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[[File:Windkessel effect.svg|thumb|The Windkessel analogy illustrated.]]
[[File:Windkessel effect.svg|thumb|The Windkessel analogy illustrated.]]
'''Windkessel effect''' is a term used in [[medicine]] to account for the shape of the [[arterial blood pressure]] waveform in terms of the interaction between the [[stroke volume]] and the [[Compliance (physiology)|compliance]] of the aorta and large elastic arteries (Windkessel vessels). Windkessel when loosely translated from [[German language|German]] to English means 'air chamber',<ref>{{cite journal |vauthors=Sagawa K, Lie RK, Schaefer J |title=Translation of Otto Frank's paper "Die Grundform des Arteriellen Pulses" Zeitschrift für Biologie 37: 483-526 (1899) |journal=J. Mol. Cell. Cardiol. |volume=22 |issue=3 |pages=253–4 |date=March 1990 |pmid=2192068 |doi= 10.1016/0022-2828(90)91459-K|url=}}</ref><ref name = "Frank 1899">{{cite journal |author=Frank O |title=The basic shape of the arterial pulse. First treatise: mathematical analysis. 1899 |journal=J. Mol. Cell. Cardiol. |volume=22 |issue=3 |pages=255–77 |date=March 1990 |pmid=21438422 |doi= 10.1016/0022-2828(90)91460-O|url=}}</ref> but is generally taken to imply an ''elastic reservoir''.<ref>Ganong M.D., William F. (2005): ''Review of Medical Physiology'', Twenty-Second Edition, page 587. The McGraw-Hill Companies, Inc.</ref> The walls of large elastic arteries (e.g. [[aorta]], [[common carotid artery|common carotid]], [[subclavian artery|subclavian]], and [[pulmonary arteries]] and their larger branches) contain elastic fibers, formed of [[elastin]]. These arteries distend when the [[blood pressure]] rises during [[Systole (medicine)|systole]] and recoil when the blood pressure falls during [[diastole]]. Since the rate of blood entering these elastic arteries exceeds that leaving them due to the [[peripheral resistance]] there is a net storage of blood during systole which discharges during diastole. The distensibility of the large elastic arteries is therefore analogous to a [[capacitor]].
'''Windkessel effect''' is a term used in [[medicine]] to account for the shape of the [[arterial blood pressure]] waveform in terms of the interaction between the [[stroke volume]] and the [[Compliance (physiology)|compliance]] of the aorta and large [[Elastic artery|elastic arteries]] (Windkessel vessels) and the [[Drag (physics)|resistance]] of the smaller arteries and [[Arteriole|arterioles]]. Windkessel when loosely translated from [[German language|German]] to English means 'air chamber',<ref>{{cite journal |vauthors=Sagawa K, Lie RK, Schaefer J |title=Translation of Otto Frank's paper "Die Grundform des Arteriellen Pulses" Zeitschrift für Biologie 37: 483-526 (1899) |journal=J. Mol. Cell. Cardiol. |volume=22 |issue=3 |pages=253–4 |date=March 1990 |pmid=2192068 |doi= 10.1016/0022-2828(90)91459-K|url=}}</ref><ref name = "Frank 1899">{{cite journal |author=Frank O |title=The basic shape of the arterial pulse. First treatise: mathematical analysis. 1899 |journal=J. Mol. Cell. Cardiol. |volume=22 |issue=3 |pages=255–77 |date=March 1990 |pmid=21438422 |doi= 10.1016/0022-2828(90)91460-O|url=}}</ref> but is generally taken to imply an ''elastic reservoir''.<ref>Ganong M.D., William F. (2005): ''Review of Medical Physiology'', Twenty-Second Edition, page 587. The McGraw-Hill Companies, Inc.</ref> The walls of large elastic arteries (e.g. [[aorta]], [[common carotid artery|common carotid]], [[subclavian artery|subclavian]], and [[pulmonary arteries]] and their larger branches) contain elastic fibers, formed of [[elastin]]. These arteries distend when the [[blood pressure]] rises during [[Systole (medicine)|systole]] and recoil when the blood pressure falls during [[diastole]]. Since the rate of blood entering these elastic arteries exceeds that leaving them via the [[peripheral resistance]], there is a net storage of blood in the the aorta and large arteries during systole, which discharges during diastole. The compliance (or [[distensibility]]) of the aorta and large elastic arteries is therefore analogous to a [[capacitor]].


