Biomolecular Hydrodynamics

Basics hemodynamics.


Here we survey modern methods for measuring the hydrodynamic properties of biological macromolecules and for calculating the hydrodynamic properties of useful models for biomolecules.

Hydrodynamic properties are quantities like sedimentation and diffusion coefficient, rotational relaxation time, intrinsic viscosity, and viscoelastic relaxation time. They enable one to determine how rapidly molecules translate and rotate in solution, and how they influence the steady state and time-dependent viscosity of solutions. These measured properties, in turn, when interpreted in terms of suitable models, give us information about the molecular weight, size, hydration, shape, exibility, conformation, and degree of association of biological macromolecules.

Like so much of science, hydrodynamics has a long history, but has been dramatically hanged by recent advances in experiment, theory, and computation. Classic textbooks tell of the experimental work of Fick in 1855 on diffusion, Svedberg in the 1920s on sedimentation, and Poiseuille in 1846 on viscosity. Theories for simple shapes such as spheres and ellipsoids were worked out by Stokes (1847), Einstein (1906), and Perrin (1936). However, recent experimental developments such as dynamic laser light scattering, nanosecond fluorescence depolarization measurements, fluorescence recovery after photoblea hing,.uorescence correlation spectroscopy, pulsed field gradient NMR, and improved analytical ultracentrifuge design have made measurements of hydrodynamic properties more convenient, precise, and refined. At the same time, development of new theoretical and computational tools have enabled calculation of the properties of complex, realistic molecular models.

We present these new techniques, along with the older but still useful ones, in a way that we hope is suitable for an audience of students and researchers in biophysics in biochemistry. We forego the heavier experimental and mathematical details and focus on physical principles, results, and applications. We also provide links to Web-based resources and computer programs that an be used to calculate hydrodynamic properties of atomic-level structures.


Fluids In motion


Until now we have been discussing fluids which were at rest. Moving fluids are also of great importance. We can learn a great deal by examining the flow of fluids through pipes.Typical behavior is shown in

Fig. 1.Examples of various features of flow in tube: (a) velocity profile; (b) streamlines; (c) turbulent flow.

As we see in part a of the figure, fluid does not simply move down a pipe in a pluglike form.


Figure 1.

Instead, the fluid close to the walls of the pipe moves scarcely at all. As shown by the velocity profile, the fluid near the center of the pipe moves most swiftly. This variation in speed across the pipe's cross section causes the fluid near the center of the pipe to rub past the outer portion of the fluid. As a result, friction energy loss occurs in the flow process. More will be said about this later.

Part b of the figure shows the streamlines of simple flow. These lines show the path a tiny particle in the fluid follows as it moves along the pipe. Flow such as this is termed laminar flow. Notice also in this figure that the flow velocity changes as the cross-sectional area changes. The fluid velocity is lower in the large cross-sectional region. Part c of the figure shows what happens if the flow becomes too swift past an obstruction. The smooth flow lines no longer exist. As the fluid rushes past the obstacle, it starts to swirl in erratic motion. No longer can one predict the exact path a particle will follow. This region of constantly changing flow lines is said to consist of turbulent flow. As you might expect, considerably more energy is lost to friction effects in turbulent flow than in laminar flow. Turbulent flow also occurs in a uniform tube even when an obstacle is not present provided the flow rate is large enough.




We saw in the last section that friction effects occur in a flowing fluid. This effect is described in terms of the viscosity of the fluid Viscosity measures how much force is required to slide one layer of the fluid over another layer. Substances which do not flow easily, such as thick tar or syrup, have large viscosity. Substances (like water) which flow easily have small viscosity.

To give quantitative meaning to viscosity, we refer to the hypothetical shear-type experiment shown in Fig. 2. There we see two parallel plates, each of area A, separated by a distance /. The region between the plates is tilled with a fluid whose viscosity we shall denote by h (Greek eta). In order to move the top plate with speed v relative to the bottom one, a force is required.

As the upper plate moves, layers of the fluid slide over each other. The force will be large if the fluid has a large viscosity.We defineh by In the SI, the units of viscosity [from Eq. (1.)] are newton-seconds per square meter.


Figure 2.

Most frequently, h is tabulated in terms of a unit called the poise (P).

