Bloedsomloop - Biofysica

• symbols
  • $r$ radius
  • $h$ hight
  • $l = \ell$ length
  • $A$ area
  • $V$ volume
  • $F$ force
  • $P$ pressure
  • $Q = F$ flow
  • $\rho$ density
  • $v$ flow speed
  • $\eta$ viscosity
  • $\gamma$ shear rate (schuifsnelheid) $\gamma = - \dfrac{dv(r)}{dr} = \dfrac{\Delta P r}{2\eta l}$
  • $\tau$ shear stress (schuifspanning) $\tau = \eta \gamma$ [Pa]
  • $R$ resistance
  • $L$ inertance
  • $C$ compliance
  • $Z$ impedance
  • $f$ frequency
  • $c$ wave speed
  • $\varepsilon$ strain (rek/Elongatie) $\varepsilon = \dfrac{L - L_0}{L_0}$
  • $\sigma$ stress (wandSpanning) $\sigma = \dfrac{F}{A}$ [Pa]
  • $Y$ Young's modulus: $\sigma(t) = Y \varepsilon(t)$
  • $T$ wall tension ("spanningskracht") [Pa m]

17 Hemodynamics

Organization of the Cardiovascular System

Hemodynamics

• sphygmomanometer: measure blood pressure
• area $A = \pi r^2$ [m2]
  • e.g., $A_{aorta} = \pi 1.5^2 \approx 7 cm2$
• density $\rho$
  • blood: 1050 kg/m3 = 1.05 g/ml
• pressure $P = \dfrac{F}{A}$ [N/m2 = Pa or mmHg or cmH2O]
  • hydrostatic pressure $P = \rho gh$
    • use $\rho_{\ce{Hg}}$ to compute in mmHg$
    • 1 mmHg = 133 Pa
• flow speed $v$ [m/s]
• flow $Q = F = \dfrac{dV}{dt} = Av$ [l/min or m3/s]
  • e.g., cardiac output Q = CO = 6 l/min = 100 cm3/s
  • e.g., $v_{aorta} = \frac{CO}{A} \approx 15 cm/s$
• viscosity $\eta$ [Poise: 1 P = 0.1 Pa.s]
  • $\eta_{water} = 0.01 P$
  • $\eta_{plasma} = 0.015 P$
  • $\eta_{blood} = 0.032 P = 0.0032 Pa.s$ (formularium)
    • at 45% hematocit
  • kinematic viscosity: $\dfrac{\eta}{\rho}$
• resistance $R$ [Pa.s / m3]
  • series: $R_s = \sum_i R_i$
  • parallel: $\frac{1}{R_p} = \sum_i \frac{1}{R_i}$
  • conditions: ...
• $\Delta P = QR = FR$
  • examples
    • $MAP \approx MAP - CVP = CO \cdot SVR$
    • $MPAP \approx MPAP - MLAP = CO \cdot PVR$
  • cf. Ohm's law $V = IR$
• $CO = Q = HR \cdot SV$
  • cardiac output flow $CO$
    • 5-6l/min
  • heart rate $HR$
  • stroke volume $SV$
  • cf. $Q = \dfrac{dV}{dt}$
• MAP = $\dfrac{1}{3}$ systolic BP + $\dfrac{2}{3}$ diastolic BP
• blood flow in smaller vessels
  • Fahraeus-Lindqvist effect
    • RBCs near center
    • plasma near walls
    • branch: plasma skimming -> lower hematocrit = bad
      • solution: arterial cushions near branches
  • capillaries: RBCs deform
• Bernoulli equation
  • cf. inertia
  • cf. conservation of energy
    • $E_{pot} + E_{kin} =$ const
  • assumptions
    • no friction
    • steady flow
  • $P + 0.5\rho v^2 + \rho gh =$ const
  • $P_1 - P_2 = 0.5\rho(v_2^2 - v_1^2) + \ldots$
  • implication
    • measure pressure with invasive catheter
      • upstream: decceleration, higher pressure, incorrect
      • downstream: acceleration, lower pressure, incorrect
      • opening to side: correct measurement
  • application: estimate valve stenosis surface area
    • measure $P_{out} - P_{valve}$
    • assume $\dfrac{v_{out}}{v_{valve}} \approx 0$
    • $\Delta P = 0.5\rho v_{valve}^2 \iff v^2 = \dfrac{2\Delta P}{\rho}$
    • $Q = Av \iff A = \dfrac{Q}{\sqrt{v^2}} = Q \sqrt{\dfrac{\rho}{2\Delta P}}$
  • application: arterial stenosis
    • semi-empirical law
    • $\Delta P = R_P Q + a Q^2$
      • $a$: empirical variable
    • 80% -> 90% stenosis: 5x higher pressure drop
• continuity
  • $Q=F$ constant in closed system
  • $A_1 v_1 = A_2 v_2$
• measurements
  • pressure $\Delta P$
    • non-invasive
      • indirect
        • sphygmomanometer
        • applanation tonometry
        • venous pressure of v. jugularis
        • VCI diameter
          • ultrasound
          • P = f(VCI diameter, collapse, breathing)
      • direct
        • N/A?
    • invasive
      • indirect
        • catheder via v. jugularis -> right heart
          • "pull back"
          • wedge pressure -> LA
      • direct
        • catheder
          • via a. radiualis
          • via v. jugularis -> right heart
            • upstream vs downstream difference
            • RA, RV, pulmonary artery
          • via a. iliaca -> left heart
  • flow $Q$
    • strategies
      • direct
      • $Q = Av$
      • $Q = SV \times HR$
    • non-invasive
      • indirect
      • direct
        • ultrasound (also speed with Doppler)
        • nuclear imaging (only volume)
          • SPECT
          • PET
        • MRI (also speed)
        • CT (only volume)
      • ?
        • cardiac ultrasound
          • 1D, 2D: assume ventricular geometry
          • 3D: no assumptions
    • invasive
      • indirect
      • direct
        • perivascular methods
          • electromagnetism
            • movement of conductor (blood) induces voltage in magnetic field
          • ultrasound
      • ?
        • Fick's method
          • add known quantity to fluid
            • O2
          • measure concentration before and after
          • golden standard
        • thermodulution: ~Fick, but with temperature instead of concentration
        • LV angiogram
          • SV
          • EF = ejected fraction
  • resistance
    • strategy: $R = \dfrac{\Delta P}{Q}$

