View source for Hemodynamics

{{short description|Dynamics of blood flow}}
'''Hemodynamics''' [[American and British English spelling differences#ae and oe|or]] '''haemodynamics''' are the [[Fluid dynamics|dynamics]] of '''blood flow'''. The [[circulatory system]] is controlled by [[homeostasis|homeostatic]] mechanisms of [[autoregulation]], just as [[hydraulic circuit]]s are controlled by [[control system]]s. The hemodynamic response continuously monitors and adjusts to conditions in the body and its environment. Hemodynamics explains the [[physical law]]s that govern the flow of blood in the [[blood vessel]]s.

Blood flow ensures the transportation of [[nutrient]]s, [[hormone]]s, metabolic waste products, [[oxygen]], and [[carbon dioxide]] throughout the body to maintain cell-level [[metabolism]], the regulation of the [[pH]], [[osmotic pressure]] and temperature of the whole body, and the protection from microbial and mechanical harm.<ref name="Principles of Anatomy & Physiology">{{cite book |title=Principles of Anatomy & Physiology |edition=13th |last1=Tortora |first1=Gerard J. |last2=Derrickson |first2=Bryan |year=2012 |publisher=John Wiley & Sons |isbn=978-0-470-56510-0 |chapter=The Cardiovascular System: The Blood |pages=729–732}}</ref>

Blood is a [[non-Newtonian fluid]], and is most efficiently studied using [[rheology]] rather than hydrodynamics. Because blood vessels are not rigid tubes, classic hydrodynamics and fluids mechanics based on the use of classical viscometers are not capable of explaining haemodynamics.<ref>{{cite book |title=Biology and Mechanics of Blood Flows, Part II: Mechanics and Medical Aspects |last1=Fieldman |first1=Joel S. |last2=Phong |first2=Duong H. |last3=Saint-Aubin |first3=Yvan |last4=Vinet |first4=Luc |year=2007 |publisher=Springer |isbn=978-0-387-74848-1 |chapter=Rheology |pages=119–123}}</ref>

The study of the blood flow is called hemodynamics, and the study of the properties of the blood flow is called [[hemorheology]].

==Blood==
{{main|Blood}}
Blood is a complex liquid. [[Blood]] is composed of [[Blood plasma|plasma]] and [[Blood#Constituents|formed elements]]. The plasma contains 91.5% water, 7% [[proteins]] and 1.5% other solutes. The formed elements are [[platelet]]s, [[white blood cell]]s, and [[red blood cell]]s. The presence of these formed elements and their interaction with plasma molecules are the main reasons why blood differs so much from ideal Newtonian fluids.<ref name="Principles of Anatomy & Physiology"/>

===Viscosity of plasma===
Normal [[blood plasma]] behaves like a Newtonian fluid at physiological rates of shear. Typical values for the [[viscosity]] of normal human plasma at 37&nbsp;°C is 1.4&nbsp;mN·s/m<sup>2</sup>.<ref>{{cite journal |url=http://jap.physiology.org/content/jap/19/1/117.full.pdf |title=Human blood under normothermic and hypothermic conditions |last=Rand |first=Peter |date=31 May 1963 |journal=Journal of Applied Physiology |volume=19 |pages=117–122 |access-date=16 September 2014 |doi=10.1152/jappl.1964.19.1.117 |pmid=14104265 }}{{Dead link|date=July 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> The viscosity of normal plasma varies with temperature in the same way as does that of its solvent water<ref name=errill>{{cite journal|author-last=Errill|author1-first=Edward W.|title=Rheology of Blood|journal=[[Physiological Reviews]]|date=October 1969|volume=49|issue=4|pages=863–888 |quote=In normal blood, at levels of shear rate such that equation 15 is applicable, i.e., Newtonian flow, it has been shown that the temperature coefficient of viscosity is identical with that for water over the range of 10-37°C|doi=10.1152/physrev.1969.49.4.863|pmid=4898603 }}</ref>;a 3°C change in temperature in the physiological range (36.5°C to 39.5°C)reduces plasma viscosity by about 10%.<ref>{{cite journal|author1-last=Cinar|author1-first=Yildirim|author2-last=Senyol|author2-first=A. Mete|author3-last=Duman|author3-first=Kamber|title=Blood viscosity and blood pressure: role of temperature and hyperglycemia|journal=[[American Journal of Hypertension]]|date=May 2001|volume=14|issue=5|pages=433–438 |doi=10.1016/S0895-7061(00)01260-7|doi-access=free|pmid=11368464 }}</ref>

===Osmotic pressure of plasma===
The osmotic pressure of solution is determined by the number of particles present and by the [[temperature]]. For example, a 1 molar solution of a substance contains {{val|6.022|e=23}} molecules per liter of that substance and at 0&nbsp;°C it has an osmotic pressure of {{cvt|2.27|MPa|atm}}. The osmotic pressure of the [[Blood plasma|plasma]] affects the mechanics of the circulation in several ways. An alteration of the osmotic pressure difference across the membrane of a blood cell causes a shift of water and a change of cell volume. The changes in shape and flexibility affect the mechanical properties of whole blood. A change in plasma [[osmotic pressure]] alters the hematocrit, that is, the volume concentration of red cells in the whole blood by redistributing water between the intravascular and extravascular spaces. This in turn affects the mechanics of the whole blood.<ref name="autogenerated3"/>

===Red blood cells===
The [[red blood cell]] is highly flexible and biconcave in shape. Its membrane has a [[Young's modulus]] in the region of 106&nbsp;[[Pascal (unit)|Pa]]. Deformation in red blood cells is induced by shear stress. When a suspension is sheared, the red blood cells deform and spin because of the velocity gradient, with the rate of deformation and spin depending on the shear rate and the concentration.
This can influence the mechanics of the circulation and may complicate the measurement of blood [[viscosity]]. It is true that in a steady state flow of a viscous fluid through a rigid spherical body immersed in the fluid, where we assume the [[inertia]] is negligible in such a flow, it is believed that the downward [[gravitational]] force of the particle is balanced by the viscous drag force. From this force balance the speed of fall can be shown to be given by [[Stokes' law]] {{citation needed|date=March 2021}}