These units are related to the SI unit by 1 SI viscosity unit = 1 N • s/m2 = 10 P = 1000 cP The poise is obtained from Eq.(l.) when the cgs system is used.Typical viscosities are given in 

Table 1.


Table 1.

Viscosities of liquids and gases at  30° c
















Blood plasma


SAE No.10 0il








* 1000 cP= 1N  s/m2 ,the SI unit.

We can gain further insight into the meaning of viscosity by examining Fig. 2b. Notice that the fluid layers next to the two plates remain attached to the plates. We can think of the fluid between the plates as consisting of many thin layers, many more than shown. As the upper plate moves, these layers must slide over each other. In a high-viscosity fluid, the layers do not slide easily. A large amount of friction work is done as the layers are made to slide past each other. It is for this reason that work done against viscous forces is equivalent to friction work.

According to the theory first developed by Stokes , the Mowing equation (known as Stake's law) gives the force needed to pull a sphere of radius a through a fluid of viscosity t] with a speed u,


To determine the size of tiny spherical particles, their sedimentation speed is often measured. Show that a particle of density d which reaches a terminal speed u while falling through a fluid of density df and viscosity t| has a radius b given by

Make use of Stake's law.

This method may serve for determination of sizes of erythrocytes.


Applying of the Bernoulli theorem for analysis of an aneurysm. Let's esteem artery a dia d1= 2,5 cm (one of arteries of abdominal cavity), which has a puff-up (aneurysm) a dia 5 cm (fig. 8). Knowing, that the average speed of a blood in norm compounds v1= 30sm * c-1, and the relative blood pressure is equal 120 torr (we shall remind, that 760 torr = 105 Pa), we shall calculate pressure P2 in an aneurysm. How will the aneurysm develop? At calculus’s we shall accept, that the artery is posed horizontally.

Figure 8


The applying of a Bernoulli relation to a horizontal lease A1 A2, gives

The condition of a continuity of current results equals






         The substitution of numerical values gives magnitude, equal approximately 42 Pa, or

The overpressure is insignificant as contrasted to absolute (760 + + 120 = 880 torr), however, it tends even more to expand puff-up, that in turn results in further pressure increase and so on. If, as it frequently happens, the walls of artery are resized by the pathological process, the breaking of an aneurysm and internal hemorrhage, quite often resulting death, is possible.

3. Clottage, arterial noise. In case of partial clogging of artery it is usually said, that clottage takes place (formation an atheromatosis plaque). On carotid artery, which average diameter dt = 1 cm (fig. 9), the blood circulates with an average speed v1=20cM x c-1.

For simplicity, it is possible to accept, that the artery is posed horizontally, the gravity of a blood is peer to density of water (103 kg x m-3), and the overpressure inside an unclogged lease of artery


Figure 9

compounds P1-P2 = 100 torr (P0 — pressure outside).

1°. What is the minimum diameter d2, at which blood flow is still possible (let's consider, that the

fractionally clogged lease of artery is cylinder)?

2°. What will happen, if diameter becomes less, than d2?


1 . Let's examine dots a1  and A2, posed on the same horizontal, in which artery diameter is equal accordingly dt, (when absence of a pathology) and d2 (partly clogged artery). Assume that P1,v1  and P2, v2 are pressure and speed of blood in these dots; then, according to the Bernoulli theorem,

Blood flow will exist till P2 is bigger then outside pressure P0 or

Considering that artery is undeformable and blood is incompressible, we shall write a condition of flow conservation:


The condition of existence of blood flow will be rewritten as


The substitution of numerical values gives d2 >= 2 mms.

2°. When d2 becomes less than this magnitude, the artery flattens under outside pressure. The pressure pi prolongs thus to increase owing to incessant heart activity (which thus works in conditions of heightened load); blood starts to leak by jerks, and with the help of a stethoscope the discontinuous noise telling about failure of circulation is being listened.


Blood-pressure measurement




Measuring of blood pressure.


Aim: To Acquaint the students with physical blood pressure measuring methods and with devices which adapt in clinic for measuring of arterial blood pressure.

Professional students orientation.

Measuring of arterial blood pressure helps to diagnose diseases  and renders assistance to improve upon effectiveness of medical process. Pressures systolic and diastolic determine fortune of heart work and vascular system, and also describe reologic blood properties.


