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Relationship between Pressure and flow

• In 1846, Poiseuille, a French physician-scientist, described the relationship between pressure and flow in a rigid glass tube: Q ~ ΔP = Pi-Po

• Poiseuille's Law: Q = ΔP pi r^4 / 8 n L In a strict sense, this equation can only be applied to a steady flow with constant viscosity (ideal fluid; e.g., water).

• Poiseuille's and Ohm's law: Current (Q) = ΔVoltage (ΔP) / R

1/R = pi r^4 / 8 n L -> R = 8 n L / pi r^4

Conductance-->Resistance

• Q = ΔP/R (Q=flow?)

Factors that determine Resistance

R=8nL/pi r^4

The major factors that determine the resistance:

• Radius of the tube (vessel) - 4th power relationship

• Viscosity of the fluid (blood)= n (eta?)

Vascular resistance and pressure distribution

R=(Pi-Po)/Q (Pi=inflow; Po=outflow)

• Taking into account the entire vascular bed, the arterioles as a group provide the greatest resistance to blood flow.

• Poiseuille's Law: Q = ΔP pi r^4 / 8 n L In a strict sense, this equation can only be applied to a steady flow with constant viscosity (ideal fluid; e.g., water).

• Poiseuille's and Ohm's law: Current (Q) = ΔVoltage (ΔP) / R

1/R = pi r^4 / 8 n L -> R = 8 n L / pi r^4

Conductance-->Resistance

• Q = ΔP/R (Q=flow?)

Factors that determine Resistance

R=8nL/pi r^4

The major factors that determine the resistance:

• Radius of the tube (vessel) - 4th power relationship

• Viscosity of the fluid (blood)= n (eta?)

Vascular resistance and pressure distribution

R=(Pi-Po)/Q (Pi=inflow; Po=outflow)

• Taking into account the entire vascular bed, the arterioles as a group provide the greatest resistance to blood flow.

Local Changes in Arteriolar resistance and Blood pressure

• When ΔP between aorta and vena cava remains nearly constant, local changes in arteriolar radius can influence local pressure profile

• arteriolar dilation: decreases resistance (decreases intravascular pressure above dilation site) and increases pressure & blood flow downstream in capillaries

• arteriolar constriction: increases resistance (increases intravascular pressure above constriction site) and decreases pressure & blood flow downstream in capillaries

Local Changes in Blood Flow and Velocity

• When ΔP between aorta and vena cava remains nearly constant, local changes in arteriolar radius can influence local blood flow and velocity

Velocity (v) = Q / pi r^2 in a blood vessel

If we substitute for Q from the Poiseuille equation (Q = ΔP pi r^4 / 8 n l) we obtain:

v = (ΔP pi r^4 / 8 n L) / (pi r^2) = ΔP r^2 / 8 n L

Only when ΔP, n and L constant, local changes in arteriolar radius (r) will alter both flow (Q, by r^4) and velocity (by r^2)

• local arteriolar dilation in a tissue: increases local blood flow and velocity

• local arteriolar constriction in a tissue: decreases local blood flow and velocity

• arteriolar dilation: decreases resistance (decreases intravascular pressure above dilation site) and increases pressure & blood flow downstream in capillaries

• arteriolar constriction: increases resistance (increases intravascular pressure above constriction site) and decreases pressure & blood flow downstream in capillaries

Local Changes in Blood Flow and Velocity

• When ΔP between aorta and vena cava remains nearly constant, local changes in arteriolar radius can influence local blood flow and velocity

Velocity (v) = Q / pi r^2 in a blood vessel

If we substitute for Q from the Poiseuille equation (Q = ΔP pi r^4 / 8 n l) we obtain:

v = (ΔP pi r^4 / 8 n L) / (pi r^2) = ΔP r^2 / 8 n L

Only when ΔP, n and L constant, local changes in arteriolar radius (r) will alter both flow (Q, by r^4) and velocity (by r^2)

• local arteriolar dilation in a tissue: increases local blood flow and velocity

• local arteriolar constriction in a tissue: decreases local blood flow and velocity

Resistance

Series Resistance

Pi-Po=(Pi-P1)+(P1-P2)+(P2-Po)

Pi-Po/Q=(Pi-P1)/Q + (P1-P2)/Q + (P2-Po)/Q

Rt=R1 + R2 + R3

120-30/5=120-110/5 + 110-60/5 + 60-30/5

18 mmHg/L/min=2 + 10 + 6 mmHg/L/min

The total resistance equals the sum of the individual resistances.

