Lumped parameter model Lumped parameter models describe the - - PDF document

Lesson 4 Lumped parameter models of the cardiovascular system: introduction and clinical applications.

Ettore Lanzarone March 18, 2019

MEDICAL SUPPORT SYSTEMS FOR CHRONIC DISEASES

Engineering and Management for Health University of Bergamo

LESSON 4

Lumped parameter model

Lumped parameter models describe the fluid-dynamic behavior within a vessel segment. Let us consider a vessel segment i. We model the pressure pi and the flow Qi based on:

• the geometrical properties of the vessel
• (see picture)
• the mechanical properties of the vessel

(Young modulus of the wall Ei).

• the rheological properties of the fluid

(the blood) Flow Qi+1 exits from segment i and enters in segment i+1

Lesson 4

Lumped parameter model

They are lumped because the concentrate the pressure and the flow in all points of the vessel into

• nly two variables: pressure pi and the flow Qi.

Blood rheology

Blood is a suspension of particles (erythrocytes, leukocytes and thrombocytes) in aqueous solution (the plasma). In general it can not be considered a homogeneous fluid. It can be assumed homogeneous when it flows into vessels whose lumen is much greater than the size of the erythrocytes: up to small arteries with an internal diameter of 0.3 mm. In these cases, the blood is characterized simply by its density and its viscosity. When blood flows in smaller vessels, it is no longer correct to use a homogeneous fluid model, but it is still possible to define an apparent viscosity to avoid distinguishing the different components.

Lesson 4

Blood rheology

Blood density is given by the average between the density of the plasma and that of the corpuscular part, weighed on the volume fraction of the two components. The fraction in volume of the corpuscular part is the Hematocrit Ht. Thus:

• the density of the corpuscular part is equal to 1095 kg/m3
• the density of plasma is equal to 1030 kg/m3

The density is independent of temperature.

 

Ht ρ Ht ρ ρ

plasma r corpuscola blood

     1

Blood rheology

When it is considered homogeneous, blood is treated as a Newtonian fluid (viscosity independent of motion conditions). The viscosity of plasma is given by the Lightfoot relationship: The viscosity of water in function of the temperature is given by the Poiseuille relation: T is the temperature expressed in Celsius [oC] μ0 is the viscosity of water at oC, equal to 1.888 cP

water plasma

μ μ   8 . 1 T . T . μ μwater

2

00022 0337 1     

Lesson 4

Blood rheology

Finally, the viscosity of whole blood is given by the Bull relationship: Ht is once again the hematocrit Summing up, the viscosity of the whole blood is expressed in function of temperature and hematocrit as:

 

Ht μ μ

plasma blood

    5 . 2 1

 

cP . T . T . Ht . . μblood 808 1 00022 0337 1 5 2 1 8 1

2

        Blood rheology

In small-diameter vessels, the corpuscular fraction tends to occupy the central part of the vessel, with the red blood cells arranged perpendicular to the capillary axis, while the plasma is placed in the part closest to the wall. In this configuration, the plasma has a lubricating effect: the internal friction

• f the blood is reduced and the viscosity is therefore lower.

By further reducing the diameter, the erythrocytes are forced to deform to advance; this increases energy losses and thus the viscosity.

Lesson 4

Blood rheology

Progressively reducing the diameter, the viscosity first decreases and then tends to rise again: Fahareus-Lindqvist effect

Blood rheology

In this case, we can anyway define an “apparent” viscosity that defines the

• verall behavior in function of the vessel diameter.

with: D internal diameter in μm Ht hematocrit in percentage μBull is the viscosity according to the Bull relationship

     

2 1 1 2 1 1 1 100 45 1 1 100 1 1 45 1                                    , D D , D D C C Ht μ plasma μ apparent μ

 

12 11 12 11 075

10 1 1 10 1 1 1 8 D D e , C

D ,

               

    645

06 085 45

44 2 6

,

D ,

• D

,

• plasma

Bull

e ,

• e

μ μ μ

 

   

Lesson 4

Blood rheology Vessel model with electrical analogy

The state variables are the average pressure pi and the flow Qi in the i-th segment. The parameters are:

• Resistance Ri: viscous hydraulic resistance which opposes the advancement of the blood.
• Compliance Ci: capacitive effect due to the elastic behavior of the walls.
• Inertance Li: inertial effect linked to the motion of blood in the vessels.
• Viscous resistance of the wall RVi: dissipative effect linked to the viscous behavior of the walls.

All these parameters are calculated starting from the rheological properties

• f the blood (density and viscosity) and the geometrical and mechanical

characteristics of the vessel (internal radius, thickness of vessel walls, length and radial Young modulus of the walls).

