New algorithm of cuff-tissue-artery system modeled as the space axisymmetric problem
aa r X i v : . [ phy s i c s . c o m p - ph ] J u l New algorithm of cu ff -tissue-artery system modeled as thespace axisymmetric problem Jiacheng Xu, Dan Hu ∗ Institute of Natural Sciences, Shanghai Jiao-Tong University, 800 Dongchuan Road, Shanghai 200240, ChinaDepartment of Mathematics, Shanghai Jiao-Tong University, 800 Dongchuan Road, Shanghai 200240, China
Abstract
In this paper, mathematical models for cu ff -tissue-artery system are developed andsimplified into an axisymmetric problem in space. It is nonlinear properties of cu ff and artery wall that make it di ffi cult to solve elastic equations directly with the finiteelement method, hence a new iteration algorithm derived from principle of virtual workis designed to deal with nonlinear boundary conditions. Numerical accuracy is highlysignificant in numerical simulation, so it is necessary to analyze the influence di ff erentfinite elements and grid generation on numerical accuracy. By dimensional analysis, it isestimated that numerical errors must be O (10 − ) cm or less. To reach desired accuracy, thenumber of grids using higher order elements becomes one-fourth as large as that usinglow order elements by convergence rate analysis. Moreover, dealing with displacementproblem under specific blood pressure needs much small grid size to make numericalerrors su ffi ciently small, which is not taken seriously in previous papers. However,it only takes a quarter of grids or less for displacement change problem to guaranteenumerical accuracy and reduce computing cost. Keywords: finite element method, principle of virtual work, numerical accuracy
1. Introduction
Blood pressure is considered as a critical reference index for diagnosing and predict-ing cardiovascular diseases in the clinical field [1]. The accurate measurement of bloodpressure has always been one of the most heated topics in both industry and academia.It is well-known that the current mainstream approach to measure blood pressure isnon-invasive methods including auscultation method and oscillometric method, whichboth use a cu ff in the measurement.Auscultation method [2, 3, 4] is detecting the appearance and disappearance of Ko-rotko ff sound generated by dynamic deformation of the blood vessel when cu ff pressurechanges. Based on the change of Korotko ff sound, diastolic and systolic pressure can be ∗ Corresponding author
Email addresses: [email protected] (Jiacheng Xu), [email protected] (Dan Hu)
Preprint submitted to TeX.sx July 14, 2020 etermined. Nowadays, auscultation method is widely accepted as the golden standarddue to high accuracy. However, Owing to strict technique and inconvenience to carry amercury sphygmomanometer, this method cannot be commonly used in each family.Oscillometric method, which was proposed firstly by Marey [5], can measure cu ff pressure oscillations when cu ff pressure changes from above systolic to below diastolicpressure. The current oscillometric algorithms are mainly classified into amplitude-based method and slope-based method. Amplitude-based method usually uses max-imum amplitude algorithm (MAA). It is now generally believed that this method canaccurately estimate mean blood pressure, which can be explained by compliance ofartery in some papers [6, 7, 8]. However, this method lacks of proper and precisemethods to estimate systolic and diastolic pressure. Some researchers [9, 10, 11] ob-tain systolic and diastolic pressure by characteristic ratios while errors may be muchlarge. The reason why the errors becomes large sometime is that characteristic ratios areacquired empirically even fixedly. Slope-based method is another way to get systolicand diastolic pressure by amplitude envelope slopes of oscillometric waveforms. Thisway can work greatly in clean data but fail in ”dirty” data in reality because it is muchsensitive to noise [12].Due to the limitations of traditional algorithms, alternative improved algorithmsbased on conventional oscillometric methods are presented, including Gaussian mix-ture regression approach [13], multiple linear regression and support vector regressionapproaches [14], neural networks [15, 16, 17], the least squares method [18] and so on.Previously, a large number of scholars have focused on oscillometric algorithms whiletheoretical