The Windkessel effect helps in [[damping]] the fluctuation in [[blood pressure]] ([[pulse pressure]]) over the [[cardiac cycle]] and assists in the maintenance of organ perfusion during diastole when cardiac ejection ceases. The idea of the Windkessel was alluded to by [[Giovanni Borelli]], although [[Stephen Hales]] articulated the concept more clearly and drew the analogy with an air chamber used in fire engines in the 18th century.<ref>Stephen Hales ''Statical Essays: Haemastaticks'', 1733</ref> [[Otto Frank (physiologist)]], an influential German physiologist, developed the concept and provided a firm mathematical foundation.<ref name = "Frank 1899" /> Frank's model is sometimes called a two-element Windkessel to distinguish it from more recent and more elaborate Windkessel models (e.g. three- or four-element Windkessels).<ref>Westerhof N, Lankhaar JW, Westerhof BE.The arterial Windkessel.Med Biol Eng Comput. 2009 Feb;47(2):131-41. Epub 2008 Jun 10.</ref>
The Windkessel effect helps in [[damping]] the fluctuation in [[blood pressure]] ([[pulse pressure]]) over the [[cardiac cycle]] and assists in the maintenance of organ perfusion during diastole when cardiac ejection ceases. The idea of the Windkessel was alluded to by [[Giovanni Borelli]], although [[Stephen Hales]] articulated the concept more clearly and drew the analogy with an air chamber used in fire engines in the 18th century.<ref>Stephen Hales ''Statical Essays: Haemastaticks'', 1733</ref> [[Otto Frank (physiologist)]], an influential German physiologist, developed the concept and provided a firm mathematical foundation.<ref name = "Frank 1899" /> Frank's model is sometimes called a two-element Windkessel to distinguish it from more recent and more elaborate Windkessel models (e.g. three- or four-element and non-linear Windkessel models).<ref>Westerhof N, Lankhaar JW, Westerhof BE.The arterial Windkessel.Med Biol Eng Comput. 2009 Feb;47(2):131-41. Epub 2008 Jun 10.</ref><ref>{{Cite journal|last=Cappello|first=A.|last2=Gnudi|first2=G.|last3=Lamberti|first3=C.|date=1995-3|title=Identification of the three-element windkessel model incorporating a pressure-dependent compliance|url=https://www.ncbi.nlm.nih.gov/pubmed/7605053|journal=Annals of Biomedical Engineering|volume=23|issue=2|pages=164–177|issn=0090-6964|pmid=7605053}}</ref>


⚫
=== The Two-Element Windkessel Model ===
We can assume the air pressure to air volume ratio is constant and thus the water flowing through the chambers follows [[Poiseuille's Law]] and is proportional to the fluid pressure. The [[differential equation]] that describes this proportionality is described below.
⚫
[[File:2-Element Windkessel model.svg|thumb|2-Element Windkessel Circuit Analogy Illustrated]]

It is assumed that the ratio of pressure to volume is constant and that outflow from the Windkessel is proportional to the fluid pressure. Volumetric inflow must equal the sum of the volume stored in the capacitive element and volumetric outflow through the resistive element. This relationship is described by a [[differential equation]]:


<math>I(t)={P(t)\over R}+C{dP(t)\over dt}</math>
<math>I(t)={P(t)\over R}+C{dP(t)\over dt}</math>


The flow out of the pump is described by I(t) and measured in volume per unit time, while P(t) is the pressure with respect to time measured in force per unit area, C is the ratio of air pressure to air volume, and R is the proportionality constant. This model also describes the relationship between current, I(t), and [[Electric potential|electrical potential]], P(t), in the following two-element Windkessel model.
''I(t)'' is volumetric inflow due to the pump (heart) and is measured in volume per unit time, while ''P(t)'' is the pressure with respect to time measured in force per unit area, ''C'' is the ratio of pressure to volume for the Windkessel, and ''R'' is the resistance relating outflow to fluid pressure. This model is identical to the relationship between current, ''I(t)'', and [[Electric potential|electrical potential]], ''P(t)'', in an electrical circuit equivalent of the two-element Windkessel model.