:<math>U_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, a^2</math><ref name="autogenerated3"/>
Where ''a'' is the particle radius, ''ρ<sub>p</sub>'', ''ρ<sub>f</sub>'' are the respectively particle and fluid density ''μ'' is the fluid viscosity, ''g'' is the gravitational acceleration. From the above equation we can see that the [[sedimentation velocity]] of the particle depends on the square of the radius. If the particle is released from rest in the [[fluid]], its sedimentation velocity ''U<sub>s</sub>'' increases until it attains the steady value called the terminal velocity (U), as shown above.{{citation needed|date=March 2021}}

===Hemodilution===
Hemodilution is the dilution of the concentration of red blood cells and plasma constituents by partially substituting the blood with [[colloid]]s or [[Volume expander|crystalloids]]. It is a strategy to avoid exposure of patients to the potential hazards of [[Blood transfusion|homologous]] blood transfusions.<ref>{{Cite journal |last1=Naqash |first1=Imtiaz A. |last2=Draboo |first2=M. A. |last3=Lone |first3=Abdul Qayoom |last4=Nengroo |first4=Showkat H. |last5=Kirmani |first5=Altaf |last6=Bhat |first6=Abdul Rashid |date=2011 |title=Evaluation of Acute Normovolemic Hemodilution and Autotransfusion in Neurosurgical Patients Undergoing Excision of Intracranial Meningioma |journal=Journal of Anaesthesiology Clinical Pharmacology |volume=27 |issue=1 |pages=54–58 |doi=10.4103/0970-9185.76645 |issn=0970-9185 |pmc=3146160 |pmid=21804707 |doi-access=free }}</ref><ref>{{Cite journal |last1=Saito |first1=Chihiro |last2=Kamei |first2=Tetsuya |last3=Kubota |first3=Shoji |last4=Yoshida |first4=Kiyoshi |last5=Hibiya |first5=Makoto |last6=Hashimoto |first6=Shuji |date=December 2018 |title=Association of Primary Hemodilution and Retrograde Autologous Priming with Transfusion in Cardiac Surgery: Analysis of the Perfusion Case Database of the Japanese Society of Extra-Corporeal Technology in Medicine |journal=The Journal of Extra-corporeal Technology |volume=50 |issue=4 |pages=231–236 |doi=10.1051/ject/201850231 |issn=0022-1058 |pmc=6296452 |pmid=30581230}}</ref>

Hemodilution can be normovolemic, which implies the dilution of normal blood constituents by the use of expanders. During acute normovolemic hemodilution (ANH), blood subsequently lost during surgery contains proportionally fewer red blood cells per milliliter, thus minimizing intraoperative loss of the whole blood. Therefore, blood lost by the patient during surgery is not actually lost by the patient, for this volume is purified and redirected into the patient.{{citation needed|date=March 2021}}

On the other hand, hypervolemic hemodilution (HVH) uses acute preoperative volume expansion without any blood removal. In choosing a fluid, however, it must be assured that when mixed, the remaining blood behaves in the microcirculation as in the original blood fluid, retaining all its properties of [[viscosity]].<ref>{{cite journal |url=http://journals.lww.com/anesthesiology/Abstract/2001/03000/Efficacy_of_Acute_Normovolemic_Hemodilution.13.aspx |publisher=the journal of American society of anesthsiologist inc |title=Efficacy of Acute Normovolemic hemodilution, Accessed as a Function of Blood lost |journal=Anesthesiology |date=March 2001 |volume=94 |issue=3 |doi=10.1097/00000542-200103000-00017 |access-date=5 April 2011 |archive-date=29 June 2012 |archive-url=https://web.archive.org/web/20120629102929/http://journals.lww.com/anesthesiology/Abstract/2001/03000/Efficacy_of_Acute_Normovolemic_Hemodilution.13.aspx |url-status=live |last1=Dembinski |first1=Rolf |last2=Max |first2=Martin |last3=López |first3=Frank |last4=Kuhlen |first4=Ralf |last5=Kurth |first5=Roland |last6=Rossaint |first6=Rolf |pages=461–467 |pmid=11374607 }}</ref>

In presenting what volume of ANH should be applied one study suggests a mathematical model of ANH which calculates the maximum possible RCM savings using ANH, given the patients weight ''H<sub>i</sub>'' and ''H<sub>m</sub>''.{{citation needed|date=March 2021}}

To maintain the normovolemia, the withdrawal of autologous blood must be simultaneously replaced by a suitable hemodilute. Ideally, this is achieved by isovolemia exchange transfusion of a plasma substitute with a colloid [[osmotic pressure]] (OP). A [[colloid]] is a fluid containing particles that are large enough to exert an oncotic pressure across the micro-vascular membrane.
When debating the use of colloid or crystalloid, it is imperative to think about all the components of the starling equation:
:<math>\ Q = K ( [P_c - P_i]S - [P_c - P_i] )</math>
To identify the minimum safe hematocrit desirable for a given patient the following equation is useful:{{citation needed|date=March 2021}}
:<math>\ BL_s = EBV \ln \frac{H_i}{H_m} </math>
where EBV is the estimated [[blood]] volume; 70&nbsp;mL/kg was used in this model and ''H<sub>i</sub>'' (initial hematocrit) is the patient's initial hematocrit.
From the equation above it is clear that the volume of blood removed during the ANH to the ''H<sub>m</sub>'' is the same as the ''BL<sub>s</sub>''.
How much blood is to be removed is usually based on the weight, not the volume. The number of units that need to be removed to hemodilute to the maximum safe hematocrit (ANH) can be found by
:<math>ANH = \frac {BL_s}{450}</math>
This is based on the assumption that each unit removed by hemodilution has a volume of 450&nbsp;mL (the actual volume of a unit will vary somewhat since completion of collection is dependent on weight and not volume).
The model assumes that the hemodilute value is equal to the ''H<sub>m</sub>'' prior to surgery, therefore, the re-transfusion of blood obtained by hemodilution must begin when SBL begins.
The RCM available for retransfusion after ANH (RCMm) can be calculated from the patient's ''H<sub>i</sub>'' and the final hematocrit after hemodilution(''H<sub>m</sub>'')
:<math> RCM = EVB \times (H_i - H_m) </math>
The maximum SBL that is possible when ANH is used without falling below Hm(BLH) is found by assuming that all the blood removed during ANH is returned to the patient at a rate sufficient to maintain the hematocrit at the minimum safe level
:<math> BL_H = \frac {RCM_H} {H_m}</math>
If ANH is used as long as SBL does not exceed ''BL<sub>H</sub>'' there will not be any need for blood transfusion. We can conclude from the foregoing that ''H'' should therefore not exceed ''s''.
The difference between the ''BL<sub>H</sub>'' and the ''BL<sub>s</sub>'' therefore is the incremental surgical blood loss (''BL<sub>i</sub>'') possible when using ANH.
:<math>\ {BL_i} = {BL_H} - {BL_s} </math>
When expressed in terms of the RCM
:<math> {RCM_i} = {BL_i} \times {H_m} </math>
Where ''RCM<sub>i</sub>'' is the red cell mass that would have to be administered using homologous blood to maintain the ''H<sub>m</sub>'' if ANH is not used and blood loss equals BLH.{{citation needed|date=March 2021}}

