Biomechanical analysis of penile erections: penile buckling behaviour under axial loading and radial compression

Authors


Gerald W. Timm, Professor of Urologic Surgery, Academic Health Center, D510 Mayo Memorial Building, MMC394, 420 Delaware Street Southeast, Minneapolis, MN 55455, USA.
e-mail: timmx025@tc.umn.edu

Abstract

OBJECTIVE

To characterize the biomechanics of erectile function, as contrary reports have modelled the penis as an isotropic material and state that only axial buckling tests can effectively predict penile rigidity; that assumption is questioned and an alternative structure proposed and validated.

METHODS

Three experimental physical cylindrical models of diameters 1.9, 2.54 and 3.81 cm were fabricated and the relationship between axial loading and radial compression was measured for cylindrical pressures of 8–20 kPa. A finite element analysis (FEA) computer model of the penis was constructed to simulate the response of the corpora cavernosa to axial and radial loading for differing diameters and lengths of the penile shaft. The stresses developed in the tunica albuginea of the corporal bodies of the penis during buckling were assessed using a mathematical analysis.

RESULTS

From the analysis of surface stresses under variable axial loading, as the angle of an applied load changes on an isotropic shaft, the magnitude of surface stresses varies up to 50 kPa, and for a pressure vessel the magnitude of surface stresses varies up to 100 kPa. The FEA model showed that nodal displacements were greatest around a ring under radial compression, and for the axially loaded model displacements were greatest at the vessel tip under the force gauge. All displacements were 0.1–1.0 mm. There was an exponential relationship between internal pressure and the axial force required to cause buckling in a thin-walled pressure vessel. There was a nearly constant relationship between circumferential displacement and internal pressure under uniform radial compression. The displacement values on the FEA analysis were approximately equal outside of the areas of high stress which were under the load of the external device (compressive ring or force gauge) in both cases. Physical modelling shows that when a pressurized vessel is under either axial or radial load the internal pressure increases. Vessels at high internal pressure require more force to cause buckling than vessels at lower internal pressure. The circumferential displacement of a vessel under radial compression is higher in vessels of lower internal pressure and less in vessels of high internal pressure. The size of a vessel also contributes to its ability to be buckled. Smaller vessels buckle under smaller load, but the ratio of force required to buckle vs. diameter of the cylinder remained constant.

CONCLUSIONS

The computer simulations show that with slight deviations from perfectly aligned axial loading the stresses felt on the walls of cylindrical columns vary considerably, whether they are isotropic beams or pressurized vessels. The material properties of the tissues within the corpora cause it to behave as a thin-walled pressurized vessel, in which the hoop stress and axial stress have a constant relationship independent of the length to diameter ratio rather than as an isotropic beam where this relationship varies. Patient discomfort and high operator dependency further contribute to the inconsistencies of axial loading methods to determine penile buckling. Based on the constant relationship between hoop stress and axial stress in thin-walled pressurized vessels this study confirms the validity and desirability of using radial compression methods to assess penile rigidity in lieu of axial loading methods.

Abbreviations
ED

erectile dysfunction

FEA

finite element analysis.

INTRODUCTION

Erectile dysfunction (ED) is estimated to affect 30 million men and their partners in the USA, with an additional 617 715 new cases annually [1]. ED, also known as impotency, is characterized by the inability to produce or sustain an erection adequate for vaginal intromission. The prevalence of ED increases with age. In a Boston community study conducted from 1987 to 1989, 1290 men aged 40–70 years were surveyed, and the prevalence rate of mild, moderate or complete ED at age 40 years was ≈40%; this rate increased to almost 70% by age 70 years. Secondary to age, other factors that are associated with ED include heart disease, hypertension, diabetes, and associated medications. Cigarette smoking, when associated with heart disease and hypertension, also greatly increases the probability of ED [1].

Treatments for ED vary from noninvasive remedies such as psychiatric therapy, oral or topical remedies, and vacuum erection devices, to minimally invasive methods, e.g. injection therapy, and to invasive procedures necessitated by penile prosthetic implants. Lifestyle changes might also be prescribed, e.g. reducing fat, cholesterol, alcohol and tobacco consumption, and losing weight and taking regular exercise. Treating psychological causes of ED might involve visits to a psychologist, psychiatrist, sex therapist, or marriage counsellor. Some men and their partners might prefer to seek therapy together. Even if the ED is a physical problem, psychological side-effects might be present, so these patients might also be recommended for counselling [2].

