shape.

The area of the figure AA'CC' is called modulus of toughness and corresponds to the energy of plastic deformation (fig. 2.2.4). This energy is a measure of the tensile strength of the body. This energy can not be recovered as it is irreversibly consumed in breaking the old bonds and creating new bonds between the molecules of the body.

Fig. 4. 2. 2. Mechanical model of elastic type of

deformation.

Solids can differ very strongly in their mechanical

properties. The steel has a very high modulus of elasticity and, hence, a very high stiffness (rigidity) but very small area of elasticity. Most polymers (e.g. rubber) have a very low modulus of elasticity, but a very wide region of elasticity. Steel and rubber both have no region of plastic deformation. Upon tensile strain the steel ruptures at a low deformation of about 0.3%, while rubber and some polymers at about 300%. Materials that have a very narrow zone of plasticity (glass, steel, ceramics, bones, oxalate kidney stones) are termed brittle ones. Conversely, some metals (copper, aluminum, aluminum alloys, wrought iron) have a wide area of plasticity and are designated as forgeable or ductile metals. Ductility of a material enables it to be drawn out into thin wire on application of a load. The ductile material can deform a lot more than a brittle material, while a brittle material typically can withstand large loads before it breaks. Higher temperatures will normally cause a material to become more ductile, while a colder temperature can cause materials to become more brittle. Malleability of a material is its ability to be flattened by hot or cold working into thin sheets without cracking.

The strain-stress curve for the compressive deformation is similar to that displayed in fig. 2.2.3. At this deformation the property hardness is exhibited; those bodies possess hardness which strongly resist against plastic deformation of compression. In this case, the point B is called a limit of hardness and B - hardness or compressive strength. Thus, stiffness (rigidity) characterizes the resistance of the deformed body against elastic deformation, while the hardness characterizes its resistance to plastic deformation. Tooth enamel is the hardest substance in human body.

The steel spring is used as a model for pure elastic deformation (Figure 2.2.5.). In this ideal case, the deformation is reversible, equilibrational and occurs without delay. According to Hooke's law, the ideal elastic deformation of the spring is = /E. The energy of the elastic deformation is retrievable; there is no conversion of energy into heat.

Exept the pure elastic, there is another, alternative type of deformation as well. It is exemplified by the initial, pre-equilibrational part of the plastic deformation of the crystal bodies and some polymer materials. In plastic deformation the equilibrium state is reached after a certain time which is less at polymers and longer at the ductile metals. The reason for this time dependence of deformation is that at the initial step the arisen internal stress rearranges and slithers the molecules of the deformed body. As a


consequence, the internal stress finally relaxes. The shifting and slipping of the molecules during the initial stage of plastic deformation resembles the internal movement of molecules during the flow of viscous fluids. Therefore, the initial non-equilibrium stage of plastic deformation is regarded as a deformation of pure viscous type.

Fig. 2. 2. 5. Mechanical model of pure viscous deformation.

To describe the deformation of pure viscous type a suitable physical model is used consisting of a

hydraulic shock absorber (Fig. 2.2.5). The pure viscous deformation increases over time according to the expression = .t/, where is the viscosity of the fluid in the shock absorber, is the internal stress and t is time. It is non-equilibrium and irreversible and lacks elasticity. Energy consumed in the deformation is not restorable; it is converted entirely into heat.

The deformation of polymers, biopolymers and tissues is of mixed type, and contains elements of both pure elastic and pure viscous types of deformations. For example, similar to the pure elastic deformation, the deformation of the tissues attains an equilibriational value, however, after a considerable time lag. In addition, the internal stress, which has arised in the deformed tissue, progressively decreases in a time interval called relaxation time. This mixed type of deformation, characteristic for the human tissues, is referred to as viscoelastic, while the deformed body is called elastomer. Other examples of elastomers are the amorphous bodies above their glass transition temperature. The difference between the viscoelastic deformation of elastomers and the cold creep of plastic bodies is that the latter occurs only when the deforming force is greater than a certain threshold, while the former could be produced by any force including the smallest one.

In case the deformation force is of periodic type with a given frequency, the produced deformation and the internal resistive stress both vary periodically with the same frequency. If the internal stress varies in phase with the deformation, the body behaves like a spring and deformation it is perfectly elastic. If the internal stress varies in phase with the rate of deformation, then the body behaves like a viscous medium and the deformation is ideally viscous.

2.3. Passive mechanical properties of biological tissues. I mpact of composition and structure. E ndoprosthesis

The passive mechanical properties of biological tissues demonstrate itself during the deformation of the tissue under the action of external force. They have to be distinguished from the biological mobility of the tissues which includes muscle contraction, cell motility, displacement of chromosomes etc. and requires the chemical energy of ATP.

In general, the tissues possess an apparent, although small ability to "flow" (viscoelasticity). Therefore, to reach equilibrium the deformation of tissues needs a significant time interval. In contrast to the plasticity of some solids, the viscoelasticity of tissues takes place even at very small force – there is no threshold limit for the magnitude of the deforming force.

The internal stress, caused by the deformation of tissues, relaxes over time, and this relaxation is partial or complete. This is a second distinction of the passive deformation of tissues compared to that in solids.

There is yet another distinction for the equilibrium deformation of tissues - it depends on the time interval, Δt, over which the force, F, is exerted. The deformation, produced by weak force acting over a long Δt, reaches a larger value and has partially irreversible character. Conversely, strong forces acting over short Δt, produce smaller and fully reversible (elastic) deformation.

A fourth distinction of the tissue deformation is that the elastic stretching and contraction are


described by two different, not coinciding curves both having common beginnings and common ends. In other words, upon the completion of the cycle stretching - contraction, the tissue deformation depicts a closed loop and the tissue returns to its original size. This type of behaviour is called hysteresis. The appearance of histeresis means that a part of the energy, consumed during the stretching will return back during the contraction because of the tissue elasticity, however, the remainder will be converted into heat due to viscose rearrangement of the constituent macromolecules of the tissue.

Fig. 2.3.1. Maxwell element serving as a mechanical model of smooth muscle tissue (above). The plot below describes the relaxation of internal stress, , arising after the imposition of permanent deformation, L.

Depending on the type and composition of tissues their deformation is described by different mechanical models. These models constitute suitable combinations of the elements for pure elastic and pure viscous deformations, respectively, springs and hydraulic shock absorbers. In general, there are two simplest possible models - the model of Maxwell and the model of Kelvin - Voigt. The

Maxwell's model (Figure 2.3.1) is applied for tissues with more pronounced viscose properties (including the viscous liquids and tissues), while the model of Kelvin - Voigt (Fig. 2.3.2) is applicable for the elastic tissues (including the solids).

Figure 2.3.2. Kelvin - Voigt model (left) describing the isotonic strain of cross-striated muscle (right). F - deforming force, L - length of the muscle.

Tissues have different structure and composition and contain different types of biopolymers. For example, visco-elastic properties of smooth muscles are best described by the Maxwell's model (Fig. 2.3.1). Assume the tissue is rapidly elongated by L (isometric stretching) at a moment, marked with an arrow. The elongation gives rise to a strong internal stress, , which sharply reachs a maximum and gradually relaxes to zero. According to this model, the initial stretching, L, is immediately taken up by the spring (the elastic components of the tissue), followed by the relaxation of the internal stress due to the movement of the piston (i.e., the displacement of the viscous components of tissue - the actin and myosin filaments slide relative to each other). This model describes why tissues containing smooth muscle (veins, urinary bladder, uterus, alveoli) are able to sustain significant deformations for a long time without


expending energy as the internal stress relaxes completely.

Fig. 2.3.3. Stress – strain curve of different tissues.

The Kelvin-Voigt model contains a spring with elasticity coefficient, α, connected in parallel with the shock absorber (damper) with viscosity, η. In this model, the stress-strain relationship is given by the expression σ = αε + η. ∂ε / ∂t, where σ is the internal stress, ε is the deformation of tissue, and t is the time. This model is

suitable to describe the viscoelastic properties of bones and cross-striated muscles, included in the arteries and skeletal muscles (Fig. 2.3.2). After applying a constant force, F, (isotonic stretching), the deformation reaches an equilibrium value after a while (time delay). Simultaneously, the internal stress relaxes, but not completely. A beter model of the skeletal muscles could be obtained if a spring is placed at the point labeled by "Q". This model is called an element of Ziner.

Fig. 2.3.4.

Uniform aligning and elongation of the polymer molecules during the deformation of skeletal muscles.

The experimental stress-strain curve for the equilibrium deformation of various tissues is shown in Fig. 2.3.3. The deformation curve of bones looks like that of solids. The curve contains an initial steep linear section corresponding to the area of elastic deformation in solids, which goes directly into the area of the rupture. Thus, there is no horizontal part, corresponding to the plastic zone for most solids, so the bones (like steel and glass) belong to brittle bodies. The bones contain a large amount of inorganic material (calcium hydroxylapatite, about 2/3 by weight) and of fibrillar protein collagen (1/3 by weight). The first component gives rise to the high tensile strength, D, and the second component to the high modulus of elasticity (E or tg ) of bones. The protein forms long elastic threads capable of elastically bending and stretching, and the space between them is filled by crystalline bodies of calcium hydroxyapatite, which provides the bone strength. The teeth contain greater amount of calcium hydroxyapatite, up to 95 %, and correspondingly have greater stiffness. The bone can be elastically deformed up to 3% of its length. In children, the collagen content of bones is higher and, relatedly, the bones are highly elastic. In adults, the calcium hydroxyapatite prevails over collagen and the bones are stiff and brittle - osteoporosis. If a bone is placed in a solution of hydrochloric acid, the inorganic material dissolves and the bone becomes flexible. If the organic material of the bone is removed by burning the bone becomes rigid and brittle. In both cases, the bone does not substantially alter its shape.

