 Journal of Biomaterials and Nanobiotechnology, 2011, 2, 369-377 doi:10.4236/jbnb.2011.24046 Published Online October 2011 (http://www.SciRP.org/journal/jbnb) Copyright © 2011 SciRes. JBNB 369Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure Amy Cypher1, Mohamed B. Elgindi2, Hatem Kouriachi3, David Peschman4, Reba Shotwell5 1University of Wisconsin-Eau Claire, Eau Claire, USA; 2University of Wisconsin-Eau Claire and Texas A & M University-Qatar, Doha, Qatar; 3University of Wisconsin-Eau Claire, Eau Claire, USA; 4University of Wisconsin-River Falls, River Falls, USA; 5University of Wisconsin-Madison, Madison, USA. Email: elgindmb@uwec.edu, elgindmb@gmail.com Received August 29th, 2011; revised September 26th, 2011; accepted October 4th, 2011. ABSTRACT In this paper, we present the mathematical equations that govern the deformation of an imbedded blood vessel under exter-nal uniform pressure taking into consideration the nonliner behavior of th e soft tissue surrounding the vessel. We presen t a bifurcation analysis and give explicit formulas for the bifurcation points and the corresponding first order approximations for the non-trivial solutions. We then present the results of a MATLAB program that integrates the equilibrium equations and calculates the blood flow rate through a deformed cross section for given values of the elasticity parameters and pres-sure. Finally, we provide (numerical) verification that the flow rate as a function of the elasticity parameters of the soft tis-sue surrounding the blood vessel is convex, and theref ore validate the invertibility of our model. Keywords: Blood Vessel, Deformation of Elastic Tube 1. Introduction Stability analysis for the buckling, post-buckling shapes and flow rate through an imbedded blood vessel under uniform external pressure were considered in . In that paper, the soft tissues surrounding blood vessels are mo- deled by numerous linear independent springs. However, biological tissues are well known to respond in a non- linear fashion to applied forces [2-6]. Since the support provided by the perivascular tissue is an important con- tributor to the in vivo structural stiffness of arteries, which will in turn affect the pressure-flow rate rela- tionship, there is a critical need for further studies. In this paper, we examine the effect of replacing the linear spring in  by nonlinear ones on the post-buckling shapes and on the pressure-flow rate relationship. Fur- thermore, we verify (numerically) the convexity of the flow rate as a function of the elasticity parameters. This convexity of the (direct) problem is important to ensure its invertibility. That is, to ensure the solvability of the (more important) inverse problem, namely, to determine the elasticity parameters of the soft tissue surrounding the blood vessel from measurements of the deformation, the pressure, and the flow rate. The paper  assumes: 1) The tethering can be represented by numerous in- dependent springs. 2) The springs are linear. Motivated by the fact that biological tissues are known to respond in a nonlinear fashion to applied forces, we begin our series of studies to improve previous results by replacing Equation (8) of  by a nonlinear function 12=FkBCAC kgBCAC . It is expected that this nonlinearity will have no effect on the stability analysis; however, it will alter the post-buckling shapes and flow rates through them. Interests in these post- buckling computations will make the present studies necessary and useful steps in the direction of describing tethered vessels more precisely. Furthermore, our (nu- merical) validation of the solvability of the inverse pro- blem gives the simple physical model used in this project advantages over more complicated ones. The rest of this paper is organized in five sections. In Section 1, we de- fine the variables and formulate the equilibrium equa- tions. In Section 2, we give a bifurcation analysis of the equilibrium equations that lead to explicit formulas for the bifurcation points, dependent on spring stiffness, and the corresponding first order approximations for the bifurcation solutions. In Section 3, we show the numeri- cal formulation . In Section 4, we present our numeri- cal results. In Section 5, we give some concluding remarks. 