analysis is extraordinarily deficient. To the best of the authors knowledge,few investigators have begun to do theoretical research. Jeon et al. used arterial-pressure-volume model to simulate oscillation waveform and proposed oscillometricalgorithm using FFT [19]. Babbs incorporated artery wall model with cu ff model tomodel cu ff pressure oscillations and extracted systolic and diastolic pressure by theregression procedure [20]. These papers didn’t take e ff ects of tissue into considerationdespite they simulated cu ff pressure oscillations well. In fact, mechanical properties oftissue can have a significant influence on stress transmission e ffi ciency. Mauck et al.presented a simple one-dimensional cu ff -arm-artery model to identify that the point ofmaximum oscillations can be used to determine mean blood pressure [7]. Ursino etal. developed a mathematical model of cu ff -arm-artery system to study the influenceof biomechanical factors on the measurement [21]. Kim et al. combined vein modelwith cu ff -arm-artery system to simulate the deformation of the arm tissue [22]. Liang etal. constructed three-dimensional cu ff -arm-artery system incorporated with the bone toobtain stress distribution further explaining clinical observation [23].In the present study, we develop a comprehensive cu ff -tissue-artery coupled modeland also simplify this model as a space axisymmetric problem. It is nonlinear propertiesof cu ff and artery wall that make solving elastic equations more di ffi cult. Inspired byprinciple of virtual work, we present a novel iterative algorithm to calculate elasticequations based on finite element method. Based on the problem we studied, wefind numerical errors produced by mesh refinement should be O (10 − ) cm or less bydimensional analysis. However, it is well-known that both mesh generation and elementtypes can a ff ect numerical accuracy. Therefore, choosing proper finite element and gridsize is so important in numerical simulations. It is found that compared with low orderelements, the number of grids using higher order elements can reduce to one fourth2ince convergence rate is raised to about twice. What’s more, it only takes a quarterof grids for displacement change problem compared with displacement problem underspecific blood pressure. That is, only small enough grid size can guarantee accuracy ofsolutions of displacements under specific blood pressure, which was always ignored inprevious papers. However, dealing with displacement change just need fewer grids tomeet precision requirements, which is beneficial to guarantee accuracy and save spacein numerical research on cu ff pressure oscillation later.
2. Model building
As we know, it is geometry, mechanical properties and load conditions of the cu ff -tissue-artery system that make it di ffi cult to make a full analysis of the system. However,it is necessary to construct simple models under proper assumptions in order to under-stand the mechanism of the system. In this problem, the e ff ect of bone and body forceare ignored. Although fat, muscle and other soft tissues have di ff erent mechanical prop-erties such as elastic modulus and Poisson ratio, they can be seen as the same kindof isotropic and homogeneous elastic body with the same mechanical properties, thatis, elastic modulus and Poisson ratio remain unchanged. Artery, tissue and cu ff areaxisymmetric with the same axis of symmetry, in which the artery is in the tissue andthe tissue is wrapped with the cu ff . ff modeling In the process of inflation and deflation of the cu ff , the external wall of the cu ff deforms much slightly, therefore, it is seen to be rigid. The internal wall of the cu ff clingsto the tissue tightly during the deformation of the cu ff so that the compliance of it is seento be large enough. Hence, cu ff pressure is constantly the same as stress of the externaltissue. The air inside the cu ff is considered as ideal gas. Moreover, both inflation anddeflation are seen as adiabatic processes satisfying( P c + P atm ) V kc = Q (1)where P c is cu ff pressure, P atm is atmosphere pressure, V c is cu ff volume, parameter k = . Q denotes quantity of theremaining air inside the cu ff at a specific time. It is found experimentally that the cross-sectional area of artery will increase slowlywith the increase of blood pressure until it reaches the