The model used assumes ANH used for a 70&nbsp;kg patient with an estimated blood volume of 70&nbsp;ml/kg (4900&nbsp;ml). A range of ''H<sub>i</sub>'' and ''H<sub>m</sub>'' was evaluated to understand conditions where hemodilution is necessary to benefit the patient.<ref name="autogenerated1">{{cite book |publisher=IEEE |title=Hemodilution:Modelling and clinincal Aspects |volume=1 |pages=177–179 |doi=10.1109/IEMBS.2001.1018881 |chapter=Hemodilution: Modeling and clinical aspects |year=2001 |last1=Kaya |first1=M. |last2=Li |first2=J.K.-J. |isbn=978-0-7803-7211-5 |s2cid=73295413 }}</ref><ref>{{cite journal |url=http://www.anesthesia-analgesia.org/content/80/1/108.full.pdf+html |title=maximum blood savings by acute Normovolemic hemodilution |journal=Anesthesia & Analgesia |volume=80 |issue=1 |pages=108–113 |access-date=5 April 2011 |date=January 1995 |last1=Feldman |first1=Jeffrey M. |last2=Roth |first2=Jonathan V. |last3=Bjoraker |first3=David G. |doi=10.1097/00000539-199501000-00019 |pmid=7802266 |s2cid=24131754 |archive-date=23 February 2022 |archive-url=https://web.archive.org/web/20220223065145/https://journals.lww.com/anesthesia-analgesia/pages/articleviewer.aspx?year=1995&issue=01000&article=00019&type=Fulltext |url-status=live |doi-access=free }}</ref>

====Result====
The result of the model calculations are presented in a table given in the appendix for a range of ''H<sub>i</sub>'' from 0.30 to 0.50 with ANH performed to minimum hematocrits from 0.30 to 0.15. Given a ''H<sub>i</sub>'' of 0.40, if the ''H<sub>m</sub>'' is assumed to be 0.25.then from the equation above the RCM count is still high and ANH is not necessary, if BL<sub>s</sub> does not exceed 2303&nbsp;ml, since the hemotocrit will not fall below H<sub>m</sub>, although five units of blood must be removed during hemodilution. Under these conditions, to achieve the maximum benefit from the technique if ANH is used, no homologous blood will be required to maintain the ''H<sub>m</sub>'' if blood loss does not exceed 2940&nbsp;ml. In such a case, ANH can save a maximum of 1.1 packed red blood cell unit equivalent, and homologous blood transfusion is necessary to maintain ''H<sub>m</sub>'', even if ANH is used.{{citation needed|date=March 2021}}
This model can be used to identify when ANH may be used for a given patient and the degree of ANH necessary to maximize that benefit.{{citation needed|date=March 2021}}

For example, if ''H<sub>i</sub>'' is 0.30 or less it is not possible to save a red cell mass equivalent to two units of homologous PRBC even if the patient is hemodiluted to an ''H<sub>m</sub>'' of 0.15. That is because from the RCM equation the patient RCM falls short from the equation giving above.
If ''H<sub>i</sub>'' is 0.40 one must remove at least 7.5 units of blood during ANH, resulting in an ''H<sub>m</sub>'' of 0.20 to save two units equivalence. Clearly, the greater the ''H<sub>i</sub>'' and the greater the number of units removed during hemodilution, the more effective ANH is for preventing homologous blood transfusion. The model here is designed to allow doctors to determine where ANH may be beneficial for a patient based on their knowledge of the ''H<sub>i</sub>'', the potential for SBL, and an estimate of the ''H<sub>m</sub>''. Though the model used a 70&nbsp;kg patient, the result can be applied to any patient. To apply these result to any body weight, any of the values BLs, BLH and ANHH or PRBC given in the table need to be multiplied by the factor we will call T
:<math> T = \frac  {\text{patient's weight in kg}} {70} </math>
Basically, the model considered above is designed to predict the maximum RCM that can save ANH.{{citation needed|date=March 2021}}

In summary, the efficacy of ANH has been described mathematically by means of measurements of surgical blood loss and blood volume flow measurement. This form of analysis permits accurate estimation of the potential efficiency of the techniques and shows the application of measurement in the medical field.<ref name="autogenerated1"/>

== Blood flow ==
===Cardiac output===
[[File:Diagram of human circulatory system.gif|thumb|upright=1.8|Diagram of the circulatory system. SVC/IVC - [[superior vena cava|Superior]]/[[inferior vena cava|Inferior]] vena cava. ]]

The heart is the driver of the circulatory system, pumping blood through rhythmic contraction and relaxation. The rate of blood flow out of the heart (often expressed in L/min) is known as the cardiac output (CO).

Blood being pumped out of the heart first enters the [[aorta]], the largest artery of the body. It then proceeds to divide into smaller and smaller arteries, then into [[arterioles]], and eventually [[capillaries]], where oxygen transfer occurs. The capillaries connect to [[venules]], and the blood then travels back through the network of veins to the [[venae cavae]] into the [[right heart]]. The micro-circulation — the arterioles, capillaries, and venules —constitutes most of the area of the vascular system and is the site of the transfer of O<sub>2</sub>, [[glucose]], and [[enzyme substrate]]s into the cells. The venous system returns the de-oxygenated blood to the right heart where it is pumped into the [[lungs]] to become oxygenated and CO<sub>2</sub> and other gaseous wastes exchanged and expelled during breathing. Blood then returns to the left side of the heart where it begins the process again.

In a normal circulatory system, the volume of blood returning to the heart each minute is approximately equal to the volume that is pumped out each minute (the cardiac output).<ref name=brs>{{cite book |last1=Costanzo |first1=Linda S. |title=Physiology |date=2003 |publisher=Lippincott Williams & Wilkins |location=Philadelphia |isbn=978-0781739191 |pages=73–113 |edition=3rd|series=Board Review Series }}</ref> Because of this, the velocity of blood flow across each level of the circulatory system is primarily determined by the total cross-sectional area of that level.

