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Using the spring constant method to analyze arterial elasticity in type 2 diabeticpatients



This study tests the validity of a newly-proposed spring constant method toanalyze arterial elasticity in type 2 diabetic patients.


The experimental group comprised 66 participants (36 men and 30 women) rangingbetween 46 and 86 years of age, all with diabetes mellitus. In the experimentalgroup, 21 participants suffered from atherosclerosis. All were subjected to themeasurements of both the carotid-femoral pulse wave velocity (cfPWV) and thespring constant method. The comparison (control) group comprised 66 normalparticipants (37 men and 29 women) with an age range of 40 to 80 years who did nothave diabetes mellitus. All control group members were subjected to measurement bythe spring constant method.


Statistical analysis of the experimental and control groups indicated asignificant negative correlation between the spring constant and the cfPWV(P < .001; r = - 0.824 and – 0.71). Multivariateanalysis similarly indicated a close relationship. The Student’s ttest was used to examine the difference in the spring constant parameter betweenthe experimental and control groups. A P- value less than .05 confirmedthat the difference between the 2 groups was statistically significant. Inreceiver operating characteristic curve (ROC), the Area Under Curve (AUC, = 0.85)indicates good discrimination. These findings imply that the spring constantmethod can effectively identify normal versus abnormal characteristics ofelasticity in normal and diabetic participants.


This study verifies the use of the spring constant method to assess arterialelasticity, and found it to be efficient and simple to use. The spring constantmethod should prove useful not only for improving clinical diagnoses, but also forscreening diabetic patients who display early evidence of vascular disease.


Arterial elasticity is a significant predictor of cardiovascular risk. Many studies haveconsistently linked a decrease in arterial elasticity to both Type 1 and Type 2 diabetes [13]. This finding may indicate an important pathway linking diabetes to increasedcardiovascular risk and mortality [4, 5]. Researchers have proposed several methods of measuring arterial elasticity,including the pulse wave velocity (PWV) method, the arterial ultrasonography method,compliance, augmentation index (AIx), and the second derivative of photoplethysmogram(SDPTG). The PWV method measures the pulse propagation speed, whereas arterialultrasonography relates the change in diameter of an artery to its distending pressure.The method of compliance, AIx, and SDPTG all evaluate arterial elasticity using arterialpressure waveforms [68]. Different methods are suitable for assessing arterial elasticity indifferent regions. The PWV method can be used to assess regional stiffness. Theultrasonography method is appropriate for assessing local stiffness, and the compliancemethod is suitable for assessing systemic stiffness [9]. Aortic PWV measures the speed at which a pressure wave is transmitted fromthe aorta to the vascular tree. The carotid-femoral PWV (cfPWV) is related to the aorticPWV and is considered the gold standard for measuring aortic stiffness. The newguidelines for the management of arterial hypertension presented by the European Societyof Hypertension and European Society of Cardiology indicate that a threshold ofcardio-femoral PWV exceeding 12 m/s is an indicator of subclinical organ damage [10, 11]. Furthermore, the cfPWV method can also be used for the analysis of thearterial elasticity of diabetic patients [1215]. However, all these methods are essentially indirect methods of assessingarterial elasticity, and include inconveniences in signal measurement.

Of all the methods of assessing arterial elasticity, the arterial pulse plays animportant role. The pulse is generated by the left ventricle and propagates from theaorta to the peripheral arteries, such as the radial and femoral arteries. The arterialpulse is distorted from its original pulse waveform in whatever region the artery ishardened, and then propagates to the end of the artery. As a result, the peripheralarterial pulse waveform provides information on the arterial elasticity of the centralor peripheral artery. Among peripheral arterial pulses, the pulse at the radial arteryis the most easily and conveniently detected signal. Thus, an increasing number ofstudies have focused on the radial arterial pulse, with the goal of finding itstime-domain and frequency-domain features to detect certain early cardiovascular-relateddiseases. For example, researchers have attempted to estimate the aortic artery pulseusing radial tonometry by a transfer function [6, 1618].

