mathematical modelling for cardiovascular system, using

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 15, Number 1 (2019), pp. 41-54

© Research India Publications

http://www.ripublication.com/gjpam.htm

Mathematical Modelling for Cardiovascular System,

Using Arterial Tree and Lumped Parameter Models

Mr. M.V. Surseh1, Dr. P.Sekar2

1Research Scholar, SPIHER, Avadi, Chennai-600 054, India.

2Dean, Faculty of Science and Humanities, SRM University, Ramapuram, Chennai, India.

Abstract

The cardiovascular framework is described by complex communications

between different control systems and physiological procedures. Diverse

methodologies are utilized to give better diagnostics and physiological under-

standing, heart prosthesis and medicinal arranging. The scientific portrayal

and displaying of the human cardiovascular framework assumes these days a

critical job in the understanding of the beginning and create mint of

cardiovascular issue by giving computer based reproduction of dynamic

procedures in this framework. This paper expects to give a review on lumped

parameter models that have been created by numerous scientists everywhere

throughout the world, to mimic the blood stream in foundational supply

routes. Looking over different references we make a survey of various ways to

deal with blood vessel tree demonstrating and examine on the uses of such

models.

Keywords: Fluid Dynamics; Mathematical Modelling; Cardiovascular

System; Arterial Tree; Lumped Parameter Models

1. INTRODUCTION

The fundamental drivers of passings in mechanical nations are vehicle diovascular

maladies. Likewise, it is in the most productive age in which the general population

are influenced by such disfacilitates. The cardiovascular framework (CVS) fills in as a

moving vehicle for the supplements, gases and waste to and from the cells, and also

for the warmth through blood convection. It comprises of heart (with its four

chambers and four valves); fundamental dissemination (veins in the body); aspiratory

course (veins in the lungs); anxious and biochemical controllers (which modify the

vessels' parameters with the end goal to direct the blood stream rate and weight in the

mailto:[email protected]

42 Mr. M.V. Surseh, Dr. P.Sekar

organs as per the real interest). In this framework different organs or instruments, for

example, kidneys, endocrine and sensory systems, breath, digestion, physical

exercise, and so forth, may impact too. These components demonstration in different

time interims. Despite the fact that most of the exploration with respect to the

physiological frameworks depends on trials on creatures, these have their

impediments. Understanding the intricate cooperation’s and physiological elements of

the CVS isn't simple, yet the utilization of modern innovation is a useful device for

this. Late advancement in imaging innovation, and in addition all the more amazing

and more affordable PCs, has open more prominent potential outcomes for growing

more modern demonstrating techniques for CVS. A superior understanding can be

accomplished by an organized investigation of the models which depict this

framework, organizing the learning and the trial perceptions in worldly and spatial

request. Besides, a computer based reproduction of dynamic procedures in the CVS

could be connected in various undertakings.

This paper portrays a few classifications and strategies for displaying the CVS,

expecting to exhibit an audit of contrast ent dimensions of demonstrating. At last, the

application fields of the explored methodologies are talked about and their focal

points and disservices.

The stream of blood in the CVS obeys to the conservation laws portrayed by their

particular conditions. The portrayal of dynamical models, some portion of whom is

the CVS, is normally done by an arrangement of differential conditions. Contingent

upon the extent of its utilization and the exactness re-quested by the model, the

correct measurements are chosen, from zero-to three-dimensional. The measure of

information given about the inward condition of the framework distinguishes three

sorts of numerical displaying techniques:

(i) White box technique portrays the interior structure of the framework and the

qualities of change from contributions to yields. In this technique every one of

the parameters and factors of the framework are resolved and additionally

their relationship. A genuine restriction of this strategy is the gigantic measure

of procedures and collaborations that assume a job in the body.

(ii) Black box strategy considers just the change scorch acteristics, which implies

just the relationship between the information and yield of a framework,

without expert viding any more profound data in regards to the inward

structure. An impediment of this technique is its straightforwardness and the

way that they ignore the nonlinearities and particular unsettling influences.

(iii) Gray box technique, which consolidates the two previously mentioned

strategies, utilizes the information on the internal structure of the framework to

characterize the inner structure of the model.