How Blood Flows

• Reynolds number $Re = \dfrac{2r \overline{v}\rho}{\eta}$
  • vessel radius $r$
    • $r_{aorta} = 1.5 cm$

Laminar

• $Re < 2200$
• concentric layers
• parabolic front
• silent
• Poiseuille law
  • conditions
    • incompressible
    • rigid cylindrical tube with radius $r_i$
    • no slippage at wall
    • laminar, steady flow
    • constant $\eta$
      • Newtonian fluid
        • water, plasma: OK
        • blood: not at lower velocity due to RBC
  • flow speed $v(r) = \dfrac{\Delta P (r_i^2 - r^2)}{4\eta l}$
    • $\eta$: viscosity
    • $l$: length
    • $r_i$: radius
    • parabolic profile
    • $v(0) = v_{max}$ (center)
    • $v(r_i) = 0$ (near wall)
    • $v(\frac{r_i}{\sqrt 2}) = \overline v = \dfrac{\Delta P r_i^2}{8\eta l}$
  • resistance $R_P$
    • $Q = A \overline v = \dfrac{\Delta P \pi r_i^4}{8\eta l}$
    • $\Delta P=QR_P \iff R_P=\dfrac{\Delta P}{Q}=\dfrac{8\eta l}{\pi r_i^4}$

Turbulent

• $Re > 2200$
  • to be sure: $Re \gg 3000$
• blunted front
• noisy -> murmurs
• when
  • large vessels
  • high $v$
    • e.g., arterial stenosis or exertion
  • low $\eta$
    • e.g., anemia

Origins of Pressure in the Circulation

• hydrostatic pressure
  • in direction of gravity
  • $P = \rho gh$
  • heart: height $h = 0$
  • recumbent vs upright
    • recumbent: no hydrostatic pressure
    • upright: hydrostatic pressure
      • more on feet
      • less on head (relative to heart)
    • $P_a - P_v$ is same in both scenarios
      • driving pressure not impacted
      • so $Q$ remains constant
• axial/driving pressure
  • causes blood flow
  • viscous resistance
• inertia
  • Bernoulli: higher $v$, lower $P$
  • pulsating flow
    • fluid briefly keeps moving forward even under negative pressure
    • inertance $L = \dfrac{\rho l}{A}$
    • $\Delta P = L \dfrac{dQ}{dt}$
    • more important than $R_P$ in large vessels
• compliance
  • transmural pressure
  • perpendicular to axis/wall
  • governs vessel diameter
  • $C = \dfrac{\Delta V}{\Delta P}$
  • $C=0 \implies$ rigid, no change in volume possible

18 Blood

Blood viscosity

• shear rate (schuifsnelheid) $\gamma = - \dfrac{dv(r)}{dr} = \dfrac{\Delta P r}{2\eta l}$
• shear stress (schuifspanning) $\tau = \eta \gamma$ [Pa]
  • $= \dfrac{\Delta P r}{2l}$
  • $= \dfrac{QRr}{2l}$
  • $= \dfrac{Qr 8\eta l}{2l \pi r^4}$
  • $= \dfrac{4\eta Q}{\pi r^3}$
• viscosity $\eta = \dfrac{\tau}{\gamma}$ [P = Pa s]
  • Newtonian fluid: constant
    • e.g., water, plasma, ~blood in large vessels
    • Einstein: $\eta = \eta_{plasma} (1 + 2.5H)$
      • $H$: hematocrit
      • not very accurate
  • non-Newtonian fluid: variable
    • e.g., blood