Blood pressure and its size have the large importance for ability to live organism, and also is the important integrated parameter gemodinamics, and so and diagnostic parameter. Work of heart and action of forces of elasticity of walls  of aortaa result in periodic change of size blood of pressure. Distinguish maximal (systaltic) pressure, which represents by itself pressure of blood on walls artery to time systole (contractions) ventricle of heart and minimal (diastolic) pressure - too in time diastoie (relaxation) ventricle. Distinction between them name pulse as pressure. The important parameter is average pressure, which represents by itself of average all instant meanings blood of pressure during an intimate cycle. This size characterizes expenses of energy for maintenance of real meanings  blood of pressure on an extent of cardiocycle.

The pressure of blood can be measured by a direct method (catheterethetion, at which with the help of a polyethylene probe in the large vessels is entered tiny manometer). This method is used in surgical practice, or in experiments on animals. In clinical practice the indirect method (bloodless), measurements of pressure of blood known under the name of a method Korotkovs and oscilometrical a method is used.


The purpose of work:

To learn physical bases of methods of measurement of pressure of blood and principle of work of devices, which are applied in clinic. To learn to measure systalticand diastolic pressure of blood by a method Korotkovs and oscilometrical by a method.



Devices and materials:

Sphigmomanometer membrane (tonometer), Stetophonendoscope, barometer.


The theoretical items of information

Описание: bp006Mean Arterial Pressure

As blood is pumped out of the heart into the resistance network of the systemic circulation, pressure is generated.  In reality, this pressure is pulsatile because of the the cardiac output is intermittent.  If we were to assume that the cardiac output were continuous (i.e., non-pulsatile), then the mean arterial pressure (MAP) is determined by the cardiac output (CO), systemic vascular resistance (SVR), and central venous pressure (CVP) according to the following relationship which is based upon the relationship between flow, pressure and resistance:

MAP = (CO × SVR) + CVP (eq. 1)

Because CVP is usually at or near 0 mmHg, this relationship is often simplified to:

MAP = CO × SVR (eq. 2)

Therefore, changes in either CO or SVR will affect MAP. If CO and SVR change reciprocally and proportionately, then MAP will not change. It is important to note that variables found  in equation 1 are all interdependent.  This means that changing one variable changes all of the others. This interdependency is best depicted using cardiac and systemic function curves (see Cardiac-Vascular Coupling). Although cardiac output is pulsatile rather than continuous, the above relationship is still a valid approximation.

In practice, MAP is not determined by knowing the CO and SVR, but rather by direct or indirect measurements of arterial pressure. From the aortic pressure trace over time, the shape of the pressure trace yields a mean pressure value (geometric mean) that is less than the arithmetic average of the systolic and diastolic pressures as shown to the right.

At normal resting heart rates, MAP can be approximated by the following equation:

Описание: MAP equation.gif (1466 bytes)

For example, if systolic pressure is 120 mmHg and diastolic pressure is 80 mmHg, then the mean arterial pressure will be approximately 93 mmHg. At high heart rates, however, MAP is more closely approximated by the arithmetic average of systolic and diastolic pressure because of the change in shape of the arterial pressure pulse (it becomes narrower). Therefore, to determine mean arterial pressure with absolute accuracy, analog electronic circuitry or digital techniques need to be employed to arrive at the mean value.

Ejection of blood into the aorta by the left ventricle results in a characteristic aortic pressure pulse (also see Cardiac Cycle). The peak of the aortic pressure pulse is termed the systolic pressure (Psystolic), and the lowest pressure in the aorta is termed the diastolic pressure (Pdiastolic). The difference between the systolic and diastolic pressures is the aortic pulse pressure. The mean aortic pressure (MAP) is the average pressure (geometric mean) during the aortic pulse cycle.The arterial Описание: BP002_aortic_pressure_pulsepulse pressure is the difference between the systolic and diastolic arterial pressures.

Pulse Pressure = Systolic Pressure - Diastolic Pressure


As the aortic pressure pulse travels down the aorta and into distributing arteries, there are characteristic changes in the systolic and diastolic pressures, as well as in the mean pressure.  As the pressure pulse moves away from the heart, the systolic pressure rises and the diastolic pressure falls.  There is also a  small decline in mean arterial pressure as the pressure pulse travels down distributing arteries due to the resistance of the arteries.  Therefore, when arterial pressure is measured using a sphygmomanometer (i.e., blood pressure cuff) on the upper arm, the pressure measurements represent the pressure within the brachial artery, which will be slightly different than the pressure measured in the aorta or the pressure measure in other distributing arteries.