Parallel Resistance

Qt=Q1 + Q2 + Q3

Qt/Pi-Po= Q1/Pi-Po + Q2/ Pi-Po + Q3/Pi-Po

1/Rt= 1/R1 + 1/R2 + 1/R3

• The reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances.

• The total resistance is always less than the individual resistances.

Arrangement of vessels in serioes and in parallel.

Series: artery-> arteriole-> capillary, etc

Parallel to organs

Renal and GI have highest blood flow and lowest resistance.

Pi-Po=(Pi-P1)+(P1-P2)+(P2-Po)

Pi-Po/Q=(Pi-P1)/Q + (P1-P2)/Q + (P2-Po)/Q

Rt=R1 + R2 + R3

120-30/5=120-110/5 + 110-60/5 + 60-30/5

18 mmHg/L/min=2 + 10 + 6 mmHg/L/min

The total resistance equals the sum of the individual resistances.

Parallel Resistance

Qt=Q1 + Q2 + Q3

Qt/Pi-Po= Q1/Pi-Po + Q2/ Pi-Po + Q3/Pi-Po

1/Rt= 1/R1 + 1/R2 + 1/R3

• The reciprocal of the total resistance equals the sum of the reciprocals of the individual resistances.

• The total resistance is always less than the individual resistances.

Arrangement of vessels in serioes and in parallel.

Series: artery-> arteriole-> capillary, etc

Parallel to organs

Renal and GI have highest blood flow and lowest resistance.

Effect of Viscosity on Resistance

• Definition of Viscosity: the internal friction to flow in a fluid.

Blood is a complex fluid (non-ideal) because of the presence of cells and proteins.

Variation in hematocrit, the percentage of blood that is occupied by red blood cells, is the major factor that changes the viscosity of blood, but this does not occur quickly.

As the blood viscosity increases, a greater pressure gradient is required to overcome viscous resistance and provide sufficient blood flow.

Laminar and Turbulent Flow

NR<2000 for streamline (laminar) and NR>3000 for turbulent.

Reynolds Number (NR) = rho D v / n

Anemia: lower n and raise v -> increase NR (bruit sound)

Atherosclerotic plaque: lower cross-sectional area and increase v -> increase NR (bruit sound)

Turbulence: much more resistance--heart must work harder to create greater pressure

Anemia: lower viscosity. Increase reynolds number. Increase velocity. This increases the chance for the audible sound (Bruit)

Atherosclerosis: reduction in cross section are will increase velocity. Velocity is the major factor to determine if there is turbulence. It overrides size

Aneurysm: artery getting larger.

Blood is a complex fluid (non-ideal) because of the presence of cells and proteins.

Variation in hematocrit, the percentage of blood that is occupied by red blood cells, is the major factor that changes the viscosity of blood, but this does not occur quickly.

As the blood viscosity increases, a greater pressure gradient is required to overcome viscous resistance and provide sufficient blood flow.

Laminar and Turbulent Flow

NR<2000 for streamline (laminar) and NR>3000 for turbulent.

Reynolds Number (NR) = rho D v / n

Anemia: lower n and raise v -> increase NR (bruit sound)

Atherosclerotic plaque: lower cross-sectional area and increase v -> increase NR (bruit sound)

Turbulence: much more resistance--heart must work harder to create greater pressure

Anemia: lower viscosity. Increase reynolds number. Increase velocity. This increases the chance for the audible sound (Bruit)

Atherosclerosis: reduction in cross section are will increase velocity. Velocity is the major factor to determine if there is turbulence. It overrides size

Aneurysm: artery getting larger.