maximal cross-sectional area in thestate of expansion. That is, the compliance of artery decrease rapidly as the transmuralpressure becomes larger when artery expands. In addition, the compliance of arterydrops dramatically as the transmural pressure becomes a little smaller when the arterywall collapses. According to these experimental findings, some paper [19] presentedartery wall model described by compliance of artery, see equ. (2). ( A α = A exp( C a P t ) , P t < A α = A m − ( A m − A ) exp( C b P t ) , P t ≥ . (2)3here A is cross-sectional area of artery when transmural pressure becomes zeros, A m isthe maximal cross-sectional area of artery, C a and C b are unknown parameters, describingthe mechanical properties of artery. In simulations later, we use C a = . mmH − , b = . mmH − which are the same as the paper [19]. In this problem, the geometry, external force distributions and constraints of tissue aresymmetrical to an axis, therefore, this problem is seen as a space axisymmetric problem.In general, polar coordinates r , θ, z are much convenient than cartesian coordinates x , y , z to solve this problem. Due to symmetry, the stress, strain and displacement at any pointare independent of θ , which are functions of r and z . What’s more, to keep symmetric,we can get that τ r θ = , τ z θ = , u θ = ∂σ r ∂ r + ∂τ zr ∂ z + σ r − σ θ r = ∂τ rz ∂ r + ∂σ z ∂ z + τ zr r = . (3)Geometrical equations: ε r = ∂ u r ∂ r ε θ = u r r ε z = ∂ w ∂ z γ zr = ∂ w ∂ r + ∂ u r ∂ z (4)Physical equations: σ r = E + µ ( µ − µ θ + ε r ) σ θ = E + µ ( µ − µ θ + ε θ ) σ z = E + µ ( µ − µ θ + ε z ) τ zr = E + µ ) γ zr (5)where θ = ε r + ε θ + ε z , E is elastic modulus and µ is Poisson ratio.
3. Numerical algorithm
The aim of this section is to solve coupled equations (3, 4, 5) numerically with thefinite element method. However, it is nonlinear boundary conditions that increasesthe di ffi culty of solving elastic equations. In order to deal with this problem, a properiterative algorithm based on principle of virtual work is designed here.From previous assumptions, body forces are ignored, hence an elastic body is justacted by surface forces. For the elastic body with volume τ and surface area S , S isseparated into force surface domain S σ and displacement surface domain S u . Surfaceforces, whose component is X i , act on S σ . Displacements, whose component is u i , areprescribed on S u . Assumed that there is any arbitrary virtual displacement in the body,the virtue work done by surface forces on S u is zero since the displacements are fixed4n S u . Without loss of generality, the area in which surface forces work can be set to S .Therefore, the variation of work done by surface forces is δ W = Z S X i δ u i d S (6)Due to deformation, the variation of elastic potential energy of the body is δ U = Z τ σ ij δε ij d τ. (7)It is noted that stress tensor has a symmetrical property, then substituting equ. (4)into equ. (7), the variation of elastic potential energy of the body can be rewritten as Z τ σ ij δε ij d τ = Z τ σ ij δ ( u i , j + u j , i )d τ = Z τ σ ij (cid:16) ( δ u i ) , j + ( δ u j ) , i (cid:17) d τ (8) = Z τ σ ij ( δ u i ) , j + σ ji ( δ u j ) , i d τ = Z τ σ ij ( δ u i ) , j d τ. Then, the variation of total energy is δ E = δ U + δ W (9) = Z τ σ ij ( δ u i ) , j d τ + Z S X i δ u i d S Due to the equilibrium equations ( σ ij , j = σ ij ( δ u i )) , j = σ ij ( δ u i ) , j + σ ij , j δ u i (10) = σ ij ( δ u i ) , j Using Gauss’ theorem, we have Z τ ( σ ij ( δ u i )) , j d τ = Z S σ ij δ u i l j d S . (11)where l j is direction cosine of surface normal.Therefore, we rewrite equ. (9) as δ E = Z τ σ ij ( δ u i ) , j d τ + Z S X i δ u i d S = Z S σ ij δ u i l j d S + Z S X i δ u i d S = Z S (cid:16) σ ij l j + X i (cid:17) δ u i d S . (12)Total energy is the minimum when a elastic body is in the equilibrium state. To maketotal energy reach minimum, we choose δ u i = − c (cid:16) X i + σ ij l j (cid:17) , c > u i so that δ E <
0. Therefore, in each iteration we update displacements by u i = u i − c (cid:16) X i + σ ij l j (cid:17) , c > . . Results Before we make numerical experiments, we preset values of some parameters: radiusof external wall of cu ff R = . cm , cu ff pressure P c = mmH , lengths of tissue andcu ff are L = cm and L c = cm , elastic modulus of tissue E = ∗ Pa , Poisson ratioof tissue µ = .