Non-invasive treatments include drug therapy in the form of oral medications that improve blood flow to the penis such as sildenafil, tadalafil and vardenafil. Vacuum pumps mechanically enhance the flow of blood to the penis, and suppositories are absorbed by erectile tissue and help the penis to become turgid. More invasive methods include injection therapy, in which a smooth muscle relaxant is injected into the base of the penis, and surgically implanted prostheses to control the erectile movement of the penis [2].

The increasing number of treatment options for men with ED requires the development of progressively more sophisticated diagnostic tests to evaluate erectile status. Earlier methods of assessing nocturnal penile tumescence [3–5] to aid with the differential diagnosis of ED were modified to include radial rigidity assessment [6–8]. Subsequent development of various pharmacological therapies necessitated the addition of real-time rigidity assessment [9,10]. The ability to monitor real-time and nocturnal tumescence and rigidity is provided by the RigiScan rigidity and tumescence monitoring system (Timm Medical Technologies, Inc. Minneapolis, MN, USA; a division of Plethora Solutions Holdings, Ltd., UK). This device measures tumescence and rigidity through loops placed at the base and tip of the penis. Most physically healthy men have three to six erections per night of restful sleep [11]. This makes the RigiScan system an important tool in determining whether someone complaining of erectile difficulty is permanently physically impaired and showing no nocturnal erections, or is temporarily psychologically impaired and achieving good nocturnal erections. The RigiScan system can also help to determine if the patient is responsive to oral or injected medications. Knowing the severity of the ED by using an objective, quantifiable test leads to individualized treatment and better therapeutic outcomes. The system can also be used to evaluate possible side-effects of new therapies for other than ED.

The RigiScan device works by periodically applying a 113 g traction force to the loops to assure contact with the penile shaft during a tumescence measurement, and applying a 280 g traction force when the penile circumference increases by 1 cm above the baseline. The displacement in the loops in response to the 280 g force is converted to a rigidity percentage. If the loops displace by ≥2.2 cm the patient has 0% rigidity during tumescence; 100% rigidity corresponds to no displacement of the loops in response to the 280 g force. Based on clinical observation, 55–60% base rigidity and a 50% tip rigidity has been found to be adequate for satisfactory vaginal intromission [10]. Above this rigidity level, the penis is considered ‘unbuckle-able’ meaning that its turgid state is strong enough to resist collapse during vaginal intromission.

A few studies have tried to disprove the efficacy of this device, which uses radial compression to measure resistance to buckling, by arguing that buckling can only be measured by axial loading. Axial rigidity, not radial penile deformation, is the physical variable that best objectively defines the capability of the erect penis to resist deformation from vaginally mediated compressive forces during vaginal intromission, and during pelvic thrusting after penetration, and thus be of sufficient functional quality for satisfactory sexual intercourse [12].

A key assumption of the Boston group is that the penis can be modelled as an isotropic shaft. Because axial stresses are the principle stresses in governing column buckling, they are the only stresses measured in the study of Udelson et al.[12]. Being a biological organ which can change in volume by up to 300% during tumescence, this represents an inherently false assumption. Rather, the penis behaves as a thin-walled pressurized vessel that becomes rigid when its walls reach their elastic limit. The RigiScan system measures the hoop stresses in the wall of the penis. The thin-walls of this physiological pressure vessel are represented by the tunica albuginea, the body’s strongest tissue membrane, second to the dura matter that surrounds the spinal cord. The tunica albuginea covers the two cavernosal bodies within the penis. The corpora cavernosa consist of erectile tissue and take up most of the volume within the penis. In response to sexual stimuli venous spaces within the erectile tissue of the corpora fill with blood, causing the penis to enlarge and increase its rigidity to produce an erection [13,14]. Because the corporal bodies represent the most important structure in producing erections, properties of the corpora are used in the models of the present study. The walls of the models have a thickness consistent with the average value of that for the tunica albuginea taken from cadaveric studies [15]. The nature of the venous channels which fill with blood during tumescence show that is it more appropriate to assume that the penis acts as a thin-walled pressure vessel, as opposed to an isotropic shaft. The following analysis confirms that radial compression measurements are a valid approach to determining penile rigidity and ability to withstand buckling.