In the blood circulation, the passive elements in the circuit are assumed to represent elements in the [[Circulatory system|cardiovascular system]]. The resistor, ''R'', represents the total peripheral resistance and the capacitor, ''C'', represents total arterial compliance.<ref name=":1">{{Cite book|url=https://www.worldcat.org/oclc/676701119|title=Snapshots of hemodynamics : an aid for clinical research and graduate education|last=N.|first=Westerhof,|date=2010|publisher=Springer|others=Stergiopulos, Nikos, 1962-, Noble, Mark I. M.|isbn=9781441963635|edition=2nd ed|location=New York|oclc=676701119}}</ref>
⚫
=== The Two-Element Windkessel Model ===
⚫
[[File:2-Element Windkessel model.svg|thumb|2-Element Windkessel Circuit Analogy Illustrated]]


During [[diastole]] there is no blood inflow since the aortic (or pulmonary valve) is closed, so the Windkessel can be solved for ''P(t)'' since ''I(t) = 0:''
The passive elements in the circuit represent elements in the [[Circulatory system|cardiovascular system]]. The Resistor R represents the total peripheral resistance and capacitor C stands for compliance of veins. During [[Diastole]] the Windkessel can be solved for P(t) since I(t)=0, no blood flow. Td is the start of [[diastole]] and P(td) is the blood pressure at the start of diastole. This model is only a rough approximation of the arterial bed a more realistic model adds more elements and are discussed below.


<math>P(t)=P(t_d)e^{-(t-t_td)\over (RC)}</math>
<math>P(t)=P(t_d)e^{-(t-t_td)\over (RC)}</math>

where ''t<sub>d</sub>'' is the time of the start of [[diastole]] and ''P(t<sub>d</sub>)'' is the blood pressure at the start of diastole. This model is only a rough approximation of the arterial circulation; more realistic models incorporate more elements, provide more realistic estimates of the blood pressure waveform and are discussed below.


=== The Three-Element Windkessel Model ===
=== The Three-Element Windkessel Model ===
The three-element Windkessel compares to the two-element by incorporating another resistive element to simulate resistance to blood flow due to the aortic valve. The [[differential equation]] for the 3-element model can be found below.
The three-element Windkessel improves on the two-element model by incorporating another resistive element to simulate resistance to blood flow due to the characteristic resistance of the aorta (or pulmonary artery). The [[differential equation]] for the 3-element model is:


<math>(1+{R_1\over R_2})I(t)+CR_1{dI(t)\over dt}= {P(t)\over R_2}+C{dP(t)\over dt}</math>
<math>(1+{R_1\over R_2})I(t)+CR_1{dI(t)\over dt}= {P(t)\over R_2}+C{dP(t)\over dt}</math>
[[File:3-Element Windkessel Model.svg|thumb|3-Element]]
[[File:3-Element Windkessel Model.svg|thumb|3-Element]]


R1 represents the resistance due to the aortic valve or pulmonary valve while R2 represents the peripheral resistance. In order to solve this differential equation assumptions are made. R1=0 & R2=R which reduces the 3-Element to a 2-Element model as shown above. This model is useful in evaluating the general characteristics of the system and has been employed to evaluate blood pressure and flow in the aorta of a chick embryo <ref name=":0">{{Cite web|url=http://www.civilized.com/mlabexamples/windkesmodel.htmld/|title=Solving Windkessel Models with MLAB|website=www.civilized.com|access-date=2018-11-14}}</ref>, and pulmonary artery in a pig<ref name=":0" />.
where ''R<sub>1</sub>'' is the characteristic resistance (this is assumed to be equivalent to the characteristic impedance),<ref name=":1" /> while ''R<sub>2</sub>'' represents the peripheral resistance. This model is widely used as an acceptable model of the circulation.<ref>{{Cite journal|last=Westerhof|first=Nico|last2=Lankhaar|first2=Jan-Willem|last3=Westerhof|first3=Berend E.|date=2009-2|title=The arterial Windkessel|url=https://www.ncbi.nlm.nih.gov/pubmed/18543011|journal=Medical & Biological Engineering & Computing|volume=47|issue=2|pages=131–141|doi=10.1007/s11517-008-0359-2|issn=1741-0444|pmid=18543011}}</ref> For example it has been employed to evaluate blood pressure and flow in the aorta of a chick embryo <ref name=":0">{{Cite web|url=http://www.civilized.com/mlabexamples/windkesmodel.htmld/|title=Solving Windkessel Models with MLAB|website=www.civilized.com|access-date=2018-11-14}}</ref> and the pulmonary artery in a pig<ref name=":0" /> as well as providing the basis for construction of physical models of the circulation providing realistic loads for experimental studies of isolated hearts.<ref>{{Cite journal|last=Westerhof|first=N.|last2=Elzinga|first2=G.|last3=Sipkema|first3=P.|date=1971-11|title=An artificial arterial system for pumping hearts|url=https://www.ncbi.nlm.nih.gov/pubmed/5117196|journal=Journal of Applied Physiology|volume=31|issue=5|pages=776–781|doi=10.1152/jappl.1971.31.5.776|issn=0021-8987|pmid=5117196}}</ref>