Relationship between pressure and flow (Poiseuille's Law)

One of the major determinants of fluid flow is pressure. In 1846, Poiseuille, a French physicianscientist, described the relationship between pressure and flow in cylindrical tubes. He conducted a series of experiments with rigid glass tubes of various sizes and fluids of different viscosities. He found that flow is directly proportional to the difference between inflow (Pi) and outflow (Po) pressures [Q ∝ ΔP = (Pi - Po)]. His work led to Poiseuille's Law: Q = ΔPπr4 / 8ηL. The formula indicates that flow is directly proportional to the intravascular pressure gradient (ΔP) and to the fourth power of the tube radius (r) and is inversely proportional to the length (L) of the tube and to viscosity (η). The π/8 in the formula is the constant of

proportionality.

In a strict sense, this equation can only be applied to ideal fluids (homogeneous fluid such as water), which undergo steady (nonpulsatile) laminar flow (all elements of the fluid move in streamlines parallel to the axis of the tube) with constant viscosity. In contrast, blood is not an ideal fluid because it is a suspension of plasma, blood cells, and proteins.

Although most of the assumptions of Poiseuille's Law are violated in certain parts of the cardiovascular system (pulsatile flow in arteries, laminar or turbulent, elastic/compliant vessels with multiple branching and bends, and non-homogeneous blood), the important principles relating to flow, pressure gradient and resistance remain applicable and the equation tells us the most important factors in determining flow.

proportionality.

In a strict sense, this equation can only be applied to ideal fluids (homogeneous fluid such as water), which undergo steady (nonpulsatile) laminar flow (all elements of the fluid move in streamlines parallel to the axis of the tube) with constant viscosity. In contrast, blood is not an ideal fluid because it is a suspension of plasma, blood cells, and proteins.

Although most of the assumptions of Poiseuille's Law are violated in certain parts of the cardiovascular system (pulsatile flow in arteries, laminar or turbulent, elastic/compliant vessels with multiple branching and bends, and non-homogeneous blood), the important principles relating to flow, pressure gradient and resistance remain applicable and the equation tells us the most important factors in determining flow.

cont'd

Since conductance (reflects ease of flow) can be determined by the formula πr4 / 8ηL, which is the reciprocal of resistance (1/R), Poiseuille's Law can be simplified to the Bulk Flow Law: Q = ΔP/R. Important formula to remember!!

The Bulk Flow Law provides the basic law of hemodynamics and is analogous to the electrical theory of Ohm's Law (I = ΔV/R), where current (analogous to flow) = Δvoltage (analogous to ΔP)/ hemodynamic resistance (analogous to electrical resistance).

Important: Acute changes in flow are determined primarily by alterations in vessel radius. There are very little changes in vessel length and viscosity from moment to moment and arterial pressure is tightly regulated (only sustained 20% increase needed to be regarded as hypertensive), but cardiac output (blood flow ejected from heart) can increase 4- or 5-fold in ordinary individuals during intense exercise. According to the Poiseuille equation the only factor left to change is vessel radius to the fourth power. A 20% increase in vessel radius would double the flow through that vessel. Another advantage of having the vessel radius as the key variable in regulating blood flow is that it allows the cardiovascular system to vary blood flow selectively to different organs. If blood flow were controlled by arterial pressure or blood viscosity, the cardiac output would have to be altered in all organs in equal proportion, which would be very inefficient and increase the work load on the heart.

The Bulk Flow Law provides the basic law of hemodynamics and is analogous to the electrical theory of Ohm's Law (I = ΔV/R), where current (analogous to flow) = Δvoltage (analogous to ΔP)/ hemodynamic resistance (analogous to electrical resistance).