METHODS

The present study consists of four interrelated parts: (i) A mathematical analysis of equivalent stresses on the wall of a pressure vessel and an isotropic shaft under variations of applied axial loading. This simulation shows how stresses on the surface of a vessel or shaft vary when loads are not completely along the neutral axis. (ii) A computerized finite element analysis (FEA) representing the penis as a pressure vessels under an axially applied axially applied and radially compressive loads. This model describes the Von Mises stresses and nodal displacements of the vessels after loading, and shows the relationship between material behaviour of a thin-walled vessel under axial loading and under radial compression. (iii) Physical models were constructed and measurements made to observe curvature and buckling in pressurized vessels (a) to determine how pressure within the penis changes under axial loading and radial compression, and (b) to find a relationship between the forces required to cause buckling and the internal pressure changes in pressure vessels. (iv) The inflection point at which the walls began to collapse was measured for three diameters 1.9, 2.54 and 3.81 cm and for internal pressures of 8–20 kPa, first using radial compression by the RigiScan loops, and then using axial loads of 0.5–2.5 kg. The models were constructed and analysed using MATLAB (MathWorks, Natick, MA, USA).

In the present experiments, several different equations were used, including the calculation of Poisson ratios, Von Mises stresses, and Young’s modulus. The Poisson ratio is the measure of the tendency of a sample of material which is stretched in one direction to become thinner in the other two directions. It uses Young’s modulus to apply this to the thin-walled pressure vessel model. The Von Mises stress is a scalar function of the components of the stress tensor that gives the overall magnitude of the tensor. It allows deformation to be predicted from simple tensile tests. This test was used to prove that the properties of the models described here were similar to those of the actual penis. Young’s modulus allows the calculation of the buckling behaviour under axial loading, thereby predicting the load at which a thin-walled pressurized vessel will buckle under compression; it relates stress to strain to find the rigidity of a vessel.

RESULTS

Model 1 (the analysis of surface stresses under slight variations of axial loading) was developed and analysed to determine the sensitivity of buckling to small deviations from the direction in which an axial force is applied. In thin-walled pressurized vessels, axial stresses and hoop stresses are related by a constant relationship [16].

image(1)
image(2)

This simulation describes the changes in localized wall stresses due to variations in applied load for both a thin-walled pressure vessel and for an isotropic cylindrical column. Figure 1 shows the shape and parameters of the cylindrical model (bar) used to represent both the pressure vessel and the isotropic shaft. The bar has one fixed end and two elements shown on its wall, K and J. Element K is on the neutral axis and element J is orthogonal to element K, yet it is on the bottom surface of the shaft. The MATLAB code which analyses the stresses felt on elements K and J uses the following equations for axial stress in an isotropic shaft.

Figure 1.

(A) Axial buckling model. P is the magnitude of the applied force. Px = Pcos(θ) and Py = Psin(θ), where θ is the angle of the force relative to the neutral axis of the isotropic shaft or the pressure vessel. The offset is the positive or negative distance from the neutral axis where P is applied. The axial bending stresses at elements K and J was analysed as è and the offset vary. (B) θ vs σX. These graphs show the variability of bending stresses felt and elements K and J as θ and the offset change slightly for an isotropic shaft. θ is the angle of the applied force, and the offset refers to the position of the applied force from the neutral axis. (C) θ vs σX for a pressure vessel, showing the variability of bending stresses felt and elements K and J as θ and the offset change slightly for a pressure vessel.

image(3)

where F is the force along the neutral axis, M is the bending moment, c is the distance from the centre of bending (neutral axis), A is the cross-sectional area of the shaft and I is the moment of inertia. Based on the position of elements K and J, in the conditions described in Fig. 1A, for an isotropic shaft, the bending stress at K and J are:

image(4)
image(5)

The stresses at element K and J, when plotted, look like the plot in Fig. 1B.

Table 1 quantitatively shows the difference among the bending stresses as the position (location and angle) of the applied force varies for an isotropic shaft. In this simulation, P is equal to 2.21 N in the negative x-direction, the radius is 19.05 mm and the length is 152.4 mm.