Previous research proposes a direct measurement method of peripheral arterialelasticity. This method, which is based on the highly elastic structure of the arterialwall, uses an elastic spring to model the radial vibration of the peripheral arterialwall at the radial artery [19]. The characteristic parameter of the spring is the spring constant, whichrepresents the ratio between exerted force and displacement according to linearity andHooke’s law. Unlike the value of the spring constant of a normal spring, a lowervalue is often used to simulate the result of aging or damage to the spring in amechanical system [2022]. In other words, a lower value for the spring constant denotes that theelasticity of the spring has deteriorated. Similarly, a lower value for the springconstant implies deterioration in the elasticity of the arterial wall. Previous researchverifies this inference [19].

Through careful verification and correlation analysis with the cfPWV, previous studyconfirms that the spring constant method can effectively evaluate the elasticity of thearterial wall. Because the elasticity of the arterial wall has important implicationsfor complications such as cardiovascular-related diseases in diabetes mellitus,long-term diabetic patients require an effective and convenient index to monitor thecharacteristics of the arterial wall. Therefore, this study evaluates the elasticity orstiffness of the radial arterial wall in type 2 diabetic patients using both the springconstant and cfPWV methods, and compares these characteristics with those of controlparticipants.


Basic theory of the spring constant method

This section briefly reviews the basic theory of the spring constant modeling. Intraditional hemodynamics, the major kinetic energy of blood moves in a longitudinaldirection; thus, the radial dilation is assumed to be small or negligible. TheMoens-Korteweg equation and the related pulse wave velocity are based on theseassumptions. Pulse wave velocity is further used as an indirect index of arterialstiffness. However, over 90 % of the energy for an in situ artery is stored in thearterial wall, and less than 10 % is stored in the circulating blood [23, 24]. Thus, the vibration in a radial (transverse) direction should beemphasized over the axial (longitudinal) direction. In circulation physiology, theleft ventricle ejects blood fluid into the aortic arch, which comprises a large turn,and transforms most of the axial kinetic energy into radial potential energy. Becausethe arterial wall (especially the media layer) consists mainly of smooth muscle cellsand elastic tissue, it behaves like a spring and can transform the pumping force fromthe left ventricle into radial vibration of the artery. Thus, the assumptions oftraditional hemodynamics should be modified, and the radial dilation should not beviewed as small or negligible. The corresponding pressure wave equation of thisimportant concept can be derived as follows [25, 26]:

2 P ( z , t ) t 2 + b P ( z , t ) t + ν 0 2 P ( z , t ) = V 2 2 P ( z , t ) z 2

Where P (z t) is the radial pressure defined as thedifference between the internal fluid pressure and the fluid pressure in the staticcondition. The attenuation term b is related to the kinetic viscosity of theartery wall, the adherent fluid in the radial direction, and the stretching andcontraction of the arterial wall itself. The characteristic angular frequencyv o is related to Young’s modulus, arterialcompliance, the mass of the wall, the adherent fluid, and the radius of the tube. Thesymbol V denotes the high-frequency phase velocity related to theshear modulus of the wall. V 2 d 2 P ( t ) d z 2 is the term resulting from the Windkessel effect [25].

To determine the propagation pressure wave in the peripheral artery at a fixedposition, Eq. (1) is simplified to the following equation:

d 2 P ( t ) d t 2 + b d P ( t ) d t + ν 0 2 P ( t ) = - k p 2 V 2 P ( t )

Where kp is the wave number [25]. In a relatively small range of pressure variation, the relationshipbetween pressure and vessel diameter can be approximated to be linear; thus,x(t), the displacement of the arterial wall is also linearlyrelated to p(t) [27, 28]. Figure 1 shows the assumption of linearapproximation. Consequently, Eq. (2) can be transformed into the following:

d 2 x ( t ) d t 2 + b d x ( t ) d t + k x ( t ) = - k p 2 V 2 x ( t )
Figure 1
figure 1

The concept of linear approximation of the pressure-diameterrelationship.

Equation (3) is analogous to describing a unit mass spring system with external force( F external = k p 2 V 2 x ( t ) ), damping force ( b d x ( t ) d t ), and a spring constant ( k = ν 0 2 ).