Cardiac output is determined by two methods. One is to use the Fick equation:

<math>CO = VO2/C_aO_2-C_vO_2 </math>

The other thermodilution method is to sense the temperature change from a liquid injected in the proximal port of a Swan-Ganz to the distal port.

Cardiac output is mathematically expressed by the following equation:

: <math>CO = SV \times HR </math>
where
* CO = cardiac output (L/sec)
* SV = stroke volume (ml)
* HR = heart rate (bpm)
The normal human cardiac output is 5-6 L/min at rest. Not all blood that enters the left ventricle exits the heart. What is left at the end of diastole (EDV) minus the stroke volume make up the end systolic volume (ESV).<ref>{{Citation |last1=King |first1=Jordan |title=Physiology, Cardiac Output |date=2022 |url=http://www.ncbi.nlm.nih.gov/books/NBK470455/ |work=StatPearls |place=Treasure Island (FL) |publisher=StatPearls Publishing |pmid=29262215 |access-date=2022-05-16 |last2=Lowery |first2=David R.}}</ref>

==== Anatomical features ====
Circulatory system of species subjected to orthostatic blood pressure (such as arboreal snakes) has evolved with physiological and morphological features to overcome the circulatory disturbance. For instance, in arboreal snakes the heart is closer to the head, in comparison with aquatic snakes. This facilitates blood perfusion to the brain.<ref>{{Cite journal|date=2014|title=Heart place and tail length evaluation in Naja oxiana, Macrovipera lebetina, and Montivipera latifii|journal=Asian Pacific Journal of Tropical Medicine|doi=10.1016/S1995-7645(14)60220-0|doi-access=free|last1=Nasoori|first1=Alireza|last2=Taghipour|first2=Ali|last3=Shahbazzadeh|first3=Delavar|last4=Aminirissehei|first4=Abdolhossein|last5=Moghaddam|first5=Sharif|volume=7|pages=S137–S142|pmid=25312108}}</ref><ref>{{Cite journal|date=2012|title=Gravity and the evolution of cardiopulmonary morphology in snakes|url= |journal=Comparative Biochemistry and Physiology|doi=10.1016/j.cbpa.2011.10.029|pmc=3242868|last1=Lillywhite|first1=Harvey B.|last2=Albert|first2=James S.|last3=Sheehy|first3=Coleman M.|last4=Seymour|first4=Roger S.|volume=161|issue=2|pages=230–242|pmid=22079804}}</ref>

=== Turbulence ===
Blood flow is also affected by the smoothness of the vessels, resulting in either turbulent (chaotic) or laminar (smooth) flow. Smoothness is reduced by the buildup of fatty deposits on the arterial walls.

The Reynolds number (denoted NR or Re) is a relationship that helps determine the behavior of a fluid in a tube, in this case blood in the vessel.

The equation for this dimensionless relationship is written as:<ref name="Munson 2009 725"/>
:<math>NR=\frac{\rho v L}{\mu}</math>
:* ''ρ'': density of the blood
:* ''v'': mean velocity of the blood
:* ''L'': characteristic dimension of the vessel, in this case diameter
:* ''μ'': viscosity of blood

The Reynolds number is directly proportional to the velocity and diameter of the tube. Note that NR is directly proportional to the mean velocity as well as the diameter. A Reynolds number of less than 2300 is laminar fluid flow, which is characterized by constant flow motion, whereas a value of over 4000, is represented as turbulent flow.<ref name="Munson 2009 725"/> Due to its smaller radius and lowest velocity compared to other vessels, the Reynolds number at the capillaries is very low, resulting in laminar instead of turbulent flow.<ref>{{cite journal|last=Fung|first=Yuan-cheng|author2=Zweifach, B.W.|title=Microcirculation: Mechanics of Blood Flow in Capillaries|journal=Annual Review of Fluid Mechanics|year=1971|volume=3|pages=189–210|doi=10.1146/annurev.fl.03.010171.001201|bibcode=1971AnRFM...3..189F}}</ref>

===Velocity===
Often expressed in cm/s. This value is inversely related to the total cross-sectional area of the blood vessel and also differs per cross-section, because in normal condition the blood flow has [[Laminar flow|laminar characteristics]]. For this reason, the blood flow velocity is the fastest in the middle of the vessel and slowest at the vessel wall. In most cases, the mean velocity is used.<ref name="TortoraDerrickson">{{cite book |title=Principles of Anatomy & Physiology |edition=13th |last1=Tortora |first1=Gerard J. |last2=Derrickson |first2=Bryan |year=2012 |publisher=John Wiley & Sons |isbn=978-0470-56510-0 |chapter=The Cardiovascular System: Blood Vessels and Hemodynamics |page=816}}</ref> There are many ways to measure blood flow velocity, like videocapillary microscoping with frame-to-frame analysis, or [[Laser Doppler velocimetry|laser Doppler anemometry]].<ref>{{cite journal |doi=10.1006/mvre.1996.0054 |title=Capillary Blood Cell Velocity in Human Skin Capillaries Located Perpendicularly to the Skin Surface: Measured by a New Laser Doppler Anemometer |year=1996 |last1=Stücker |first1=M. |last2=Bailer |first2=V. |last3=Reuther |first3=T. |last4=Hoffman |first4=K. |last5=Kellam |first5=K. |last6=Altmeyer |first6=P. |journal=Microvascular Research  |volume=52 |issue=2 |pages=188–192 |pmid=8901447}}</ref>
Blood velocities in [[arteries]] are higher during [[Systole (medicine)|systole]] than during [[diastole]]. One parameter to quantify this difference is the '''pulsatility index''' ('''PI'''), which is equal to the difference between the peak systolic velocity and the minimum diastolic velocity divided by the mean velocity during the [[cardiac cycle]]. This value decreases with distance from the heart.<ref>{{cite book |title=Principles of Anatomy & Physiology |edition=13th |last1=Tortora |first1=Gerard J. |last2=Derrickson |first2=Bryan |year=2012 |publisher=John Wiley & Sons |isbn=978-0470-56510-0 |type=Laminar flow analysis |chapter=The Cardiovascular System: Blood Vessels and Hemodynamics |page=817}}</ref>