2. Mathematical Formulation We consider the deformation of a thin-walled elastic Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure 370 cylinder tethered by continuously distributed nonlinear springs to a rigid outer cylinder (see Figure 1, below). The interior cylinder is subjected to internal pressure i, and external pressure e. This cylinder will remain circular until a bifurcation pressure difference is exceeded. PPTo formulate the mathematical equations governing the equlibrium, we consider the forces acting on an ele- mental length of the interior cylinder (see Figure 2, below). In Figure 3, below, we analyze changes in the coor- dinates of an element length due to a displacement from point A to point B. 2.1. Notations In the rest of this paper, we use the following notations for our variables: iPP: Internal pressure eS: External pressure : Shearing force s: Arc length tqq: Tangential stress n: Normal stress M: Moment T: Tension : Local curvature of vessel from x axis 2.2. Assumptions We make the following assumptions: 1) The flow of blood through a tethered blood vessel is slow and steady. 2) The cross section does not vary much along a segment, so that the internal pressure is taken as constant (locally). 2.3. Remark From the assumptions above, we conclude that we can solve for the deformed shape first, then calculate the flow rate afterward. 2.4. Equilibrium Equations Balancing forces in the normal direction gives: =nTdq dsdS (1) Balancing forces in tangential direction gives: =0tqds SddT (2) Balance of moments gives: =dM Sds (3) Assuming the wall thickness to be small compared to the radius, it follows that the moment M is proportional to the local curvature, where E and I are material constants : d=dMEIs (4) Figure 1. Elastic cylinder tethered to a rigid cylinder. Figure 2. An element length. Figure 3. A small displacement from A to B. Copyright © 2011 SciRes. JBNB Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure371 From which 22dd=ddMEIss (5) From (2) we get 22ddd=0dddtTqEI sss (6) From (1) we get dd=dd nSTssq (7) And this gives 22222ddddd =dd dddnqTSTEIsssssR (8) From (6) we get 22222ddddd=0ddddtEI EITqsssRRs ds (9) From (8) and (9) we get 3423 24232ddd dddddddd dddd=0dddnntqss2sss sqqsss  (10) where =ssR (11) 3=nnqRqEI (12) 3=ttqRqEI (13) We also have d=cosdxs (14) d=sindys (15) Let 0F be the tension per unit length per unit area of a spring at spring length of 1a (). >1aWe define 0=eiPPPF (16) We assume that the extra force due to deformation is F and is given by: 12=FkBCAC kgBCAC  (17) where and are spring constants and g is a nonlinear function. 1k2k=1ACa (18) 2=cossin =BCas xas yd2 (19) 222222sin1 costan== sec1cos cosxxxxxx (20)  222222sinsec=tan1 =1cossin cos=cosasyasxasy asxasx    (21)  222222coscos =sin coscos=asxasy asxasxd  (22) coscos= asdx (23) sinsin= asyd (24) But cos cossin sin=cos =coscos sincossin= cosasx asydd    (25) =coscossinsintFZqasxad sy (26) =sincoscossinnFZqasxad sy (27)  12=FZkZkgZ (28) =1Zda (29) 3. Bifurcation Analysis For low values of the pressure difference, the interior cylinder remains circular. As the pressure difference in- creases byond some critical values, non-circular solu- tions occur (see Figure 4, below). These critical values of the pressure difference are called bifurcation points while the corresponding non-circular solutions are called bifurcation solutions. It is well known that bifurcation may occur only at pressure difference values that corre- spond to a singular linearized problem about the circular solution. Copyright © 2011 SciRes. JBNB Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure 372 Figure 4. Bifurcation points with solutions for N = 2, 3, and 4. In this section, we present the calculations to find the bifurcation points: the critical values of the pressure difference at which the vessel deforms into non-circular shapes. First, we set 1ys (local angle) 2Momentys 3Shearys 4Tensionys 5ysx 6ysy where 2π0,sN is the arclength. We define the state vector:    123456=,,,,,Ysysysysysysys (30) Then the equations of equilibrium can be written as:  23423211=,= =cossinntyyyy qPYs FYPyy qyy0 (31) with boundary conditions: 1π0=2y 360= 0=0yy 12ππ2π=2yNN  32π=0yN  22 2256 562π2π00=yyyyNN    ,nPq=P, and defined by: t Pressure difference, q151=sincos cossinnFZqyasyyasd6y , 151=coscos sinsintFZqyasyyasd6y , where n and t are the normal and tangential com- ponents of stress per unit length, q qFZ is the force due to the springs, which is given by: 12=FZkZkgZ (32) where =1Zda (33) And 2256=cos sindasy asy . (34) In the circular case, the Basic Solution is given by: 1230456π210==cossinys sysysYs ys Pyssyss (35) where =1da, , , and =0Z=0tq=0nqg. Assuming that 0g0, we have 00=,YPYs F, P, and furthermore 0Ys satisfies the boundary conditions for formula (31). The Fréchet derivative of ,FYP at the basic solution is given by: 421503215110100000010000=0sin00000cos00 000nnYttqqyyyyFY qqyyyyyy66ntqyqy (36) where: =00001156====YYntttYYYqqqqyyyy0 (37) 015=cosnYqksy (38) Copyright © 2011 SciRes. JBNB Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure373 016=sinnYqks (39) yTherefore, the linearized problem about the solution is: (40) Which may be written as: basic 