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The effects of exercise on blood flow with reference to the human cardiovascular system: a finite element study
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1992 Phys. Med. Biol. 37 1033 (http://iopscience.iop.org/0031-9155/37/5/001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.173.72.87 This content was downloaded on 24/06/2015 at 00:58
Please note that terms and conditions apply.
Phys. Med. Biol., 1992, Val. 37, No 5, 1033-1045. Printed in the UK
The effects of exercise on blood flow with reference to the human cardiovascular system: a finite element study Virender K Sudt, R Srini Srinivasanz, John B Charles§ and Michael W Bung05 t Biophysics Department, All India Institute of Medical Sciences, New Delhi, India
*5
KRUG Intemational, Houston, Texas, USA Space Biomedical Research Institute, NASAJohnsan Space Center, Houston, Texas, USA
Received 11 June 1991, in final form 16 December 1991 Abstract. This paper reports on a theoretical invertigation into the effects of vasomotion on blood through the human cardiovascular system. T h e finite element method has been used to analyse the model. Vasoconstrinion and vasodilation may be effected either through the action of the central nervous system or autoregulation. One afthe conditions responsible for vasomotion is exercise. The proposed model has been solved and quantitative results of Rows and pressures due IO changing the conductances ofspecific networks of arferioles, capillaries and venules comprising the arms, legs, stomach and their combinations have been obtained.
1. Introduction
The central nervous system and localized autoregulation are the chief mechanisms responsible for effecting changes in the peripheral resistance of the human cardiovascular system (Guyton 1963). Autoregulation is a local mechanism which pervades the entire vascular bed in order to meet the local demand of nutrients, whereas the central nervous system triggers vasodilation and vasoconstriction in response to various inputs received from baroreceptors. The above mentioned processes of autoregulation and action of the central nervous system are brought into play during bodily exercise by a human subject. It was determined b y White (1983) that heavy exercise increases the cardiac output by as much as four times from resting condition and the supply of blood flow in the lower extremities increases by about ten times when exercising the muscles. Such large changes in blood flow are attributed to very appreciable increases in conductance values of various vascular segments, especially those of arterioles, capillaries and venules, etc. The mammalian cardiovascular system is a complex arrangement of arteries, arterioles, capillaries, venules and veins. The major motive force for sending the blood lo various organs and other parts of the body is provided by the pumping action of the left and right ventricles. Since the number of individual branches and segments in the system is very large, the possibility of constructing an exact replica in tenns of an analog or digital computer simulation model appears quite remote. Electrical/electronic analog computer oriented models were constructed and used in experiments by Warner (1959), Noordergraaf (1963). Weigel (1964), Beneken (1964, 19651, De Pater (1964), Jager et al (1965), Dick and Rideout (1965) and Snyder et al (1968). Mathematical 0031-9155/92/051033+13$04.50
@ 1992 IOP Publishing Ltd
1033
1034
V K Sud et a1
models derived from hydrodynamics equations were suggested by Jochim and McDonnel (1947) and Guyton (1963). More recently, digital computer simulation models ofthe arterial and arterial-venous system have been reported by many researchers. These include the models by Hyndeman (1972). Avolio (1980), White el a/ (1983) Dagan (1982). Hardy et a1 (1982), Leaning et a1 (1983). Jaron et a/ (1984) and Sud and Sekhon (1986). In the present analysis the finite element method is employed to construct a computer simulation model for analysing blood flow through a closed system when the conductance values of specific networks of arterioles, venules and capillaries change as a result of exercise of that specific limb. Computational results on blood flow are also presented when exercising various parts of the body, i.e., the upper extremities, stomach, lower extremities and a combination of these. The finite element method has been used because it is direct and more versatile in its applicability. The method is capable of taking into account any prescribed boundary conditions. The proposed model can be modified to simulate inexpensively cardiovascular abnormalities such as hypotension, hypertension, stenosis and haemorrhage as compared with other mechanical, electrical and electronic models.