There are cavities in some bones (femurs, bones of birds). Surprisingly enough, these cavities do not reduce the strength of bones because the ratio bone diameter / diameter of the cavity is average 11: 8,


which appears to be optimal. The same ratio is proved optimal and for the various iron pipes used in the construction of modern bridges, buildings and more.

The stress-strain curve for the deformation of cross-striated muscles (skeletal muscles, arteries) and smooth muscles (veins and alveoli) showed that, compared to bones, these tissues have a lower strength, but are capable of much larger deformations (Fig. 2.3.3). This is due to the constituent material of these tissues, containing only fibrillar proteins, primarily collagen, and the lack of hard inorganic material.

With skeletal muscles and arteries the initial steep portion of the deformation curve is composed of two sections each having its own inclination angle relative to the X axis. At the section with smaller inclination, the deformation is due to the rearrangement of the helical collagen molecules (Fig.2.3. 4 - the first stage of deformation). The next, steeper section of the curve corresponds to deformation due to the stretching of collagen molecules (fig.2.3.4 - the second stage of deformation). Based on above reasons, the modulus of elasticity of these tissues depends on the magnitude of deformation.

Besides collagen, many biological tissues contain yet another structural protein, elastin, whose modulus of elasticity is about

20 times smaller than that of collagen (Table. 2.2.1). Elastin can be reversibly stretched to about 200-300 %, while the collagen can be stretched to only about 10 %. Different blood vessels contain collagen and elastin at different proportions. For example, elastin predominates in the aorta and in the large arteries, located close to the heart, hence, the strength of the walls of these blood vessels is lower, but their deformability is higher. Increased deformability and elasticity of the aorta and adjacent arteries play an important role in the movement of blood to the capilaries and for the reduction of blood pressure pulsations, caused by the periodic contractions of the heart.

Endoprosthesis means replacing various internal organs and tussues like joints, tendons, heart valves, ocular lenses, blood vessels, bones, intestines, teeth, etc by mechanical analogues (prostheses, dentures). The prosthesis is made of biocompatible materials i.e., which correspond to the requirement of tissue compatibility. Such materials include ceramics, alloy steel and polymers. Prosthesese must have mechanical properties (elasticity, hardness, strength, viscosity) similar to those of the replaced tissues and organs.

2.4. Basic concepts describing the motion of fluids in pipes. Static and dynamic pressure. Viscosity. Hydraulic resistance. Transition from laminar to turbulent flow

In medical physics, the movement of fluids is studied in order to obtain a quantitative view of the flow of blood in circulatory system. This is necessary to assess the energy loss during the flow of blood and to describe the factors disturbing this flow.

Fluids (liquids and gases) do not have their own form and take the form of the container in which they are placed. Each force, F, acting perpendicularly to an area, S, of the surface of the fluid, produces a pressure, P, on this area, designated as static pressure, P = F / S. Due to thermal motion of the molecules of the fluid, this locally exerted pressure is distributed in the whole volume of fluid and allowed to act on the entire surface of the container (Pascal's principle). The external force can be generated by a pump or can be the mere weight of the fluid, or could arise from the inertia properties of fluid. Thus, each fluid (water, mercury) placed in a vertically elongated container, exerts a pressure at the bottom of the

Table 2.2.1. Modulus of elasticity (Young's modulus) of some materials.

Material Young modulus, Е (Pa)

Elastin 105 - 106

Collagen 107 - 108

Bones 2. 109

Veins 8,5. 105

Arteries 5. 104

Muscles at rest 9. 105

Smooth muscle 104

Tendons 1,6. 108

Wood 1,2. 1010

Rubber 5. 106

Steel 2. 10 11


container. This pressure is called hydrostatic pressure, Phst = gH, where H is the height of the container, is the density of fluid and g is the earth gravity acceleration. In fact, the gH is the weight of a column of this liquid with a cross-section equal to a unit (usually 1 cm 2). Similarly, the atmospheric air exerts a pressure on the surface of the Earth, called atmospheric pressure, Patm, due to the weight of the overlying column of air. Consider a U-shaped tube with two shoulders, one filled with a liquid (water, mercury) and the other one is open to the atmosphere (Fig. 2.4.1). In this case the hydrostatic pressures of the two columns will equalize, i.e., Patm = gH. This enables us to measure the atmospheric pressure and express it as the height of water or mercury column.

Fig. 2.4.1. U-shaped liquid manometer. It allows the atmospheric pressure to be expressed by the height of the water or mercury column, exerting an equivalent hydrostatic pressure.

The SI unit of static pressure is N/m2 (Newton per square meter), called Pascal (Pa). We are still allowed to use some old units, outside the SI, as bars, atmospheres, mm Hg and so on.

Atmospheric pressure is a quite large quantity. Under standard conditions (at the sea level altitude, temperature 20°C), the atmospheric pressure is equal to 1 bar = 100 kPa = 0.987 atm = 750 mm Hg = 10200 mm water column. Hence, it is equal to the hydrostatic pressure of about 10 meters water layer. All fluids in human body have a pressure equal to or near the atmospheric

pressure. Any pressure that is created by compression in addition to the atmospheric pressure is called a positive or compression pressure. Such are the blood pressure, intraocular pressure, respiratory pressure. Any decrease in pressure below the level of atmospheric pressure, which is created by swelling is called negative pressure (e.g., the pressure in the pleura of the lung).

Fig. 2.4.2. Flow of incompressible fluid in a pipe with variable cross-section. The changes in pressure and velocity are expressed by the continuity equation and Bernoulli's law.

Hyperbaric oxygenation is a curative treatment, applied to patients with insufficient oxygen concentration in blood (coronary disease, radiculitis, bleeding in the brain). The patient is placed in a pressure chamber with a positive pressure of oxygen thus increasing the saturation of blood with oxygen.

Consider a fluid flowing in a given direction at a velocity, V. In addition to their thermal motion, the molecules of the fluid move translationally in the same direction and with the same averaged velocity as the fluid. Therefore, when these molecules meet any surface on their way, they produce a new pressure on that surface, called dynamic pressure, Pdyn = (.V2) / 2.

Blood is incompressible fluid, because it does not change its density, , while flowing. Let such incompressible fluid flows with velocity, V, through a pipe with variable cross-section, S (fig.2.4.2). Due to the cross-section variation the fluid velocity will also change. For example the cross-section and fluid


velocity will be S1, V1 at the point (1) and S2, V2, at the point (2). Due to some conservation laws in physics, the changes in S, V and p are interrelated. The conservation of matter law gives rise to the so called continuity equation, meaning that J = S1.V1 = S2.V2 = const along each cross-section of the pipe. Here, J, is the flux rate through the pipe. The conservation of energy law gives rise to the Bernoulli's equation, meaning that the total amount of the static pressure, P, hydrostatic pressure, gH, and dynamic pressure, (V2)/2 is constant along the length of the pipe, i.e., P + gH + 0.5 V2 = const.

Based on above equations it follows that at any point where the cross section of the pipe is smaller, the velocity will be higher and the static pressure will be lower (fig. 2.4.2).

Fig. 2.3.4. Separation of a liquid into different layers when a plate floating on its surface is drawn by a force, F.

Consider a plate with area, S, placed on a liquid layer with thickness, x, and let the plate be pulled by a forse, F, parallel to the surface (Fig. 2.4.3). After the initial acceleration, the plate will continue to move with

a constant velocity, V, proportional to the force, F. The F/S ratio is called shear stress. As the fluid adheres to the plate it will also start to flow together with the plate. If the velocity of the plate has low value, the fluid under the plate will flow in separate layers which do not mix to each other. This is a laminar (i.e., layered, streamline or sheet-like) flow. The layer lying under the moving plate has the same velocity as the plate, while the bottom layer is immobile. Therefore, the velocities of the separate intermediate layers are different and a velocity gradient (called shear rate), V/x, could be defined, approximately equal to V/x (s-1). In 1687 the English physicist Isaac Newton found that the shear stress is proportional to the shear rate, ie, F/S = .V/x. Here is called coefficient of dynamic viscosity, dynamic viscosity or simply viscosity.

The viscosity is a measure for the friction between two adjacent layers of the fluid that are moving at different velocities. Based on this, the dynamic viscosity is also called an internal or tr a nsversal friction and also molecular or intrinsic friction. This internal friction slows down the movement of fluid and causes dissipation of energy. The viscosity is a property of liquids and gases expressing their resistance to flow.

In SI, the viscosity is measured in Pa.s, but the old unit poise (P) is also used. 1 P = 0.1 Pa.s. Of practical use is the unit cP (santi poise), 1cP = 0.01 P = 0.001 Pa.s. The ratio / is referred to as a kinematic viscosity, where is the density of the liquid. 1/ is called fluidity, it is greater in fluids with low viscosity.