Table 1.  Selected results from axial model, showing how bending stresses and Von Mises equivalent stresses on elements K and J on an isotropic shaft vary as the angle of the force applied (θ) varies
θ, radσK, kPaOffset 0 mm σJ, kPaOffset −9.025 mm σJ, kPaChange in stress as θ changes by π/20, kPa
ΔσKΔσJ (0 mm)ΔσJ (−9.025 mm)
Bending stresses
 −1.57080.000062.03062.030
 −1.4137−0.303260.96362.538−0.3032−1.0670.508
 −1.2566−0.599058.39559.530−0.2958−2.568−3.008
 −1.0996−0.880054.38953.057−0.2810−4.006−6.473
 −0.9425−1.139449.03351.203−0.2594−5.356−1.854
 −0.7854−1.370742.49145.089−0.2313−6.542−6.114
 −0.6283−1.568234.89237.864−0.1975−7.599−7.225
 −0.4712−1.727226.43429.707−0.1590−8.458−8.157
 −0.3142−1.843617.32520.818−0.1164−9.109−8.889
 −0.1571−1.91467.78911.417−0.0710−9.536−9.401
 0.0000−1.9384−1.9381.735−0.0238−9.727−9.682
 0.1571−1.9146−11.618−7.9900.0238−9.680−9.725
 0.3142−1.8436−21.012−17.5180.0710−9.394−9.528
 0.4712−1.7272−29.888−26.6150.1164−8.876−9.097
 0.6283−1.5682−38.029−35.0570.1590−8.141−8.442
 0.7854−1.3707−45.233−42.6350.1975−7.204−7.578
 0.9425−1.1394−51.323−49.1640.2313−6.090−6.529
 1.0996−0.8800−56.149−54.4820.2594−4.826−5.318
 1.2566−0.5990−59.593−58.4580.2810−3.444−3.976
 1.4137−0.3032−61.570−60.9950.2958−1.977−2.537
 1.57080.0000−62.030−62.0300.3032−0.460−1.035
Von Mises stresses
 −1.570887.9060218.310218.310
 −1.413786.9750215.410216.700−0.9310−2.900−1.610
 −1.256686.0750209.170211.700−0.9000−6.240−5.000
 −1.099685.2260199.760203.480−0.8490−9.410−8.220
 −0.942584.4490187.470192.250−0.7770−12.290−11.230
 −0.785483.7610172.660178.360−0.6880−14.810−13.890
 −0.628383.1780155.800162.240−0.5830−16.860−16.120
 −0.471282.7110137.470144.410−0.4670−18.330−17.830
 −0.314282.3710118.400125.560−0.3400−19.070−18.850
 −0.157182.164099.520106.510−0.2070−18.880−19.050
 0.000082.095082.09088.380−0.0690−17.430−18.130
 0.157182.164067.93072.7600.0690−14.160−15.620
 0.314282.371059.32061.7900.2070−8.610−10.970
 0.471282.711057.86057.5700.3400−1.460−4.220
 0.628383.178062.61060.2300.46704.7502.660
 0.785483.610070.70067.3700.43208.0907.140
 0.942584.449079.47076.1000.83908.7708.730
 1.099685.226087.24084.3700.77707.7708.270
 1.256686.075093.02090.9600.84905.7806.590
 1.413786.975096.25095.1800.90003.2304.220
 1.570887.906096.68096.6800.93100.4301.500

The central equations for the pressure vessel model take into account the internal pressure and the thickness of the vessel wall. In a thin-walled vessel, the hoop and axial stresses are separate principle stresses. To compare the two values, equivalent Von Mises stresses were determined for elements K and J. From Eqn 1 and Eqn 2, the equivalent stress is:

image(6)

For the pressure vessel model under the parameters described in Fig. 1A:

image(7)
image(8)
image(9)
image(10)
image(11)
image(12)

where ‘p’ is the internal pressure, which was equated to 9333 Pa, or 70 mmHg. Higher pressures might be used to represent average blood pressure values within the corpora during tumescence. The stresses at element K and J, when plotted, are shown in Fig. 1C.

Table 1 quantitatively shows the difference among the bending stresses as the position (location and angle) of the applied force varies for a pressure vessel. In this simulation, the applied force, radius and the length are the same as for the isotropic shaft model.

The goal of Model 2, computing the FEA of a pressure vessel under radial compression and axial loading mimicking current diagnostic methods, was to verify whether there is a relationship between hoop and axial displacements in a thin-walled pressurized vessel such that the stresses developed during axial loading and radial compression are related.