In physics, elasticity is the physical property of a material that returns to itsoriginal shape after an external deforming force is removed. The stiffness,k', of a body is a measure of the resistance offered by an elastic bodyto deformation. For an elastic body with a single degree of freedom, stiffness isdefined as k ' = F δ . The force applied to the body is denoted byF, and δ represents the displacement produced by the force. A springconstant can be used as a measure of the force required to achieve a particularextension. For a given displacement, x(t), the larger the value ofk, the greater the restoring force; that is, the elastic muscle fiber ofthe artery wall produces greater force to restore its original shape. Accordingly,using the spring constant to assess elastic force makes it possible to evaluate thecharacteristics of the elastic muscle fiber. These characteristics are related to theso-called elasticity or stiffness of the arterial wall [29].

Pulse Acquiring System

Figure 2 illustrates the proposed pulse-acquiring instrument.We measured the radial arterial pulse at the wrist of the right hand, adjacent to theventral surface of the radial styloid process (Figure 2). Thepulse recorder was used to record the radial arterial pulse. The peak and valley ofthe radial pulses were respectively calibrated using the SBP (Systolic BloodPressure) and DBP (Diastolic Blood Pressure) of an oscillometric cuff system (OmronHealthcare Co., Ltd.). Measurements were taken immediately after the radial pulses inthe same arm. Three supine blood pressure measurements were taken 3 min apart, andthe average of the 3 readings was used.

Figure 2
figure 2

Schematic diagrams of the pulse-acquiring instrument and the detectionposition of the radial arterial pulse at the wrist of the right hand.

The pressure sensor was mounted in a holder that allowed it to travel in 3dimensions. The patient’s wrist was held motionless and adjusted in such amanner that the sensor made direct contact with the skin at the desired position onthe radial artery. The measurement and analysis procedure took approximately3–5 min when executed by an experienced operator. A pulse sequence of 10 s(roughly 10 pulses) was selected. Finally, we removed a continuous, piecewise lineartrend to reduce the respiratory effect on pulse waveform. Within-operator andbetween-operator analyses revealed significantly high reproducibility [19].

To measure the arterial pulse, we gently compressed the radial artery against thebone to detect the pulse waveform of the radial artery with low distortion. Adjustingthe pressure sensor vertically or horizontally obtained the maximum amplitude of thearterial pulse under various contact pressures, and then the acquired pulse wassupposed to be optimum. In other words, the sensor was positioned exactly over theradial artery. With the optimum contact pressure (i.e., transmural pressure = 0), thesensor was considered well matched with the vibration of the arterial wall. Greateror lesser contact pressures would caused a distortion in the arterial pulse waveformand the spring constant [3032]. The optimum pulse accurately reflects the vibration of the arterial wall,and can therefore be used to evaluate arterial elasticity. Figure 3 shows the acquired pulse sequence on the radial artery at the wrist.Some physiological factors, such as heart rate or respiration, may exert a smallinfluence on arterial displacement. Thus, the arterial pulse waveforms for each pulseare not identical. Consequently, the values of the calculated spring constants foreach pulse vary slightly. To reduce these influences, the analysis in this study usesthe average of the spring constants of 5 steady pulse signals.

Figure 3
figure 3

The acquired optimum pulse sequence.

Spring constant calculation

The arterial pulse is separated by points A, B, C, and G, and divided into 3segments, A → B, B → C, and C → G, as Figure 4(a) shows. The points A and C represent the valley and peak of thearterial pulse, respectively. While the arterial wall dilates from A to B, it isdriven by the force because of the Windkessel effect, the restoring force of thespring, and damping force. The force caused by the Windkessel effect acts on thearterial wall mainly between points A and B, since the peak of blood flow in theperipheral artery occurs approximately within this period [33]. The total blood flow gradually decreases between points B and G, andgreatly decreases the driving force generated by the Windkessel effect. At point C,the driving force caused by the Windkessel effect can be disregarded. B is defined asthe maximum velocity point, that is, the equilibrium point while the arterial walldilates from A to C [29].

Figure 4
figure 4

(a) The oscillation procedure of the arterial pulse is separated by pointsA, B, and C, and divided into two segments A → B and B → C. (b) The first-order differential curve of the arterial pulse waveform. (c)The second-order differential curve of the arterial pulse waveform.