Liquids in which has constant value, independent on the shear rate, are termed Newtonian liquids (e.g., water, blood plasma). In other liquids depends on the shear rate, and they are referred to as non-Newtonian. The non-Newtonian fluids typically contain large, non-spherical particles which deform and aggregate during the flow. Blood is a typical non-Newtonian fluid due to the red blood cells which have the shape of biconcave discs. Furthermore, while moving erythrocytes deform and aggregate. The dependence of the viscosity of a non-Newtonian fluid on the shear rate is called dilatant behavior. Fig. 2.4.4 shows examples of liquids with different dilatant behavior.

The aqueous solutions of polymers, such as nucleic acids, polysaccharides and some proteins are non-Newtonian fluids. At rest the polymer molecules are intertwined with one another which reduce the fluidity of their solutions during slow movement. With increasing the velocity of flow the polymer molecules progressively deform and separate from each other, whereby the viscosity decreases. With increasing the concentration the polymer solution converts into gel, whereat the flow property is lost. The gel state can be also reached in another way – by sufficiently reducing the temperature of concentrated polymer solution.


Fig. 2.4.4. Flow curves illustrating Newtonian (A) and non-Newtonian (B, C, D) fluid behavior. B – pseudo-plastic material, C, D – dilatant materials.

The polymer solutions have a viscosity, , which largely depends on the concentration, C, of the polymer: = о. (1 + К1.С + К2.С2 + К3.С3 +... ). Here о is the viscosity of pure solvent (water), and K1, K2, etc. are constant coefficients. The first coefficient, K1, has the greatest value and is called characteristic viscosity. It depends on the molecular weight of the polymer, M: K1 = b.Ma. This equation allows the molecular weight of the polymer to be determined measuring the viscosity of the dilute polymer solution.

Consider a fluid with viscosity, , flowing laminarly through a pipe with the length, L, and constant radius, r. The difference in static pressure, P, at both ends of the pipe, i.e., P1 - P2 = P is the force driving the flow. The amount of fluid flowing through a given cross-section for a unit time, 1 s, is referred to as flux of the fluid or flux rate, J. According to the law of Poiseuille, J = P/R, where R = 8L / r4 is called hydraulic resistance of the pipe. R is proportional to the energy loss spent for the passage of a certain amount of fluid through the pipe. On the other hand, at a constant resistance, R, larger flux can be transmitted applying greater driving force, P. Hence, P is also a measure of the energy loss during the flow of fluids.

Fig. 2.4.5. Constriction of an artery at atherosclerosis (above), and formation of blood clot as a result of turbulent flow (below).

The blood supply to internal organs and brain is also a subject to the law of Poiseuille. Let us indicate the blood flux feeding a given organ by J and P input

and Poutlet are the blood pressures at the input and output of the organ. The driving force for blood supply should be P = Pinput - Poutlet P, because the

Poutlet coincides with the venous pressure, which is very low. Therefore, J P/R. If the hydraulic resistance, R, of the organ increases, e.g. due to atherosclerosis, the maintenance of normal blood supply requires the blood pressure to increase, i.e. a hypertension condition must take place. Using proper drugs the blood pressure will be lowered to normal, lowering and the blood supply to the organ, but this is the smaller trouble.

The experimental studies have shown that the linear relationship between the flux, J, of a fluid and the driving force, P, is maintained up to a certain limit of P. The laminar character of the flow is also maintained within the same limits of P. The blood flow in the entire circulation system of human is also laminar at norm. Laminar flow of blood inhibits the formation of thrombuses and plaques in blood vessels. Above a certain limit of P, however, the individual layers of the blood flow begin to mix. This motion is called turbulent (vortex). The appearance of turbulent flow is very dangerous, because this kind of flowing stimulates blood clotting and plaque formation.

Consider a blood vessel with a radius, ro, and a blood flux, Jo, necessary for the metabolism of a given tissue. In atherosclerosis, a plaque of condenced cholesterol and calcium is formed at a given location on the vessel wall reducing the cross-section radius from ro to r1. In case this constriction is strong enough the laminar flow turns into turbulent after this plaque. Turbulent flow of blood gives rise to blood


clots at the site of constriction. At a given moment a part of the clot can detach and will start to float along the arteries. Finally, this clot can plug an important brain vessel causing initially transient ischemic attack in the brain. If the further formation of clots is not prevented, bigger clots can be released causing permanent brain ischemia and stroke, which can have severe consequences for the patient.

The transition from laminar to turbulent flow can be predicted calculating the dimensionless Reynolds number, Re = 2r..V/. In this expression V is the velocity of the fluid in the pipe, r is the radius of the pipe, and are the density and viscosity of the fluid, respectively. After the pipe is constricted at a given place, the Reynolds number, Re, increases due to the increased velocity, V, at that place. In case the Re becomes larger than 1100 a transition from laminar to turbulent motion takes place. When the flow becomes turbulent, the viscosity increases. This results from the appearance of a new, viscous contribution, called eddy viscosity, in addition to the intrinsic viscosity. Depending on the Reynolds number, the viscosity of turbulent flow can be as high as 106 times greater than the viscosity of laminar flow, thus strongly increasing the hydraulic resistance.

The formula of Poiseuille can be rewritten as J = К. r4. P/L, where K is a constant and the quantity, P /L is called pressure gradient or pressure drop. The pressure drop also characterizes the energy loss, for example, during the flow of blood through a blood vessel. Consider several blood vessels with different radius, connected in series, for instance an artery and arteriole. Since the flux of blood is the same in these vessels, it follows that the expression r4.P/L = const for each of these vessels. Concequently, the pressure drop and therefore energy loss in vessels of smaller radius will be larger. For this reason, the biggest pressure drop is obtained across the blood capillaries - about 50 mm Hg column.

2.5. Measurement of flux rate and pressure in fluids. Velocity and pressure of blood in the circulatory system of human. Role of the visco-elastic properties of blood vessels

In medicine, the flux, J, and pressure, P, are measured in the diagnostics of circulatory and respiratory systems, in the apparatus for inhalation anesthesia, heart-lung apparatus, in machines for dialysis and so on.

1. Measurement of pressure. Usually, if not stated otherwise, the term pressure means the static type of pressure. The pressure of a fluid, confined in a vessel, can be precisely measured by directly connecting an appropriate gauge, called manometer, to the vessel (invasive method). Different types of manometers are used:

a) deformation manometers – the measured pressure is fed through a pipe to a suitable hollow deformable spring or bellows, which are stretched according to the magnitude of the pressure. By means of a rotating gear (pinion), this stretch deflects an errow that points to a scale, graduated in the units of pressure.

b) electrical manometers – the pressure deforms a piezoelectric crystal, generating electrical voltage, or sets at a distance the plates of a capacitor, changing the capacity, ect.

Blood pressure in humans is most often measured noninvasively using the compensation method. Firstly, an inflatable cuff is wrapped around the upper part of the arm pressing the arteria brachialis. By increasing the pressure in the cuf, which is measured by a manometer, the blood pressure inside the arteria is fully compensated blocking the passage of blood. The measurement commences by slow deflation of the cuff, allowing the blood vessel to expand. When the pressure in the cuff becomes equal to the systolic blood pressure, Ps, the vessel is unpluged at each heartbeat and the blood starts to flow turbulently giving rise to a characteristic noise detected by a headphones. When the pressure in the cuff becomes equal to the diastolic blood pressure, Pd, the cross-section of the arteria becomes sufficiently large. At this moment the turbulent motion passes into laminar and the noise disappears. The difference Ps - Pd is referred to as pulse pressure. It becomes greater in elderly when the elasticity of the large blood vessels is reduced. A narrower cuff is used for children, because the cuff for adults should give incorrect, reduced blood pressure values. In some emergency cases, the blood pressure is measured invasively with a manometer, directly connected to the artery through a cannula.

2. Measurement of the flux of fluid. The amount of fluid that passes through the cross-section of


the pipe per unit time is referred to as flux (fluence rate, flow rate), J. A mass flux (Jm) or volume flux (Jv) could be determined according to whether the mass or volume of the transported fluid is measured. In fact, if is the density of the fluid, Jm = Jv. . We should always bring in mind that the density, , depends on the temperature and the pressure of fluid! The flux density j = J/S, where S is the area of pipe cross-section.

The following methods, marked by their invasive nature, are most frequently used to measure the flux rate of fluids:

a) F l ux meter with variable pressure drop (Fig. 2.5.1). The tested fluid flows in a vessel equipped with a narrowing device (diaphragm). According to the continuity equation, the velocity of fluid is smaller in front of the diaphragm compared to that after the diaphragm. According to the law of Bernoulli, this velocity difference on both sides of the diaphragm gives rise to a static pressure difference, P, (pressure drop) which progressively increases with the fluid flux. Hence, measuring the pressure drop, P, with a differential manometer we can find the fluid flux with sufficient accuracy.

Fig. 2.5.1. Flux rate measurement of fluids by narrowing device (left) and by a rotameter (right).

b) F l ux meter with constant pressure drop (rotameter - Fig. 2.5.1). This device contains a vertical conical tube equipped with internal conical float. The fluid flows upwards through the

tube and the float hangs somewhere in the middle of the tube. The velocity of the fluid is higher at the upper part than at the bottom of the float. This velocity defference gives rise to a drop in static pressure, displacing the float up or down in order to find a balance between the lift and the weight of float. The vertical location of the equilibrated float will depend on the flux rate - the higher the flux, the higher the float position, indicating the flux value on a nearby scale.

c) Tachometric fluxmeters. The fluid flows through a pipe equipted with an internal small turbine (a propeller, oval wheels etc.). The number of rotations per unit time is proportional to the flux rate. This type of devices are the most accurate fluxmeters.