=== The Four-Element Windkessel Model ===
=== The Four-Element Windkessel Model ===
[[File:2-element, 3-element and 4 element Windkessel models.svg|thumb|4-Element compared to the 2- and 3-Element Windkessel models]]
[[File:2-element, 3-element and 4 element Windkessel models.svg|thumb|4-Element compared to the 2- and 3-Element Windkessel models]]
The four-element model includes an [[inductor]], L, into the branch of the circuit to simulate the [[inertia]] of blood flow that is neglected in the two- and three- element models. We can find the governing differential equation the same way as the 2- and 3- element by taking the electrical potential drop across the inductor as, L [d I(t) /dt], which has units of mass per length, <math>{M\over L^4}</math>.
The three-element model overestimates the compliance and underestimates the characteristic impedance of the circulation.<ref name=":1" /> The four-element model includes an [[inductor]], ''L'', which has units of mass per length, (<math>{M\over l^4}</math>), into the proximal component of the circuit to account for the [[inertia]] of blood flow. This is neglected in the two- and three- element models. The relevant equation is:


<math>(1+{R_1\over R_2})I(t)+(R_1C+{L\over R_2}){dI(t)\over dt}+LC{d^2I(t)\over dt^2}={P(t)\over R_2}+C{dP(t)\over dt}</math>
<math>(1+{R_1\over R_2})I(t)+(R_1C+{L\over R_2}){dI(t)\over dt}+LC{d^2I(t)\over dt^2}={P(t)\over R_2}+C{dP(t)\over dt}</math>


=== Use of Windkessel models ===
This equation can be simplified by assuming L=0, which reduces it the three-element model. The equation can also be reduced by evaluating during diastole, which I(t) and its derivatives go to zero, and P(t) can be solved as in the three-element.
⚫
These models relate blood flow to blood pressure through parameters of ''R, C (''and, in the case of the four-element model, ''L)''. These equations can be easily solved (e.g. by employing MATLAB and its supplement SIMULINK) to either find the values of pressure given flow and ''R, C, L'' parameters, or find values of ''R, C, L'' given flow and pressure. An example for the two-element model is shown below, where ''I(t)'' is depicted as an input signal during systole and diastole. Systole is represented by the ''sin'' function, while flow during diastole is zero. ''s'' represents the duration of the cardiac cycle , while ''Ts'' represents the duration of systole, and ''Td'' represents the duration of diastole (e.g. in seconds).

⚫
These three models all lead to equations that relates blood flow to blood pressure through parameters of R,L, and C. These equations can be easily solved by employing MATLAB and it's supplement SIMULINK to either find the values of pressure given flow and R,C parameters, find values of flow given pressure and R,C parameters, or find values of R,C given flow and pressure. An example to solve for pressure given flow and parameters can be found below, where I(t) is depicted as an input signal during systole and diastole. Systole is represented by the sin function while flow during diastole is zero. s represents the period the cardiac cycles in seconds while Ts represents the period of systole during that cardiac cycles in seconds, while Td represents the period of diastole during that cardiac cycle in seconds.


