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2012. 09. 25. Fluid flow, circulation, cardiac biophysics 19. September 2012. Tamás Huber Main properties of fluids • Fluid is a substance that can flow. • Fluids are either liquids or gases (states of matter: solid, liquid, and gas). • Liquid: A state of matter in which the molecules are relatively free to change their positions with respect to each other but restricted by cohesive forces so as to maintain a relatively fixed volume • Gas: a state of matter in which the molecules are practically unrestricted by cohesive forces. A gas has neither definite shape nor volume. 1 2012. 09. 25. Hydrostatic pressure in a liquid The hydrostatic pressure is independent of the form of the container and is proportionally to the height (h) and the density (ρ) of the fluid column. Pressure is always perpendicular to the surface of an object. p= × = × × = × × × Pascal’s law: Any external pressure applied to an enclosed fluid at rest is transmitted undiminished throughout the liquid and onto the walls of the containing vessel. Fluids are uncompressible: p = F1/A1 = F2/A2. F1 « F2 If the height of the fluid's surface above the bottom of the five vessels is the same, in which vessel is the pressure of the fluid on the bottom of the vessel the greatest ? The pressure at a given depth does not depend upon the shape of the vessel containing the liquid or the amount of liquid in the vessel. 2 2012. 09. 25. Archimedes' principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. A crane lowers an iron container into sweet water (water = 1000 kg/m3) which is suspended on a rope. The container has a mass of 0.5 tons and its density is 7850 kg/m3. What is the tension in the rope? Vsubmerged = m/iron T= G-Fbuoyant= mg - fluid*g*Vsubmerged T= 4905 – 625 = 4280 N 3 2012. 09. 25. Continuity equation Fluids are incompressible, the intensity of current is constant in both position and time. The cross section of the tube (A) is inversely proportional to the velocity of flow (v). IV V A v t Av t t Iv = Q = A* v = constant, stationary flow Bernoulli’s law Upon flow in a curved tube: the potential energy: mgh1 = mgh2 p1 V + mgh1 + (mv 12/2) = p2 V + mgh2+ (mv22/2) p1 + ρgh1 + (ρv 12/2) = p2 + ρgh2 + (ρv 22/2) p1 v 12 v 22 g h1 p 2 g h 2 const . 2 2 The general form of Bernoulli’s low: Static pressure Hydrostatic pressure Dynamic pressure 4 2012. 09. 25. Water flows from a large pressurized container into the open air. The pressure difference p is measured between the cross sections A1 and A2. p g pa h A2 A1 A3 p A1 = 0,8 m2, A2 = 0,2 m2, A3 = 0,4 m2, h = 0,9 m, ρ = 103 kg/m3, pa = 105 N/m2, p = 0,5 · 105 N/m2, g = 10 m/s2. Calculate a) the velocities v1, v2 and v3, b) the pressures p1, p2, p3 and the pressure „p” in the tank above the water surface. b) p1= 1,1*105 Pa; p2=6*104 Pa; p3= 105 Pa; p=1,04*105 Pa Bernoulli’s law: p1 Continuity equation : 2 v1 p2 v22 p3 v32 2 2 2 v1 A1 v2 A2 v3 A3 v1 a) v2 A2 A1 p 2 2 p 2 p1 v1 v2 p2 p1 p2 v12 v22 2 2 2 2 2 2 p 2p v v12 v22 v22 v12 2 1 A22 A2 1 A 2p v 2 A2 v22 2 2 2 v22 1 22 A1 A1 2p 2p 1 2p v22 v2 2 A22 A22 A 2 1 2 1 2 1 2 A1 A1 A1 2 1 2p A22 1 2 A1 a) v 1=2,58 m/s; v2=10,33 m/s; v3=5,17 m/s 5 2012. 09. 25. b) I.: pa p3 II.: p2 III.: p gh p3 2 v2 p3 v32 p2 p3 v32 v22 2 2 2 2 2 v3 p p3 v32 gh 2 2 b) p1= 1,1*105 Pa; p2=6*104 Pa; p3= 105 Pa; p=1,04*105 Pa Laminar flow of real fluids Newton’s law of friction F A Ns m2 F Viscosity (kinematic): Viscosity (dynamic): v h =/ Pa s Viscosity depends on: • Quality of material • Concentration • Temperature (↑temp , η ↓) • Pressure 6 2012. 09. 