Venous Pressure



Venous pressure is a term that represents the average blood pressure within the venous compartment. We sometimes use the term "central venous pressure" (CVP) to describe the pressure in the thoracic vena cava near the right atrium. CVP is influenced by a number of factors, including cardiac output, respiratory activity, contraction of skeletal muscles (particularly legs and abdomen), sympathetic vasoconstrictor tone, and hydrostatic forces (i.e., gravity). All of these factors, however, ultimately affect CVP (D PV) by changing either venous blood volume (D V) or venous compliance (CV) as shown in equation 1.

Описание: Venous press equation.gif (1217 bytes)  Equation 1

Therefore, an increase in venous volume will increase PV by an amount determined by CV. Furthermore, a decrease in venous compliance, as occurs during sympathetic activation of veins, will increase PV. Several important factors or mechanisms can increase CVP as summarized below. The initial pressure necessary for movement of blood on vessel to system is created by work of heart. In this plan heart represents by itself ritmical the working pump, at which the working phase (contractions to a muscle - systole) alternates with a single phase (relaxation to a muscle - diastole). At each contractions left ventricale of heart in aortaa, which filled by blood under the appropriate pressure, is pushed out so-called shock volume of blood, which on the average equal 60-70gg. After that the valves aortaa are closed. Additional volume of blood, that has arrived in aortaa raises in it pressure and will cause a stretching of walls of vessels, increasing thus their volume. This pressure in aortaa refers to as systaltic. The wave of the increased pressure of blood is quickly distributed lengthways arterial of a part vessel of system and will cause fluctuations of its walls. This wave of pressure refers to as pulse wave, speed of its distribution depends from elasticity both density of walls of vessels and equal 6-8 m/sec.

In the period diastoie of a wall aorta are gradually reduced to an initial situation and thus additional volume of blood in environmental artery forces the way. The walls of these vessels being in turn stretched, and being then reduced push blood in the following sites vessel of system. In result the flow of blood accepts continuous character with speed in the large vessels about 0,3-0,5 m/sec

The quantity of blood, which proceeds through cross section of a vessel for a time unit, refers to as as volumetric speed bloodstream. This speed depends on distinction of pressure in the beginning and at the end of a site vessel of system and general resistance of a flow of blood. Volumetric speed determine behind the formula Puaselya, though resistance of a flow of blood in vessel to system greater, than taken into account in the formula, owing to losses of energy during deformation it elastic of walls, and also turbulent of current in branchings.                     


Change of speed and pressure of blood on different sites vessel of system .

In artereries and capillaries Blood the pressure strongly falls, that is caused by the large resistance owing to friction in artereries and capillaries. For an explanation it we shall simulate capillary system with n identical in parallel connected tubs in radius . Hydraulic resistance by one tubs  back proportional , and for n in parallel-connected tubs - back proportional . It is possible to tell total cross section S proportional , that, obviously, that it is possible to assert resulting resistance of system back proportional  We may an image, what even in case of large S, if  small enough -  too small. So, resulting resistance large. Such phenomenon also is observed in a case artereries and capillaries. In venouses vessels with the area of section approximately in 2 times of the greater area of section artereries, speed of current of blood small and recession of pressure insignificant. In wide vein the pressure near heart becomes on some millimeters lowest from atmospheric, thus blood moves for the account absorbtion of action chest of a crate during a breath.

The movement of blood in vessel to system and distribution it between different sites of this system depends on work of heart, section of vessels, their elasticity, quantity(amount) of circulating blood, it reological of properties, tonus of vessels, and is adjusted by the central nervous system.

Vessels the system is not connected to an atmosphere. The vessels are placed in different directions. Consider, that in artereries and venouses vessels connected by capillaries, the hydrostatic pressure of blood is mutual balance. If the walls of vessels are damaged, can be connections of a vessel with an atmosphere and then the action of hydrostatic pressure of blood is shown.