FEA methods were applied to pressure vessels which had pressure loads applied in two cases: (i) as a circumferential ring around the middle of the vessel to represent a radial compressing loop of the RigiScan system; and (ii) through a device that represents a force gauge similar to those used in other penile buckling experiments, which apply axial loading to measure rigidity [12].

Determining the shape to best represent the penis has its challenges. First, a thin-walled hollow cylinder was used. One end was fixed in the vertical and horizontal directions representing the penile shaft on the assumption that if it was not fixed in the horizontal as well as vertical direction, the structure would be unstable. Fixing the base in all directions might not be physiological correct as the penis walls have the ability to spread radially at the shaft even if constrained axially. Also, the step which applies the radial compression used nonlinear geometry arguments which seem to cause a problem in obtaining a converged solution.

To keep the original boundary condition of a vertical constraint only, without under-constraining the problem, a half cylinder was used with symmetry arguments along the meridianal cross-section (BC-1: U3 = 0, BC-2: XSYMM (U1 = UR2 = UR3 = 0). However, the half-cylinder model’s nodal displacement results were unstable around the shaft end. Therefore, a quarter-model with symmetrical conditions in both horizontal directions (x and y) was used (BC-1: U3 = 0, BC-2: XSYMM (U1 = UR2 = UR3 = 0), BC-3: YSYMM (U2 = UR1 = UR3 = 0)).

However, the quarter-cylinder model did not follow the physiological properties of the penis. A quarter-bullet shape was used to better represent the male anatomy. This model gave more accurate results because in the quarter-cylinder model without a domed end, the height of the vessel shortened after deformation, whereas with the top dome cover, the vessel wall expanded in all directions. The quarter-bullet model had the same symmetry arguments as the quarter-cylinder model but the neutral axis of the vessel was in the y-direction (BC-1: U2 = 0, BC-2: XSYMM (U1 = UR2 = UR3 = 0), BC-3: ZSYMM (U3 = UR1 = UR2 = 0)). Moreover, the dome-shaped end is important for the strength of the penis. Pressure vessels such as hulls have a curved, dome-like ends rather than flat ends, because flat plates have no meridianal curvature and have to resist the effects of pressure in flexure. For a flat plate to have a strength equal to the cylindrical shell it is attached to, it would require a wall thickness 10 times greater than that of the bullet [17].

An internal pressure of 20 kPa was used, and the dimensions of the model were taken from the same average values for an adult male penis during tumescence. Because the RigiScan system uses a 280 g traction in its rings, 280 g was used for the applied force in both cases for easier comparison of the stresses. The material properties of the wall, mainly the elasticity modulus and Poisson’s ratio, were based on values presented in previous research involving the tensile properties of the tunica albuginea [15].

Figure 2A shows the principal maximum stresses and the Von Mises stresses of the axially loaded model, Fig. 2B the axial and radial displacements of the axially loaded model, Fig. 2C the principal maximum stresses and the Von Mises stresses of the radially compressed model and Fig. 2D the axial and radial displacements of the radially compressed model.Table 2 shows the displacements in both models at selected positions.

Figure 2.

(A) The principal maximum stresses and the Von Mises stresses, and (B) the axial and radial displacements of the axially loaded model. (C) The principal maximum stresses and the Von Mises stresses, and (D) the axial and radial displacements of the radially compressed model.

Table 2. 
Total displacements at select nodes of the radially compressed model vs axially loaded model
Selected node and positionMagnitude of displacement, mm
Radially compressedAxially loaded
1 (tip)1.241.32
100 (top, middle, side-edge)1.161.35
850 (top, near ring)0.7421.07
1010 (top. middle)0.1041.31
200 (base-edge, bottom)0.5120.643
5 (ring, edge-side)0.5121.02
142 (centre of ring)0.5981.00
1672 (bottom, middle0.5830.760
112 (top, edge, left)1.091.39
92 (top, edge, left)1.071.37
169 (bottom, edge, left)0.5730.744
221 (bottom, edge, right)0.5710.736

Model 3a (the physical modelling of elastic-walled pressure vessels under axial loading) was devised to observe buckling behaviour in a pressure vessel under axial loading, assessing how buckling curvature occurs and whether or not internal pressure changed during loading. Figure 3A shows the testing apparatus. The materials used in the physical model included and water-column manometer, clips and bands, condoms, first-aid rolls, bandage rolls and a digital force gauge. The metal clips and rubber bands were used for securing the fixed end of the vessel, condoms provided an inner vessel to the model to prevent water leakage, and the bandages provided a sleeve which constrained the expansion of the condoms. The sleeves were made in three different sizes of 1.9, 2.54 and 3.81 cm, and were constructed of materials with tensile properties similar to the tunica albuginea. Tensile tests were conducted to verify similar elasticity moduli, Poisson’s ratios and fibre alignment between the bandages and actual tunica albuginea.