The CVS can be viewed as a wide pressure driven system under the activity of a

pulsatile siphon. Diverse conduct can be seen at different areas of the shut circle. The

0D models (named lumped parameter models) consider as uniform the dispersion of

principal factors (weight, volume and stream rate) in each and every compartment

Mathematical Modelling for Cardiovascular System, Using Arterial Tree… 43

(organ, vein, and so forth.) of the model, in each snapshot of time. The models of

higher measurements represent the spatial variety of these parameters, as showed in

Figure 1.

The 0D models that speak to the principle parts of the framework, (vasculature

compartments and the heart with its valves), are proper for the investigation of

worldwide distribution of the weight, stream rate and blood volume, for particular

physiological conditions. The spatial circulation of the parameters can be

approximated by setting up the alleged multi-compartments models, each

compartment of which should be homogenous and de-scribed by a lumped parameter

display.

The 1D model, where the wave transports impact is effortlessly spoken to, portray the

variety of the speed of move through the length of the vein. The 2D models are

suitable to utilize when the outspread changes of the stream speed in a cylinder with

pivotal symmetry are re-quested. The utilization of 3D models is required for the

unpredictable portrayal of the blood stream in bifurcations of the vessels, through

heart valves, inside ventricles, and so on. The 1D, 2D and 3D (circulated parameter

models) and lumped models of the venous framework are not researched in this audit.

Table 1 thinks about a portion of the uses of various sorts of models.

Figure 1. Schematic representation of decoupling by segmental dimension reduction.

Table 1. Comparison of modeling techniques for the study of cardiovascular

dynamics.

The model The application

Distributed

parameters

3D Study of the local flow in 3-dimensional areas.

2D Study of local flow in vessels with axial symmetry.

1D Pulsed wave reflection effect in systemic circulation.

Lumped

parameters

0D Cardiovascular dynamics in the whole CVS.

Pressure & flow changes in local areas of circulation.


2. DISCUSSIONS

Models are a useful apparatus to comprehend the usefulness of a genuine framework

by improving it. The demonstrating strategies and their item ought to dependably be

assessed by their applications. The two most imperative applications of the

physiological procedures reproduction are: the plan of new therapeutic gadgets and,

gathering more data about the perplexing working of these frameworks with

numerous factors. A superior comprehension of CVS leads toward more compelling

and more affordable medicinal treatment. On the opposite side, the reproduction of

the CVS may fill in as a testing and train-ing stage for the prescription understudies

and their effect in decreasing the tests on creatures is noteworthy.

The simple incorporation of the regular enhancements to a current model is another

imperative viewpoint. Of a similar significance is the model approval, which

comprises in assessing the exactness of the suspicions through comparison of trial

information with those reenacted.

In 0D models that mimic the worldwide hemodynamics of the CVS, is connected the

similarity between the blood stream in the foundational corridors and the flow in an

electric circuit. The similarity begins at the left ventricle where the circulatory strain

differs from a low (about zero) to a high (roughly 120 mmHg), at that point proceeds

into the aorta and into the fundamental supply routes, where the weight variety is

altogether lower on account of the versatility of the substantial foundational conduits.

The inclination of the circulatory strain in blood vessel tree drives the blood against

the hydraulic impedance like the slope of the voltage in an electric circuit pushes the

flow against the electric impedance. The aggregate misfortune because of grinding,

the capacitance of the vessels because of their versatility and the inactivity of blood

speak to the water powered impedance like the electric impedance is spoken to by the

capacitance, opposition and inductance of the circuit. The stream of blood is portrayed

by the condition of coherence for the mass protection, by the Poiseuille's law for the

enduring stream and by the Navier-Stokes condition for the non-relentless state. Ana-

consistently, the current in a circuit is portrayed by Kir-chhoff's laws, Ohm's law and

the line transmission condition for the voltage-current connection for high

frequencies. Subsequently, as appeared in Table 2, the erosion because of consistency,

the inactivity of the stream, the capacitance of the vessel, the pulse and the stream

rate, can be respectively depicted by the opposition R, inductance L, capacitance C,

voltage and the flow in an electric circuit. As outcome the strategies utilized for the

investigation of these circuits can be utilized to dissect the cardiovascular elements as

well. The demonstrated that 0D models depict the vessel with an arrangement of two

normal differential conditions (Eq.1) for every compartment, (which speak to the

mass and force preservation), included by one logarithmic balance condition (which

relates the volume of the compartment with the weight in it).


