On the elastic properties of arteries

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1 Jounal of Bioechanics 36 (2003) Shot counication On the elastic popeties of ateies C.G. Stephanis a, *,D.E. Mououas a,d.g. Tsagadopoulos b a Depatent of Physics, National Technical Univesity of Athens, Zogafou Capus, Athens, Geece b Cadiac depatent, Athens Chest Hospital Sotiia, Geece Accepted 15 Apil 2003 Abstact A new coefficient of elasticity is poposed that elates to the elastic state of the blood vessels. This easue is poposed as a esult of the ealization,fo pesonal expeience as well as fo the intenational liteatue,of the difficulty in easuing the thickness of the blood vessels in vivo with acceptable pecision. The easueent of E being dependent on the easueent of the thickness of the vessels becoes a highly uneliable poposition. Its elation to E (Young odulus) and to the pulse wave velocity (PWV) is established. We give thee exaples showing how the poposed coefficient can be easued Elsevie Ltd. All ights eseved. Keywods: Ateies; Mechanical popeties elasticity 1. Intoduction It is well known that the elasticity of ateials is expessed in Hooke s law as follows: Tensile stess of a solid is popotional to the aount of defoation, povided the defoation is not peanent. Hooke s law applied to an elastic ba of length l can be expessed as P ¼ E Dl l ; ð1þ whee E is the coefficient of elasticity of the ateial (Young odulus),and P is the stess of the ba. In the case of a thin-walled pipe of intenal adius and thickness h; open at both ends,we have fo Eq. (1) DP h ¼ E D ; ð2þ whee is the intenal adius of the tube and h the thickness,and DP the change of intenal pessue. If the atio h= is vey sall then we have that intenal and extenal adii ae appoxiately equal. Although we have esevations in applying the above to ateies because they ae not elastic ateials in an absolute sense,since they ae viscous to a cetain degee,and because of the existence of unknown foces fo *Coesponding autho. Tel.: ; fax: E-ail addess: cossteph@cental.ntua.g (C.G. Stephanis). uscula fibes suounding the,we believe that,by calculating E using Eq. (2),we can obtain ipotant infoation elated to thei condition. The coefficient E can also be calculated fo the Moens Koteweg equation sffiffiffiffiffiffiffi Eh CðPWVÞ ¼ ð3þ 2 by easuing the velocity CðPWVÞ and the othe paaetes. In both Eqs. (2) and (3) eos ae intoduced fo the geoetical diensions of the blood vessels, especially the thickness h that cannot be easued with high pecision. The eo fo easueents in vivo can be as high as 100% (Peteson et al.,1960; Ruthefod, 1995),a fact that causes popagation of the eo in the calculation of E: This is the eason why any eseaches use siple equations that do not eploy thickness h (Buton,1951). So that they eithe oit fo elation (2) the h eoneously,since the esulting units of easue ae wong,o they oit the entie atio =h in ode to define a new coefficient E p (Pessue stain elastic odulus) (Peteson et al.,1960) D DP ¼ E p ; ð4þ wheeas othes use only the pulse wave velocity (PWV) as a eans to estiate stiffness. Alost all the papes /03/$ - see font atte 2003 Elsevie Ltd. All ights eseved. doi: /s (03)00188-x