19 Arteries and Veins

The arterial distribution and venous-collection systems

• distribution
  • 84%: systemic
    • 14% in arteries
    • 6% in capillaries
    • 64% in veins
  • 9%: pulmonary
  • 7%: heart
• pressures (mmHg)
  • systemic
    • aorta: 95
    • arterioles: 60
    • capillaries: 25
    • venules: 15
    • veins: 3-15
  • pulmonary
    • artery: 15
    • capillaries: 10
    • veins: 5
  • largest drop in arterioles
• control of capillary pressure
  • cf. voltage splitter
  • $P_c = \dfrac{R_{post} P_a + R_{pre} P_v}{R_{post} + R_{pre}}$
  • $R_{pre} < R_{post}: P_c \to P_a$
  • $R_{pre} > R_{post}: P_c \to P_v$
    • meestal voorkomend

Elastic properties of Blood Vessels

• vessel wall
  • 4 layers
    • endothelium
      • only layer in small vessels
      • no active components
    • intima
    • media
      • smooth muscle
        • active component
        • mostly in arterioles + sphincters
    • adventitia
  • passive components
    • elastine
    • collagen I/III: less elastic
  • arteries vs veins
    • arteries
      • relatively more elastine -> less rigid
      • resistance vessels
      • ~constant compliance $C = \frac{\Delta V}{\Delta P}$
    • veins
      • high compliance at very low pressures
        • not because of elastine
        • because of geometry: flat -> filled tube
      • low compliance at very high pressures
      • capacity vessels
• rigid vs elastic vessel
  • rigid: Poiseuille (see above)
    • $\Delta P = QR$ (linear relation)
    • $R_P=\dfrac{8\eta l}{\pi r_i^4}=$ constant
  • elastic
    • non-linear relation between $Q$ and $\Delta P$
    • also depends on activity of smooth muscles
      • "sympathetic stimulation"
      • increases "critical closing pressure"
        • more $\Delta P$ needed to have any flow at all
• vasoconstriction and vasodilation
  • regulate blood distribution
  • strain (rek) $\varepsilon = \dfrac{L - L_0}{L_0}$
  • strain rate $\dfrac{d\varepsilon}{dt}$
  • stress (wandspanning) $\sigma = \dfrac{F}{A}$ [Pa]
  • stress-strain: $\sigma(t) = Y \varepsilon(t)$
    • cf. Hooke's law: $F = k \Delta L$
    • $Y$: Young's modulus = elastic modulus [Pa]
  • wall tension $T$ ("spanningskracht")
    • force per unit length
    • $T = \Delta P r = (P_t - P_i) r$ [Pa m]
      • $P_t$: tissue pressure (at outside tube wall)
      • $P_i$: intravascular pressure
      • $r$: tube radius (without wall?)
    • does not include wall thickness
    • highly correlated with elastine presence
    • different between aorta and vena cava due to elastine
  • Laplace stress (wandspanning) $\sigma$ [Pa]
    • alternative for wall tension
    • force per area
    • tube
      • length $L$
      • radius $r$
      • wall thickness $w$
      • $F_{cohesion} = \sigma A_{2 slices} = \sigma \cdot 2wL$
      • $F_{pressure} = \Delta P A_{plane} = \Delta P (2rL)$
      • $F_c = F_p \iff \sigma = \Delta P \dfrac{r}{w}$
    • sphere: $\sigma = \Delta P \dfrac{r}{2w}$
  • age
    • stiff vessels (more collagen)
• impact of smooth muscles
  • adapts wall tension $T$
  • peaks at 190% relative radius
  • -> stable vessels (no blowout, no collaps)
• pulsating flow
  • $P$ curve $\neq Q$ curve
  • effect of impedance $Z$ (= resistance + compliance + inertance)
  • frequency dependent
    • $f=0 \implies Z=R$
    • lower frequencies -> compliance, negative phase angle
    • higher frequencies -> mostly $R$
    • windkessel effect
      • heart has intermittent output
      • vessels have pulsative yet continues forward flow
      • how? aorta's compliance acts as buffer
  • clinical relevance
    • stents have little compliance
      • careful with usage in large vessels close to heart
• waves
  • background info for chapter 22
  • pressure wave
  • flow wave
  • wave speed $c$
    • $c$ > flow speed $v$
    • frequency dependent
    • lower C ~ higher c
      • e.g. in older people
      • e.g. in peripheral vessels
  • reflections
    • e.g. at aorta bifurcation
    • superposition: changes upstream pressure wave
    • augmentation index (AI)