    When blood in sufficient volume passes through the narrowed gleam artery, speed of current of a liquid back proportional to the area cross section of a vessel:

         Where  - speed in a bottleneck,

 - speed in a wide place of a pipe,

 - area cross section in a bottleneck,

 - area cross section in a wide place.




fig. 2

In the narrowed place artery the current of blood is accelerated and types turbulent character, which is accompanied by occurrence of noise. When the gleam shoulder artery reaches normal size, the movement of blood becomes quieter, accepts laminar character, the noise disappear and again is precisely listened tones. At the moment of a sharp indulgence of tones or at the moment of disappearance of last tone, pressure in of cuff so also indications manometerа are equaled to the minimal pressure of blood.

The further contractions of pressure in of cuff enables artery to leave from the intense condition and wall it will not change with sound frequency, the current of blood in artery becomes laminar that is why neither tones, nor noise is not listened.

For listening the sound phenomena, which arise in vessels is applied Stetophonendoscope. Stetophonendoscope is imposed in area elbow pit from its internal party, where passes the bottom piece shoulder artery, which branches something below than place of listening on two branches - elbow and beam (Fig. 2).

Techniques of measurement

Rough estimates without using any equipment at all

It is not possible to derive a numerical value for blood pressure without some equipment, but a crude assessment of the circulation can still be obtained. If you can feel a radial pulse the systolic blood pressure is usually at least 80 mmHg. The character of the pulse, i.e. bounding or thready, gives a further clue. In most cases, shocked patients have cold hands and feet. (The most important exception to this is a patient who is shocked because of severe sepsis). Capillary refill time is another simple test of circulatory adequacy: press firmly on the patient's nail bed with your thumb; release your thumb and see how long it takes for blood to return. A refill time of greater than 2 seconds suggests an inadequate circulation.

Manual non-invasive blood pressure measurement

This requires, at the very least, an inflatable cuff with a pressure gauge (sphygmomanometer). Wind the cuff round the arm (which should be at about heart level) and inflate it to a pressure higher than the expected blood pressure. Then deflate the cuff slowly. With a stethoscope, listen over the brachial artery. When the cuff reaches systolic pressure, a clear tapping sound is heard in time with the heart beat. As the cuff deflates further, the sounds become quieter, but become louder again before disappearing altogether. The point at which the sounds disappear is the diastolic pressure. If you have no stethoscope, the systolic blood pressure can be found by palpating the brachial artery and noting the pressure in the cuff at which it returns.

The sounds heard while measuring blood pressure in this way are called the Korotkoff sounds, and undergo 5 phases:

1.     initial 'tapping' sound (cuff pressure = systolic pressure)

2.     sounds increase in intensity

3.     sounds at maximum intensity

4.     sounds become muffled

5.     sounds disappear

Most inaccuracies result from the use of the wrong size of cuff. A narrow cuff wrapped round a fat arm will give an abnormally high reading, and vice versa. The World Health Organisation recommends a 14cm cuff for use in adults. Smaller cuffs for infants and children are available. In occasional patients, the reading obtained from one arm can be different from that obtained from the other arm. An appropriate size of cuff can be applied to the calf, and pressure estimated by palpation of the posterior tibial pulse.


The Von Recklinghausen Oscillotonometer is a device which allows both systolic and diastolic blood pressure to be read without a stethoscope. It consists of two overlapping cuffs (one large, one small) a large dial for reading pressure, a bleed valve and a control lever. The large cuff performs the usual function of the sphygmomanometer cuff. The job of the smaller cuff is basically to amplify the pulsations which occur as the larger cuff is deflated, so that instead of listening for the Korotkoff sounds, they are seen as oscillations of the needle on the pressure gauge. The lever simply switches the dial between the two cuffs.

Wrap the cuff round the arm in the usual way, and inflate it. Adjust the bleed valve so that the pressure falls slowly. Pull the control lever towards you. The needle will jump slightly in time with the pulse. As the cuff pressure approaches systolic, the needle suddenly starts to jump more vigorously. At this point, let go of the lever, and the needle will display systolic pressure. Pull the lever forward again. As the pressure is reduced, the needle jumps more vigorously. If the lever is released at the point of maximum needle oscillations, the dial will read the mean arterial pressure. If it is released at the point when the needle jumps get suddenly smaller, the dial reads diastolic pressure.