2. Formulation of the model
The cardiovascu!ar system is made!!ed based :pox tho oxtexsio:: of 2 previans ar?e:ia! model given by Sud and Sekhon (1986). The model is updated by attaching parallel veins to the corresponding arteries and appropriately connecting arterioles, capillaries and veins with the terminal arteries and to their appropriate veins. There are 262 segments which represent the arteries, the veins and the two ventricles or the heart. Each network of arterioles, capillaries and venules (abbreviated as the ACV network) connecting a peripheral artery to the appropriate vein is also represented as a branch. The model contains 63 such ACV networks. Thus the human cardiovascular system is modelled by as many as a total of 325 elements consisting of arteries, veins and networks of arterioles, capillaries and venules through which blood is made to flow by the pumping action of the heart. The junction of one segment with one or more other segments is called a node. There are 262 nodes connecting the 325 elements in the model. Figure 1 is a schematic diagram of the entire model with the cardiopulmonary section shown in detail. The number of segments and nodes for each region of the circulatory sysrem is clearly shown. The data regarding the lengths and radii of the arteries and veins are known, as are the density and viscosity of blood. Further, the equivalent conductance of each ACV network is also assumed to be given a priori. 2.1. Solution approach It is convenient to solve the proposed model of the cardiovascular system by visualizing it to consist of two portions; figure 1 contains both portions. The first portion is referred to as the systemic portion (main flow system); it comprises all segments of the complete model excluding the left and the right ventricles and pulmonary system. The second portion is referred to as the cardiopulmonary system. The finite element method is employed first to analyse the flow through the systemic portion and then to solve the flow problem associated with the cardiopulmonary portion. Application of the finite element method to either of the above flow problems involves: (i) determination of
Effects of exercise on bloodflow Ciavlolion IRophl)
112,3&~101
c,lc"lot,onlLclfl
128,lLI
w
-Arl~r#oI
----vl"o",
1035
Sq"
KV W.llO,l NOM
Figure 1. Schematic diagram of the cardiovascular model showing the cardiopulmonary portion in detail. The numbers in circles refer to either ACV networks or segments, i.e., arteries, veins, and the two cardiac chambers; numbers not circled refer to nodes. The three numben shown in brackets for each circulation are, respectively, the number of segments, the number of nodes and the number of ACV networks associated with that circulation. The total numben for the segments, nodes and ACV networks in the entire system are 262, 264 and 63, respectively. Segments numbered 316 and 317 represent the vena cavae (superior and inferior) connecting to the right atrium.
the pressure-Row relationships of individual segments, (ii) synthesis of the behaviour of individual segments to form a global system of equations based upon the pressureRow relationship, (iii) identification of applicable boundary conditions, and (iv) solution of the global system of equations subject to the specified boundary condition.
2.2. Element conductance matrix In the finite element method for each element, a mathematical relationship describing its behaviour has to be established. In the case of blood Row, the relationship is the one connecting the flow rate through a segment with the pressure at the inlet and outlet of that segment. The flow rate, qa, through the segment a whose end nodes are i and j can therefore he expressed as
where c. is a constant depending upon the dimensions of the arterial and venous segments. pi and fi are the values of Ruid pressures at the nodes i and j , respectively.
1036
V K Sud et a1
The quantity c. corresponds to the so-called ‘element conductance’ in literature on the finite element method. The matrix comprising all the ‘element conductances’ is usually referred to as the ‘global conductance matrix’. In the present study, following Womersley (1955), the conductance in the case of pulsatile flow can be written as
where R. and I. are the radius and length of the segment a, respectively;
M,~~(l+h~o-2h,o~0sS~~)’’~ =( 2 / a ) ( M 1 ( a ) / M d a ) )
S10=3/4?r - & ( a + ) &(a)
M o ( a ) , M , ( a ) , Bo and 8, are the amplitudes and phase angles which are related by the following Bessel functions; Jo( i’l2a)= MO(a ) exp[ iO0(a)I J1(i”’a ) = M , ( a ) exp[iO0(a)l
a =%(pn/p)1/2
where = i ? r f is the anguizr frequency (rad s-’j of rhe puisariir iiuw, ; ‘is iiic puke frequency (Hz)and p is the density of blood. R, and 1. are the radius and length of the segment a respectively. The phase angle between the motion of blood and pressure gradient is given by q0= tan-’
hlosin S,, 1 - h,o COS 61,
(3)
2.2.1. Conductance of an ACV network. A terminating artery is connected to its corresponding vein through a network of a very large number of arterioles, capillaries and venules. Therefore, it is almost impossible to calculate, theoretically, the conductances of the ACV networks of the entire cardiovascular system. In the present study, the equivalent conductances of all the ACV networks are determined indirectly when the flow rates through the terminating arteries and the pressure drops between the ACV junctions are known. The total known flux through terminal segments is divided amongst the terminal segments in direct proportion to their respective conductances.