Important: Acute changes in flow are determined primarily by alterations in vessel radius. There are very little changes in vessel length and viscosity from moment to moment and arterial pressure is tightly regulated (only sustained 20% increase needed to be regarded as hypertensive), but cardiac output (blood flow ejected from heart) can increase 4- or 5-fold in ordinary individuals during intense exercise. According to the Poiseuille equation the only factor left to change is vessel radius to the fourth power. A 20% increase in vessel radius would double the flow through that vessel. Another advantage of having the vessel radius as the key variable in regulating blood flow is that it allows the cardiovascular system to vary blood flow selectively to different organs. If blood flow were controlled by arterial pressure or blood viscosity, the cardiac output would have to be altered in all organs in equal proportion, which would be very inefficient and increase the work load on the heart.

Resistance to flow

a. Resistance (R) = (Pi - Po)/Q = 8ηL/πr4 (Hydraulic Resistance Equation) (R is expressed in mmHg/L/min or mmHg/ml/sec)

i. Resistance is the impediment to flow in blood vessels due to frictional forces (causes loss of energy or pressure in form of heat), and the primary contributing factors are the radius (r) of the vessels - 4th power relationship - and viscosity (η) of the blood.

ii. The length (L) of the vessels is generally fixed so this does not influence changes in vascular resistance.

iii. Total peripheral resistance (TPR): TPR is the resistance of the entire systemic vasculature. This can be measured with the flow, pressure and resistance relationship by submitting cardiac output for flow (Q) and the difference in pressure between the aorta and the vena cava for ΔP (detail will be provided in Arterial System lecture).

iv. Resistance in a single organ: This can be calculated by substituting the specific organ flow (e.g. flow in heart) for Q and the difference in pressure between the artery (Pi) and vein (Po) in the organ for ΔP.

i. Resistance is the impediment to flow in blood vessels due to frictional forces (causes loss of energy or pressure in form of heat), and the primary contributing factors are the radius (r) of the vessels - 4th power relationship - and viscosity (η) of the blood.

ii. The length (L) of the vessels is generally fixed so this does not influence changes in vascular resistance.

iii. Total peripheral resistance (TPR): TPR is the resistance of the entire systemic vasculature. This can be measured with the flow, pressure and resistance relationship by submitting cardiac output for flow (Q) and the difference in pressure between the aorta and the vena cava for ΔP (detail will be provided in Arterial System lecture).

iv. Resistance in a single organ: This can be calculated by substituting the specific organ flow (e.g. flow in heart) for Q and the difference in pressure between the artery (Pi) and vein (Po) in the organ for ΔP.

Effect of vessel radius on resistance

i. The greatest upstream to downstream drop in internal pressure (energy) in the circulation occurs in the arterioles and small arteries as a group, as flowing blood overcomes resistance (shown below representing the systemic circulation). Because total flow is the same through the various series components of the circulatory system, and taking into account the entire vascular bed, it follows that the greatest resistance to flow resides in the arterioles when using the formula R = (Pi - Po) / Q.

ii. The highest resistance in the body does not reside in the capillaries because of the larger number of parallel capillaries compared with parallel arterioles (see below, cii). Although there is only one aorta and only few elastic arteries, the radii of these vessels are so large in comparison to those of arterioles that they offer relatively insignificant resistance to flow in systemic circulation.

iii. In addition to having the largest structural resistance, the arterioles also have the greatest capacity to alter their resistance. Since the arterioles contain mostly smooth muscle fibers in their vascular wall, which allow for the lumen radius to be varied, it is clear from the hydraulic resistance formula that small changes in the vessel radius can dramatically influence local (within each tissue) vascular resistance and pressure without altering driving pressure gradient between aorta and vena cava. This allows for distribution of blood flow to the organs where it is in greater demand under normal resting conditions.

ii. The highest resistance in the body does not reside in the capillaries because of the larger number of parallel capillaries compared with parallel arterioles (see below, cii). Although there is only one aorta and only few elastic arteries, the radii of these vessels are so large in comparison to those of arterioles that they offer relatively insignificant resistance to flow in systemic circulation.

iii. In addition to having the largest structural resistance, the arterioles also have the greatest capacity to alter their resistance. Since the arterioles contain mostly smooth muscle fibers in their vascular wall, which allow for the lumen radius to be varied, it is clear from the hydraulic resistance formula that small changes in the vessel radius can dramatically influence local (within each tissue) vascular resistance and pressure without altering driving pressure gradient between aorta and vena cava. This allows for distribution of blood flow to the organs where it is in greater demand under normal resting conditions.