Automatic non-invasive blood pressure measurement

Automatic devices which essentially apply the same principle as the oscillotonometer have been produced (e.g. the 'Dinamap' made by Critikon). They require a supply of electricity. A single cuff is applied to the patients arm, and the machine inflates it to a level assumed to be greater than systolic pressure. The cuff is deflated gradually. A sensor then measures the tiny oscillations in the pressure of the cuff caused by the pulse. Systolic is taken to be when the pulsations start, mean pressure is when they are maximal, and diastolic is when they disappear. They can produce fairly accurate readings and free the hands of the anaesthetist for other tasks. There are important sources of inaccuracy, however. Such devices tend to over-read at low blood pressure, and under-read very high blood pressure. The cuff should be an appropriate size. The patient should be still during measurement. The technique relies heavily on a constant pulse volume, so in a patient with an irregular heart beat (especially atrial fibrillation) readings can be inaccurate. Sometimes an automatic blood pressure measuring device inflates and deflates repeatedly "hunting" without displaying the blood pressure successfully. If the pulse is palpated as the cuff is being inflated and deflated the blood pressure may be estimated by palpation and reading the cuff pressure on the display.






Invasive arterial pressure measurement


This technique involves direct measurement of arterial pressure by placing a cannula in an artery (usually radial, femoral, dorsalis pedis or brachial). The cannula must be connected to a sterile, fluid-filled system, which is connected to an electronic monitor. The advantage of this system is that pressure is constantly monitored beat-by-beat, and a waveform (a graph of pressure against time) can be displayed. Patients with invasive arterial monitoring require very close supervision, as there is a danger of severe bleeding if the line becomes disconnected












Equipment for megement of Blood Pressure.


Digital Blood Pressure Monitor, Automatic Pump    Digital Blood Pressure Monitor, Manual Pump



Blood Pressure Monitors Mercurial Monitors       Aneroid Blood Pressure Monitors

Описание: Sphygomanometers
Blood Pressure Monitors
, Clock Aneroid Sphygmomanometr


Determinants of blood viscosity


1. Hematocrit (htc,Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\fi.gif):                 Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\htc.gif

Normal value: 0.4-0.5.Viscosity of blood as a suspension (in the physiologically relevant range of htc):

Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\viscformula.gif

Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\eta.gif       = viscosity of suspension A, B = empirical constants 

2. Plasma viscosity  Depends on plasma proteins. In paraproteinaemias (e.g. myeloma multiplexor. plasmocytoma) the concentration of immunoglobulins is high, leading to increased viscosity.

3. Plasticity of red blood cells65% suspension of blood-cell-size particles is rock hard. In contrast, a 95% blood suspension if fluid, with viscosity of ~20 mPas! Deformation of red blood cells: droplet, parachute, arrowhead shapes.

 4. Aggregation of red blood cells is stack or roleaux formation. More pronounced at low flow rates. 

5. Flow rate, velocity gradient

Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\viscvelocity.gif


6. Diameter of blood vessels With a dicrease of vessel diameter, the anomalic (non-newtonian) behavior of blood becomes more pronounced.

Axial migration: the red blood cells line up in the axis of the vessel. In the axis the velocity gradient decreases, and near the vessel wall it increases. Increase in velocity gradient decreases apparent viscosity (Fåhraeus-Lindquist effect).

The circulatory system

 Vessel type


 Total cross-sectional area (cm2)

 Fraction of total blood volume(%)

 Mean pressure (Hgmm/kPa)

 Flow rate (m/s)


 25 mm






 4 mm





 30 µm





 8 µm






 20 µm






 5 mm




 Vena cava

 30 mm





Physical parameters in different parts of the circulatory system

Описание: C:\Vaculenko\Metod\biophiz\Transport in macroscopic systems\Blood Circulation\pote\parameters.gif


NB: -Pressure=pressure that sustains flow, "blood pressure". Reason of pressure drop: flow resistance; most of energy is converted to heat. Flow rate and total cross-sectional area change inversely (based on equation of continuity, Av=constant). Flow rate typically does not exceed the critical (see Reynolds number),and flow remains laminar. (But: behind aortic valve, constricted vessels, low-voscosity conditions, Korotkoff sound). Arterioles (vessels containing smooth muscle, under vegetative innervation) are pressure-regulators: "resistance vessels." Most of blood volume in veins: "capacitance vessels."