:<math>PI = \frac{v_{systole} - v_{diastole}}{v_{mean}}</math>

{| class="wikitable"
|+ Relation between blood flow velocity and total cross-section area in human
|-
!Type of blood vessels
!Total cross-section area
!Blood velocity in cm/s
|-
| Aorta || 3–5&nbsp;cm<sup>2</sup> || 40&nbsp;cm/s
|-
| Capillaries || 4500–6000&nbsp;cm<sup>2</sup> || 0.03&nbsp;cm/s<ref>{{cite book |title=Human anatomy & physiology |edition=9th |last1=Marieb |first1=Elaine N. |last2=Hoehn |first2=Katja |year=2013 |publisher=Pearson Education |isbn=978-0-321-74326-8 |chapter=The Cardiovascular System:Blood Vessels |page=712}}</ref>
|-
| Vena cavae inferior and superior || 14&nbsp;cm<sup>2</sup> || 15&nbsp;cm/s
|-
|}

==Blood vessels==
===Vascular resistance===
{{main|Vascular resistance}}
Resistance is also related to vessel radius, vessel length, and blood viscosity.

In a first approach based on fluids, as indicated by the [[Hagen–Poiseuille equation]].<ref name="Munson 2009 725">{{cite book|vauthors = Munson BR, Young DF, Okiishi TH, Huebsch WW|title=Fundamentals of Fluid Mechanics|year=2009|publisher=John Wiley &Sons, Inc.|location=New Jersey|isbn=978-0-470-26284-9|page=725|edition=Sixth}}</ref> The equation is as follows:

:<math>\Delta P = \frac{8 \mu l Q}{\pi r^4}</math>
:* ∆''P'': pressure drop/gradient
:* ''μ'': viscosity
:* ''l'': length of tube. In the case of vessels with infinitely long lengths, l is replaced with diameter of the vessel.
:* ''Q'': flow rate of the blood in the vessel
:* ''r'': radius of the vessel

In a second approach, more realistic of the [[Vascular resistance#Regulation|vascular resistance]] and coming from experimental observations on blood flows, according to Thurston,<ref name="ReferenceA">GB Thurston, Viscosity and viscoelasticity of blood in small diameter tubes, Microvasular Research 11, 133 146, 1976</ref> there is a plasma release-cell layering at the walls surrounding a plugged flow. It is a fluid layer in which at a distance δ, viscosity η is a function of δ written as η(δ), and these surrounding layers do not meet at the vessel centre in real blood flow. Instead, there is the plugged flow which is hyperviscous because holding high concentration of RBCs. Thurston assembled this layer to the flow resistance to describe blood flow by means of a viscosity η(δ) and thickness δ from the wall layer.

The blood resistance law appears as R adapted to blood flow profile :

:<math>R = \frac{c L \eta(\delta)}{(\pi \delta r^3)} </math><ref name="ReferenceA"/>
where
* R = resistance to blood flow
* c = constant coefficient of flow
* L = length of the vessel
* η(δ) = [[viscosity]] of blood in the wall plasma release-cell layering
* r = radius of the blood vessel
* δ = distance in the plasma release-cell layer

Blood resistance varies depending on blood viscosity and its plugged flow (or sheath flow since they are complementary across the vessel section) size as well, and on the size of the vessels.
Assuming steady, laminar flow in the vessel, the blood vessels behavior is similar to that of a pipe. For instance if p1 and p2 are pressures are at the ends of the tube, the pressure drop/gradient is:<ref>{{cite journal | author = Womersley JR | title = Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known | journal = Journal of Physiology | volume = 127 | issue = 3 | pages = 553–563 | year = 1955 | pmid = 14368548 | pmc = 1365740 | doi=10.1113/jphysiol.1955.sp005276}}</ref>
:<math>\frac{p_1 - p_2}{l} = \Delta P</math>

The larger arteries, including all large enough to see without magnification, are conduits with low [[vascular resistance]] (assuming no advanced [[atherosclerosis|atherosclerotic]] changes) with high flow rates that generate only small drops in pressure. The smaller arteries and arterioles have higher resistance, and confer the main blood pressure drop across major arteries to capillaries in the circulatory system.

[[File:Blausen 0092 BloodPressureFlow.png|thumb|Illustration demonstrating how vessel narrowing, or vasoconstriction, increases blood pressure|upright=1.5]]

In the arterioles blood pressure is lower than in the major arteries. This is due to bifurcations, which cause a drop in pressure. The more bifurcations, the higher the total cross-sectional area, therefore the pressure across the surface drops. This is why{{citation needed|reason=Arteries have a cross-sectional aea of 20 cm2, arterioles of 40 cm2 and cappilaries of 2500 cm2. (source: Guyton & Hall medical physiology) Thus, if we were to go by this reasoning, the highest pressure drop would be in the capillaries, but we know that the highest pressure drop is in the arterioles. The previous sentence, unless there is a direct quote from somewhere, isn't really a reason for the highest pressure drop in the circulatory system. This needs clarification|date=April 2015}} the arterioles have the highest pressure-drop. The pressure drop of the arterioles is the product of flow rate and resistance: ∆P=Q xresistance. The high resistance observed in the arterioles, which factor largely in the ∆''P'' is a result of a smaller radius of about 30&nbsp;μm.<ref>{{cite book|last=Sircar|first=Sabyasach|title=Principles of Medical Physiology|year=2008|publisher=vistasta Publishing|location=India|isbn=978-1-58890-572-7}}</ref> The smaller the radius of a tube, the larger the resistance to fluid flow.

Immediately following the arterioles are the capillaries. Following the logic observed in the arterioles, we expect the blood pressure to be lower in the capillaries compared to the arterioles. Since pressure is a function of force per unit area, (''P''&nbsp;=&nbsp;''F''/''A''), the larger the surface area, the lesser the pressure when an external force acts on it. Though the radii of the capillaries are very small, the network of capillaries has the largest surface area in the vascular network. They are known to have the largest surface area (485&nbsp;mm^2) in the human vascular network. The larger the total cross-sectional area, the lower the mean velocity as well as the pressure.<ref name="Fung 1997 571"/>

Substances called [[vasoconstrictor]]s can reduce the size of blood vessels, thereby increasing blood pressure. [[Vasodilator]]s (such as [[nitroglycerin]]) increase the size of blood vessels, thereby decreasing arterial pressure.