2.2.2. Heartpump. The pumping action of the left and the right ventricles is considered in terms of the rise of the main blood pressure caused by them. Mathematically, we can represent the pumping action as follows:
where poL,pIL,pon and pin represent the pressures at the outlet and inlet of the left and right ventricles, respectively. SP, and SP, denote pressures caused by the pumping action of the respective ventricle.
Effects of exercise on bloodflow
1037
2.3. Assembly At any node i (figure 2), let the net Aow rate be qi. From the continuity equation, we find qa + q b + q c +.
. .= qi.
(6)
Alternatively, in the case of pulsatile Row, we can write equation (6) as ) .. = q 8 , (p, -p,)c. sin(nt + e a ) + (p, -pk)cb sin(nr + E ! ~ +. Rearranging, equation (7) gives us [c. sin(Qt+ E?,,)+ cb sin(nr+ E:,,) +. . .lp,
sin(nt+E;,,)pj-cb sin(nr+EP0)pk+.. . = q # . Equation (8) can be transformed into the matrix form -c.
( k , s, Univemily of Utrecht V Internal Reporl White R 1, Craston R C and Fitzjerell D G 1983 Cardiovascular modelling: Simulating the human response to exercise, lower body negative pressure, zero gravity and clinical conditions Ad". Cardiovosc. Physinl. 5 195-229 Womersley J R 1955 Method for the calculation o f velocity, rate of Row and viscous drag in arteries when the pressure gradient is known J. Physiol. 127 553-63
Search
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About
Contact us
My IOPscience
The effects of exercise on blood flow with reference to the human cardiovascular system: a finite element study
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1992 Phys. Med. Biol. 37 1033 (http://iopscience.iop.org/0031-9155/37/5/001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 129.173.72.87 This content was downloaded on 24/06/2015 at 00:58
Please note that terms and conditions apply.
Phys. Med. Biol., 1992, Val. 37, No 5, 1033-1045. Printed in the UK
The effects of exercise on blood flow with reference to the human cardiovascular system: a finite element study Virender K Sudt, R Srini Srinivasanz, John B Charles§ and Michael W Bung05 t Biophysics Department, All India Institute of Medical Sciences, New Delhi, India
*5
KRUG Intemational, Houston, Texas, USA Space Biomedical Research Institute, NASAJohnsan Space Center, Houston, Texas, USA
Received 11 June 1991, in final form 16 December 1991 Abstract. This paper reports on a theoretical invertigation into the effects of vasomotion on blood through the human cardiovascular system. T h e finite element method has been used to analyse the model. Vasoconstrinion and vasodilation may be effected either through the action of the central nervous system or autoregulation. One afthe conditions responsible for vasomotion is exercise. The proposed model has been solved and quantitative results of Rows and pressures due IO changing the conductances ofspecific networks of arferioles, capillaries and venules comprising the arms, legs, stomach and their combinations have been obtained.