Table 2. The quantities which characterize the fluid dynamics and their electrical

analogues.

Fluid dynamics Physiology. variables Electrical analogue

Pressure P[Pa = J/m3] Blood press. [mmHg] Voltage U [V = J/C]

Flow rate Q[m3/s] Blood flow rate [L/s] Current I [A = C/s]

Volume V[m3] Blood volume [L] Charge q [C]

Viscosity η Bl. Res.

4

8

rNlR

Electrical resistance R

Elastic coefficient Vessel’s wall comply. Capacitor’s Capac. C

Inertance Blood inertia Inductor’s inertance L

Poiseuille’s law: Q=

lrP

RP

8

4

Ohm’s law: I =

RU

outinoutout

outinin

QQQRdt

dPL

QQdt

dPC

..

.

(1)

The models of the blood vessel tree, in which the left ventricle and the correct

chamber speak to the limit conditions, have advanced by ending up more perplexing

with the expansion of components that depict particular physiological or physical

wonders. An impressive number of 0D models are produced to mimic the blood

vessel framework, beginning with the basic Windkessel (WK) models that think about

the veins as stores in zero weight and the entire blood vessel arrange as a solitary

capacitor in parallel with a solitary resistor. A critical point of reference in this

improvement was Guyton's far reaching model which particularly portrays the larger

part of blood vessel bifurcations and in addition some hormonal control components.

The 0D models can be sorted as single compartment models and multi-compartments

models.

Single-Compartment Models

There are various models of the CVS that have been broadly examined as of late. It is

an arrangement of mathematical conditions between a portion of the factors of the

framework, basically blood weights and streams that portray these models. These

factors interrelate by an arrangement of dad rameters (blood consistency, vessel

breadth, vessel divider flexibility, and so forth.) which are troublesome or even


difficult to be specifically estimated. A normal esteem sensibly fitted to the trial

information is by and large given to the different parameters.

The first to begin the worldwide powerful demonstrating of the CVS. From that point

forward, scientific model-ing has turned into an essential for hemodynamics analysis,

couple with the test approach. The entire blood vessel organize is depicted by a

solitary combination resistor-inductor-capacitor (RLC). The first to advance such

model, named Wind-kessel, recommending that the weight varieties are identified

with the substantial conduits versatility. This hypothesis was quantitatively figured

later, who de-scribed the hemodynamics of blood vessel arrange regarding

consistence and opposition.

Fank's model (Figure 2(a)) includes a capacitor C (which depicts the vault properties

of substantial arteries) associated in parallel with a resistor (which portrays the

dissipative idea of fringe vessels). This 2-element WK, which portrays the weight rot

in aorta amid diastole, is still being used today to assess the aggregate blood vessel

limit when the aortic weight and fringe opposition are known. This model disregards

the veins, which are portrayed as zero weight far fields.

The consistence of the vast corridors goes about as the WK (air port). The protection

from blood stream in the fundamental blood vessel arranges is for the most part found

in the opposition vessels: the littlest corridors and the arterioles. It isn't conceivable

the strict division of agreeable (channel) conduits and obstruction vessels. This is on

the grounds that resistive vessels have some consistence too and, vast, consistent

supply routes have little resistive properties.

The 2-component WK demonstrate demonstrates the essential effect in the heap on

the core of both: add up to blood vessel consistence and also fringe obstruction (the

opposition of the en-tire foundational vascular bed). As this model portrays the entire

blood vessel tree, as far as a weight stream connection at its passage by two

parameters that have physiological importance, it isn't conceivable to think about

marvels, for example, wave travel and wave reflections, which happen inside this

system. Despite the fact that this model can't de-copyist the high recurrence segments

that are related with the impressions of the weight in the blood vessel arrange, it

speaks to great the after load on the heart. Then again, it clarifies aortic weight rot in

diastole, however fells short in systole. Subsequently trademark impedance was

presented as a third component of the WK display.

The demonstrated the quantitative discrepancies between the weight estimated in

systole and the one anticipated by the 2-component WK, yet he didn't clarify the

reason for this.


Figure 2. (a) 2-element WK model; (b) 3-element WK ; (c) 3-element WK

The advancements towards all the more incredible processing capacities together with

the handy estimation of aortic stream made conceivable the count of the information

impedance. It was discovered that the info impedance modulus at higher frequencies

was equivalent to the trademark impedance of the proximal aorta. This prompted the

expansion of aortic trademark impedance to the 2-component WK.