45, systolic pressure P s = mmH , diastolic pressure P d = mmH ,artery parameters C a = . mmH − , C b = . mmH − , the radius of external tissue r ext = . cm , the radius of artery under no force r = . cm , which is also the radius ofinternal tissue. Before numerical computation, we should do dimensional analysis to analyze nu-merical accuracy. Since the quantity of air inside the cu ff remains almost unchanged ina much short time, we can deduce from cu ff model ( P c + P atm ) V kc = Q that( P c + P atm ) V kc = ( P c + P atm + ∆ P c )( V c − ∆ V c ) k = ( P c + P atm + ∆ P c ) V kc (1 − ∆ V c V c ) k . (13)Due to the fact that ∆ V c V c <<
1, then neglecting the higher-order terms, equ. (13) canbe simplified into ( P c + P atm ) V kc = ( P c + P atm + ∆ P c ) V kc (1 − k ∆ V c V c ) . (14)Furthermore, equ. (14) is simplified as follows when neglecting small quantity ∆ V c ∆ P c ∆ P c = k ( P c + P atm ) ∆ V c V c . (15)In a practical measurement, the amplitude of cu ff pressure oscillation when cu ff pressure is 100mmHg is about 1.5mmHg. Under such circumstances, we can estimatethat relative volume change rate is ∆ V c V c = ∆ P c k ( P c + P atm ) = . . (16)To facilitate estimation, we assumed that the volume of the internal wall of the cu ff always keeps cylindrical, hence V c ≈ π R L c − π r ext L c . Moreover, Neglecting high-orderterm, it is estimated that ∆ V c ≈ π r ext ∆ r ext L c . Combining with equ. (16), we can get thatdisplacement change of external tissue is ∆ r ext = . ∗ − cm . (17)From equ.(17), we can know that the order of magnitude of displacement changeof external tissue is about O (10 − ) cm , therefore, in numerical computation, numericalerrors produced by mesh refinement should be smaller than it.In fact, in using the finite element method, we need to choose proper elements andgrid size to guarantee accuracy and reduce computational time and memory space. Here,we compare two elements, which are low order elements and higher order elements.The main di ff erence between them is higher order elements have quadratic displacementbehavior since there is an intermediate node between the two nodes.6 Distance (cm) -2-1.5-1-0.500.511.52 D i s p l a c e m en t c hange ( c m ) × -3 Diastolic pressure
Distance (cm) -2-1.5-1-0.500.511.52 D i s p l a c e m en t c hange ( c m ) × -3 Systolic pressure -2 -1 Grid size (cm) -6 -5 -4 -3 E rr o r Convergence rate
Figure 1: Left and middle: displacement change of external tissue by mesh refinement under diastolic andsystolic pressure. (Symbol → represents grid size is refined) Right: convergence rate analysis of displacementchange by mesh refinement under systolic pressure for low order element. From left and middle pictures in Fig 1, we can find that under specific blood pres-sure, the order of magnitude of displacement change of external tissue caused by meshrefinement from the center of the cu ff to the end is approximately O (10 − − − ) cm whengrid size is larger than 1 / cm , but is O (10 − ) cm or less when grid size is less than orequal to 1 / cm .Although the range of grid size meeting the predetermined accuracy is roughlyknown, there is still a question of how large grid size is needed to make numerical errorssmall enough. To tackle this question, convergent rate is needed to be estimated so thatproper grid size can be calculated.Supposed that u ∗ is true solution of elastic equations, u h and u h are numerical solutionwhen grid size are 2 h and h , respectively, then we have ( u h − u ∗ = C (2 h ) α u h − u ∗ = Ch α (18)where α is convergent rate.According to equ. (18), we can know that u h − u h = h α ( C − α C ) . (19)where e C = C − α C . Taking norms on both sides of equ. (19), we can get k u h − u h k = k e C k h α . (20)Taking logarithms on both sides of equ. (20), we can get lo ( k u h − u h k ) = α lo ( h ) + lo ( k e C k ) (21)From equ. (21), we can know that the relationship between lo ( k u h − u h k ) and lo ( h )is linear, and the slope is convergent order α . In addition, k e C k can be calculated usingthe intercept. 7rom right picture in Fig 1, we can calculate α = .