<math>I(t)=I_osin[{(pi*{t\over s})\over Ts}] for {t\over s}\leq Ts
<math>I(t)=I_osin[{(pi*{t\over s})\over Ts}] for {t\over s}\leq Ts
Line 41: Line 44:
[[File:Cardiac cycle pressure only.png|thumb|Graph Evaluating Systole and Diastole Pressure]]
[[File:Cardiac cycle pressure only.png|thumb|Graph Evaluating Systole and Diastole Pressure]]


=== Windkessel models in physiology and disease ===
The Windkessel effect becomes diminished with age as the elastic arteries become less compliant, termed ''hardening of the arteries'' or [[arteriosclerosis]], probably secondary to fragmentation and loss of elastin. The reduction in the Windkessel effect results in increased [[pulse pressure]] and elevated systolic pressure for a given [[stroke volume]]. Elevated systolic pressure ([[hypertension]]) predicts [[myocardial infarction]], [[stroke]], [[heart failure]] and a variety of other cardiovascular diseases.<ref>{{cite journal |vauthors=Lewington S, Clarke R, Qizilbash N, Peto R, Collins R |title=Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies |journal=Lancet |volume=360 |issue=9349 |pages=1903–13 |date=December 2002 |pmid=12493255 |doi= 10.1016/S0140-6736(02)11911-8|url=}}</ref>
The 'Windkessel effect' becomes diminished with age as the elastic arteries become less compliant, termed ''hardening of the arteries'' or [[arteriosclerosis]], probably secondary to fragmentation and loss of elastin.<ref>{{Cite journal|last=Greenwald|first=S. E.|date=2007|title=Ageing of the conduit arteries|url=https://www.ncbi.nlm.nih.gov/pubmed/17200940|journal=The Journal of Pathology|volume=211|issue=2|pages=157–172|doi=10.1002/path.2101|issn=0022-3417|pmid=17200940|via=}}</ref> The reduction in the Windkessel effect results in increased [[pulse pressure]] for a given [[stroke volume]]. The increased pulse pressure results in elevated systolic pressure ([[hypertension]]) which increases the risk of [[myocardial infarction]], [[stroke]], [[heart failure]] and a variety of other cardiovascular diseases.<ref>{{cite journal |vauthors=Lewington S, Clarke R, Qizilbash N, Peto R, Collins R |title=Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies |journal=Lancet |volume=360 |issue=9349 |pages=1903–13 |date=December 2002 |pmid=12493255 |doi= 10.1016/S0140-6736(02)11911-8|url=}}</ref>