Lesson 4

Vessel model with electrical analogy

The fluid dynamic equations are obtained starting from:

• continuity equation (i.e., mass conservation)

 capacitive effect deriving from the conservation of the mass

• energy balance

 resistive and inertial effects deriving from the energy balance The dissipative element relative to the viscosity of the wall is then added.

Vessel model with electrical analogy

The conservation of the mass can be written as: M is the mass contained in the segment 𝑛 ̇ is the mass flow rate (subscript 1 for mass entering the vessel segment and subscript 2 for mass leaving the vessel segment). Considering a constant density ρ and denoting by V the volume and by Q the volumetric flow, the expression becomes:

2 1

m m dt dM    

2 1 2 1

Q Q dt dV Q ρ Q ρ dt dV ρ      

Lesson 4

Vessel model with electrical analogy

We now include the definition of Compliance: Thus, the first equation for the vessel motion is:

2 1

Q Q dt dp C  

dp dV C 

Vessel model with electrical analogy

The second equation is obtained with the energy balance, under the hypothesis that the incoming mass flow equals the outgoing one. The energy balance is: where:

• E is the energy;
• v is the velocity;
• ΔH the energy dissipation per unit of time due to blood viscosity;
• β is a multiplicative coefficient linked to the motion regime.

Subscript 1 is incoming quantities and subscript 2 for outgoing quantities. The difference in altitude between input and the output is neglected.

ΔH m v β v β m ρ p ρ p m dt dE                          2 2

2 2 2 2 1 1 2 1

Lesson 4

Vessel model with electrical analogy

Energy is only kinetic energy (neglecting thermal effects): Its derivative is (with constant radius):

2 2

2 2 2

v l πr ρ v M E     dt dv l m dt dv v l πr ρ dt dE       

2

Vessel model with electrical analogy

Thus, the energy balance is: The terms related to the velocity are equal and are simplified (same input and output flow and same cross sections). Thus:

ΔH m v β v β m ρ p ρ p m dt dv l m                            2 2

2 2 2 2 1 1 2 1

ΔH ρ dt dQ r π l ρ p p ΔH ρ dt dv l ρ p p            

2 2 1 2 1

Lesson 4

Vessel model with electrical analogy

is rewritten as:

ΔH ρ dt dQ r π l ρ p p       

2 2 1

Q R dt dQ L p p    

2 1

Vessel model with electrical analogy

The two equations that describe the fluid dynamics are: Now:

• How to determine the parameters R, L and C based on mechanical

and geometrical properties.

• How to describe in terms of an electrical circuit.
• How to attach the viscosity of the vessel walls.

Q R dt dQ L p p Q Q dt dp C      

2 1 2 1 Dynamical system described by two ODEs of the first order for the two state variables p and Q

Lesson 4

Vessel model with electrical analogy

COMPLIANCE C We assume a thin-walled vessel. Under this assumption, the compliance C is determined with the Laplace law: (T is the circumferential tension in the wall) By differentiating this law we have:

r p T  

dp r dr p dT    

Vessel model with electrical analogy

dT is expressed as a function of the infinitesimal circumferential deformation dr: where E is the Young modulus of the vessel walls. Thus:

E s dT E dσ dε r dr dε    

r dr E s dT  

Lesson 4

Vessel model with electrical analogy

By replacing in the Laplace law we get: We can now express the compliance: And pr is negligible with respect to Es:

p r E s r dp dr  

r s-p E r l π p r E s r l r π dp dr dr dV dp dV C           

3

2 2

s E r l π C    

3

2

Vessel model with electrical analogy

RESISTANCE R R represents the pressure drop per unit of volumetric flow due to viscosity, while ΔH the same fall per unit of density. Resistance R is expressed using the Hagen-Poiseuille relationship: Thus:

Q r π l μ Δp

4

8    

4

8 r π l μ Q Δp R     

Lesson 4

Vessel model with electrical analogy

INERTANCE L 2

r π l ρ L   

Vessel model with electrical analogy

ELECTRICAL CIRCUIT The two equations that describe the fluid dynamics are:

Q R dt dQ L p p Q Q dt dp C      

2 1 2 1

We can replace:

• Pressure p with electrical voltage V
• Flow Q with electrical current I

Lesson 4

Vessel model with electrical analogy

Then, we can consider the following electrical components of a circuit RESISTANCE