If the blood viscosity increases (gets thicker), the result is an increase in arterial pressure. Certain [[medical conditions]] can change the viscosity of the blood. For instance, anemia (low [[red blood cell]] concentration) reduces viscosity, whereas increased red blood cell concentration increases viscosity. It had been thought that [[aspirin]] and related "[[blood thinner]]" drugs decreased the viscosity of blood, but instead studies found that they act by reducing the tendency of the blood to clot.<ref>{{cite journal |vauthors=Rosenson RS, Wolff D, Green D, Boss AH, Kensey KR | title = Aspirin. Aspirin does not alter native blood viscosity | journal = J. Thromb. Haemost. | volume = 2 | issue = 2 | pages = 340–341 | date = February 2004 | pmid = 14996003 | doi = 10.1111/j.1538-79333.2004.0615f.x | doi-access = free }}</ref>

To determine the systemic vascular resistance (SVR) the formula for calculating all resistance is used.

<math> R = (\Delta pressure) / flow. </math>

This translates for SVR into:

<math>SVR=(MAP -CVP)/CO </math>

Where
* SVR = systemic vascular resistance (mmHg/L/min)
* MAP = mean arterial pressure  (mmHg)
* CVP = central venous pressure (mmHg)
* CO = cardiac output (L/min)<ref name="auto">{{Citation |last1=Delong |first1=Claire |title=Physiology, Peripheral Vascular Resistance |date=2022 |url=http://www.ncbi.nlm.nih.gov/books/NBK538308/ |work=StatPearls |place=Treasure Island (FL) |publisher=StatPearls Publishing |pmid=30855892 |access-date=2022-05-16 |last2=Sharma |first2=Sandeep}}</ref>
To get this in Wood units the answer is multiplied by 80.

Normal systemic vascular resistance is between 900 and 1440 dynes/sec/cm−5.<ref>{{Citation |last=Naderi |first=Nasim |title=Chapter 11 - Hemodynamic Study |date=2022-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780323809153000132 |work=Practical Cardiology (Second Edition) |pages=201–216 |editor-last=Maleki |editor-first=Majid |publisher=Elsevier |language=en |isbn=978-0-323-80915-3 |access-date=2022-05-22 |editor2-last=Alizadehasl |editor2-first=Azin |editor3-last=Haghjoo |editor3-first=Majid}}</ref>

=== Wall tension ===

[[File:Circumferential stress.svg|thumb|Components of [[cylinder stress]]]]

Regardless of site, blood pressure is related to the [[wall tension]] of the vessel according to the [[Young–Laplace equation]] (assuming that the thickness of the vessel wall is very small as compared to the diameter of the [[Lumen (anatomy)|lumen]]):
:<math> \sigma_\theta = \dfrac{Pr}{t} \ </math>
where
* ''P'' is the blood pressure
* ''t'' is the wall thickness
* ''r'' is the inside radius of the cylinder.
* <math> \sigma_\theta \! </math> is the [[cylinder stress]] or "hoop stress".
For the thin-walled assumption to be valid the vessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius.

The [[cylinder stress]], in turn, is the average [[force]] exerted circumferentially (perpendicular both to the axis and to the radius of the object) in the cylinder wall, and can be described as:
:<math> \sigma_\theta = \dfrac{F}{tl} \ </math>
where:
* ''F'' is the [[force]] exerted circumferentially on an area of the cylinder wall that has the following two lengths as sides:
* ''t'' is the radial thickness of the cylinder
* ''l'' is the axial length of the cylinder

===Stress===
When force is applied to a material it starts to deform or move. As the force needed to deform a material (e.g. to make a fluid flow) increases with the size of the surface of the material A.,<ref name="autogenerated3">{{cite book |last1=Caro |first1=C.G. |last2=Pedley |first2=T.J. |last3=Schroter |first3=R.C. |last4=Seed |first4=W.A. |publisher=Oxford University Press |title=The Mechanics of Circulation |year=1978 |pages=3–60, 151–176 |isbn=978-0-19-263323-1}}</ref> the magnitude of this force F is proportional to the area A of the portion of the surface. Therefore, the quantity (F/A) that is the force per unit area is called the stress. The [[shear stress]] at the wall that is associated with blood flow through an artery depends on the artery size and geometry and can range between 0.5 and 4 [[Pascal (unit)|Pa]].<ref>{{cite journal |title=Measuring Wall Shear Stress Using Velocity-Encoded MRI |last=Potters |date=13 February 2014 |journal=Current Cardiovascular Imaging Reports |doi=10.1007/s12410-014-9257-1 |volume=7|issue = 4|s2cid=55721300 }}</ref>
:<math>\sigma = \frac{F}{A}</math>.
Under normal conditions, to avoid atherogenesis, thrombosis, smooth muscle proliferation and endothelial apoptosis, shear stress maintains its magnitude and direction within an acceptable range. In some cases occurring due to blood hammer, shear stress reaches larger values. While the direction of the stress may also change by the reverse flow, depending on the hemodynamic conditions. Therefore, this situation can lead to atherosclerosis disease.<ref>{{cite journal |last1=Tazraei |first1=P. |last2=Riasi |first2=A. |last3=Takabi |first3=B. |title=The influence of the non-Newtonian properties of blood on blood-hammer through the posterior cerebral artery |journal=Mathematical Biosciences |date=2015 |volume=264 |pages=119–127 |doi=10.1016/j.mbs.2015.03.013|pmid=25865933 }}</ref>
[[File:Laminar shear.svg|thumb|right|320px|Laminar shear of fluid between two plates. <math>v=u, \tau=\sigma</math>. Friction between the fluid and the moving boundaries causes the fluid to shear (flow). The force required for this action per unit area is the stress. The relation between the stress (force) and the shear rate (flow velocity) determines the viscosity.]]