2 1728 ARTICLE IN PRESS C.G. Stephanis et al. / Jounal of Bioechanics 36 (2003) Noenclatue P stess execised on a ba E Young odulus l length of the ba ¼ d=2 whee d is the inne diaete of the atey h thickness of the atey wall CðPWVÞ pulse wave velocity blood density pessue stain elastic odulus E p V d Y P s P d d s d d volue of the atey segent unde contol the coefficient of volue elasticity inne diaete of the atey segent c p =c v systolic pessue diastolic pessue diaete of the atey duing the systolic phase diaete of the atey duing the diastolic phase pesented at the confeence (Tenth Euopean Meeting On Hypetension,2000) efe to PWV as a eans fo easuing the stiffness of the vessels. As a consequence of the above,it is ipossible to copae esults of vaious eseaches in ode to daw consistent conclusions. The Young coefficient E is a vey useful easue, dependent only on the stuctual ateial constituting the blood vessels,and not at all on thei geoety. To use steel as an exaple,all pipes ade of steel, independently of thei diaete,have the sae E; as long as the atio of thickness to diaete is a sall nube. Thus,this is the way the eseaches took thei easueents,liteally handling saples of vessels in the lab (Davison et al.,1995). 2. The poposed coefficient We believe that in heodynaics it is also ipotant to know the ease of defoation of blood vessels,which is a function of geoetical diensions,such as thickness and diaete,and the ateial (tissue) the blood vessels consist of. In ode to study this defoation,we intoduce the definition of anothe elasticity coefficient that we call volue elasticity coefficient based on Hooke s law as well,wheeby pessue p is popotional to defoations. Fo now on,we efe to volue defoations,not length defoations DP ¼ DV V : Of couse,hee we apply the sae constaints on the ateials to be elastic,etc. and we have the sae esevations we have aleady entioned egading the value of E as applied to blood vessels. Let us see now in what way the two coefficients and E elate to each othe. Since the volue V of a blood vessel segent of length l is V ¼ lp 2 : ð5þ And the change of volue is DV ¼ l2pd and ðdv=v ¼ 2D=Þ By also consideing Eq. (2),we have ¼ Eh 2 : ð6þ And now fo Eqs. (3) and (6) we have ffiffiffi CðPWVÞ ¼ ; ð7þ whee we can see the Moens Koteweg elation geatly siplified. So in ode to easue thee is no need to easue diaete and thickness of the vessel but all is needed is to easue velocity PWV,aise it to the powe of two and ultiply by (: blood density). Hee,we would like to ephasize the following facts that ake us believe that the coefficient of volue elasticity ; as defined by Eq. (7),could be a good choice: (a) The elation is exteely siple. (b) The popagation velocity of the paticula defoation has the classical geneal fo,that is,the popagation velocity of an elastic defoation is equal to the squae oot of atio of the coesponding coefficient of elasticity ove the density,as shown by p the following well-known elations: ffiffiffiffiffiffiffiffiffi E= that is,velocity of longitudinal waves in od (E: p The Young odulus, : density), ffiffiffiffiffiffiffiffiffi G= that is,velocity of shea waves in a solid (G: the shea odulus), p CðSound VelosityÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi b ad = that is,velocity of elastic waves in fluids (liquids and gases),whee b ad is the adiabatic Bulk odulus. 3. How to easue l can be easued using ethods siila to ethods we easue E except that thee is no need to easue thickness of the vessel and in any cases we do not even need to easue the othe geoetic diensions.

3 C.G. Stephanis et al. / Jounal of Bioechanics 36 (2003) Method 1 Using a B-scan we take iages duing the systolic phase and we easue the intenal diaete d s of the vessel. Then we do the sae duing the diastolic phase and easue the diaete d d : Then we obtain the diffeence DP ¼ P s P d and by applying Eq. (5) we find ¼ DPðV d =DVÞ; whee DV ¼ V s V d and finally d 2 d ¼ðP s P d Þ ds 2 : ð8þ d2 d 3.2. Method 2 We easue PWV and apply Eq. (7) ffiffiffi CðPWVÞ ¼ : Hee we do not need to easue geoetical diensions Method 3 Fo the bachial atey we can use a oe diect ethod,by eans of an extepoaneous instuent, known in the bibliogaphy as pulse plethysogaph (Ruthefod,1995). With a sphygoanoete having the oiginal anoete equipped with an additional oe sensitive anoete with esolution of,say 0.1 ba,we easue with Standad Medical Manoete (SM) (Fig. 1) the systolic and diastolic pessues. Then we ceate a pessue salle than the diastolic,say 50 ba,and we see that the instuent Diffeential Manoete (DM) (Fig. 1) shows a fluctuation Dp at caused by fluctuations in the volue of the atey. Then we inset in the cuff a known volue of ai,say 3c 3,slowly so that we obtain isotheal copession. The pessue becoes p þ Dp: Since the pessue of the ai contained in the cuff changes by Dp; fo a change in volue by 3 c 3,when we have a fluctuation in pessue by Dp at =g what would the change of volue of ai in the cuff be. (The division by g ¼ c p =c v ¼ 1:41 fo the ai is due to the fact that the copession of the ai fo the volue fluctuations of the atey is apid,theefoe adiabatic.) What we find is the change in volue DV of the atey fo the section being coveed by the cuff (10 c). If we also obtain V d fo the B scan we have ¼ DP V d DV : Let us pesent an exaple of easueents using the ethods descibed above. We apply the to a healthy peson and we find. Fig. 1. Diaga of the laboatoy setup fo the diect easueent of the bachial atey volue fluctuations.

1730 ARTICLE IN PRESS C.G. Stephanis et al. / Jounal of Bioechanics 36 (2003) 1727 1731 3.4. Method 1 (B-scan) We easued the of the bachial atey.

The pocedue was as follows: We easued the blood pessue by eans of an odinay sphygoanoete,with the subject lying in hoizontal position.