39 and k e C k = . ∗ − . Toreach the expected accuracy, numerical errors produced by mesh refinement should beabout O (10 − ) cm , that is, k u h − u h k ∼ O (10 − ) cm . Then we can estimate roughly that10 − cm ∼ k e C k h α . Therefore, h ≈ . cm , that is, if numerical errors by mesh refinementis su ffi ciently small, grid size should be less than or equal to 0 . cm , which is consistentwith the results shown in the Fig 1.In fact, in our research, we pay more attention to cu ff pressure oscillation or cu ff volume oscillation during the change of blood pressure. Therefore, it is necessaryto analyze numerical errors in numerical simulations to meet desired precision for cu ff volume oscillation. Here, we study displacement change of external tissue from diastolicto systolic, that is, displacement amplitude of external tissue in a cardiac cycle.From the left picture of Fig 2, we can see that displacement amplitude of externaltissue from the center of the cu ff to the end in a cardiac cycle becomes closer whengrid size gets smaller. In the middle picture, we can find that the order of magnitudeof displacement amplitude change of external tissue caused by mesh refinement isis approximately O (10 − ) cm , which is a little better than that of displacement changeproduced by mesh refinement under specific blood pressure. Besides, numerical errorscannot decay to expected requirement until grid size is less than or equal to 1 / cm . Distance (cm) D i s p l a c e m en t a m p li t ude ( c m ) × -3 Cardiac cycle
Distance (cm) -1-0.500.51 D i s p l a c e m en t a m p li t ude c hange ( c m ) × -3 Cardiac cycle -2 -1 Grid size (cm) -6 -5 -4 E rr o r Convergence rate
Figure 2: Left: displacement amplitude of external tissue under di ff erent grid size. Middle: displacementamplitude change of external tissue produced by mesh refinement. (Symbol → represents grid size is refined)Right: convergence rate analysis of displacement amplitude for low order element. Then, we estimate how large grid size to achieve the precision by convergent rate.According to the right picture in Fig 2, we calculate α = .
195 and k e C k = . ∗ − fordisplacement amplitude problem. Using similar method, we get step size h ≈ . cm by 10 − cm ∼ k e C k h α . Equally, if numerical errors by mesh refinement is su ffi ciently small,grid size should be less than or equal to 0 . cm , which is also consistent with theresults shown in the middle figure in Fig 2. Now, we explore displacement change produced by mesh refinement under specificblood pressure using higher order elements. From left and middle pictures in Fig 3, itis found that the order of magnitude of displacement change of external tissue from thecenter of the cu ff to the end caused by mesh refinement is approximately O (10 − − − ) cm when grid size is larger than 1 / cm , but becomes O (10 − ) cm or less when grid size is8ess than or equal to 1 / cm . Therefore, it will need about one quarter of grids whenusing higher order elements compared with using low order elements. That is, underspecific blood pressure, convergence rate of higher order elements is about twice thanthat of low order elements. Distance (cm) -0.01-0.00500.0050.01 D i s p l a c e m en t c hange ( c m ) Diastolic pressure
Distance (cm) -0.01-0.00500.0050.01 D i s p l a c e m en t c hange ( c m ) Systolic pressure -2 -1 Grid size -6 -5 -4 -3 E rr o r Convergence rate
Figure 3: Left and middle: displacement change of external tissue by mesh refinement under diastolic andsystolic pressure. Right: convergence rate analysis of displacement change for higher order element.
Now, we can calculate convergent rate to verify the results and also estimate theproper grid size. According to right picture in Fig 3, we can get convergence rate α = .