1. Introduction
The central nervous system and localized autoregulation are the chief mechanisms responsible for effecting changes in the peripheral resistance of the human cardiovascular system (Guyton 1963). Autoregulation is a local mechanism which pervades the entire vascular bed in order to meet the local demand of nutrients, whereas the central nervous system triggers vasodilation and vasoconstriction in response to various inputs received from baroreceptors. The above mentioned processes of autoregulation and action of the central nervous system are brought into play during bodily exercise by a human subject. It was determined b y White (1983) that heavy exercise increases the cardiac output by as much as four times from resting condition and the supply of blood flow in the lower extremities increases by about ten times when exercising the muscles. Such large changes in blood flow are attributed to very appreciable increases in conductance values of various vascular segments, especially those of arterioles, capillaries and venules, etc. The mammalian cardiovascular system is a complex arrangement of arteries, arterioles, capillaries, venules and veins. The major motive force for sending the blood lo various organs and other parts of the body is provided by the pumping action of the left and right ventricles. Since the number of individual branches and segments in the system is very large, the possibility of constructing an exact replica in tenns of an analog or digital computer simulation model appears quite remote. Electrical/electronic analog computer oriented models were constructed and used in experiments by Warner (1959), Noordergraaf (1963). Weigel (1964), Beneken (1964, 19651, De Pater (1964), Jager et al (1965), Dick and Rideout (1965) and Snyder et al (1968). Mathematical 0031-9155/92/051033+13$04.50
@ 1992 IOP Publishing Ltd
1033
1034
V K Sud et a1
models derived from hydrodynamics equations were suggested by Jochim and McDonnel (1947) and Guyton (1963). More recently, digital computer simulation models ofthe arterial and arterial-venous system have been reported by many researchers. These include the models by Hyndeman (1972). Avolio (1980), White el a/ (1983) Dagan (1982). Hardy et a1 (1982), Leaning et a1 (1983). Jaron et a/ (1984) and Sud and Sekhon (1986). In the present analysis the finite element method is employed to construct a computer simulation model for analysing blood flow through a closed system when the conductance values of specific networks of arterioles, venules and capillaries change as a result of exercise of that specific limb. Computational results on blood flow are also presented when exercising various parts of the body, i.e., the upper extremities, stomach, lower extremities and a combination of these. The finite element method has been used because it is direct and more versatile in its applicability. The method is capable of taking into account any prescribed boundary conditions. The proposed model can be modified to simulate inexpensively cardiovascular abnormalities such as hypotension, hypertension, stenosis and haemorrhage as compared with other mechanical, electrical and electronic models.
2. Formulation of the model
The cardiovascu!ar system is made!!ed based :pox tho oxtexsio:: of 2 previans ar?e:ia! model given by Sud and Sekhon (1986). The model is updated by attaching parallel veins to the corresponding arteries and appropriately connecting arterioles, capillaries and veins with the terminal arteries and to their appropriate veins. There are 262 segments which represent the arteries, the veins and the two ventricles or the heart. Each network of arterioles, capillaries and venules (abbreviated as the ACV network) connecting a peripheral artery to the appropriate vein is also represented as a branch. The model contains 63 such ACV networks. Thus the human cardiovascular system is modelled by as many as a total of 325 elements consisting of arteries, veins and networks of arterioles, capillaries and venules through which blood is made to flow by the pumping action of the heart. The junction of one segment with one or more other segments is called a node. There are 262 nodes connecting the 325 elements in the model. Figure 1 is a schematic diagram of the entire model with the cardiopulmonary section shown in detail. The number of segments and nodes for each region of the circulatory sysrem is clearly shown. The data regarding the lengths and radii of the arteries and veins are known, as are the density and viscosity of blood. Further, the equivalent conductance of each ACV network is also assumed to be given a priori. 2.1. Solution approach It is convenient to solve the proposed model of the cardiovascular system by visualizing it to consist of two portions; figure 1 contains both portions. The first portion is referred to as the systemic portion (main flow system); it comprises all segments of the complete model excluding the left and the right ventricles and pulmonary system. The second portion is referred to as the cardiopulmonary system. The finite element method is employed first to analyse the flow through the systemic portion and then to solve the flow problem associated with the cardiopulmonary portion. Application of the finite element method to either of the above flow problems involves: (i) determination of
Effects of exercise on bloodflow Ciavlolion IRophl)
112,3&~101
c,lc"lot,onlLclfl
128,lLI
w
-Arl~r#oI
----vl"o",
1035
Sq"
KV W.llO,l NOM
Figure 1. Schematic diagram of the cardiovascular model showing the cardiopulmonary portion in detail. The numbers in circles refer to either ACV networks or segments, i.e., arteries, veins, and the two cardiac chambers; numbers not circled refer to nodes. The three numben shown in brackets for each circulation are, respectively, the number of segments, the number of nodes and the number of ACV networks associated with that circulation. The total numben for the segments, nodes and ACV networks in the entire system are 262, 264 and 63, respectively. Segments numbered 316 and 317 represent the vena cavae (superior and inferior) connecting to the right atrium.
the pressure-Row relationships of individual segments, (ii) synthesis of the behaviour of individual segments to form a global system of equations based upon the pressureRow relationship, (iii) identification of applicable boundary conditions, and (iv) solution of the global system of equations subject to the specified boundary condition.