=== Capacitance ===
{{main|Capacitance of blood vessels}}
Veins are described as the "capacitance vessels" of the body because over 70% of the blood volume resides in the venous system. Veins are more compliant than arteries and expand to accommodate changing volume.<ref>{{cite book|last1=Lough|first1=Mary|title=Hemodynamic Monitoring: Evolving technologies and clinical practice|date=2015-04-15|publisher=Elsevier Mosby|location=St. Louis, Missouri|isbn=978-0-323-08512-0|page=25|edition=1}}</ref>

== Blood pressure ==
The blood pressure in the circulation is principally due to the pumping action of the heart.<ref name = 'Caro'>{{cite book |author=Caro, Colin G. |title=The Mechanics of The Circulation|publisher=Oxford University Press |location=Oxford [Oxfordshire] |year=1978 |isbn=978-0-19-263323-1}}</ref> The pumping action of the heart generates pulsatile blood flow, which is conducted into the arteries, across the micro-circulation and eventually, back via the venous system to the heart. During each heartbeat, systemic arterial blood pressure varies between a maximum ([[Systole (medicine)|systolic]]) and a minimum ([[diastolic]]) pressure.<ref>{{cite web | url = http://healthlifeandstuff.com/2010/06/normal-blood-pressure-range-adults/ | title = Normal Blood Pressure Range Adults | publisher = Health and Life | access-date = 2016-02-06 | archive-url = https://web.archive.org/web/20160318195914/http://healthlifeandstuff.com/2010/06/normal-blood-pressure-range-adults | archive-date = 2016-03-18 | url-status = dead | date = 2010-06-07 }}</ref> In physiology, these are often simplified into one value, the [[mean arterial pressure|mean arterial pressure (MAP)]], which is calculated as follows:

<math>MAP = DP + 1/3(PP) </math>

where:
* MAP = Mean Arterial Pressure
* DP = Diastolic blood pressure
* PP = Pulse pressure which is systolic pressure minus diastolic pressure.<ref>{{Citation |last1=DeMers |first1=Daniel |title=Physiology, Mean Arterial Pressure |date=2022 |url=http://www.ncbi.nlm.nih.gov/books/NBK538226/ |work=StatPearls |place=Treasure Island (FL) |publisher=StatPearls Publishing |pmid=30855814 |access-date=2022-05-22 |last2=Wachs |first2=Daliah}}</ref>

Differences in mean blood pressure are responsible for blood flow from one location to another in the circulation. The rate of mean blood flow depends on both blood pressure and the resistance to flow presented by the blood vessels. Mean blood pressure decreases as the [[Circulatory system|circulating blood]] moves away from the [[heart]] through arteries and [[capillaries]] due to [[Viscosity|viscous]] losses of energy. Mean blood pressure drops over the whole circulation, although most of the fall occurs along the small arteries and [[arteriole]]s.<ref name='Klabunde2005p93-4'>{{cite book | last = Klabunde | first = Richard | title = Cardiovascular Physiology Concepts | publisher = Lippincott Williams & Wilkins | year = 2005 | pages = 93–94 | isbn = 978-0-7817-5030-1 }}</ref> Gravity affects blood pressure via [[Fluid statics|hydrostatic]] forces (e.g., during standing), and valves in veins, [[breathing]], and pumping from contraction of skeletal muscles also influence blood pressure in veins.<ref name = 'Caro'/>

The relationship between pressure, flow, and resistance is expressed in the following equation:<ref name="brs" />

:<math>Flow = Pressure/Resistance </math>

When applied to the circulatory system, we get:

: <math>CO = (MAP-RAP)/SVR </math>
where
* CO = cardiac output (in L/min)
* MAP = mean arterial pressure (in mmHg), the average pressure of blood as it leaves the heart
* RAP = right atrial pressure (in mmHg), the average pressure of blood as it returns to the heart
* SVR = [[ systemic vascular resistance]] (in mmHg * min/L)

A simplified form of this equation assumes right atrial pressure is approximately 0:

:<math>CO\approx MAP/SVR</math>

The ideal blood pressure in the [[brachial artery]], where standard [[blood pressure cuff]]s measure pressure, is <120/80&nbsp;mmHg. Other major arteries have similar levels of blood pressure recordings indicating very low disparities among major arteries. In the innominate artery, the average reading is 110/70&nbsp;mmHg, the right subclavian artery averages 120/80 and the abdominal aorta is 110/70&nbsp;mmHg.<ref name="Fung 1997 571">{{cite book|last=Fung|first=Yuan-cheng|title=Biomechanics:Circulation|year=1997|publisher=Springer|location=New York|isbn=978-0-387-94384-8|page=571}}</ref> The relatively uniform pressure in the arteries indicate that these blood vessels act as a pressure reservoir for fluids that are transported within them.

Pressure drops gradually as blood flows from the major arteries, through the arterioles, the capillaries until blood is pushed up back into the heart via the venules, the veins through the vena cava with the help of the muscles. At any given pressure drop, the flow rate is determined by the resistance to the blood flow. In the arteries, with the absence of diseases, there is very little or no resistance to blood. The vessel diameter is the most principal determinant to control resistance. Compared to other smaller vessels in the body, the artery has a much bigger diameter (4&nbsp; mm), therefore the resistance is low.<ref name="Fung 1997 571"/>

The ''arm–leg (blood pressure) gradient'' is the difference between the blood pressure measured in the arms and that measured in the legs. It is normally less than 10&nbsp;mm Hg,<ref name=Markham>{{cite journal |vauthors=Markham LW, Knecht SK, Daniels SR, Mays WA, Khoury PR, Knilans TK | title = Development of exercise-induced arm-leg blood pressure gradient and abnormal arterial compliance in patients with repaired coarctation of the aorta | journal = Am. J. Cardiol. | volume = 94 | issue = 9 | pages = 1200–1202 | date = November 2004 | pmid = 15518624 | doi = 10.1016/j.amjcard.2004.07.097 }}</ref> but may be increased in e.g. [[coarctation of the aorta]].<ref name=Markham/>

==Clinical significance==

===Pressure monitoring===
[[File:Maquet Flow-I anesthesia machine.jpg|thumb|An [[anesthetic machine]] with integrated systems for monitoring of several hemodynamic parameters, including [[blood pressure]] and [[heart rate]]]]
Hemodynamic [[Monitoring (medicine)|monitoring]] is the observation of hemodynamic parameters over time, such as [[blood pressure]] and [[heart rate]]. Blood pressure can be monitored either invasively through an inserted blood pressure transducer assembly (providing continuous monitoring), or noninvasively by repeatedly measuring the blood pressure with an inflatable [[blood pressure cuff]].