Figure 3.

(A) The testing apparatus for the axially loaded physical model. (B) The force to buckle vs the fixed initial internal pressure. (C) The testing apparatus for radially loaded physical model. (D) The circumferential displacement and initial pressures of radially loaded physical model.

Each sleeve was pressurized using the manometer to 3.92–16 kPa. Physiological low, normal and high blood pressures of 80, 100, 120 and 150 mmHg corresponding to 10.67, 13.3, 16.0 and 20.0 kPa, respectively. The initial pressure was noted as the sleeve was pressurized, and a force gauge with a cupped tip applied an axial load until there was noticeable curvature in the pressure vessel. This force, termed the force of buckling, was recorded. The final internal pressure at the force of buckling was also recorded. Figure 3B shows the results of these measurements.

Model 3b (physical modelling of elastic-walled pressure vessels under radial compression) represented the way in which the RigiScan device works. The materials are the same except an analogue force gauge was used, that was easily attached to the RigiScan loop. The variable recorded was the change in circumference of the pressurized vessel under 280 g traction, and Fig. 3C shows the experimental system; the results are shown in Fig. 3D.

Model 4 was used to determine the inflection point at which walls collapse for radial and axial loading; the purpose of this was to observe buckling behaviour under axial loading and then to measure the circumferential displacement that occurred upon buckling. After applying 280 g traction radially via a RigiScan loop, the resulting circumferential displacement was measured. The experimental system was that described in Fig. 3. Three sleeves were made, differing only in their diameters, as previously noted in Model 3a, of 1.9, 2.54 and 3.81 cm.

The cloth-covered base loop of the RigiScan is placed around the model and tightened until snug against it. Using the plastic syringe, water was added to the water-column manometer at the top. Column pressures used were considered as physiological low, medium and high pressures as described in Model 3a. The model was manipulated so that all air was removed from the system. The initial pressure was recorded; the digital force gauge with a cupped tip attached was then used to apply a force to the end of the penile model. The buckling force was recorded as the force applied by the gauge just before buckling occurred. Then 280 g of traction was applied to the circumference of the penile model and the resulting displacement recorded. Finally, after the two tests had been completed, the water was emptied from the column and the entire procedure was repeated. Ten tests per pressure per sleeve were made, resulting in 150 tests in all. The mean of both tests for each pressure and each size of sleeve is shown in Table 3. Figure 4A shows the results of the axial buckling test and Fig. 4B the results of the radial displacement test. The force and displacement are plotted against the internal pressure for each sleeve. A line of best fit is shown for each of the sets of data for each sleeve. The intersection of this line to each group of data under each pressure is termed the inflection point.

Table 3. 
The mean buckling force and displacement of each test
VariablePressure, mmHg
6080100120150
Mean force before buckling, kg
 Small model0.9631.0361.2131.4011.529
 Medium model0.9921.0741.3941.5921.773
 Large model0.9641.2751.6541.7372.029
Mean displacement with 280 g traction, cm
 Small model1.10.990.70.640.38
 Medium model1.421.240.90.630.39
 large model2.251.680.940.680.4
Figure 4.


(A) the axial test results; all values of each test are plotted and a line of best fit intersects each of these sets of data at their average point. (B) the radial test results; details as in A.

To test the hypothesis that there is a uniform relationship between radial stress and axial compression, the ratio between the force applied to the model and the pressure that correlates with this force was determined. The ratio measured is at the inflection point on the graph of displacement vs internal pressure, where no more change in the displacement can be measured, i.e. when there is zero displacement. For the 1.9 cm sleeve this point was found at an internal pressure of 17.33 kPa, equivalent to ≈1.4 kg of axial force. For the 2.54 cm sleeve this inflection point was at 18.67 kPa, ≈1.7 kg of axial force, and for the 3.81 cm sleeve it was at 20 kPa, or ≈2 kg. The respective ratios were 0.0108, 0.0121 and 0.013, which when expressed to two decimal places were ≈0.01, showing a constant relationship independent of diameter, as predicted by the theory. As the sleeve size increased the inflection points increased by 1.33 kPa (10 mmHg). Similarly, when assessing the force required for buckling as a function of pressure, each pressure increase of 1.33 kPa required an increase of force for buckling of 0.3 kg.

