Modeling Vascular Networks with Applications. Instructor: Van Savage Winter 2012 Quarter Meeting time: Monday and Wednesday, 10:00-11:50 am

Transcription

1 Modeling Vascular Networks with Applications Instructor: Van Savage Winter 2012 Quarter Meeting time: Monday and Wednesday, 10:00-11:50 am

2 Lumped Models

3 Ignore details of spatial structure and use lumped or circuit model

4 Rules for constructing differential equation for elements in series versus elements in parallel Basic circuit elements: resistor, capacitor, inductor All are considered to be independent even though they really are not for blood flow. In SERIES: Sum of pressure drops across each element. Volume flow rate is same across each element. In PARALLEL: Sum across volume flow rates for each element Pressure drop is same across each element. Each element has its own relation for pressure drop and flow rate, and since these are independent, we just sum across them in different ways for different circuit configurations.

5 Relationship for each element in isolation, as derived last class Resistor: "p = Q R (t)z Capacitor: "p = 1 # C dt Q (t) C! Inductor: "p = L d Q R dt!

6 Fluid analogy But in reality, can t really draw separate elements like this for fluid.

7 Examples of lumped models 1. Windkessel model: Resistor and capacitor in parallel 2. Viscoelasticity: Resistor in parallel with a resistor and capacitor in series 3. Inertia: Resistor-Inductor pair in series and resistor-capacitor in series with these two pairs in parallel with one another. 4. Back pressure: Resistor-back pressure pair in series and resistor-capacitor in series with these two pairs in parallel with one another.

8 Windkessel model

9 Windkessel model 1. First developed by Otto Frank in 1899 and is German for elastic reservoir 2. Heart working as pump has systolic (contraction and high pressure) and diastolic (relaxation and low pressure) 3. Represent systole by capacitor (balloon) that absorbs the high pressure and flow rate by expanding elastic vessels 4. Represent diastole by forward (axial flow down vessel with flow from discharging capacitor deflating balloon) that is resisted/dissipated so represented by resistor 5. At any time flow can go to expansion or forward movement, so these two elements are considered to be in parallel

10 Windkessel diagram

11 Equivalent impedance for whole circuit Represent pressure from heart as simple plane wave "p = "p 0 e i#t Impedance from capacitor is thus found by using Ohm s law! Q C = C d"p dt = Ci#"p $ Z C = "p Q C = 1 i#c

12 Equivalent impedance for whole circuit Since resistor and capacitor are in parallel we add by inverses 1 = Z eq Z R Z C = 1 R + i"c and thus! Z eq = R 1+ i"cr = R 1+ i"t C

13 Recall definitions for complex impedance Z = Z diss + iz ref From our equation when C=0! From our equation when R=0! Z diss = R Z ref = 1 i"c Captures some aspects of wave reflections and this term is also called reactance

14 Recall definitions for complex impedance Z = Z diss + iz ref For t C <<1 we make the approximation! Z eq = R e "i#t C so Z diss = Rcos("t C ) and Z ref = "Rsin(#t C )! Dissipation and reflection depend on both circuit elements

15 Dynamics of flow t C =0.01s

16 Dynamics of flow t C =0.2 s

17 Capacitive flow depends on slope of pressure t C =0.01 s

18 Capacitive flow depends on slope of pressure t C =0.2 s

19 Uses for Windkessel model 1. Modeling systemic blood flow 2. Modeling coronary blood flow 3. Modeling atherosclerosis by changing values of capacitance and resistance of vessels 4. Modeling effects of medication that change capacitance and resistance of vessels 5. Modeling pathologies that affect viscosity of blood (resistance) or compliance/elasticity of vessels (capacitance) 6. Modeling grafts and stents and their resistance and capacitance (which is lower than normal vessels)

20 Problems with Windkessel model In the 1950 s it became clear there are problems with this method 1. Essentially ignores wave reflections at junctions 2. There is a large capacitive flow reserve beyond what is filled/ expanded during systole 3. Multi-faceted regulatory mechanisms beside just capacitance help control flow through vascular network 4. Most waves occur during diastole No model yet deals with all of these issues successfully.

21 Viscoelasticity model

22 Viscoelastic materials 1. Viscoelastic materials that exhibit both elastic and viscous properties when pressure is applied. Their response to pressure changes in time. 2. Elastic materials respond to changes in pressure at a rate that is proportional to the volume flow rate. Due to bonds stretching in an ordered solid but without the order changing. Like a rubber band or spring. 3. Viscous materials respond to pressure at a rate proportional to the time derivative of volume flow rate (i.e., volume flow acceleration). Diffusion through amorphous solid. Like molasses or syrup. 4. All materials are viscoelastic to some degree

23 Viscoelastic model 1. Difficult to add this to structured or unlumped model without doing some serious fluid mechanics 2. A capacitor can be used to represent the elastic properties of vessel 3. A resistor can be used to represent the viscous or dissipative properties of a vessel. 4. Replace elastic vessel walls (capacitor) with viscoelastic vessel walls (capacitor and resistor in series). 5. This is the Maxwell model of viscoelasticity. Other versions, such as the Kelvin-Voigt model (resistor and capacitor in parallel) with other variations also exist

24 Viscoelastic Windkessel diagram

25 Equivalent impedance for whole circuit Represent pressure from heart as simple plane wave "p = "p 0 e i#t If resistance and capacitance are constant (approximation), then! d"p dt Q = C C + R v d Q C dt

26 Equivalent impedance for whole circuit Using Fourier transforms we derive that the impedance through the resistor-capacitor pair in series is Z eq = R! diss Z eq,c = R v + 1 i"c and thus combining with the other resistor in parallel 1+ i"r v C 1+ i"c(r v + R diss ) = R diss 1+ i"t C,v 1+ i"(t C,v + t C )

27 Recall definitions for complex impedance When t v,c and t C are small we can express this as! Z eq ~ R diss e i"t v,c e i"(t v,c +t C ) = R disse #i"t C So, in this limit this is just like the Windkessel model because there is almost no flow through capacitive element in series to start so the dissipation through that part of circuit can be ignored. Outside of this regime the dissipative (real part/resistance) and reflective (imaginary part/reactance) is more complicated but can be determined.