Because the first derivative of the arterial pulse waveform denotes the movingvelocity of the arterial wall, the maximum velocity point can be derived by findingthe peak of the first-order differential curve of the arterial pulse waveform (4(b)).At point B, the second derivative of the arterial pulse waveform (Figure 4(c)) relating to the force is 0. Thus, the net (total) forceacting on the artery wall is 0 according to Newton’s laws, and the arterialwall simultaneously approaches maximum velocity. Therefore, this point is consideredan equivalence point in spring vibration. As the dilation extends from B to C, therestoring force of the spring and the damping forces become the major forces actingon the arterial wall. However, at the end point C of this dilation process, thevelocity of arterial wall is 0, resulting from the zero value of the first derivativeof the arterial pulse waveform illustrated in Figure 4(b), andstarts to contract. Consequently, no damping force acts at point C because thedamping force is assumed to relate to the velocity.

Therefore, at point C, the damping force and the external force originating from theWindkessel effect are both 0. Equation (3) can then be simplified as follows:

d 2 x ( t ) d t 2 | at point C + k x ( t ) | at point C = 0

The spring constant k of the arterial wall is derived as

k = - d 2 x ( t ) d t 2 | at point C x ( t ) | at point C = - - E F B D = E F B D

Where k signifies the elasticity of the arterial wall. The symbol B D represents the displacement between the equivalentpoint B and point C. The symbol E F denotes the magnitude of the second derivative of thearterial pulse waveform at point C. Point C plays an important role in calculatingthe spring constant k.

Pulse measurement experiment

The research parameters were measured in the study participants between November 15,2010, and August 30, 2011, at hospital in Taiwan. The experimental protocol wasapproved by Institutional Review Board of Taichung Hospital, and written informedconsent was obtained from all participants before they enrolled in the study. Wedesigned an experiment to investigate the arterial elasticity in type 2 diabeticpatients using both the spring constant and cfPWV methods, and then compared theresults with those of the control participants. According to the World HealthOrganization (WHO) definition of diabetes, the inclusion criteria for type 2 diabeticpatients includes a fasting plasma glucose ≥ 7.0 mmol/L or an oral glucosetolerance test (OGTT) 2-h postprandial glucose ≥ 11.1 mmol/L [34]. The subjects of diabetic group and control group were randomly selected.Participants with a history of heart failure, arrhythmia (atrial fibrillation, atrialflutter, and so on) were excluded because such diseases may affect the propagationspeed of the arterial pulse. The inclusion criteria of control group included sex,age, and body mass index (BMI) that matched the diabetic patients. Subjects with ahistory of diabetes mellitus or atherosclerosis were excluded from the controlgroup.

Members of the experimental group included 66 participants with diabetes mellitus(aged 46 to 86 years; 36 men and 30 women) subjected to measurement by both the cfPWVand spring constant methods. Participants in the control group of 66 normalparticipants (aged 40 to 80 years; 37 men and 29 women) were measured only by thespring constant method. All participants were asked not to imbibe any alcoholic orcaffeinated beverages on the day of the experiment. For pulse measurement,participants were instructed to sit and relax for 5 min prior to the radial pulsemeasurement to reduce respiration interference. The patient’s right hand wasplaced on the measuring platform (Figure 3), and the operatortook 10 arterial pulses at the wrist. Five steady pulse signals were selected insequence for the spring constant calculation, and these 5 calculated values wereaveraged to represent the spring constant of the participants. For the cfPWVmeasurement, participants were instructed to rest in a supine position for 5 min,after which an automatic device (SphygmoCor system, AtCor Medical, Australia) wasused to obtain the measurement.

Statistical analysis

We used the statistical toolbox in Matlab 7.11. Data were expressed as means andstandard deviations, and the Student’s t-test was used to comparegroup differences. Simple and multiple regression analyses were performed to identifythe relationships between arterial elasticity measures and other hemodynamic andclinical variables. A value of P < .05 was considered statisticallysignificant.