0Ys0 0s12311456100 000100 0001cossin=00100 0cos00000sin00 000YysysysPksksysyssyss56d=1 cossinyPykysyss  (41) Integrating twice gives: 2w111d 1115611cossind=yPyky sysscsc   (42) ith boundary conditions:  1360= 0= 0=0yyy 132π 2π==0yyNN  552π2π2π2π0=cos sinyy yNN NN    6Therefore, To solve the fferential equation: 1=2=0c. di111561sincosyPy  kysysc s (43) We write the solution in the form: 1=12nnπ== sinysbnNs (44) 5 31=1cos1 cos==cos 11nnsnN snNyxsbnN nN(45) c 6 4=1sin1sin1==sin21 1nnbsnN snNyy scnN nN (46) where Substituting these equations into the differential equation gives: =2,3,4,N. 22 122=1sin1= 0nnkbnNsnNP (47) 22 1221=0,1nkbnNP nnN  >1 n-zero solution (a bifurcation point) to exist, must be (48) For a no1b 0. ore, Theref2121=01NP N  (49) From whice get: k h w*2 12=1 1NkPN N  (50) here *NP is the criticant, at which the vessel dwl pressure value, or bifurcation poi eforms into a shape with N axes of symmetry. This result allows us to find the bifurcation points for alugiven ves of N and 1k. Before the first bifurcation point, for *20,PP, we have the basic (trivial) solution which corr2esponds to the undeformed shape. The deformed shapes corresponding to N = 2, 3, 4, , N exist for PP, wherehe circular solution becomes unstable, ands is lost. A series expansion of the first order approximations of the bifurcation solutions are given by formula (43). Additionally, equating t nesuniqueNP with 1NP gives us the pressure where the shape with N axes of symmetry collapses. For example, the =2N case occurs for 0 ≤ k1 ≤ 24, while the =3N case occurs for 24 ≤ k1 ≤ 120. 4. Numerical Formulatiextern). , we write the equili- on Due to the assumed uniform al pressure, the vessel will deform into radially symmetric shapes, with N axes of symmetry ( N, an integer, 2Given P, a, N, 1k, and 2kbrium equations in the form: 12=yy 23=yy 34=yy 332 34222=ntnyyqqyqyy1nNThen 2yy y  51=cosyy 61=sinyy where 1=y, 2=y, 3=y, 4=y , 5=yx, 6=yy and 56coscos sinsinZasyasy=tFq (51) d56=sincos cossinnFZqasyadsy (52) Copyright © 2011 SciRes. JBNB Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure Copyright © 2011 SciRes. JBNB 374 2256=cos sindasy asy  (53) =1Zda (54) and 2π0,sN, with boundary conditions 1π0=2y 030=0y 60=0 y12ππ2π=02yNN   32π=0yN 22 2256 562π2π00 =yy yy0N    . We can then numerically solve for the shape of the deformed vessel. Using this shape, we solve the follow- g (normalized) Poisson equation in a MAT- AinL=1v B program to find the velocity of the blood, ,vxy. We can then find the flow rate through the deformed vessel by integrating the velocity function over the area of the cross sectional area of the vessel. 5. Numerical Results We created a MATLAB code that uses ,, ,ak kN, and P as inputs to solve for the shape and fl12ow rate. The this programollowing m outputs of for figures below are examplethe cases N = 2, 3, and 4. We used the fodel to represent the nonlinearality of the soft tissue surrounding the blood vessel: 12tanh=FZkZk Z (55) In each pair below, the figure on the left shows the N Figure 5. MATLAB results for N = 2. Figure 6. MATLAB results for N = 3. Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure375 Figure 7. MATLAB results for N = 4. shape of the deformed vessel, which is then meshed to create the image on the right, a representation of the velocity profile of the blood flowing through the vessel. 6. Conclusions In real situations, the pressure and deformed shape can be easily determined using medical technology (for example, by x-ray or ultrasound), hence we would seek to determine the elasticity of the tissue and) based on a given pressure and shape, sin only be determined while the tissue is in vivo. The tables of graphs below show the varying ofand independently with a constant pressure ansym ry shape of . As can be seen, all grashow a strictly increalationship. By combining the flowversus data, we created a modat alsoex curvature. This mverifiesnumericallynique minimumobtaine The uni could then be determby Newton'ving us a way to determthe elasticity param. 3D(1kce th 2kose can 1k d a phs 3D odel ined ine 2k met rate el th (d. using=2Nsing re1 and ows conv) that a minimMethod, gters, 1kk shques e2k uumi and can be 2k Figure 8. N = 2 graphs of flow rate and k1 with constant k2 and pressure. Copyright © 2011 SciRes. JBNB Flow Rate through a Blood Vessel Deformed Due to a Uniform Pressure 376 Figure 9. N = 2 graphs of flow rate and k2 with constant k1 and pressure. Figure 10. 3D graph of flow ra te, k1, and k2 with constant pressure. 7. Acknowledgemennce Foundation grant number 0552350 and UW-Eau Claire Reasearch Office for their support of the SUREPAM program. REFERENCES  C. Y. Wang, L. T. Watson and M. P. 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