28 Dynamics of flow t C =0.2s and R v /R diss =2

29 Dynamics of flow t C =0.2s and R v /R diss =0.5

30 Dynamics of flow t C =0.2s and R v /R diss =0.1

31 Uses for Viscoelastic Windkessel model Several diseases, such as cardiovascular disease, peripheral vascular disease, sickle cell anemia, diabetes, stroke, etc. can change the values of R v and C and allow us to model the effects of these various pathologies

32 Viscoelastic Windkessel model with inertia/inductance

33 Viscoelastic-Inertial models 1. Inductors are typically ignored in lumped models, probably because people are interested in steady-state conditions 2. Inductor and dissipative resistor are both effects that occur in rigid tubes and since acceleration must occur before steady state, these effects occur (partially) sequentially, so the inductor is placed in series with the dissipative resistor

34 Viscoelastic-Inertial Windkessel diagram

35 Equivalent impedance for whole circuit Represent pressure from heart as simple plane wave! "p = "p 0 e i#t For resistor and inductor in series! Z eq,l = Z diss + Z L = R diss + i"l For resistor and capacitor in series Z eq,c = Z V + Z C = R V + 1 i"c

36 Equivalent impedance for whole circuit Combining these two lumped elements in parallel into one final lump, 1 = = Z eq Z eq,c Z eq,l after some algebra 1 R diss + i"l + 1 R v + 1 i"c Z eq = R diss (1" #$ 2 t C t L ) + i$(#t C + t L ) (1"$ 2 t C t L ) + i$t C (1+ #) ; # = R v R diss

37 Resonance, normal modes, and magic norms We can check various limits, corresponding to eliminating elements of the circuit, and everything checks out. Note that in the special case where 1. λ=1 so R v =R diss 2. t C= t L so L=R diss2 C We obtain Z eq = R diss So, total flow is same as if it was just flow through single, dissipative resistor. Thus, pressure drop and volume flow rate fall right on top of one another and are in phase. More generally, resonance depends on value of frequency, ω!

38 Dynamics of flow t C =0.2s, λ=r v /R diss =2, t L =1.0s

39 Dynamics of flow t C =0.2s, λ=r v /R diss =2, t L =0.1s

40 Dynamics of flow: Normal modes tc=0.2s, λ=rv/rdiss=1, tl=0.2s

41 Back-pressure model

42 Back-pressure model 1. Coronary blood flow is highest during diastolic/low pressure phase. This is not captured by our models thus far, and seems counterintuitive. 2. When heart contracts (systole/high pressure) muscles around heart in coronary region contract and increase pressure around vessels, preventing expansion and flow. 3. Can model this by adding back pressure that works against axial flow down the vessels. (We ignore inductance.) 4. Note that this violates the notion of truly being in parallel, and is analogous to case of asymmetric branching. Could use this kind of circuit to model asymmetric branching junction. 5. Forms of impedances are unchanged.

43 Back-pressure Windkessel diagram

44 Composite of forward and back pressure

45 Dynamics of total flow with no back pressure-flow is low during diastole t C =0.1s, λ=r v /R diss =1

46 Dynamics of total flow flow is high during diastole t C =0.1s, λ=r v /R diss =1

47 Dynamics of resistive and capacitive flow with back pressure t C =0.1s, λ=r v /R diss =1

48 Waterfall analogy

49 Topics we did not cover 1. Numerical simulations: Charles Taylor, Peskin, Olufsen 2. Fourier components to represent composite pressure and flow waves 3. Plant vascular models 4. Lungs and air transport 5. Neural architecture and sleep 6. Intracellular transport

50 1. Fractal geometry What you have learned 2. Symmetric and asymmetric branching networks 3. Fluid mechanics rigid tube, Newtonian fluids and laminar flow, pulsatile flow, elastic vessel walls, pressure, volume flow rates, complex impedance 4. Energy loss to dissipation and wave reflections at junctions 5. Using energy minimization, space-filling, and constant size of terminal units to determine fractal-like structure of vascular network 6. Structure and flow of network used to explain biological scaling

51 What you have learned 7. Applications to angiogenesis and tumor growth 8. How to use circuit elements resistance, capacitance, inductance to describe properties dissipation, elastic expansion of vessel walls, inertia/mass of fluid flow through network 9. Combining circuit elements in series and in parallel 10. Common versions of lumped models Windkessel, viscoelastic, inertial, and back pressure 11. How to analyze data for vascular networks and identify various types of patterns

52 What mathematical methods did we learn? 1. Working with several coupled partial differential equations 2. Separation of variables 3.Fourier transform 4. Bessel functions 5. Dirac delta function 6. Asymptotic expansions and Taylor series 7. Complex analysis 8. Wave-like equations and wave functions 9. Diffusion-type equations 10. Navier-Stokes equations