=== Limitations of Windkessel models ===
Although the Windkessel is a simple and convenient concept, it is widely seen as in conflict with, and superseded by, modern approaches that interpret arterial pressure and flow waveforms in terms of wave propagation and reflection,<ref>McDonald D.A. (1960). ''Blood Flow in Arteries''. Monographs of the Physiological Society. Baltimore: Williams and Wilkins Company</ref><ref>Nichols W.W., O'Rourke M.F. (2005). ''McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles''. Hodder Arnold Publication</ref> although recently attempts have been made to integrate wave propagation and Windkessel approaches.<ref>{{cite journal |vauthors=Tyberg JV, Davies JE, Wang Z, etal |title=Wave intensity analysis and the development of the reservoir-wave approach |journal=Med Biol Eng Comput |volume=47 |issue=2 |pages=221–32 |date=February 2009 |pmid=19189147 |doi=10.1007/s11517-008-0430-z |url=}}</ref> The situation may even be more complex since the arterial wall contains smooth muscle cells that hypothetically contribute to the pressure wave from the heart on a beat-to-beat basis. The theory of arterial acceleration proposes that the onset of systole triggers a short-lasting depolarisation within smooth muscle cells that spreads along the arterial tree as a peristaltic wave. This mechanism is thought to amplify and distribute the pressure wave over the arterial tree allowing it to penetrate in even the most remote capillary systems<ref>{{cite journal |vauthors=Schaafsma A |title=Harvey with a modern twist: How and why conducting arteries amplify the pressure wave originating from the heart |journal=Medical Hypotheses |volume=82 |issue=5 |pages=589–94 |date=February 2014 |doi=10.1016/j.mehy.2014.02.016 |url=https://doi.org/10.1016/j.mehy.2014.02.016}}</ref>.
Although the Windkessel is a simple and convenient concept, it has been largely superseded by more modern approaches that interpret arterial pressure and flow waveforms in terms of wave propagation and reflection.<ref>Nichols W.W., O'Rourke M.F. (2005). ''McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles''. Hodder Arnold Publication</ref> Recent attempts to integrate wave propagation and Windkessel approaches through a reservoir concept,<ref>{{cite journal |vauthors=Tyberg JV, Davies JE, Wang Z, etal |title=Wave intensity analysis and the development of the reservoir-wave approach |journal=Med Biol Eng Comput |volume=47 |issue=2 |pages=221–32 |date=February 2009 |pmid=19189147 |doi=10.1007/s11517-008-0430-z |url=}}</ref> have been criticized<ref>{{Cite journal|last=Segers|first=Patrick|last2=Swillens|first2=Abigail|last3=Vermeersch|first3=Sebastian|date=2012-4|title=Reservations on the reservoir|url=https://www.ncbi.nlm.nih.gov/pubmed/22418902|journal=Journal of Hypertension|volume=30|issue=4|pages=676–678|doi=10.1097/HJH.0b013e32835077be|issn=1473-5598|pmid=22418902}}</ref><ref>{{Cite journal|last=Westerhof|first=Nico|last2=Segers|first2=Patrick|last3=Westerhof|first3=Berend E.|date=2015-7|title=Wave Separation, Wave Intensity, the Reservoir-Wave Concept, and the Instantaneous Wave-Free Ratio: Presumptions and Principles|url=https://www.ncbi.nlm.nih.gov/pubmed/26015448|journal=Hypertension (Dallas, Tex.: 1979)|volume=66|issue=1|pages=93–98|doi=10.1161/HYPERTENSIONAHA.115.05567|issn=1524-4563|pmid=26015448}}</ref> and a recent consensus document highlighted the wave-like nature of the reservoir.<ref>{{Cite journal|date=2017-06-01|title=Towards a consensus on the understanding and analysis of the pulse waveform: Results from the 2016 Workshop on Arterial Hemodynamics: Past, present and future|url=https://www.sciencedirect.com/science/article/pii/S1872931217300327|journal=Artery Research|language=en|volume=18|pages=75–80|doi=10.1016/j.artres.2017.03.004|issn=1872-9312|pmc=PMC5470638|pmid=28626494}}</ref>


==References==
==References==

Revision as of 13:29, 13 January 2019

The Windkessel analogy illustrated.

Windkessel effect is a term used in medicine to account for the shape of the arterial blood pressure waveform in terms of the interaction between the stroke volume and the compliance of the aorta and large elastic arteries (Windkessel vessels) and the resistance of the smaller arteries and arterioles. Windkessel when loosely translated from German to English means 'air chamber',[1][2] but is generally taken to imply an elastic reservoir.[3] The walls of large elastic arteries (e.g. aorta, common carotid, subclavian, and pulmonary arteries and their larger branches) contain elastic fibers, formed of elastin. These arteries distend when the blood pressure rises during systole and recoil when the blood pressure falls during diastole. Since the rate of blood entering these elastic arteries exceeds that leaving them via the peripheral resistance, there is a net storage of blood in the the aorta and large arteries during systole, which discharges during diastole. The compliance (or distensibility) of the aorta and large elastic arteries is therefore analogous to a capacitor.

The Windkessel effect helps in damping the fluctuation in blood pressure (pulse pressure) over the cardiac cycle and assists in the maintenance of organ perfusion during diastole when cardiac ejection ceases. The idea of the Windkessel was alluded to by Giovanni Borelli, although Stephen Hales articulated the concept more clearly and drew the analogy with an air chamber used in fire engines in the 18th century.[4] Otto Frank (physiologist), an influential German physiologist, developed the concept and provided a firm mathematical foundation.[2] Frank's model is sometimes called a two-element Windkessel to distinguish it from more recent and more elaborate Windkessel models (e.g. three- or four-element and non-linear Windkessel models).[5][6]

The Two-Element Windkessel Model

2-Element Windkessel Circuit Analogy Illustrated

It is assumed that the ratio of pressure to volume is constant and that outflow from the Windkessel is proportional to the fluid pressure. Volumetric inflow must equal the sum of the volume stored in the capacitive element and volumetric outflow through the resistive element. This relationship is described by a differential equation:

I(t) is volumetric inflow due to the pump (heart) and is measured in volume per unit time, while P(t) is the pressure with respect to time measured in force per unit area, C is the ratio of pressure to volume for the Windkessel, and R is the resistance relating outflow to fluid pressure. This model is identical to the relationship between current, I(t), and electrical potential, P(t), in an electrical circuit equivalent of the two-element Windkessel model.