4 1730 ARTICLE IN PRESS C.G. Stephanis et al. / Jounal of Bioechanics 36 (2003) Method 1 (B-scan) We easued the of the bachial atey. Because this atey is vey close to the skin suface,we used 7.5 MHz tansduces. The pocedue was as follows: We easued the blood pessue by eans of an odinay sphygoanoete,with the subject lying in hoizontal position. We found systolic pessue : P s ¼ 120 Hg; diastolic pessue : P d ¼ 84 Hg: Iediately afte,we pefoed a B-scan. The esolution of the B-Scan was 0.1. In ode to have a bette pespective of the atey bounday,we also used a Dopple sonogaphy (Fig. 2) so as to obtain a distinct colo of the blood in otion. We found diaete duing the systolic phase : d s ¼ 4:5; diaete duing the diastolic phase : d d ¼ 4:0: Fig. 2 shows the B-scan fo the systolic phase along with the paaetes of the easueent. We obtained a siila B-scan fo the diastolic phase. By applying the foula,we have d 2 d B-Sc ¼ðP s P d Þ ds 2 ¼ 135 Hg: d2 d 3.5. Method 2 (B-scan) Othe eseaches geneally easue the PWV with autoatic instuent such a Coplio. Since we wee not equipped with such an instuent,we ade use of two extepoaneous plethysogaphs,as descibed in Method 3. We positioned the on the uppe half of the a at a pedeteined distance,and,fo the tie diffeence in the aival of the cadiac pulses,we calculated the velocity. We easue PWV and we find 3:8 /s. Fo the equation ffiffiffi we have PWV ¼ C 2 ¼ 14; 400 Pa ¼ 108 Hg: 3.6. Method 3 (B-scan) We easue systolic pessue : P s ¼ 120 Hg; Fig. 2. B-scan illustation of the bachial atey fo the systolic phase. diastolic pessue : P d ¼ 84 Hg: We set the pessue inside the cuff to a value of 50 Hg (Fig. 1). Then we activate the sensitive diffeential anoete DM and we easue the fluctuation of the pessue Dp at caused by fluctuations in the volue of the atey (Fig. 3) Dp at ¼ 1:2 ba: Fig. 3. Pessue fluctuations in the cuff,caused by the pulsations of the atey. 1 V coesponds to 1 ba.

5 C.G. Stephanis et al. / Jounal of Bioechanics 36 (2003) The DM has sensitivity of 1 V/ba and esponse tie of 100 ns. Using a syinge,we slowly inset a volue of ai equal to 3.0 c 3 and the change in the pessue of ai inside the cuff becoes Dp ¼ 6:0 ba: So,when he have a change in the volue of the cuff equal to 3:0c 3 the pessue ises by 6:0 ba what is the change DV in the volue when the pessue changes by 1:2=1:41 ba: What we find is DV ¼ð3:01:2Þ=ð1:46:0Þ ¼ 0:42 c 3 : Theefoe,fo Eq. (5) pl ¼ DPV d DV ¼ 36 1:26 0:42 ¼ 107 Hg; whee V d is the volue of a blood vessel with a length of l ¼ 10 c V d ¼ p d2 d 10 ¼ 1:26 c3 : 4 Eventually,fo all the above ethods,we have B-Sc ¼ Hg; PWV ¼ Hg; pl ¼ Hg: All of the above thee values ae fully suppoted (justified,coveed) within the liits of the easueent eos of the available equipent. The eo in the B- scan ethod is lage because the diaete of the bachial atey is sall; theefoe the elative eo in the easueent of the diaete becoes lage. Appendix A. The estiation of l eo We conside that in elation (8) ¼ðP s P d Þðdd 2=ðd2 s d2 d ÞÞ the eo in (P s P d )=36 Hg is vey sall,wheeas the eo of d s and d d is x 2 In ½1Š x y 2 x 2; 72x 2 Out ½1Š ¼ ð x 2 þ y 2 Þ þ 72x x 2 þ y 2; In ð2þ y 36x 2 y 2 x 2; Out ½2Š ¼ 72x2 y ð x 2 þ y 2 Þ 2; 72x 3 dðþ ¼O ð x 2 þ y 2 Þ þ 72x 2 x 2 þ y 2 dx 2! þ 72x2 y ð x 2 þ y 2 Þ 2 dy ; In ½7Š ¼x ¼ d d ¼ 4; Out [7]=4, In ð4þ :¼ y ¼ d s ¼ 4:5; Out [4]=4.5, dx ¼ dy ¼ 70:1; 72x 3 In ½5Š :¼ O ð x 2 þ y 2 Þ þ 72x 2 x 2 þ y 2 0:1 2! þ 72x2 y ð x 2 þ y 2 Þ 2 0:1 ; Out [5]= The eo of is about 40 Hg. Refeences Buton,A.C.,1951. On the physical equilibiu of sall blood vessels. Aeican Jounal of Physiology 319,164. Davison,I.G.,Wight,G.M.,DeMont,M.E.,1995. The stuctue and physical popeties of invetebate and piitive vetebate ateies. Jounal of Expeiental Biology 198, Peteson,L.H.,Jensen,R.E.,Panell,J.,1960. Mechanical popeties of ateies in vivo. Ciculation Reseach 8, Ruthefod,R.B.,1995. Vascula Sugey,Vol. 1,4th Edition. W.B. Saundes Copany,London,pp Tenth Euopean Meeting On Hypetension,G.otebog (Sweden),May 29 June 3,2000; Jounal of Hypetension 18.

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