25. = ∙ ∙ A Newtonian fluid with a dynamic viscosity of 0.41 Pas and a density of 820 kg/m3 flows through a 25 mm diameter pipe with a velocity of 2.4 m/s. Is this flow laminar or turbulent? R = (2.4*820*12.5*10-3) / 0.41 = 60 Laminar flow When water is running in a round tube of radius 3 cm at a flow velocity of 2.2 m/s, is this flow laminar or turbulent? Assume that the kinematic viscosity of water is 9.11*10-7 m2/s. ∙ = / = R = (2.2*0.03) / 9.11*10-7 = 72448 Turbulent flow Osborne Reynolds (1842-1912) The world's longest continuously running laboratory experiment (Thomas Parnell, University of Queensland, 1927) The pitch has a viscosity approximately 230 billion (2.3×1011) times that of water. http://smp.uq.edu.au/content/pitch-drop-experiment 7 2012. 09. 25. Circulation The blood circulation is maintained by a pressure difference called blood pressure. The origine of this pressure difference is the pump function of the heart. The HAGEN-POISEUILLE law in a tube of circular cross section: R 4 p Q , 8 l If the radius of a tube decreases a greater pressure difference is required to maintain the previous flow rate. 8 2012. 09. 25. ANEURYSM, the devil’s circle. A positive feedback. Increased diameter of a weak part of the blood vessel A1 V1 A2 p1 A2 >A1 (continuity equation) V2 < V1 A increases V2 (Bernoulli’s law) V1 p1 A1 p2 p2 > p1 Positiv feedback p increases v decreases Continuity equation v A constant Bernoulli’s law p+ 1 2 v2 constant Blood composition I. Blood cells: - red blood cells, also called RBCs or erythrocytes (4-5 million/ 1 mm³ of blood, diameter approx. 7-8 μm, thickness 2-3 μm). - white blood cells, also called leukocytes (4000-10000/ 1 mm³ of blood, granulocytes, monocytes, lymphocytes). - platelets, also called thrombocytes (150400 thousand/ 1 mm³ of blood). Hematocrit (hct, ) is the proportion of blood volume that is occupied by red blood cells. Normalvalue: 0.4-0.5. 9 2012. 09. 25. Blood composition II. Blood plasma: - Approx. 90% watercontent. - Mineral ions (Na+, K+, Ca2+,Cl-,HCO3-) - Organic molecules (glucose, aminoacids, carbamide and uric acid) - Plasma proteins: albumins globulins fibrinogen Blood serum is blood plasma without fibrinogen or the other clotting factors. Circulatory System I. The circulatory system: • consists of the heart and the blood vessels (arteries, capillaries and veins) • closed system (the blood can not escape) Function: • carry oxygen and nutrients to tissues. • carry away the products of metabolism. 10 2012. 09. 25. Circulatory system II. Vessel type Diameter Total crosssectional area (cm2) Aorta 25 mm 2.5 Artery 4 mm 20 Arteriole 30 µm 40 Capillary 8 µm 2500 Venule 20 µm 250 Vein 5 mm 80 Vena cava 30 mm 8 Ratio of total Average Flow rate blood volume pressure (m/s) (%) (Hgmm/kPa) 100/13 15 0.33 96/12.7 85->30/ 11.3->4 5 30->10/ 4->1.3 0.0003 10/1.3 59 5/0.66 0.006 0/0 0.22 Physical parameters in different parts of the circulatory system Flow rate Tot al c rosssectiona l area Pressure Aorta Arteries Arterioles Capillaries Veins 11 2012. 09. 25. Cardiac biophysics The heart muscle • • • • • „brick” shaped cells (20 µm X 100 µm) Usually contains 1 central nucleus Striated Contains contractile proteins (actin & myosin) End to end junctions (intercalated disc: electric synapse) fast propagation of the action potential from cell to cell • Excitability: pacemaker function, automacy ( nerves (skeletal muscle)) 12 2012. 09. 25. Pulmonary and systemic circulation (functional and structural separation) Pulmonary circulation: • Heart-lung (right ventricule – lung – left atrium) • O2 uptake from the lung • low pressure Systemic circulation: • Heart - body (left ventricule – body – right atrium) • O2 to the body • High pressure Structure of the human heart Aorta A. pulmonalis Left atrium Bulbus aortae Aorta valve Mitral (bicuspid) valve Right atrium Tricuspid valve Left ventricle Right ventricle Septum 13 2012. 09. 25. Cardiac cycle Systole (contraction) •Isovolumetric contraction • Ejection 0.3 s Diastole (relaxation) • Isovolumetric relaxation • Rapid ventricular filling • Diastasis ( occurs just before contraction and 0.8 s (HR:72/min.) 0.5 s during which little additional blood enters the ventricle) Pressure – volume diagram Pressure (kPa) Aortic valve closing systole ejection Aortic valve opens 120 Hgmm = 16 kPa P=~15 kPa diastole isovolumetric relaxation Area! systole isovolumetric contraction ~ 10 Hgmm = 1-2 kPa 80 diastole 140 ventricular filling Volume (ml) V=140-80=60ml Work = (15*103) Pa x (60*10-6)m3 = 0.9 J = 900 mJ (/contraction) 14 2012. 