Hypertension is diagnosed by the presence of arterial blood pressures of 140/90 or greater for two clinical visits.<ref name="auto"/>

Pulmonary Artery Wedge Pressure can show if there is congestive heart failure, mitral and aortic valve disorders, [[hypervolemia]], shunts, or cardiac tamponade.<ref>{{Cite web |title=RHC {{!}} Vanderbilt University Medical Center |url=https://medsites.vumc.org/commodorecompendium/rhc |access-date=2022-05-16 |website=medsites.vumc.org}}</ref>

===Remote, indirect monitoring of blood flow by laser Doppler===
[[File:Eye fundus image by laser Doppler holography.gif|thumb|left|200px|[[Laser Doppler imaging]] reveals retinal blood flow]]
Noninvasive hemodynamic monitoring of eye fundus vessels can be performed by Laser Doppler holography, with near infrared light. The eye offers a unique opportunity for the non-invasive exploration of [[cardiovascular diseases]]. [[Laser Doppler imaging]] by [[digital holography]] can measure blood flow in the [[retina]] and [[choroid]], whose Doppler responses exhibit a [[pulse]]-shaped profile with time<ref>Puyo, L., M. Paques, M. Fink, J-A. Sahel, and M. Atlan. "In vivo laser Doppler holography of the human retina." Biomedical optics express 9, no. 9 (2018): 4113-4129.</ref><ref>Puyo, Léo, Michel Paques, Mathias Fink, José-Alain Sahel, and Michael Atlan. "Waveform analysis of human retinal and choroidal blood flow with laser Doppler holography." Biomedical Optics Express 10, no. 10 (2019): 4942-4963.</ref> This technique enables non invasive functional [[microangiography]] by high-contrast measurement of Doppler responses from endoluminal blood flow profiles in vessels in the posterior segment of the eye. Differences in [[blood pressure]] drive the flow of blood throughout the circulation. The rate of mean blood flow depends on both blood pressure and the hemodynamic resistance to flow presented by the blood vessels.

==Glossary==

;ANH:Acute Normovolemic Hemodilution
;ANH<sub>u</sub>:Number of Units During ANH
;BL<sub>H</sub>:Maximum Blood Loss Possible When ANH Is Used Before Homologous Blood Transfusion Is Needed
;BL<sub>I</sub>:Incremental Blood Loss Possible with ANH.(BL<sub>H</sub> – BL<sub>s</sub>)
;BL<sub>s</sub>:Maximum blood loss without ANH before homologous blood transfusion is required
;EBV:Estimated Blood Volume(70&nbsp;mL/kg)
;Hct:Haematocrit Always Expressed Here As A Fraction
;H<sub>i</sub>:Initial Haematocrit
;H<sub>m</sub>:Minimum Safe Haematocrit
;PRBC:Packed Red Blood Cell Equivalent Saved by ANH
;RCM:Red cell mass.
;RCM<sub>H</sub>:Cell Mass Available For Transfusion after ANH
;RCM<sub>I</sub>:Red Cell Mass Saved by ANH
;SBL:Surgical Blood Loss

==Etymology and pronunciation==
The word ''hemodynamics'' ({{IPAc-en|ˌ|h|iː|m|ə|d|aɪ|ˈ|n|æ|m|ɪ|k|s|,_|-|m|oʊ|-}}{{refn|{{Cite encyclopedia |url=http://www.lexico.com/definition/haemodynamic |archive-url=https://web.archive.org/web/20200322182542/https://www.lexico.com/definition/haemodynamic |url-status=dead |archive-date=2020-03-22 |title=haemodynamic |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}} }}) uses [[classical compound|combining forms]] of ''[[wikt:hemo-#Prefix|hemo-]]'' (which comes from the ancient Greek ''haima'', meaning blood) and ''[[dynamics (mechanics)|dynamics]]'', thus "the [[fluid dynamics|dynamics]] of [[blood]]". The vowel of the ''hemo-'' syllable is variously written according to the [[American and British English spelling differences#ae and oe|ae/e variation]].

{{col div|colwidth=30em}}
* [[Blood hammer]]
* [[Blood pressure]]
* [[Cardiac output]]
* [[Cardiovascular System Dynamics Society]]
* [[Electrical cardiometry]]
* [[Esophogeal doppler]]
* [[Hemodynamics of the aorta]]
* [[Impedance cardiography]]
* [[Photoplethysmogram]]
* [[Laser Doppler imaging]]
* [[Windkessel effect]]
* [[Functional near-infrared spectroscopy]]
{{colend}}

== Notes and references ==
{{reflist|30em}}

== Bibliography ==
* Berne RM, Levy MN. Cardiovascular physiology. 7th Ed Mosby 1997
* Rowell LB. Human Cardiovascular Control. Oxford University press 1993
* Braunwald E (Editor). Heart Disease: A Textbook of Cardiovascular Medicine. 5th Ed. W.B.Saunders 1997
* Siderman S, Beyar R, Kleber AG. Cardiac Electrophysiology, Circulation and Transport. Kluwer Academic Publishers 1991
* [http://www.americanheart.org American Heart Association]
* Otto CM, Stoddard M, Waggoner A, Zoghbi WA. Recommendations for Quantification of Doppler Echocardiography: A Report from the Doppler Quantification Task Force of the Nomenclature and Standards Committee of the American Society of Echocardiography. J Am Soc Echocardiogr 2002;15:167-184
* Peterson LH, The Dynamics of Pulsatile Blood Flow, Circ. Res. 1954;2;127-139
* [https://www.ncbi.nlm.nih.gov/pubmed/12024086 Hemodynamic Monitoring], Bigatello LM, George E., Minerva Anestesiol, 2002 Apr;68(4):219-25
* Claude Franceschi L'investigation vasculaire par ultrasonographie Doppler Masson 1979 ISBN Nr 2-225-63679-6
* Claude Franceschi; Paolo Zamboni Principles of Venous Hemodynamics Nova Science Publishers 2009-01 ISBN Nr 1606924850/9781606924853
* Claude Franceschi [https://angio.info/LIVRES%20Books,%20BREVETS_/Venous%20Insufficiency%204%20languages%20e-books/VENOUS%20INSUFFICIENCY%20OF%20THE%20PELVIS%20AND%20LOWER%20LIMBS%2016%2010%2020%2021.pdf Venous Insufficiency of the pelvis and lower extremities-Hemodynamic Rationale]
* WR Milnor: Hemodynamics, Williams & Wilkins, 1982
* B Bo Sramek: Systemic Hemodynamics and Hemodynamic Management, 4th Edition, ESBN 1-59196-046-0

==External links==
* [http://www.learnhemodynamics.com Learn hemodynamics]

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