The following non-invasive methods are used to measure the flux rate of blood:d) Ultrasound fluxmeter. Ultrasound with frequency, o, is directed to the tissue containg blood

vessel. The ultrasound beam is reflected by the moving red blood cells and changes its frequency to (a Doppler effect). The frequency difference, = - о, depends on the speed of red blood cells, respectively, on the flux rate of blood.

e) Electromagnetic fluxmeters. A strong, permanent magnetic field is imposed across the tissue containing a blood vessel. The field acts on the moving ions of blood plasma by a force, called Lorentz force, which is directed perpendicular to the field and to the velocity of blood. Due to their opposite charges, the cations and anions deviate in opposite directions and accumulate on the opposite walls of the blood vessel (the Hall effect). Thus, an electrical voltage, proportional to the velocity of blood, is generated across the blood vessel. This voltage is measured using a sensitive millivolt meter and two electrodes contacting the external aspects of the tissue.

f) Balistocardiography. The patient is placed upright on a flat horizontal plate that serves as a pan in a balance. The pan is allowed to oscillate alongside its vertical axis and these oscilations are measured and recorded. At each systole, the patient’s heart ejects about 70 ml (a stroke volume) of blood into the entrance of the aorta. At first, the ejected portion of blood moves almost vertically, then it deviates and goes down following the course of the aorta. In turn, the body of patient experiences two reactive recoils, the first one directed downwards and the second one upwards. These movements of the patient body are transmitted to the pan thereby they are recorded and used as a measure of the flux of blood at the output of heart.


Human vascular system (fig. 2.5.2) is a closed system of vessels with different diameters, having a viscoelastic walls. Energy for the movement of blood is taken mainly from the contractions of the heart, which works like a rhythmic mechanical pump. Hence, the blood pressure, P, which is produced by heart, has also rhythmic, pulsative character. The amplitude of its pulsations, however, is strongly reduced by the elasticity of the walls of aorta and large arteries.

Fig. 2.5.2. Anatomical arrangement of the vascular network in human.

Fig. 2.5.3 and Table 2.5.1 show how the blood pressure, P, and the mean velocity, V, of the blood vary in different-sized blood vessels. The velocity of blood is greatest in the large blood vessels, arteries and

veins. Therefore, they transmit a large quantity of blood and the disturbance of their integrity results in great blood loss. By contrast, the velocity of blood is low in the capillaries, thereby assisting the food and oxigen supply to tissues and the removal of metabolic wastes through the thin walls of capillaries.

The blood pressure in aorta, arteries and small arteries pulsates and the amplitude of pulsations decreases with the distance from the heart. How to explain this? The walls of aorta are rich in elastin and collagen, proteins which supports their elastic deformation. During the systole, the left ventricle of the heart contracts exerting a force on the valve at the entrance of aorta. The valve opens and transmits a portion of blood, normally about 70 ml, this is the so called stroke volume of the heart. This increases the local blood pressure and causes an elastic expansion of the walls of aorta. The extra volume of aorta takes up the incoming portion of blood thus preventing the blood pressure to reach too high values. The maximal level of blood pressure reached at this moment is referred to as systolic blood pressure, normally about 120 mm Hg. After the systolic contraction the heart muscle relaxes and the blood pressure in aorta begins to fall. This allows the aorta to shrink elastically to its original diameter using the energy accumulated in the elastic deformation of walls. This elastic contraction of the aorta counteracts the fall of blood pressure and maintains it at a minimal level, called diastolic blood pressure, normally about 80 mm Hg. In addition, the diastolic pressure plays an important role closing back the aortic valve and directing the stroke volume of blood towards the peripheral vascular system.

The periodic expansion and contraction of the wall of the aorta is called pulse wave. This is a mechanical wave of elastic deformation which sets at the beginning of aorta and propagates with a certain velocity in the direction of consequent large and small arteries. While the wave advances its amplitude decreases – quenching of the pulse wave. There are several reasons for this quenching; the loss of energy, the decrease in elastin/collagen ratio with related fall of elasticity of blood vessel wall, ect.

The pulse wave is a factor reducing the pulsation of blood pressure. The velocity of pulse wave is, however, greater than the velocity at which the blood flows through arteries. Therefore, there may be moments when the pressure at the distal end of some long vessel (femoral artery) is greater than that at the proximal end. This reversed pressure gradient causes a temporary backflow of blood, called reverse phase flow.

The pulsation of pressure in the arteries leads to similar periodic change in the blood flux, volume, linear dimensions and electrical resistance of tissues. For a given tissue (brain, muscle, limb), the extent of its blood filling is evaluated by recording the changes in its electrical impedance - impedance plethysmography or rheography.


Fig. 2.5.3. Blood pressure and velocity distribution over the various blood vessels in human.

The propagation of pulse wave along the large blood vessels causes their deformation with accompanying sound (noise). At a given point of the artery, this periodic sound can be detected by a sensitive microphone and converted into electric current. After its amplification by an electronic amplifier the current is recorded as a sequence of periodic peaks. This method for studying circulatory system is called sphygmography. Usually, a phonocardiograph equipped with a suitable microphone is used for this purpose. The parameters of the recorded peaks (amplitude, width, and shape) are sufficient to calculate the blood pressure and the viscosity of blood. In addition, the phonocardiograph determines the pulse wave velocity and attenuation rate which allow the calculation of the viscoelastic parameters of blood vessel wall that is clinically significant.

At the entry of the venous system, the blood pressure is too low, only about 10 mm Hg (about 100-150 mm water column). This allows the intravenous inflow of nutrient and medicinal fluids from a container via a cannula directly into the patient's vein. To overcome the intravenous blood pressure it is sufficient the container to be raised at about 50 cm above the lying patient creating enough hydrostatic pressure.

Besides the static pressure, produced by the contractions of heart muscle, the blood circulation is also affected by the pressure created by the weight of blood and, sometimes, by the acceleration produced by transport vehicles. The pressure created by the weight of blood is called hydrostatic and is expressed by the formule p = ρgh, where ρ is the density of blood, g is the earth gravity acceleration and h is the height, measured from the level of heart. Thus, the hydrostatic pressure of blood has positive values below the heart and negative values above the heart. In an adult at upright position, the weight of blood produces an additional pressure of about 100 mm Hg at the level of feet, but reduces the blood pressure in the brain by about 25 mm Hg. If the transport vehicle (airplane, space-rocket) moves with great acceleration the inertia properties (mas) of the blood of passengers (especially pilots, astronauts, motorists) create extra pressure whose positive values are in the opposite direction to the driving force (acceleration). If the acceleration is directed from the head to the feet, the blood pressure in the head will increase causing rapture of blood vessels and leakage of blood. Conversely, if the acceleration is directed from the feet to the head, the blood pressure and blood flux of the head will be reduced causing oxygen and glucose starvation of brain. This condition is, however, more sparing, especially if it has short duration. Overall, a trained person can bear accelerations not greater than 3 x g, i.e., 3 times the acceleration of earth gravity.


Table 2.5.1. Typical values for the average pressure and velocity of blood at the entrance to the indicated branch of cardio-vascular tree, and typical dimensions of the vessels.

Blood vessel

Average pressure (mm

Hg)

Diameter (mm)

Length (mm)

Wall thickness

(mm)

Average velocity

(cm/s)

Total cross-

section (cmLeft

ventricle100 - - - -

Aorta 100 20 500 2 48 2.5Arteries 95 4 500 1 45 20Arterioles 86 0.05 10 0.2 5 40Capillaries 30 0.008 1 0.001 0.1 2600Venules 10 0.02 2 0.002 0.2 250Veins 4 5 25 0.5 1 80Vena cava 3 30 500 1.5 38 8Left and

right atrium3 - - - -

Blood pressure inside the larger blood vessels is much greater than the pressure in outside media. The pressure difference inside and outside these vessels is called transmural pressure, Ptm, which creates a force that stretchs tangentialy the wall of vessels. This stretch deformation gives rise to internal tension in the wall, called tangential elastic stress, . The transmural pressure and tangential elastic stress both are measures of the forces that separate the molecules of wall and tend to rapture the wall. At normal conditions the tangential stress counterbalances the transmural pressure and the blood vessel wall is at steady stretched state. If r and b are the radius and thickness of the wall, respectively, the tangental elastic stress inside the wall is given by the law of Laplase: = Рtm. r /b. To preserve the integrity of the wall, must have subcritical value, which depends on wall tissue structure. Hence, vessels with greater radius (left ventricle, aorta, thoracic artery and arteries in the brain) should have walls with greater thickness, according to the Laplase law (table 2.5.1). Also, the law of Laplace explains why the walls of the left atrium and the large arteries get thicker at hypertension. This is an adaptation response in order to reduce the internal stress in their walls as the hypertention increases the transmural pressure. However, the thickened walls of the left atrium have lower ductility, which requires yet higher blood pressure to fill the left atrium. This steadily worsening condition is called diastolic heart failure.

The radius of capillaries is very small and the above formula indicates very low internal stress in their walls. This allows the capillary walls to be extremely thin, containing one layer of cells only. Such a type of vessel walls in turn facilitates the diffusion and filtration of metabolic substances through them.