In the blood circulation, the passive elements in the circuit are assumed to represent elements in the cardiovascular system. The resistor, R, represents the total peripheral resistance and the capacitor, C, represents total arterial compliance.[7]

During diastole there is no blood inflow since the aortic (or pulmonary valve) is closed, so the Windkessel can be solved for P(t) since I(t) = 0:

where td is the time of the start of diastole and P(td) is the blood pressure at the start of diastole. This model is only a rough approximation of the arterial circulation; more realistic models incorporate more elements, provide more realistic estimates of the blood pressure waveform and are discussed below.

The Three-Element Windkessel Model

The three-element Windkessel improves on the two-element model by incorporating another resistive element to simulate resistance to blood flow due to the characteristic resistance of the aorta (or pulmonary artery). The differential equation for the 3-element model is:

3-Element

where R1 is the characteristic resistance (this is assumed to be equivalent to the characteristic impedance),[7] while R2 represents the peripheral resistance. This model is widely used as an acceptable model of the circulation.[8] For example it has been employed to evaluate blood pressure and flow in the aorta of a chick embryo [9] and the pulmonary artery in a pig[9] as well as providing the basis for construction of physical models of the circulation providing realistic loads for experimental studies of isolated hearts.[10]

The Four-Element Windkessel Model

4-Element compared to the 2- and 3-Element Windkessel models

The three-element model overestimates the compliance and underestimates the characteristic impedance of the circulation.[7] The four-element model includes an inductor, L, which has units of mass per length, (), into the proximal component of the circuit to account for the inertia of blood flow. This is neglected in the two- and three- element models. The relevant equation is:

Use of Windkessel models

These models relate blood flow to blood pressure through parameters of R, C (and, in the case of the four-element model, L). These equations can be easily solved (e.g. by employing MATLAB and its supplement SIMULINK) to either find the values of pressure given flow and R, C, L parameters, or find values of R, C, L given flow and pressure. An example for the two-element model is shown below, where I(t) is depicted as an input signal during systole and diastole. Systole is represented by the sin function, while flow during diastole is zero. s represents the duration of the cardiac cycle , while Ts represents the duration of systole, and Td represents the duration of diastole (e.g. in seconds).

Graph Evaluating Systole and Diastole Pressure

Windkessel models in physiology and disease

The 'Windkessel effect' becomes diminished with age as the elastic arteries become less compliant, termed hardening of the arteries or arteriosclerosis, probably secondary to fragmentation and loss of elastin.[11] The reduction in the Windkessel effect results in increased pulse pressure for a given stroke volume. The increased pulse pressure results in elevated systolic pressure (hypertension) which increases the risk of myocardial infarction, stroke, heart failure and a variety of other cardiovascular diseases.[12]

Limitations of Windkessel models

Although the Windkessel is a simple and convenient concept, it has been largely superseded by more modern approaches that interpret arterial pressure and flow waveforms in terms of wave propagation and reflection.[13] Recent attempts to integrate wave propagation and Windkessel approaches through a reservoir concept,[14] have been criticized[15][16] and a recent consensus document highlighted the wave-like nature of the reservoir.[17]