Table 1 shows a summary of the means and standard deviations forthe clinical characteristics of both the diabetes-patient and normal-participant groups.The radial pulse measurement was started 5 min after the cfPWV measurement. Table 1 shows the evaluated cfPWV, spring constants, and P- valuesof the experimental and control groups. Figure 5(a) plots thespring constant versus the cfPWV of the experimental group. As the cfPWV increased, thespring constant tended to decrease. This tendency is reflected by the negative-slopeline ( y = - 1.09 x + 1701.3 ) produced by the linear regression method. TheP- values and correlation coefficients (< .001 and -.824) indicate asignificant negative correlation between the spring constant and the cfPWV. Thus, if thecalculated value of the spring constant is lower, the value of the cfPWV is higher. Thisfinding implies that the elasticity of the arterial wall deteriorates over time,possibly through aging, damage, or diabetes mellitus. The control group revealed asimilar relationship, as Figure 5(b) shows. We used theStudent’s t test to analyze the differences between the experimental andcontrol groups for the spring constant. The resulting P-value (< .05)indicated a significant difference in this parameter between the two groups. Thisfinding implies that the spring constant method can distinguish between normal andabnormal elasticity characteristics in normal and diabetes participants as effectivelyas the cfPWV [1215].

Table 1 Clinical characteristics of the diabetic patient group, normal subject group,and P- value
Figure 5
figure 5

(a) Plot of the spring constant versus the cfPWV for experimental group. (b)Plot of the spring constant versus the cfPWV for control group.

Table 2 shows the multivariate regression analysis of therelationship between the spring constant and age, height, brachial systolic bloodpressure, heart rate, and cfPWV, respectively. The spring constant is independentlyinfluenced by all 5 clinical parameters, with an adjusted R2 of 0.59. Themeasure β (standardized coefficient) indicates how strongly each predictor variableinfluences the criterion variable. These results also imply that the spring constant issignificantly related to the cfPWV (P < .001). The results in Table 2 show that as age, brachial SBP, and cfPWV increased, the springconstant significantly decreased, indicating deteriorating elasticity in the artery.These results agree with a basic physiological understanding and prior research findingson vessel elasticity, and confirm the validity of using a spring constant to modelvessel elasticity [35, 36].

Table 2 Multivariate relationship between spring constant and clinical characteristicsof the diabetic patient group

In the experimental group of 66 participants with diabetes mellitus, 21 participantssuffered from atherosclerosis; thus, they were viewed as patients with arterialstiffness. We used the receiver operating characteristic curve (ROC) to assess thesensitivity of the proposed method to evaluate arterial stiffness (Figure 6). The Area Under Curve (AUC) (= 0.85) indicates good discrimination. Themean and standard deviation of the 21 participants with atherosclerosis are 578.8 ±144.1 (g/s2). This mean value approaches the best operating point of the ROCcurve. Thus, a spring constant less than the mean (578.8 (g/s2)) maypreliminarily predict a risk of arterial stiffness for diabetic patients.

Figure 6
figure 6

The receiver operating characteristic curve.


Effect of diabetes on arterial elasticity

The possible reasons for increased arterial stiffening in type 2 diabetes includeimpaired glycemic control and the formation of advanced glycation endproducts (AGEs)that destroy the function of endothelial cells and lead to structural changes in thevessel walls [37, 38]. Arterial stiffness is related to endothelial dysfunction, which leads toan imbalance in the release of vasoactive substances from the endothelium. Smallerarterioles and branch vessels are more prone to be affected than larger arteriesbecause the thinner media of the smaller arteries responds strongly to theendothelial release of nitric oxide. Thus, the presymptomatic monitoring of thearterial elasticity of both large and small arteries in patients with type 2 diabetesshould decrease the risk of cardiovascular complications. Many studies havedemonstrated an increased arterial stiffening of the large artery in both type 1 andtype 2 diabetes using both central PWV and central pulse pressure [1, 39]. The stiffening of the peripheral artery may be more obvious than that ofthe larger central arteries in type 2 diabetic patients [40]. Nevertheless, few studies have discussed the effect of diabetes on theelasticity of small or peripheral arteries.