In some conditions, the walls of large blood vessels weaken and their transmural pressure causes local expansion of the walls - aneurysm. Other organs outside the blood circulation system also form such local expansions of tissue - diverticula. At some moment the aneurysm may rupture causing hudge hemorrhage, which in most cases ends in death. The aneurysm of cerebral vessels raptures most frequently during hypertensive crisis.

2.6. Hydraulic resistance of blood vessels, and factors affecting it. Resistive and capacitive vessels

At each systole the left ventricle of human heart pushes a certain volume of blood (about 70 ml), known as a stroke volume of the heart, Vo, into the aorta, the main vessel of the systemic part of human circulation. In this case the heart consumes energy equal to the work done to expand the walls of the aorta (p.Vo) plus the kinetic energy of the stroke volume blood. At the same time, the right ventricle sends another portion of blood to the pulmonary vessels of lung, consuming about 5 times less energy compared to that of the left ventricle. As a total the two chambers performe about 1J job per systole. Considering a heart rate 60 systole/min, we could estimate the average power of the heart activity about 1 W! For


comparison, the internal heat production of a human at rest is about 150 W. Thus, we come to the conclusion that human heart consumes too small energy to move the blood into the highly branched circulation of human. How is this done? The main reason for this is that the hydraulic resistance, R, of the circulatory system is generally very low. The basic physical factors that determine the hydraulic resistance, R, of various blood vessels and the mechanisms used to reduce it are analysed below.

There are blood vessels in which the blood flow, J, obeys the equation J = P/R. These are called resistive vessels. Here, P = P2 – P1 is the pressure drop at both ends of the vessel and R is the hydraulic resistance of the vessel, the so called vascular resistance. Aorta, arteries, arterioles, capillaries and venules are resistive blood vessels and for each of them the hydraulic resistance can be defined and determined. The mechanism for the movement of blood in the veins is different, because it is not possible to define a parameter such as hydraulic resistance for them. Such vessels are defined capacitive vessels.

Starting from the heat the blood passes the aorta and enters a system of smaller branched vessels such as arteries, arterioles and capillaries. The total hydraulic resistance, R, of latter system will depend on several factors examined in details below. These include the radius of blood vessel, the arrangement of vessels, the transverse distribution of erythrocytes across the cross-section of blood vessel, the deformability of erythrocytes. The dependence of hydraulic resistance on the blood viscosity is more complex and important and will be separately discussed in next topic.

Fig. 2.6.1. Series and parallel arrangement of vasculature. Movement of blood through blood vessels connected in series (A) and in parallel (B).

1. Dependence of Rtot on the blood vessel arrangement. Let the R1, R2, R3 and so on designate the hydraulic resistances of the diffent types of blood

vessels which conduct the blood flow (fig. 2.6.1). The total hydraulic resistance of the blood vessel system will depend on the way the vessels are arranged. There are arrangements whereat the vessels are connected in series, as in arteries plus arterioles and veins plus venules. Then, the R tot = R1 + R2 + R3 + ... In the parallel arrangement (as in system of blood capillaries), 1/R tot = 1/R1 + 1/R2 + 1/R3 + ... If the resistance of all vessels are the same, i.e., R1 = R2 = R3 = ... = R and N is the total number of vessels, then Rtot = RN for vessels connected in series and Rtot = R/N vessels connected in parallel. It is clear that in the vessels connected in parallel, as in the system of blood capillaries, the total vascular resistance strongly falls.

2. Dependence of R on the radius of the blood vessels, r . Hydraulic resistance, R, of a blood vessel is given by the formula R = 8.L / r4, where is the viscosity of the blood, L and r are the radius of the vessel, respectively. It is obvious that hydraulic resistance, R, strongly depends on the radius of the vessel. The circulatory system of a human contains arteries (r between 3 - 0.5 mm), arterioles (r between 100-10 m), capillaries (r about 2 m), venules (r between 10-250 m) and veins (r between 0.75- 7.7 mm). Hence, the least resistance has the aorta, compared to that of arteries, while that of arterioles is much greater. The resistance, R, of the individual blood capillary will be thousands of times greater than the R of the other vessels.

3. Dependence of R on the transversal distribution of erythrocytes. When a viscous fluid flows laminarly in a vessel of medium size, the fluid velocity, V, will vary across the cross-section of the vessel. V is almost zero in the layer adjacent to the vessel wall and progressively arises in the layers proximall to the axis of the vessel (fig. 2.6.2). Similar radial distribution of the velocities exists when blood flows in the small arteries and arteriols. In addition, the erythrocytes align themselves perpendicular to the axis of blood stream and concentrate in the central core of the vessel. By contrast, there are too low erythrocytes close to the wall where the flow consists mainly of blood plasma. This separation of erythrocytes from the


blood plasma is called wall effect. This effect is explained by the Bernoulli's law, according to which in a laminar flow the static pressure, P, is greater near the walls (zone of low velocity) compared to that at the axis (zone of high velocity). Therefore, the pressure difference pushes the erythrocytes towards the axis of the vessel. The wall effect reduces the friction of blood against the walls and, hence, decreases R tot.

Fig. 2. 6. 2. Velocity and erythrocyte distribution across the blood vessels of medium size.

4. Dependence of R of the blood capillaries on the elastic deformability of erythrocytes. Human erythrocytes have a mean diameter of 7-8 m. Nevertheless, they freely pass through the blood capillaries which have a typical diameter of about 3 - 4 m. This is due to the high deformability of red blood cells as indicated in the fig. 2.6.3. Upon entering the blood capillary each erythrocyte deforms so that its shape becomes like elongated droplet. During the erythrocyte passage the cytosol remains immobile while the erythrocyte membrane scrolls about the cytosol like a tank chain. This is the so called tank-treading motion of erythrocytes. Due to these two effects, the erythrocytes do not tuch the capillary wall, hence, the viscosity of the blood in the capillaries is reduced to 1.7 cP, i.e. practically to the viscosity of pure plasma. This greatly reduces the R. The erythrocytes recover their original biconcave form when they come out from the capillaries.

Fig. 2. 6. 3. Deformation of erythrocytes during their movement through a capillary.

The hydraulic resistance of all the systemic blood vasculature is called total peripheral resistance. Due to above enlisted reasons, the various parts of human circulatory system have different hydraulic resistance, R (Fig. 2.6.4). 97% of the total peripheral resistance is attributed to the resistive vessels (aorta, arteries, arterioles, capillaries and venules), while only 3 % could be due to the pure capacitive vessels (veins). In fact, the R of the capacitive vessels is assessed provisionally.

The total resistance of the capillary system (i.e. the system of microcirculation) is about 27% of that of the entire circulatory system. This is due to the great number of capillaries and to their special arrangement in parallel. The R of the system of microcirculation is subject to continuous change. For example, the thermoregulation in humans includes change in the number of open capillaries, in order to increase or reduce the outflow of heat to the body surface.


The resistance of the arterial part of the circulatory system is about 19 % of the total one. The large resistive blood vessels vary their resistance in broad limits by changing their radius. The R of arteries varies depending on the blood pressure, because the walls of arteries are highly elastic. The arteries are constantly maintained in an open state due to the transmural pressure, while the elasticity of the wall counteracts this expansion. Upon the increase in blood pressure arteries widen and R diminishes, facilitating the rapid draining of the blood towards the venous system and reducing the blood pressure. The ability of blood vessel walls to expand and contract passively with changes in pressure is an important function of large arteries and is called wall compliance or vascular compliance . Plaques or elastin degradation reduce the compliance of a diseased blood vessel.

Fig. 2. 6. 4. Hydraulic resistance of different parts of the human blood circulatory system. The total resistance of the entire system is taken as 100%.

Despite blood pressure, there are nervous and humoral factors which also affect the radius of large vessels and, respectively, their resistance. The walls of these vessels contain cross-striated muscle fibers. Under the impact of nerve impulses these muscles contract and maintain the vessel walls in a tense, partially compressed state (muscle tonus). The higher the frequency of the impulses, the greater is muscle contraction and vessel shrinkage and, respectively, the greater will be the

vessel resistance. Muscle fibers are also contracted by specific bioactive substances (serotonin, prostaglandins) released in the blood plasma. By contrast, other substances (nitric oxide) relax these muscles and dilate blood vessels (vasodilation) reducing their resistance.

The penitration of air bubbles in a small blood vessel can clog it (gas embolism) and, consequently, could cause necrosis of respective organ or tissue.

The inlet pressure of the venous system is insufficient to move the venous blood to the heart. Blood flow in the veins is driven by other mechanisms, different to the pressure drop, and hence, it does not obey the law of Poiseuille. There are three such mechanisms, the first one is the contraction of the body muscles (skeletal muscle pump) and the second one is the negative pressure produced in the pleural cavity by the breathing muscles (respiratory pump). The third mechanism relies on the valves, distributed along the length of each vein, which admit the blood flow in one direction only, to the heart. The walls of the veins contain large amounts of smooth muscles, which allow huge expansion of veins with full relaxation of the internal stress. The normal state of these valves is closed, so when the blood flows towards a given valve, the vein extends, allowing accumulation of large amounts of blood in front of the valve. When the blood accumulation reachs the critical mass the valve opens, transferring the excess blood behind the valve until the valve closes again. Thus, the venous blood flows unidirectionally to the heart, the reverse flow is not allowed by valves. In view of their role, the veins are reffered to as capacitive vessels and the concept of hydraulic resistance can be defined only in a conventional sense for them. The term "capacitive vessel" reflects the ability of the veins to expand significantly in compliance with the volume of blood they could collect. This allows the veins to be considered as a receptacle for blood.