Resistance in Series and Parallel

In the cardiovascular system, the vascular beds are arranged in series or in parallel. The arrangement of blood vessels within a given organ supplied by a large artery, smaller arteries, arterioles, capillaries, and veins are arranged in series. Individual members of a given category of vessels (i.e., arterioles or venules) are usually arranged in parallel. The main large arteries (branches of the aorta) supplying the different organs in the systemic circulation are also examples of parallel resistance. Formulas have been derived for the series and parallel arrangements.

i. Series resistance: In the schematic below, three hydraulic resistances are arranged in series, and the inflow and outflow for the system remain constant. The pressure drop across the entire system, Pi and Po, consists of the sum of the pressure drops across each of the individual resistances. The flow through any cross-section should remain constant, so dividing each component in (a) by flow it is evident that the total resistance of the entire system of tubes in series equals the sum of individual resistances.

Example in the circulation: Rtotal = Rartery + Rarterioles + Rcapillaries

Each blood vessel (e.g., the largest artery) or set of blood vessels (e.g., all of the capillaries) in series receives the same total blood flow.

As blood flows through the series of blood vessels, the pressure decreases.

i. Series resistance: In the schematic below, three hydraulic resistances are arranged in series, and the inflow and outflow for the system remain constant. The pressure drop across the entire system, Pi and Po, consists of the sum of the pressure drops across each of the individual resistances. The flow through any cross-section should remain constant, so dividing each component in (a) by flow it is evident that the total resistance of the entire system of tubes in series equals the sum of individual resistances.

Example in the circulation: Rtotal = Rartery + Rarterioles + Rcapillaries

Each blood vessel (e.g., the largest artery) or set of blood vessels (e.g., all of the capillaries) in series receives the same total blood flow.

As blood flows through the series of blood vessels, the pressure decreases.

cont'd

ii. Parallel resistance: For resistances in parallel (see schematic below), the inflow and outflow pressures are the same for all tubes. The total flow equals the sum of the flows through the individual parallel elements (a). Because the pressure gradient is identical for all parallel elements, each term in (a) may be divided by the pressure gradient to yield equation (b).

- The reciprocal of the total resistance of tubes in parallel equals the sum of the reciprocals of the individual resistances.

- For any parallel arrangement, the total resistance (Rt) will always be less than that of any individual component. Thus, parallel arrangement of vessels in the circulation strongly reduces the resistance to blood flow. When an artery is added in parallel, the total resistance decreases. Although individual capillaries have a high resistance owing to their small diameter, the capillary bed as a whole only represents a small fraction of the total vascular resistance because there are many more capillaries than arterioles in a vascular bed and they are associated in parallel.

- The reciprocal of the total resistance of tubes in parallel equals the sum of the reciprocals of the individual resistances.

- For any parallel arrangement, the total resistance (Rt) will always be less than that of any individual component. Thus, parallel arrangement of vessels in the circulation strongly reduces the resistance to blood flow. When an artery is added in parallel, the total resistance decreases. Although individual capillaries have a high resistance owing to their small diameter, the capillary bed as a whole only represents a small fraction of the total vascular resistance because there are many more capillaries than arterioles in a vascular bed and they are associated in parallel.

Effect of viscosity on resistance

i. Viscosity (η) is the internal friction to flow in a fluid such as blood or "thickness" of blood. Blood is a complex fluid because of the presence of cells (primarily red blood cells - RBCs) and proteins. Variation in the hematocrit, which is the percentage of the blood that is occupied by RBCs (normal level of 40-45% in adults), is the major factor that changes the viscosity of blood.

ii. As the blood viscosity increases, a greater pressure gradient is required to overcome viscous resistance and provide sufficient blood flow. However, since it takes several days to make RBCs and they last for several months, this is not a means of changing vascular resistance quickly.

ii. As the blood viscosity increases, a greater pressure gradient is required to overcome viscous resistance and provide sufficient blood flow. However, since it takes several days to make RBCs and they last for several months, this is not a means of changing vascular resistance quickly.