I R V  

 V

Vessel model with electrical analogy

Then, we can consider the following electrical components of a circuit INDUCTANCE (equivalent of INERTANCE)

dt dI L V  

 V

Lesson 4

Vessel model with electrical analogy

Then, we can consider the following electrical components of a circuit CAPACITY (equivalent of COMPLIANCE)

dt dV C I  

 V

Vessel model with electrical analogy

Then, we can consider the Kirchhoff's circuit laws

CURRENT LAW

Lesson 4

Vessel model with electrical analogy

Then, we can consider the Kirchhoff's circuit laws

VOLTAGE LAW

Vessel model with electrical analogy

Equivalent equations:

Q R dt dQ L p p Q Q dt dp C      

2 1 2 1

I R dt dI L V V I I dt dV C      

2 1 2 1

Lesson 4

Vessel model with electrical analogy

Possible circuits for a vessel:

I R dt dI L V V I I dt dV C      

2 1 2 1

Vessel model with electrical analogy

VISCOSITY OF THE VESSEL WALLS The viscous component involves internal sliding of material with associated conversion of mechanical energy into heat and temperature increase. Considering vessel walls, the viscous dissipative effect attenuates the oscillations of the walls. In the absence of dissipation, the oscillations generated during a cardiac cycle would not be damped and would be added to those of the following cycle. This would lead to instability. The presence of a dissipative effect eliminates this problem, as well as responding to a need for the model to relate to physiological reality.

Lesson 4

Vessel model with electrical analogy

The viscous resistance is assigned according to Bergel (1961), which gives the value of the time constant of the analog RC of the vessel wall. According to this work: In a π shape we have:

sec 002 . R C

V 



• D. H. Bergel

“The static elastic properties of the arterial wall” Journal of Physiology, 1961, 150, 445–457

Vessel model with electrical analogy

Equations:

2 2 002

1 1 1 1    

                

i i i i i i i i i i i i i i i

C C Ceq L Q R p p dt dQ Ceq dt dQ dt dQ , Q Q dt dp

Lesson 4

Circuit layouts

SOME EXAMPLES OF LAYOUTS

1899 Otto Frank Windkessel model of the whole circulation With only two parameters

Circuit layouts

SOME EXAMPLES OF LAYOUTS

1971 Westerhof Model with two resistances

Lesson 4

Circuit layouts

SOME EXAMPLES OF LAYOUTS

1982 Burattini and Gnudi Entire systemic circulation with segments and

peripheral resistances

• R. Burattini – G. Gnudi

“Computer identification of models for the arterial tree input impedance: comparison between two new models and first experimental results” Medical & Biological Engineering & Computing, 1982, 20, 134–144

Circuit layouts

SOME EXAMPLES OF LAYOUTS

How to get the values of mechanical and geometrical properties of each segment?

Westerhof provided cadaver measures

Lesson 4

Circuit layouts

SOME EXAMPLES OF LAYOUTS

2007, 2009 Lanzarone et al. Entire systemic circulation in which the peripheral

resistances are not constant to model the peripheral control of the circulation.

• E. Lanzarone, P. Liani, G. Baselli, M.L. Costantino.

Model of arterial tree and peripheral control for the study of physiological and assisted circulation. Med Eng Phys 2007; 29(5):542-55

• E. Lanzarone, G. Casagrande, R. Fumero, M.L. Costantino.

Integrated model of endothelial NO regulation and systemic circulation for the comparison between pulsatile and continuous perfusion. IEEE T Bio-Med Eng 2009; 56(5):1331-40

Circuit layouts

SOME EXAMPLES OF LAYOUTS

1999 Sharp and Dharmalingam Approach to estimate the coeffients based on the number of elements in the circuit

MK Sharp, RK Dharmalingam. Development on an hydraulic model of the human systemic circulation. ASAIO J 1999; 45: 535-540.

Lesson 4

Input flow

INPUT FLOW

The input flow of the arterial systems (trhough the aortic valve) is usually modeled considering the expression

• f Swanson and Clark:

with:

       

c s s sc in

T t per T T t per Q Q                                        t T π , t T π , t T π , T T Q , Q

s s s s c mean sc

3 sin 092 2 sin 23 sin 924 65 1

Input flow

INPUT FLOW

The systolic period Ts can be expressed in function of the cardiac rate (Katz and Feil):

Also the cardiac period Tc can be expressed in function of the same rate f is in Hz

f . Ts 096  f Tc 1 

Lesson 4

Input flow

INPUT FLOW

Thus, the input flow is expressed only in function of mean flow and cardiac frequency.

5 10 15 20 25 0,2 0,4 0,6 0,8 t [s] Qin [l/min]

Practical example

I provide you a Simnon T file with:

• model of the circulation including 4 parameters (RLRC)
• Swanson and Clark input flow

Next lesson: practical lesson on lumped parameter models starting from this file and other models.