2.7. Viscosity of blood and methods for its measurement. Factors determining the viscosity of blood. Deformability of erythrocytes

Blood viscosity originates from the friction between the molecules of the plasma and suspended cells, primarily red blood cells, as the blood flows. Viscosity is an important parameter of blood, directly affecting the hydraulic resistance of blood vessels. Blood is non-Newtonian fluid with apparent dilatant behaviour, because its viscosity is shear-rate dependent and has different values depending on the flow rate in different blood vessels. Viscosity, , of blood containing separate, non-aggregated erythrocytes is 4.5 cP. In some diseases (anemia, cirrhosis of the liver, tuberculosis), it is lowered, while at polycythemia, atherosclerosis, myocardial infarction, diabetes, cancer of the bone marrow and increased plasma density it may increase significantly. As the hydraulic resistance of the blood vessels is proportional to it will also be increased at such pathologic conditions. This will severely reduce the blood supply to tissues and will increase the energy expenditure of heart. These facts all determine the diagnostic significance of blood viscosity and require more knowlage about the factors which affect it.

Fig. 2. 7. 1. Schematic diagram of rotational viscometer.

There are three frequently used classical methods (apparatus) for determinination of the viscosity of liquids and blood;

a) Capillary method - suitable for Newtonian liquids of low viscosity (water, blood plasma, cerebrospinal fluid). The tested liquid is pushed to pass through a tiny capillary with diameter about 100 m and length L at a constant pressure difference. The fluid with volume, W, will poor out for the flow time, t. According to the law of Poiseuille, W = K.t/, where K is a constant. This equation allows the determination of through measurements of t and W. K is separately determined testing a liquid with known viscosity. In

practice, the determination of K is avoided testing the liquid under study and a reference liquid (distilled water) with known viscosity. At 20°C, the viscosity of the water is 0.01 P = 1 cP (centipoise). In this case the so called kinetic or relative viscosity, /water, is determined.

Table 7. 1. 2. Viscosity of some liquids at 20 °C.

Liquid

Viscosity (cP)

Liquid Viscosity (cP)

Water 1.0 Blood plasma

1.64 – 1.69

Ethyl alcohol

1.2 Mercury

1.55

Blood 4.2 - 6.0

Glycerin

1500


b) Rotational viscometer - suitable for non-Newtonian fluids with a high viscosity (blood, polymer solutions) - Fig. 2.7.1. It consists of immobile container with proper shape (cup, cylinder, plate, cone) which contains the sample fluid. Driven by an electric motor a rotor with the same shape is submerged and rotates inside the sample fluid. A helical spring is fixed on the shaft that rotates the submerged rotor. The more viscous the liquid, the greater resistance it will exersice on the rotation of the rotor and the spring will twist at larger angle, . The angle, , is indicated by an arrow and its value directly points out the viscosity of liquid. To assess the dilatant behaviour of tested fluid, the viscosity measurements are carried out at various shear rates (velocity gradients), i.e., at different distances between the rotor and outer container, and at various angular speeds of electric engine. The rotary viscometer is another similar type of viscosimeter containing rotor, submerged in a sample liquid. The torque necessary for the rotation of its rotor is proportional to the viscosity.

Fig. 2. 7. 2. Dependence of the viscosity, , of an erythrocyte suspension on the hematocrit and on the erythrocyte deformability. The suspension medium is blood plasma homologous to erythrocytes.

c) The falling ball viscometer – the viscosity of media with high density (glycerin, pitch) is

determined using a spherical ball with higher density than that of the tested medium. The ball is allowed to fall accross the tested medium. Ball weight accelerates the falling, while medium viscosity resists the falling by the force of Stokes which increases with the square power of ball velocity. The acceleration continues untill the two forces equalize. Since then the ball falls with a constant velocity which depends inversely to the viscosity. Thus, the determination of the viscosity is reduced to measuring the velocity of ball falling. The method is not good for viscoelastic fluids.

Fig. 2.7.3. Blood viscosity dependence at laminar flow with different shear rates (left). Right: Formation of erythrocyte aggregates type "stacked coins" 1 -single cells at high shear rates, 2

– aggregated erythrocytes at low shear rates.


In addition there are more recent methods to measure viscosity. The ultrasound viscometer uses ultrasound wave propagating through the sample fluid. Measured are the speed and attenuation of the wave, which are used to derive the viscosity of fluid. These viscometers are able to obtain the frequency dependence of fluid viscosity and the relaxation time between the pure elastic and viscoelastic state of the fluid. The method is good for high viscosity fluids and sample fluids with small volume.

The vibration viscometers find broad application. An elastic plate is immersed into the tested sample fluid and vibrated with a constant displacement magnitude. By detecting the power necessary to maintain the vibration, the viscosity of the liquid is measured. The physical quantity to be measured is expressed as “viscosity × density.”

The viscosity of blood is indirectly determined measuring the erythrocyte sedimentation rate (ESR in mm/hour). An anticoagulated blood sample is placed in a vertical tube for a certain time to allow the sedimentation of erythrocytes. The latter obeys the same mechanism described for the falling ball viscosimeter. Under these conditions, the erythrocytes are agglutinated into the so called “stacks of coins”, nevertheless, the parameter ESR is widely used in the clinical study of the blood of patients. The higher the viscosity of blood, the lower is ESR.

Blood is a suspension of blood cells (mostly erythrocytes) in a solution, called blood plasma. Blood plasma is an isotonic aqueous solution of salts (mainly NaCl) and proteins (mainly albumin). It is a Newtonian fluid having a viscosity of about 1.7 cP, which is independent of flow rate. By contrast, blood is non-Newtonian fluid, because its viscosity is affected by its velocity of flow. This is due to the ability of erythrocytes to deform and aggregate.

The following experiment illustrates which are the chief parameters impacting the viscosity of blood. Let us progressively add isolated human erythrocytes to an isotonic solusion which contains NaCl and albumin at the same concentrations as those of the blood plasma. The volume of erythrocytes to the volume of suspension ratio is called hematocrit value. The intact erythrocytes possess high elastic deformability. They do not aggregate in such suspension, i.e., they freely flow as single cells, well separated from one another. The viscosity of the obtained suspension could be measured by a rotational viscosimeter. With increasing the hematocrit value, the resulting viscosity, , of erythrocyte suspension increases as well (Fig. 2.7.2). At a hematocrit of 45%, equal to that of human blood, the erythrocyte suspension obtains a viscosity of 4.5 cP, exactly coinciding with that of native human blood. However, if the added erythrocytes are devoid of their elastic deformability by pre-fixation with glutaraldehyde, the viscosity of the resulting suspension will be much greater than in previous case (Fig. 2.7.2). This experiment shows that the viscosity of blood, containing non-aggregated single cells, depends mainly on two factors: the concentration of erythrocytes in blood and erythrocyte deformability.

Fig. 2. 7. 4. Principle of thromboelastography (left). Thromboelastogram of blood sample (right).


Unlike the above erythrocyte suspension the viscosity of native blood depends on a third factor, the flow rate. In vessels where the flow rate is low (as in veins and venules), the erythrocytes aggregate forming the so called “stacked coins” and viscosity is very high (Fig. 2.7.3). An obligatory factor for the formation of such cell aggregates is the presence of some fibrillar proteins (fibrinogen and gamma-globulin) in the blood plasma. At higher flow rates (in the arteries and arteriols), the erythrocyte aggregates disintegrate and the blood viscosity drops to the normal value of 4.5 cP.

The change in the viscosity of blood is used to determine the parameters of the blood clotting system. For this purpose the method of thromboelastography is used which senses the change in viscosity during the intentional clotting of blood sample. A metal disk, suspended on a thin torsion thread, is immersed into the blood sample which is placed in a small cuvette. The cuvette is made to swing around an equilibrational position. The swinging motion of the cuvette is transmitted through the blood to the metal disk and, thereby to the torsion thread (fig.2.7.4). Swinging of the thread is converted into alternating electric signals, which are stored into computer memory. At a given moment a clotting agent is added to the blood sample. Starting at the addition of coagulation agent, the amplitude of disk swinging initially is small. Later on it progressively increases with the initiation of clotting to reach a maximum. Next, the fibrinolysis is activated causing disintegration of blood clotts and reduction of swinging amplitude. The method allows measuring all important parameters of hemostasis; the response time (R), the coagulation time (T), the rate of clotting (angle α), and the elasticity of resulting clot (characterized by the amplitude ma). The slope of curve after complete clotting corresponds to the activity of thrombolytic enzymes (fibrinolysis).

Erythrocytes have a biconcave shape and resist to its deformation. In the large blood vessels and blood capillaries erythrocytes sustain elastic, fully reversible, deformations. The reversible deformability of red blood cells is a major factor for the normal blood circulation. The following three factors strongly contribute to erythrocyte deformability:

a) Viscosity of the cytoplasm of erythrocytes . Upon increasing the hemoglobin concentration in cytoplasm, erythrocyte deformability worsens.

b) Deformability of erythrocyte plasma membranes . The trilaminar structure of erythrocyte membranes facilitates the high deformability and elasticity of erythrocytes. In many diseases, the erythrocyte membrane deformability deteriorates.

c) The plasma membrane area to cell volume ratio. Erythrocyte membrane itself is not capable of area expansion however, it practically has no resistance to bending deformation. In respect to the volume, the membrane of erythrocytes has excess area allowing significant changes in the shape and volume of erythrocytes without tearing the membrane apart. Thus, the large cell surface area to cell volume ratio renders excess stability to erythrocytes during the frequently encountered mechanical deformations in blood circulation. Erythrocyte stability is determined using the hypotonic fragility test.