The alleged Westkessel show was produced, who embedded an extra resistor ZC

(Figure 2(b)), which speaks to the aortic characteristic impedance of the blood vessel

tree. Trademark impedance interfaces this 3-component WK and wave parts of the


blood vessel framework, depicted in, to transmission marvels. The aggregate

obstruction gives the aggregate opposition of the foundational vascular system in the

previously mentioned R-C demonstrates, while the capacitor C gives the storehouse

properties of the blood vessel arrange because of the blood vessels' versatility. The

trademark impedance is rep-disdained as a resistor (since it has indistinguishable

measurements from a resistor), yet it ought to be underscored that it's anything but an

opposition and ought to be translated regarding oscillatory wonders.

By investigating the stream and the aortic weight amid diastole, when the 2-

component and 3-component WK act comparably, The demonstrated that the

expansion of the trademark impedance to the 2-component WK is necessary to

portray stream and weight all through the whole heart cycle.

In vivo ponders have demonstrated that in spite of its straightforwardness, this 3-

component WK display fundamentally enhanced the reproduction of the high

recurrence parts by predicting precisely the systolic and diastolic aortic pres-beyond

any doubt, the siphoned blood volume by the left ventricle and stroke work. Despite

the fact that this model is to a great extent utilized in CVS recreations, similar

investigations have demonstrated that it doesn't give a reasonable waveform of aortic

weight and belittles the mean blood vessel weight and the maxi-mal aortic stream rate.

Another arrangement of the Westkessel display is developed. This portrays the blood

vessel includes by putting a little resistor ZC in arrangement to the capacitor C. The

components ZC and C associated in such way, portray the elasto-thick highlights of

the vessel's dividers, contrasting from the arrangement presented, referred to above,

where the ZC depicts the wave reflection amid the stream. Despite the fact that these

two 3-component WK models are proportionate in a specific dimension, regardless it

remains the exchange of the need of wave reflection against the elasto-thick properties

of the vessel dividers in the blood vessel qualities.

With the end goal to build the model's exactness in depicting the impedance of the

veins and diminishing the errors in the low frequencies go, included an inductive

component L (Figure 3(a)) which de-copyists the inertial impact of the blood stream,

presenting so the 4-component WK. This fourth component speaks to the aggregate

blood vessel inertance (the entirety of all inertances in the blood vessel portions). The

utilization of aggregate blood vessel inertance amends the conduct of the information

impedance in low frequencies where the 3-component WK is off base.

Another design (Figure 3(b)) with the inductive component is contemplated. This

arrangement inertance influences the blood vessel input impedance just at high

frequencies and not at low frequencies.


Figure 3. (a) 4-element WK ; (b) 4-element WK

Figure 4 plots the deliberate aortic info impedance together with the impedances

anticipated by the 2-, 3-and 4-component WK models. The 2-component WK plainly

misses the mark, particularly in the high frequencies extend. The 3-component WK is

less exact at low frequencies. The more components are included a model; more

troublesome it moves toward becoming to distinguish its parameters. The expansion

of a solitary component in the 4-component WK presents substantially more

challenges than in the 2-and 3-component WK.

Figure 4, Comparison of measured and predicted input impedances.


With the reason to additionally enhance the foundational vessels tree models, have

included different resistive and inductive components (Figure 5) to reproduce the

impedance of the laminar stream (the sleeve impact), yet this model isn't additionally

investigated by different specialists as a result of its high multifaceted nature.

Figure 5. Electrical circuit representing the laminar oscillatory flow impedance in a

straight circular tube. The corrective network, enclosed by broken lines, accounts for

the sleeve effect.

The previously mentioned models portray fundamentally the attributes of the stream

and weight in aorta and in the courses where the weight beats in the venous side are

not noteworthy. Be that as it may, on account of the coronary and aspiratory course,

the beats of the weight in veins assume a critical job. Hence forth, including other R,

L, and C components empowers the depiction of the stream singe acteristics in veins,

presenting so the 5-and 6-component models.

The in vivo investigations of have demonstrated that the 5-component display offers

a superior portrayal of the elements of blood microcirculation and, the 6-component

models better depict the contribute of the venous sys-tem in the fundamental course,

contrasted and the 2-, 3-and 4-component WK.