 Auxiliary factors of circulation

1.     Arterial elasticity (storage of potential energy)

2.      Venous valves (Harvey's experiment)

3.      Muscle action

4.      Negative intrathoracic pressure

5.      "Up-and-down" movement of atrioventricular plane.

The cardiac cycle

1.     Systole: contraction

2.      Diastole: relaxation





 0,1 s

 0,7 s


 0,3 s

 0,5 s



 The work of heart


pdV = static (volumetric) component, 1/2mv2 = dynamic component, p = pressure, dV = stroke volume, m = mass of blood, v = flow rate. 


Translational diffusion coefficient


Molecules in solution undergo brownian motion under thermal bombardment from their surroundings, which causes them to move even when not subject to an external force.This thermal motion, called diffusion, is random in magnitude and direction, but can be measured by the techniques described below. Diffusion is also observed when a concentration gradient is set up in solution or when a barrier between solutions at two different concentrations is removed. The tendency toward equalization of concentrations is attributable at the macroscopic level to a gradient of concentration or chemical potential; but at the molecular level it can also be understood as the random motion of molecules, with more molecules in the concentrated region of solution available to move randomly into the more dilute region.

The rate of brownian motion or evening-out of concentration gradients is proportional to the translational diffusion coefficient Dt, which in turn is inversely proportional to ft. Here we summarize the basic laws characterizing translational diffusion, and describe some of the most important modern techniques whereby Dt can be measured.

We note that molecules also undergo rotational brownian motion. This will be discussed later.


Arterial Ultrasound Scan

High frequency ultrasound in the 7-12 MHz region is used for high resolution imaging of arteries which lie close to the surface of the body, such as the carotid arteries. Using a nominal sound velocity of 1540 m/s in tissue, the sound wavelength in tissue for a 7 MHz sound wave can be obtained from the wave relationship v = f.

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\ultransducer.gif

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\ultrawave.gif

Using the general principle of imaging that you can't see anything smaller than the wavelength suggests a 0.2 mm ultimate resolution limit.

In addition to imaging the arterial walls, the ultrasound techniques can measure the blood flow velocity by making use of the Doppler effect. The reflected ultrasound is shifted in frequency from the frequency of the source, and that difference in frequency can be accurately measured by detecting the beat frequency between the incident and reflected waves. The beat frequency is directly proportional to the velocity of flow, so continuous recording of the beat frequencies from the different parts of the the arteriy gives you an image of the velocity profile of the blood flow as a function of time.

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\dopcarotid.gif

The sketches above are conceptual only; no attempt has been made to accurately scale velocities and colors. But hopefully it illustrates the use of false color imaging to give an instant view of the distribution of velocities present. The lower part of the illustration contains an intensity modulated power spectrum in which the beat frequency and therefore the speed of the blood is on the vertical axis. Such spectra are produced by analysis of the reflected ultrasound using a mathematical process called a fast Fourier transform (FFT) in which the distribution of reflected power as a function of frequency is extracted. This is done repetitively and the results plotted as a function of time (horizontal axis). The distance vertically from the axis indicates the beat frequency and therefore the velocity of flow. The relative amount of power reflected at a given velocity value is indicated by the brightness of the display at that point. A single uniform flow speed would give a single bright line, so the display indicates a considerable range of velocities present in the flow at any time. Note that at the time of the peaks, essentially all of the blood has a fairly high velocity since the part of the spectrum near the horizontal axis is dark.

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\carotidimg.jpg

Carotid Image

This example of a clinical carotid image is taken during the relatively quiet period between the peaks. Note that red indicates a slower flow than blue, but in the same direction since carotid flow does not reverse. So the implication of red and blue is not the same as in the astrophysical red shift of light from stars. Note the background grayscale image which is formed from the pulse-echo ranging data.

Taken literally, the above image would suggest a fairly uniform flow velocity over the cross-section of the artery. But that is not the nature of the expected laminar flow, in which there is expected to be a velocity profile with the highest velocity on the center line and dropping toward zero velocity at the walls. I'm guessing that the red range of the false color image is set to include a wide range of low velocities.

A higher ultrasound frequency like 12 MHz gives a shorter wavelength and therefore higher resolution, but that advantage is partially canceled by the fact that the higher frequency is attenuated more in tissue. So judgments must be made about the relative merits of deeper penetration (low frequency) vs higher resolution (higher frequency).