The deformability of erythrocytes and erythrocyte membranes is measured using the following methods:

1) Under the action of a certain pressure drop a portion of blood is compell to flow through a filter with pore size slightly smaller than the diameter of the erythrocytes. The volume of blood filtered out for a given time interval indicates the deformability of cells.

Fig. 2. 6. 4.

Measurement of the deformability of

erythrocytes by their aspiration into a

micropipette.

2) Under microscope and using a

micromanipulator, a micropipette is touched to a chosen erythrocyte (Fig. 2.6.4). A negative pressure, produced in the micropipette, causes a portion of the erythrocyte to be sucked. The length of the sucked portion (so called language) is measured and considered as indicative of the deformability of erythrocyte membrane.

3) The deformability of erythrocytes is frequently measured by ektacytometer. The blood


sample is passed through a thin, transparent cuvette. Depending on the flow velocity of blood sample (i.e., on the shear rate), erythrocytes become extended. In older equipments the elongated cells are fixed using glutaraldehyde, and the degree of deformation is measured under a microscope. In modern optical techniques the degree of cell deformation is measured by the diffraction pattern of the laser light scattered by the suspended cells.

2.8. Physical mechanism of pulmonary respiration. Elastic and viscous energy losses at breathing. Role of surface pressure and surfactants in the volume stability of lung alveoli

Lung represents a cluster of numerous alveoli, which are connected with each other and with the atmosphere through a system of tubes (bronchi and trachea). Alveoli are the active part of the lung because they can change their volume, taking up and giving off air to and from the atmosphere. The walls of alveoli are very thin, in addition their outside aspect is in constant contact with the blood. This facilitates the diffusion of gases (O2 and CO2) across the wall of alveoli. Pulmonary respiration is a cyclic process of contraction and expansion of the volume of the alveoli during which a continuous ventilation of the alveoli takes place. This process can be modeled and best described with the laws and concepts of physics.

The inside aspect of each alveolus is covered with a thin layer of water that plays a major role in the mechanism of breathing. This role is explained by the forces of surface tension of this layer and the pressure they create. Strong forces of attraction act between the molecules of a liquid. For a molecule located inside the volume of liquid, these forces counterbalance each other. For a molecule placed on the surface of the liquid a part of forces remain unbalanced. The resultant force is directed inward, perpendicular to the surface of the liquid, causing the liquid to shrink to a volume corresponding to the minimum surface area. These surface forces are characterized by the quantity surface tension, , which is

equal to the work done against these forces in effort to increase the surface area by unit. For a spherical surface with a radius r, the surface tension creates a centripetal pressure, p, which according to the Laplace's formula is p = 2. /r.

Fig. 2. 8. 1. Anatomical view of human lung.

The surface tension of a liquid, , can be strongly reduced dissolving surface-active agents or surfactants (soaps, detergents, flocculants) in the liquid. The molecules of the surfactant largely concentrate at the liquid surface, making it more hydrophobic, thereby reducing .

The expansion and contraction of the lung and alvioli can be modeled using the cyclic work of a

bellows. The latter works in two stages, inspiration and expiration (Fig. 2.8.2). The first stage - inspiration is the active phase in which the external force F extends the skin bag of the bellows overcoming the elastic resistance of the spring. During this phase, the spring is stretched out and a large portion of mechanical energy is accumuled in the form of energy of elastic deformation. During this stage the bellows swells and the created inside negative pressure (respiratory pressure) attracts atmospheric air through the hydraulic resistance, R, of the duct. Thus, except the elastic resistance of the spring, the force, F, must overcome and the hydraulic resistance of the duct during this phase. Accordingly, the energy consumed is divided into two parts - an elastic energy loss to overcome the elastic forces of the spring and viscous energy loss for transferring the air molecules from the atmosphere into the bellows. The second phase of expiration is passive one, because the force, F, ceaces and the contraction of the bellows is


performed at the expense of the elastic energy collected in the stretched spring. This energy is spent entirely for overcoming the hydraulic resistance of the duct.

A more realistic model of the lung is shown in Fig. 2.8.2. Similar to the bellows, the lung works in two stages, inspiration and expiration, with an intermediate state of rest and relaxation between them. The lung is located inside the so called pleural cavity. The latter is surrounded by outside airtight wall, pleura, and contains air. The gass law of Boyle - Marriott applies for pleural cavity whereat at a constant temperature the product of the pressure, P, and volume, V, of an ideal gas is constant: pV = const. If the mean radius of alveoli is r, the water layer on the inside aspect of alveoli gives rise to surface tension, , and corresponding centripetal pressure, p = 2. /r, which causes shrinkage of the alveoli. When the lung is at rest, this pressure shrinks the alveoli and thereby the pleural cavity is enlarged producing a negative pressure of about -5 cm water column, called intrapleural pressure - Ppl. This is the main role of the water layer of the alveoli in the mechanism of breathing.

Fig. 2.8.1. Bronchi, bronchial tree, and lungs.

A portion of the pleura is attached to the diaphragm and ribcage of the thorax. This portion represents the moving part of the pleura. When the diaphragm and intercostal muscles contract the moving parts of the pleura expand the pleural cavity (Fig. 2.8.2). Addressing the mechanical model in Fig. 2.8.1, this expansion can be defined as the first stage of pulmonary respiration when the muscle force, F, is activated. The force further broadens the pleural cavity and reduces the interpleural pressure even more to about -5 to -10 cm water column. The deapened negative intrapleural pressure causes expansion of alveoli and lowers the intraalveolar (intrapulmonary) pressure. The latter drives the air to flow into the lungs, down its pressure gradient, until intrapulmonary pressure equalises atmospheric pressure. The hydraulic

resistance, R, of the trachea restricts the magnitude of the incoming air flow.

Fig. 8. 2. 2. Mechanical model of

human lung.


As in the bellows, the muscular force, F, overcomes two resistances in this stage. The first one is the purely elastic resistance originating from the elastic deformation of the thorax tissue, while the second one is a viscous resistance. The latter includes the hydraulic resistance, R, of the trachea, the surface tension of the aqueous layer of the alveoli, and the resistance corresponding to the viscous deformation of the thorax tissues.

In the second stage (expiration), muscle force, F, ceases off. As with the bellows, the energy stored in the elastic deformation of tissue is used to compress the pleural cavity, causing shrinkage of the alveoli and expulsion of their air to the atmosphere. During this stage the centripetal pressure of the alveoli helps the contraction of the alveoli and extruding their air to the atmosphere.

Fig. 2. 8. 3. Physical model of the lung. For convenience, only one alveolus is shown.

The work, E, done by the thorax muscles during a small change in the lung volume, V, is given by the expression E = p.V, where p is the respiratory pressure. The total energy consumed in one cycle of inspiration - expiration is equal to the work done

by the muscles for this cycle. It can be determined from the pV plot of lung (Fig. 2.8.4), where V is the volume of alveoli (lung volume) and p is the respiratory pressure. The pV plot is determined by the so called apparatus "iron lung" used to diagnose lung function. The dashed line shows the p – V relationship during one cycle of expansion - contraction of the alveoli, provided the total resistance is purely elastic one. Indeed, this resistance has both an elastic and viscous nature. Upon expansion of the alveoli in the initial area A on the PV-diagram of the lung, the muscles overcome mainly the centripetal forces of the surface tension of aqueous water layer. In the final zone B, the forces generated by the elastic deformation of the walls of the alveoli sharply increase. The contraction of the alveoli is caused by the same forces, taken in reverse order. The curves corresponding to the inspiration and expiration do not coincide; this is reffered to as hysteresis. The irreversible viscous loss of energy during the breathing is the reason for this hysteresis.

Fig. 2. 8. 4. Relationship between the lung volume and the respiratory pressure during one

cycle of inspiration - expiration.

The total work produced during the inspiration against the elastic resistance is given by the dotted area on the pV-plot in Fig. 2.8.4. This work represents the elastic energy loss, supplied by the muscles during the active phase of the inspiration. This energy is stored in the elastic deformation of lung tissues. In lung, as well

as in bellows, this energy is returned and consumed during the next stage, the expiration. Another part of the muscle energy is spent for overcoming the viscous resistance. It is given by the shaded area on the pV plot and corresponds to the viscous energy loss. The total energy consumed by the muscles in one cycle is


equal to the sum of the elastic and viscous energy losses. The elastic energy, viscous energy and total energy are to be determined in order to correctly diagnose some pulmonary diseases.

Fig. 2.8.5. shows the dependence of the elastic and viscous energy loss on the frequency of breathing. The total energy loss has minimum at about 15 breaths per minute, which coincides with the normal respiratory rate. In certain lung diseases one of the two types of losses is increased, which changes the frequency and the depth of breathing.