2.2. Element conductance matrix In the finite element method for each element, a mathematical relationship describing its behaviour has to be established. In the case of blood Row, the relationship is the one connecting the flow rate through a segment with the pressure at the inlet and outlet of that segment. The flow rate, qa, through the segment a whose end nodes are i and j can therefore he expressed as
where c. is a constant depending upon the dimensions of the arterial and venous segments. pi and fi are the values of Ruid pressures at the nodes i and j , respectively.
1036
V K Sud et a1
The quantity c. corresponds to the so-called ‘element conductance’ in literature on the finite element method. The matrix comprising all the ‘element conductances’ is usually referred to as the ‘global conductance matrix’. In the present study, following Womersley (1955), the conductance in the case of pulsatile flow can be written as
where R. and I. are the radius and length of the segment a, respectively;
M,~~(l+h~o-2h,o~0sS~~)’’~ =( 2 / a ) ( M 1 ( a ) / M d a ) )
S10=3/4?r - & ( a + ) &(a)
M o ( a ) , M , ( a ) , Bo and 8, are the amplitudes and phase angles which are related by the following Bessel functions; Jo( i’l2a)= MO(a ) exp[ iO0(a)I J1(i”’a ) = M , ( a ) exp[iO0(a)l
a =%(pn/p)1/2
where = i ? r f is the anguizr frequency (rad s-’j of rhe puisariir iiuw, ; ‘is iiic puke frequency (Hz)and p is the density of blood. R, and 1. are the radius and length of the segment a respectively. The phase angle between the motion of blood and pressure gradient is given by q0= tan-’
hlosin S,, 1 - h,o COS 61,
(3)
2.2.1. Conductance of an ACV network. A terminating artery is connected to its corresponding vein through a network of a very large number of arterioles, capillaries and venules. Therefore, it is almost impossible to calculate, theoretically, the conductances of the ACV networks of the entire cardiovascular system. In the present study, the equivalent conductances of all the ACV networks are determined indirectly when the flow rates through the terminating arteries and the pressure drops between the ACV junctions are known. The total known flux through terminal segments is divided amongst the terminal segments in direct proportion to their respective conductances.
2.2.2. Heartpump. The pumping action of the left and the right ventricles is considered in terms of the rise of the main blood pressure caused by them. Mathematically, we can represent the pumping action as follows:
where poL,pIL,pon and pin represent the pressures at the outlet and inlet of the left and right ventricles, respectively. SP, and SP, denote pressures caused by the pumping action of the respective ventricle.
Effects of exercise on bloodflow
1037
2.3. Assembly At any node i (figure 2), let the net Aow rate be qi. From the continuity equation, we find qa + q b + q c +.
. .= qi.
(6)
Alternatively, in the case of pulsatile Row, we can write equation (6) as ) .. = q 8 , (p, -p,)c. sin(nt + e a ) + (p, -pk)cb sin(nr + E ! ~ +. Rearranging, equation (7) gives us [c. sin(Qt+ E?,,)+ cb sin(nr+ E:,,) +. . .lp,
sin(nt+E;,,)pj-cb sin(nr+EP0)pk+.. . = q # . Equation (8) can be transformed into the matrix form -c.
( k , s, Univemily of Utrecht V Internal Reporl White R 1, Craston R C and Fitzjerell D G 1983 Cardiovascular modelling: Simulating the human response to exercise, lower body negative pressure, zero gravity and clinical conditions Ad". Cardiovosc. Physinl. 5 195-229 Womersley J R 1955 Method for the calculation o f velocity, rate of Row and viscous drag in arteries when the pressure gradient is known J. Physiol. 127 553-63