27 and k e C k = . h ≈ . cm by 10 − cm ∼ k e C k h α . Therefore,grid size is more than twice as big as that calculated with low order elements, which isconsistent with the previous analysis. Hence, as long as gird size is less than or equal to0 . cm , numerical errors by mesh refinement for displacement problem under specificblood pressure is about O (10 − ) cm , which is similar as the results in the left and middlepictures in Fig 3.In the same way, we need to explore amplitude problem for higher order elements.As can be seen in Fig 4, displacement amplitude in a cardiac cycle becomes much closerwhen grid size gets a little smaller from left picture. In the middle picture, the order ofmagnitude of displacement amplitude change by mesh refinement reaches O (10 − ) cm fast as long as grid size is 1 / cm or less. That is, the number of grids will be quarteredcompared with low order elements for amplitude problem.Now, we also estimate convergence rate under such circumstances. From the rightpicture in Fig. 4, we can get α = .
76 and k e C k = . ∗ − . Subsequently, we estimategrid size h ≈ . cm by 10 − cm ∼ k e C k h α , which is also twice larger than that computedwith low order elements. In the same way, we see that if grid size is less than or equal to0 . cm , numerical errors by mesh refinement for displacement amplitude will attain O (10 − ) cm or less. No matter displacement problem under specific blood pressure or displacementchange problem in a time period it is, it will take a quarter of grids or less using higherorder elements to achieve desired accuracy compared with using low order elements.That is the advantage of higher order elements so that we will use them in numericalsimulation later. 9
Distance (cm) D i s p l a c e m en t a m p li t ude ( c m ) × -3 Cardiac cycle
Distance (cm) -1-0.500.51 D i s p l a c e m en t a m p li t ude c hange ( c m ) × -3 Cardiac cycle -2 -1 Grid size (cm) -7 -6 -5 -4 E rr o r Convergence rate
Figure 4: Left: displacement amplitude of external tissue under di ff erent mesh size. Middle: displace-ment amplitude change of external tissue produced by mesh refinement.Right: convergence rate analysis ofdisplacement amplitude change for higher order element. For displacement problem under specific blood pressure, mesh generation is greatlyimportant because it needs a large number of grids to reach expected precision. In ouranalysis, to get desired accuracy, grid size tends to be much small even if higher orderelements are used, i.e. h . cm . In fact, this problem is not taken seriously inprevious papers.However, for displacement change problem, there is an interesting result that we justuse large grid size rather small grid size to meet the precision requirement. Specificallyspeaking, it is enough to reach the requirement when h . cm .In fact, the focus of our research is cu ff pressure oscillation of oscillometric method.Above all, the grid size of higher order elements is chosen to be 1 / cm for displacementchange problem, so much less grids will be used and most importantly high accuracycan be attained in numerical simulation.
5. Conclusion
This paper presents space axisymmetric mathematical models of cu ff -tissue-arterysystem based on proper assumptions and simplifications. Although elastic equationscan be solved by the finite element method, nonlinear boundary conditions make itdi ffi cult to solve equations straightforwardly.To deal with this tricky problem, this paper develops a new iterative algorithm in-spired with principle of virtue work. Before we use the new iterative algorithm to dosimulations, it is necessary for us to analyze numerical accuracy for the problem we stud-ied. By dimensional analysis, the important finding is that numerical errors producedby mesh refinement for displacement change problem need to be much small, about O (10 − ) cm or less. Therefore, the chosen of finite element types and mesh generation isof greatly significance in numerical computation.On the one hand, it is found that the number of grids using higher order elements canreduce to one fourth of that using low order elements no matter displacement problemunder specific blood pressure or displacement change problem in a time frame it is.Hence, higher order elements will be chosed in numerical simulation later from thisimportant result. 10n the other hand, it will take very small grid size to meet precision requirements fordisplacement problem under specific blood pressure. In fact, few paper paid more at-tention to mesh generation when studying displacements under specific blood pressure,which may lead to large numerical errors using improper mesh generation. However,grid size can be more than doubled for displacement change problem. As a matter offact, displacement change problem is exactly what we focus on. 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