The expansion of an alveolus is explained in the same way as the stretching of the blood vessel. A transmural pressure, Ptm, is also imposed across the wall of the alveolus. It is equal to the pressure difference on both sides of the wall, i.e., P tm = Ppl - Pres, where Ppl is the negative pressure inside the pleura and Pres is the respiratory pressure inside the alveolus. To expand the alveolus, the transmural pressure must overcome the sum of the centripetal pressures of the aqueous layer (the Laplace pressure 2. /r) and the pressure, Pel, generated by the elastic deformation of the wall of alveolus. Contrary to transmural pressure, these pressures both act to shrink the alveolus. At the points of maximal extension and maximum contraction of the alveolus all forces acting on its wall are in equilibrium, as expressed by the following equation:

Рtm = Рpl - Рrep = 2./r + Pel

When the pleura expands the intrapleural pressure, Ppl, becomes more negative causing an increase in the Ptm and swelling of alveolus, i.e. the radius r increases. Conversely, upon relaxation of muscles, the pleura cavity shrinks leading to increase in intrapleural pressure Ppl, decrease in the Ptm and shrinkage of alveolus, i.e. the radius r decreases. The above equation is obeyed because Pel is a variable - it increases with the expansion of alveolus and decreases with its contraction. In the state of full contraction the transmural pressure is counterbalanced essentially by the Laplas pressure of the aqueous layer, while in the state of full expantion, this pressure is equilibrated mainly by the Pel.

Fig. 2. 8. 5. Elastic, viscous and total energy losses of lung as dependent on the breathing frequency.

As we see the water layer and related Laplas pressure are important factor in lung activity. However, the lung contains alveoli with different radius which gives rise to a serious problem because these alveoli are interconnected to each other. The forces of surface tension, , produce greater Laplace pressure, 2/r, in the alveoli with smaller radius compared to that in the alveoli with greater radius. The higher Laplace pressure of small alveoli will overcome

the lower Laplace pressure in large alveoli causing transfer of air from the small alveoli to larger ones. As a result, the larger alveoli will expand even more and their Laplas pressure will drop even further. This self-accelerating process will continue until the final collapse of the small alveoli and their drop out from the lung function. Such an outcome will strongly reduce the total lung area. This dangerous tendency is called mechanical instability of small alvioli.

However, there is a mechanism counteracting to this mechanical instability. This mechanism uses special surface active substancies, surfactants, which have the same concentration in the water layers of all alveoli, big and small ones. Surfactants lower Laplace pressure and remove the tendency of small alveoli to collapse. How do they manage to do this? When a given alveolus begin to shrink (or swell) its Laplace pressure increases (decreases) proportional to 1/r, while the area of its water layer decreases (increases) more strongly in proportion to r2. The same is valid and for the surface concentration of the surfactant which strongly concentrates at the shrinking alveoli and dilutes in swelling alveoli equalizing the Laplas


pressure in the two groups of alveoli. As a result the presence of surfactants prevents the tendency for shrinkage of small alveoli and preserves the active area of lung.

In addition, surfactants make the breathing easier because decreasing , they reduce the resistance in the initial section of the p-V plot. In some cases, however, as in some newborns, the amount of surfactants or their effectiveness are reduced, hence, the smallest alveoli are not involved in respiration and the newborn can not obtain enough oxygen.

2.9. С едиментация и центрофугиране. Р азделяне на биомакромолекули и клетки чрез центрофугиране. Д иференциално, равновесно и скоростно

центрофугиране. С едиментационен коефициент.

Частици, суспендирани в определена среда, могат да бъдат отделени от нея чрез седиментация (утаяване) и центрофугиране, при условие че имат плътност различна от тази на плътността o на средата. При частици, по-малки от около 0.5 м, брауновото движение е съществено и пречи на тяхното утаяване, ето защо те могат да се отделят само със центрофугиране. При утаяването, движеща сила за отделянето е теглото на частиците G = .V.g, а при центрофугирането - центробежната сила Fцф = .V.2.r, породена от въртенето на суспензията (фиг. 2.9.1). Тук g е земното ускорение, V е обемът на частиците, - ъгловата скорост на въртене и r – разстоянието от частицата до оста на въртене. На движещата сила противодейства подемната сила на Архимед Fa, която при утаяването е равна на o.V.g, а при центрофугирането на o.V.2.r. Резултантната сила, действаща върху частиците ще бъде:

Fсед = G - Fa = .V.g - o.V.g = ( - o).V.g при седиментацията и

Fцф = Fцб - Fa = .V. 2.r - o.V. 2.r = ( - o).V. 2.r при центрофугирането.

Фиг. 2. 9. 1. Сили действащи на суспендираните частици при утаяване и центрофугиране.

Съотношението на ускоренията, създавани при центрофугирането (aцф) и при седиментацията (aсед) е същото, както на съответните резултантни сили: aцб/ aсед = Fцф / Fсед = 2.r / g. Оттук, ускорението при центрофугиране

се изразява обикновенно в единици за земно ускорение ( х g). При ценрофугиране с високи обороти, разделянето става много по-бързо спрямо утаяването. Времето необходимо за утаяване на частиците зависи обратно пропорционално от квадрата на оборотите. Така, ако оборотите се увеличат два пъти, времето за отделяне на частиците се намалява четири пъти. Суспензията се поставя в подходящи гнезда на центрофужен ротор, който може да бъде ъглов ротор или летящ ротор (фиг. 2.9.2). За да се избягнат вибрациите по време на центрофугирането, роторът трябва да бъде добре балансиран, а оста на въртене е препоръчително да бъде деформируема (иглена ос). Балансирането на ротора става, като кюветата съдържаща суспензията и една друга кювета със същата форма и тегло се поставят в гнезда симетрично разположени спрямо оста на въртене.


Фиг. 2. 9. 2. Разрез на ъглов и летящ ротор.

Извесни са няколко начина за разделяне на суспендирани частици чрез центрофугиране:

1. Ако в суспензията има няколко фракции частици с различна плътност, разделянето става чрез няколко последователни центрофугирания. Първото центрофугиране е с ниски обороти при което се отделят най-плътните и едри частици като утайка (седимент). Надутаечната среда (супернатанта) се центрофугира отново при по-високи обороти за да се отделят следващите по плътност частици и т.н.- диференциално центрофугиране. Така в суспензия, съдържаща клетъчни ядра и митохондрии, най-напред се отделят ядрата чрез нискооборотно центрофугиране (1000 об/мин), след което от супернатанта се отделят митохондриите чрез центрофугиране със средни обороти (15 хиляди об/мин).

2. Суспензията се надслоява върху носеща среда, чиято плътност нараства отгоре надолу равномерно или на скокове. След центрофугирането, отделените частици застават на местата, където тяхната плътност се изравнява с тази на средата. Това е градиентно или зонално центрофугиране. Нарича се още равновесно или препаративно центрофугиране. Извършва се с летящ ротор при много високи обороти (над 50 000 об/мин) в т.н. ултрацентрофуги.

Фиг. 2. 9. 3. Принципна схема на ултрацентрофуга за определяне на седиментационния коефициент S и молекулна маса M на частици.

3. Ако разтворените частици са макромолекули, те могат да се разделят в зависимост от големината на тяхната

молекулна маса като се използва неравновесно центрофугиране, наречено скоросно утаяване. В процеса на центрофугиране, на всяка една частица действат три сили -силата на триене между частицата и околната среда (нарича се сила на Стокс), центробежната сила и подемната сила на Архимед. Тези сили взаимно се уравновесяват, поради което частиците се движат равномерно и праволинейно със скорост B в посока надалеч от оста на въртене. Разтвора на биополимера се поставя в прозрачна кварцова кювета и се центрофугира със свръхвисоки обороти (100 хиляди об/мин) във вакуумна хладилна центрофуга (фиг. 2.9.3). При утаяването се образува граница между разтворените частици и разтворителя, която също се движи със скорост В в посока надалеч от оста на въртене. От двете страни на тази граница ще имаме различен коефициент на пречупване на светлината. Тази граница се установява, като през кюветата се пропуска светлинен лъч, който след това попада върху фотографска плака. На получената снимка, границата се очертава като една тъмна ивица, ако в разтвора имаме молекули от един вид, или няколко ивици ако молекулите са от


различни фракции. Скоростта В се определя чрез няколко фотографии, снети през известен интервал от време.

По големината на скоростта В се определя молекулната маса M на частиците. На центробежната сила Fцб=M. 2.r, противодействат архимедовата сила Fарх = M.v. o. 2.r и силата на триене (сила на Стокс) Fтр = k.B. Тук k е коефициент на триене, а v е парциалният специфичен обем на разтвореното вещество, т.е., нарастването на обема на разтвора след добавянето на 1 g от разтворените частици. От своя страна, M.v представлява парциалния молекулен обем (обема на една макромолекула). Заместваме тези изрази за силите и получаваме

M. 2.r = M . v. 2. r . o + k . B

След преобразуване на това уравнение се получава

M .(1 - v. o) . 2 . r = k . B

Полагаме S = B /(2 . r) (седиментационен коефициент) и за молекулната маса получаваме

M = k.S/(1-v.о).

Седиментационният коефициент има смисъл на подвижност, т.е., скорост с която се движат частиците под действие на стандартна по големина сила, в случая единица центробежно ускорение. За различни частици (биомакромолекули, полимери, рибозоми и др.), седиментационният коефициент се измерва като се измери скоростта В и по горната формула се изчислява молекулната им маса.

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