References

  1. ^ Sagawa K, Lie RK, Schaefer J (March 1990). "Translation of Otto Frank's paper "Die Grundform des Arteriellen Pulses" Zeitschrift für Biologie 37: 483-526 (1899)". J. Mol. Cell. Cardiol. 22 (3): 253–4. doi:10.1016/0022-2828(90)91459-K. PMID 2192068.
  2. ^ a b Frank O (March 1990). "The basic shape of the arterial pulse. First treatise: mathematical analysis. 1899". J. Mol. Cell. Cardiol. 22 (3): 255–77. doi:10.1016/0022-2828(90)91460-O. PMID 21438422.
  3. ^ Ganong M.D., William F. (2005): Review of Medical Physiology, Twenty-Second Edition, page 587. The McGraw-Hill Companies, Inc.
  4. ^ Stephen Hales Statical Essays: Haemastaticks, 1733
  5. ^ Westerhof N, Lankhaar JW, Westerhof BE.The arterial Windkessel.Med Biol Eng Comput. 2009 Feb;47(2):131-41. Epub 2008 Jun 10.
  6. ^ Cappello, A.; Gnudi, G.; Lamberti, C. (1995-3). "Identification of the three-element windkessel model incorporating a pressure-dependent compliance". Annals of Biomedical Engineering. 23 (2): 164–177. ISSN 0090-6964. PMID 7605053. {{cite journal}}: Check date values in: |date= (help)
  7. ^ a b c N., Westerhof, (2010). Snapshots of hemodynamics : an aid for clinical research and graduate education. Stergiopulos, Nikos, 1962-, Noble, Mark I. M. (2nd ed ed.). New York: Springer. ISBN 9781441963635. OCLC 676701119. {{cite book}}: |edition= has extra text (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
  8. ^ Westerhof, Nico; Lankhaar, Jan-Willem; Westerhof, Berend E. (2009-2). "The arterial Windkessel". Medical & Biological Engineering & Computing. 47 (2): 131–141. doi:10.1007/s11517-008-0359-2. ISSN 1741-0444. PMID 18543011. {{cite journal}}: Check date values in: |date= (help)
  9. ^ a b "Solving Windkessel Models with MLAB". www.civilized.com. Retrieved 2018-11-14.
  10. ^ Westerhof, N.; Elzinga, G.; Sipkema, P. (1971-11). "An artificial arterial system for pumping hearts". Journal of Applied Physiology. 31 (5): 776–781. doi:10.1152/jappl.1971.31.5.776. ISSN 0021-8987. PMID 5117196. {{cite journal}}: Check date values in: |date= (help)
  11. ^ Greenwald, S. E. (2007). "Ageing of the conduit arteries". The Journal of Pathology. 211 (2): 157–172. doi:10.1002/path.2101. ISSN 0022-3417. PMID 17200940.
  12. ^ Lewington S, Clarke R, Qizilbash N, Peto R, Collins R (December 2002). "Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies". Lancet. 360 (9349): 1903–13. doi:10.1016/S0140-6736(02)11911-8. PMID 12493255.
  13. ^ Nichols W.W., O'Rourke M.F. (2005). McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles. Hodder Arnold Publication
  14. ^ Tyberg JV, Davies JE, Wang Z, et al. (February 2009). "Wave intensity analysis and the development of the reservoir-wave approach". Med Biol Eng Comput. 47 (2): 221–32. doi:10.1007/s11517-008-0430-z. PMID 19189147.
  15. ^ Segers, Patrick; Swillens, Abigail; Vermeersch, Sebastian (2012-4). "Reservations on the reservoir". Journal of Hypertension. 30 (4): 676–678. doi:10.1097/HJH.0b013e32835077be. ISSN 1473-5598. PMID 22418902. {{cite journal}}: Check date values in: |date= (help)
  16. ^ Westerhof, Nico; Segers, Patrick; Westerhof, Berend E. (2015-7). "Wave Separation, Wave Intensity, the Reservoir-Wave Concept, and the Instantaneous Wave-Free Ratio: Presumptions and Principles". Hypertension (Dallas, Tex.: 1979). 66 (1): 93–98. doi:10.1161/HYPERTENSIONAHA.115.05567. ISSN 1524-4563. PMID 26015448. {{cite journal}}: Check date values in: |date= (help)
  17. ^ "Towards a consensus on the understanding and analysis of the pulse waveform: Results from the 2016 Workshop on Arterial Hemodynamics: Past, present and future". Artery Research. 18: 75–80. 2017-06-01. doi:10.1016/j.artres.2017.03.004. ISSN 1872-9312. PMC 5470638. PMID 28626494.{{cite journal}}: CS1 maint: PMC format (link)
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