The ultrasound sources are generally tuned ceramic wafers of a material such as PZT which are driven by applying an AC voltage at the design frequency. The voltage causes mechanical vibration by the piezoelectric effect.

The ultrasound scans can detect the buildup of plaque in the arteries. Besides the direct imaging of the narrowed vessel, the Doppler information can be converted into false color images which profile the flow velocity. Flow in a region of obstruction must be at a higher velocity to maintain the flowrate, and that velocity information is confirmation of a narrowing of the vessel.


Implementation Methods of practical work

Poiseuille's Law Calculation

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\poic.gif

Poiseuille's law can be used to calculate volume flowrate only in the case of laminar flow.

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\poic2.gif

Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\poise.gif

Any of the parameters below can be changed. When you have finished entering data, click on the quantity you wish to calculate in the formula above. The different parameters will not be forced to be consistent until you click on the quantity to calculate.

For pressure difference: x 10^ Pa = kPa
= atmos = mmHg = lb/in^2 = cm water = inches water

applied to a tube of radius = cm = inches (corresponding to area =cm^2 = in^2)
and length = cm = inches,
for a fluid with viscosity = poise = x the viscosity of water,
the volume flowrate will be = cm^3/s =in^3/s=liters/min=ft^3/min= U.S. gal/min.

Default values will be assigned to parameters to which you have not given values. You can change any of those default values as part of your exploration.

Bernoulli Calculation

The calculation of the "real world" pressure in a constriction of a tube is difficult to do because of viscous losses, turbulence, and the assumptions which must be made about the velocity profile (which affect the calculated kinetic energy). The model calculation here assumes laminar flow (no turbulence), assumes that the distance from the larger diameter to the smaller is short enough that viscous losses can be neglected, and assumes that the velocity profile follows that of theoretical laminar flow. Specifically, this involves assuming that the effective flow velocity is one half of the maximum velocity, and that the average kinetic energy density is given by one third of the maximum kinetic energy density. Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\berc.gif

Now if you can swallow all those assumptions, you can model* the flow in a tube where the volume flowrate is Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\sf.gif= cm^3/s and the fluid density is Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\rho-2.gif= gm/cm^3. For an inlet tube area Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\a1.gif= cm^2 (radius Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\r1.gif=cm), the geometry of flow leads to an effective fluid velocity of Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\v1.gif=cm/s. Since the Bernoulli equation includes the fluid potential energy as well, the height of the inlet tube is specified as Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\h1.gif= cm. If the area of the tube is constricted to Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\a2.gif=cm^2 (radius Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\r2.gif= cm), then without any further assumptions the effective fluid velocity in the constriction must be Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\v2.gif= cm/s. The height of the constricted tube is specified as Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\h2.gif= cm.

The kinetic energy densities at the two locations in the tube can now be calculated, and the Bernoulli equation applied to constrain the process to conserve energy, thus giving a value for the pressure in the constriction. First, specify a pressure in the inlet tube:
Inlet pressure =
Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\p1.gif = kPa = lb/in^2 = mmHg = atmos.
The energy densities can now be calculated. The energy unit for the CGS units used is the erg.








The pressure energy density in the constricted tube can now be finally converted into more conventional pressure units to see the effect of the constricted flow on the fluid pressure:

Calculated pressure in constriction =
Описание: D:\..\mater biophisics\Hiper_Phizics\hyperphysics\p2.gif= kPa = lb/in^2 = mmHg = atmos.

This calculation can give some perspective on the energy involved in fluid flow, but it's accuracy is always suspect because of the assumption of laminar flow. For typical inlet conditions, the energy density associated with the pressure will be dominant on the input side; after all, we live at the bottom of an atmospheric sea which contributes a large amount of pressure energy. If a drastic enough reduction in radius is used to yield a pressure in the constriction which is less than
atmospheric pressure, there is almost certainly some turbulence involved in the flow into that constriction. Nevertheless, the calculation can show why we can get a significant amount of suction (pressure less than atmospheric) with an "aspirator" on a high pressure faucet. These devices consist of a metal tube of reducing radius with a side tube into the region of constricted radius for suction.

*Note: Some default values will be entered for some of the values as you start exploring the calculation. All of them can be changed as a part of your calculation.




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