DISCUSSION

From the Model 1 MATLAB simulations, it is apparent that slight deviations from perfectly aligned axial loading can greatly influence the stresses felt on the walls of columns, whether they are isotropic beams or pressurized vessels. The FEA results show that the magnitude of stresses on pressure vessels under axial or radial loading is similar. The same is true for the radial and axial displacements calculated on each model.

To connect the results between the two physical Models 3a and 3b, an additional experiment was done. The mean axial force which produced failure for each sleeve diameter and initial pressure was applied radially and the circumferential displacement was noted. When assessing the present results vs. those of Udelson et al.[12] there are two different factors that must be considered when confirming that radial compression is a valid way of measuring erectile function. One factor discussed in the previous study [12] is that ‘Radial rigidity values vary with tunical surface wall tension forces; RigiScan radial deformation values reach a finite value and becomes insensitive with increasing intracavernosal pressures. In contrast, axial rigidity values are dependent upon erectile tissue mechanical properties and penile geometry as well as intracavernosal pressure. With increasing intracavernosal pressures axial rigidity values increase towards infinity and become more sensitive’[12]. As shown by the present results, the values for the displacement with 280 g traction applied decay to zero (the inflection point) and the value for the amount of force required to buckle the penis steadily increases. In the study of Udelson et al., this is referred to as a negative feature in diagnosing ED, when it is really a positive physiological feature provided by the body. When 280 g of traction is applied around the penis during tumescence and there is no displacement, it suggests that the individual has an ‘unbuckle-able’ penis and can generate penile rigidity adequate for vaginal intromission, whereas when there is displacement above a certain value, he cannot. The present study shows that as intracavernosal pressure increases the amount of circumferential displacement decreases. However, when testing axial buckling, the amount of force required to cause buckling will increase without bound as the internal pressure of the penis increases, causing extreme discomfort to the patient. That the radial compression decays to zero, and the amount of force required to buckle the penis increases without bound, gives a positive reason why radial testing provides a much more comfortable and practical way to ascertain erectile capability.

Udelson et al.[12] mention in the experimental trial that ‘a second observer was used to help confirm that no buckling force measurements were made at an angle to the erect penile shaft’. This is a feature of axial testing which can affect the results. If the test for axial compression is taken at any angle other then 0°, there will be a discrepancy in the results. Testing at any angle offset from the neutral axis of the shaft will require trigonometric conversions to determine the axial stress. When using the radial rigidity test there is no dependence on operator skill.

In conclusion, as the MATLAB models showed that slight variations in the way an axial load is applied can lead to dramatic differences in the stresses induced in the wall of a beam, whether the beam is modelled as an isotropic material or a pressurized vessel. Therefore, using axial loading methods to measure rigidity can easily be very inconsistent and highly operator-dependent to ensure that the load is always perfectly applied along the neutral axis. This variability is not present when the vessel is radially compressed.

The primary conclusion of the physical modelling is that as a pressurized vessel is loaded, in either an axial or radial direction, the internal pressure rises. Vessels of high internal pressure require more force to cause buckling than do vessels of lower internal pressure, whereas in an isotropic beam, the buckling force is determined solely by the mechanical properties of material. The size of a vessel also contributes to its ability to be buckled; smaller vessels buckle under lesser loads. As seen by comparing the axial buckling with the radial compression experiments, a smaller sleeve diameter has a smaller average change in pressure between initial and buckling pressures.

The physical modelling results can be compared with the results of previous human experiments [12,18]; these studies also show a change in internal pressure under axial loading in human subjects. The volume ratio vs change in pressure plots in these earlier studies can be compared to the physical modelling described here, to prove the assumption that the penis behaves like a thin-walled pressure vessel rather than an isotropic beam. This study confirms that radial rigidity measurements are physically equivalent to axial loading measurements without the added disadvantages of requiring an external assistant, variations in application direction and potential discomfort to the patient when the penis becomes unbuckle-able.

CONFLICT OF INTEREST

Gerald W. Timm is a Patent Inventor for Mentioned Product. Source of funding: Undergraduate Directed Study Course.

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