Validity of spring constant method

Some noninvasive modalities, such as MR imaging (MRI), computed tomography (CT), anddigital subtraction angiography (DSA) can provide more accurate information aboutplaque, stenosis, or stroke [41]. In addition, the circulating level of microparticles, oxidizedlow-density lipoproteins, and intima-media thickness are associated with the arterialelasticity of diabetes patient [4244]. However, these techniques are expensive and inconvenient, and aretherefore not suitable for the early and fast diagnosis of arterial stiffness.

Because the radial pulse is easily detected at the peripheral artery, it is feasibleto quickly evaluate the elasticity of the peripheral artery of diabetic participantusing the radial pulse waveform. Using the proposed spring constant method, thisstudy analyzes the peripheral arterial elasticity of diabetes and normalparticipants. Results show that the spring constants of diabetes participants aresignificantly lower than those of normal participants. This implies that the arterialstiffening of diabetes participants is more serious than that of normal participants.These results agree with prior research. This study also evaluates the sensitivity ofthis method based on the receiver operating characteristic curve, and the value ofAUC shows good discrimination. Although more subjects are required to define the riskvalue of spring constant for arterial stiffness, a spring constant below 578.8(g/s2) may imply a risk of arterial stiffness for diabetic patients inthis study. This suggests that the spring constant method can effectively distinguishbetween peripheral arterial elasticity in the experimental and control groups. Inaddition, the spring constants of diabetic participants and normal subjects aresignificantly correlated with their cfPWV values. This confirms the accuracy of thespring constant method for assessing peripheral arterial elasticity.

The pressure-diameter or stress–strain relationship of the artery wall isnon-linear when pressure and diameter are observed over a wide range [27]. However, the dynamic variation in arterial pulse pressure is relativelysmall compared with the overall blood pressure. Thus, a linear approximation of thepressure-diameter or stress–strain relationship is feasible. Because of thecomplexity of a non-linear physiological system, the assumption of linearity is anessential approach to find the average effect or tendency of a complex system.Although approximations may cause some distortion, the significant correlationbetween the cfPWV and the spring constant in these results confirms the feasibilityof a linear approximation.

Currently available methods are basically qualitative and indirect in principle. Forexample, some problems still occur with PWV. First, arterial stiffness does notaffect PWV independently. The contraction ability of the left ventricle, the bloodviscosity, and the resistance of the terminal vascular bed may also affect PWV.Second, it is difficult to estimate the actual wave travel distances [45, 46]. In contrast, the spring constant method directly and quantitativelyevaluates peripheral arterial elasticity. It is theoretically based on the pressurewave equation, and can simply derive the spring constant using 1 optimum pulse at asingle measurement position. The proposed method requires only a few minutes tomeasure a radial pressure pulse and calculate the spring constant, making itconvenient and practical for clinical use.

Although the radial pulse is detected at the radial artery, it propagates from theheart, through the aortic region, and to the peripheral artery. Thus, this pulsemeasurement may include information about systemic, regional, or local arterialstiffness, and it can be used to assess arterial elasticity [10, 47]. Stiffened arteries may cause sudden death from events such as stroke ormyocardial infarction, which often occur in diabetic patients. Thus, effectivelong-term monitoring of the status of arterial elasticity is an important issue fordiabetic patients. The method proposed in this study has the advantage of directlymeasuring peripheral arterial elasticity. The spring constant method should prove tobe useful not only for improving clinical diagnosis, but also for screening diabeticpatients for early signs of vascular disease.



pulse wave velocity


carotid-femoral pulse wave velocity


Augmentation index


Second derivative of photoplethysmogram


Advancedglycation endproducts.


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Correspondence to Ching-Chuan Wei.

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The authors declare that they have no competing interests.

Authors’ contributions

C-C W carried out the research design, performed the statistical analysis, and preparedthe final manuscript. S-W H participated in the design of this study and resultinterpretation. C-T B made contributions to the acquisition of data. All authors readand approved the final manuscript.

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Wei, CC., Huang, SW. & Bau, CT. Using the spring constant method to analyze arterial elasticity in type 2 diabeticpatients. Cardiovasc Diabetol 11, 39 (2012).

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  • Spring constant
  • Arterial elasticity
  • Type 2 diabetes
  • Carotid-femoral pulse wave velocity