Laminar vs. Turbulent Flow

Laminar vs. Turbulent Flow: Under normal conditions, blood normally flows through all vessels (like in narrow rigid tubes) in the cardiovascular system in an orderly streamlined manner (straight line) called laminar flow. The elements of the fluid flow in a parabolic fashion with the fastest velocity in the center of tube and falls to zero at the wall due to viscous resistance. In contrast, turbulent flow is a result of irregular motions of the fluid elements and rapid, radial mixing occurs (see ure below). The radial vibrations of blood hitting the vessel wall (only in large arteries or heart chambers) causes generation of audible sound waves.

a. Laminar (streamline) flow is equivalent to the pressure gradient (Q ∝ ΔP) up until it reaches a critical velocity. At or above this velocity, flow is turbulent and is proportional to the square of the pressure gradient (see figure above). Because energy is wasted in propelling blood radially and axially, more energy (pressure) is required to drive turbulent flow than laminar flow. This results in reduced flow for a given pressure gradient and causes the heart (produce larger pressure gradient) to work harder to provide adequate blood flow. Therefore, in turbulent flow the flow resistance is greater than that predicted by the Poiseuille's equation.

b. The Reynolds number (NR) can predict whether blood flow will be streamline or turbulent flow. The number (dimensionless value) equals the density of the fluid (ρ) x tube (vessel) diameter (D) x velocity (v)/ viscosity (η): NR = ρDv / η. - For values <2000, the flow is laminar and for values >3000 the flow is turbulent. In the cardiovascular system, the major influences on the Reynolds number are changes in blood velocity and viscosity.

a. Laminar (streamline) flow is equivalent to the pressure gradient (Q ∝ ΔP) up until it reaches a critical velocity. At or above this velocity, flow is turbulent and is proportional to the square of the pressure gradient (see figure above). Because energy is wasted in propelling blood radially and axially, more energy (pressure) is required to drive turbulent flow than laminar flow. This results in reduced flow for a given pressure gradient and causes the heart (produce larger pressure gradient) to work harder to provide adequate blood flow. Therefore, in turbulent flow the flow resistance is greater than that predicted by the Poiseuille's equation.

b. The Reynolds number (NR) can predict whether blood flow will be streamline or turbulent flow. The number (dimensionless value) equals the density of the fluid (ρ) x tube (vessel) diameter (D) x velocity (v)/ viscosity (η): NR = ρDv / η. - For values <2000, the flow is laminar and for values >3000 the flow is turbulent. In the cardiovascular system, the major influences on the Reynolds number are changes in blood velocity and viscosity.

Clinical Examples of turbulent flow

i. Severe anemia: when RBC count (hematocrit) decreases, there is also a decrease in viscosity of blood and high flow velocity (high cardiac output that usually prevails in these patients); this increases the Reynolds number and can create turbulent flow that can be detected as an audible vibration called a bruit (sound from peripheral artery) or murmur (cardiac origin). In contrast, normal laminar blood flow is silent.

ii. Atherosclerotic plaque: when the lumen (cross-sectional area) of aorta or large arteries is significantly narrowed by atherosclerotic plaque. This can increase the velocity of blood flow through the constriction, produce turbulence, and consequently sound.

iii. Korotkoff sounds heard in large brachial artery during blood pressure cuff measurement (will discuss further in Arterial System lecture).

ii. Atherosclerotic plaque: when the lumen (cross-sectional area) of aorta or large arteries is significantly narrowed by atherosclerotic plaque. This can increase the velocity of blood flow through the constriction, produce turbulence, and consequently sound.

iii. Korotkoff sounds heard in large brachial artery during blood pressure cuff measurement (will discuss further in Arterial System lecture).