Multi-Compartments Model

The multi-compartments models are produced to redress the powerlessness of the

single-compartment models to calculate the stream and the weight in various sections

of the vessel since they regard the foundational tree as a solitary square. In the multi-

compartments models, the foundational circulation is parceled in various fragments

(compartments), each portrayed by a blend RLC as per the neighborhood qualities of


the vessel. These compartments are joined to make the entire model of the entire

blood vessel arrange (e.g. Figure 6).

Figure 6. Example of multi-compartment model (ao: aorta; at: arteries; al: arterioles;

cp: capilaries; vn: veins).

The foundational dissemination can be isolated, contingent upon the particular

motivation behind its utilization and the exactness asked for by the model, to give

insights about a specific region, while alternate fragments can be assembled by

utilizing less advanced components. Such adaptability in the model setup is an

amazing and supportive instrument for the cardiovascular reenactments.

For the most part, the attributes of the sections of the vessels are not the same as those

of the entire framework. There is an itemized portrayal of the four run of the mill

fitting setups for the fragments of the veins, whose names and plans are given by

Figure 7.

Figure 7. Typical configurations for blood vessel segments, used as blocks in the

multi-compartments models (a) L element (b) Inverted L element (c) π element

(d) T element.


4. CONCLUSIONS

The CVS is extremely unpredictable and an excessive number of inquiries concerning

it are still with no answer. This audit explores the blood vessel 0D distributed models

and makes a depiction of their advancement and applications.

The numerical demonstrating is an amazing device for developing the comprehension

of the CVS usefulness and for enhancing the therapeutic gear innovation. In the

meantime it offers a promising strategy for aiding the non-intrusive diagnostics and in

the preparation of the cardio-specialists. Scientific models of the CVS fluctuate

fundamentally in their unpredictability and their targets. They run from the

straightforward WK show, for example, to complex portrayals of the vascular system,

for example, or even limited component models of numerous degrees of opportunity.

The lumped parameter models empower worldwide portrayal of the stream in

foundational blood vessel arrange. Henceforth the WK extends its appropriateness in

general cardiovascular elements ponders.

A model can't be named "terrible" or "great", however its quality ought to be assessed

relying upon the solid reason for its utilization. All models have distinctive

utilizations or objectives, yet they all offer the shared objective of comprehend ing

non-intrusively the CVS work. The improvement of a trustful model incorporates the

reflection of a genuine sys-tem, exploratory reproduction, the examination of the

reenactment information with the genuine framework and the decrease of the possible

irregularities.

Numerous analysts display the fundamental vasculature in portions which speak to the

different veins (top illaries and veins, arterioles, courses and aorta), Since the stream

in aorta and in the conduits (which are versatile) is pulsatile, it ought to be viewed as

the blend RLC of the three components. As in arterioles and vessels (with moderately

unbending dividers) the stream is relentless and the grating is prevailing, a resistive

component is sufficient. The inertial impact is disregarded in veins (since they have

extraordinary limit and the stream is moderately unfaltering), by making so the RC

mix sufficient to rep-detest the stream attributes. Some multi-compartments models,

for example, depict the distribution of the weight and the stream attributes in every

bifurcation of the vessels. Models of this intricacy as, despite the fact that depict

precisely the framework; present down to earth challenges in characterizing the

correct qualities for the vast majority of the parameters.

In this survey the principle perspectives and utilization of the WK are talked about.

This lumped parameter model can be utilized for the foundational blood vessel

framework and the pneumonic blood vessel bed of all well evolved creatures [44]. As

constraints of WK model of the blood vessel framework we notice its lack of ability

to think about the wave transmission marvels, the adjustments in the blood stream

conveyances, the impacts of nearby vascular changes, and so on [44].

Explores committed to complex reproductions which give numerous subtle elements

on the CVS are very costly (regarding HR), however such points of interest not

generally are required. Subsequently, less point by point models have evil presence


started to be more valuable as they give helpful information at a sensible

computational expense.

More profound investigations of the co-operations of the CVS with the respiratory,

apprehensive and stomach related frame works, will prompt wards improvement of

more essential models. Up until this point, the 0D models are to a great extent utilized

in the clinical diagnostics and treatment, yet perhaps it's time that in a joint effort with

the restorative network these models move toward becoming personalized as

indicated by the particular physiological and neurotic states of the patients.

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