09. 25. The work of the heart • Static component = p * ΔV • Dynamic component = ½ m * v2 Work = p * ΔV + ½ m * v2 ~98% ~2% Work = 15x103 N/m2 * 60x10-5 m3 + ½ 0.07kg * (0.5 m/s)2 = 0.9 + 0.0175 = ~ 0.92 Joule The static (volumetric) component dominates over the dynamic. Cardiac performance Cardiac output: the volume of blood pumped out in each minute. stroke volume (the amount of blood pumped out in one contraction (~60-70 ml)) Depends on: • preload • afterload • contractility CO = HR x SV cardiac output expressed in l/min (normal ~5 l/min) the number of beats per minute (~70-80/min.) 15 2012. 09. 25. Cardiac preload • the load to which the cardiac muscle is subjected before shortening. • the initial stretching of the cardiac myocytes before the contraction. • altered end-diastolic pressure and volume. preload sarcomere length Sarcomere length – tension relationship preload sarcomere length sarcomere length tension, force ? 16 2012. 09. 25. Sarcomere length – isometric tension Myosin Actin filaments Gordon AM, Huxley AF, Julian FJ. The variation in isometric tension with sarcomere length in vertebrate muscle fibres. J Physiol. 1966 May;184(1):170-92. Force development during muscle contraction Tension or force The final, resultant force passive force active force Muscle length 17 2012. 09. 25. Measuring the CO • Non-invasive (do not enter the body (circulatory system)) – Transoesophageal echocardiography – 2D echocardiography (Doppler US) – MRI – Arterial pulse contour analysis (based on following the pressure pulsation) • Invasive (a part of the body is entered, as by puncture or incision) – The Fick principle – The dilution technique The Fick principle the volume of blood flowing through an organ in a minute Q M VA the number of moles of a substance added to the blood by an organ in one minute the venous and arterial concentrations of that substance. To measure the blood flow through an organ that adds substances to, or removes substances from, the blood. 18 2012. 09. 25. Determination of stroke volume according to Fick's principle a. The quantity of O2 taken up during one ventilation cycle (inspiration + expiration) is equivalent to the quantity of O2 used for the oxygenisation of blood during this period. b. Inspired air contains 21% O2. Expired air contains 16% O2. The difference is 5%. c. Since the average volume of one inspiration is 500 ml, 25 ml O2 was absorbed in the blood. d. The O2 contents of arterial and venous blood are 20% and 12%, respectively. The difference is 8%. That is, 8% of volume of blood flowing through the lungs during one ventilation cycle (x) is 25 ml. Therefore, x=(100/8)*25=312.5 ml. e. Since there are ~4 cardiac cycles for each ventilation cycle, the stroke volume is ~ 78 ml. Dilution technique cc. • Dye dilution — A known amount of dye (indocyanine green, lithium) is injected into the pulmonary artery — its concentration is measured at the periphery. — CO can be calculated from the injected dose, the under curve area and its duration (Short duration high CO). time • Thermodilution — Small amount of cold saline (5-10 ml ) injected through the port of a pulmonary artery catheter. — Temperature changes are measured by a distal thermistor. (e.g.: PiCCO Monitoring) 19 2012. 09. 25. PiCCO Monitoring Pulse Contour Cardiac Output • A combination of transpulmonary thermodilution and arterial pulse contour analysis. • Able to a) assess cardiac function b) assess volume status c) evaluate treatment e.g. inotropes (an agent that alters the force or energy of muscular contractions) • Measuring the CO with PiCCO 20 2012. 09. 25. The End! 21