Agerelated dynamics of elasticity of deep dorsal vein of human penis according to results of direct measurements

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 The venous system of penis is assigned an important role in the mechanisms of development of and supporting erection [1]. The diagnostics and surgical correction of pathological venous outflow is not always successful [2]. The rate of erectile disorders is associated with different factors, but it progressively grows with age. Changes in the venous vessels of different localization in norm and in pathology were studied in several works [3, 4]. However, among numerous research works in the field of phlebology there are no publications concerning direct measurements of the elasticity of veins with quantitative evaluation of this parameter [5].

In works [6, 7] the authors propose a physical model of a biological tissue and a method of its mathematical description permitting to determine reduction in the elasticity of the fibrous tunic and cavernous arteries of penis with age. The aim of our research is experimental study of age-related dynamics of elasticity of the major venous vessel of penis – the deep dorsal vein (DDV).

Materials and Methods

Research was conducted on samples of DDV received in autopsy of 30 male individuals who suddenly died from traumas or acute diseases at the age of 18 to 83 years. The DDV of penis was isolated by acute method without surrounding tissues from a standard incision used in autopsy. A fragment of vein 2.5-3.5 cm in length was isolated distally to the retaining ligament served as a landmark. The obtained material was transported with strict observance of the temperature and humidity requirements. The period from the moment of death to the study did not exceed 18 hours.

In the experiment the samples of veins were subject to discretely increasing stretching force F in the longitudinal direction with record of the corresponding absolute increments of the length Δl on a specially designed setup [6, 7].

Subsequent mathematical processing of experimental results consisted in transformation of the mass of discrete data into analytical dependences of the kind:

$\frac{\Delta \iota }{\iota }=f\left(\frac{F}{S}\right)$. (1)

where l and S – initial length and cross-sectional area of the samples.

Transformation was carried out using the general approximating function [6]:

 $\frac{\Delta \iota }{\iota }=[\frac{1}{a}+\frac{1}{b+c{\displaystyle \frac{F}{S}}}]\times \frac{F}{S}$. (2)

where a, b and c – constant coefficients characterizing the properties of a specific sample. Coefficients a and b have dimension of Young’s module (N/m²), coefficient c is a dimensionless value.

According to the definition of elasticity α, its numeric value equals the slope of the curve to dependence (2) and, consequently, is determined by the expression:

 $\alpha =\underset{\Delta (F/S)\to 0}{lim}\frac{\Delta (\Delta {\rm I}/{\rm I})}{\Delta (F/S)}=\frac{d(\Delta {\rm I}/{\rm I})}{d(F/S)}$. (3)

 Thus, after differentiation of the Equation (2), we receive:

$\alpha =\frac{1}{a}+\frac{b}{(b+c{\displaystyle \frac{F}{S}}{)}^2}$. (4)

As it is seen, the value of elasticity is not a constant since it considerably depends on force load F/S. Therefore, it is possible to speak only about its initial value α₀ as some limit to which α tends at F →0.

Results and Discussion

Processing of experimental data was performed using the extension package Curve Fitting Toolbox of the computational environment Matlab, and consisted in determination of coefficients a, b and c in equation (2). The results of approximation, with indication of the age are given in Table 1. The values of reliability of approximation R–square given in the Table, evidence practically ideal coincidence of experimental results with their presentation in the analytical form using approximating function (2). This also confirms the correctness of the ratio following from it (4) that considerably extends possibilities of the analysis of the experimental results.

Thus, use of coefficients a and b permits to calculate the initial (maximal) value of the elasticity of the samples α₀ and to determine its dependence on age. The results of calculations by formula (4) are given in Figure 1.

No less informative is the parameter of dependence of the average value of elasticity ᾱ₀ on the force load F/S (Fig. 2) calculated by formula (4) on the basis of the average values of coefficients  ,  and  :

 $\overline{a}=\sum _{i=1}^N\frac{a_i}{N}=8,150\times {10}^7 N/m^2$;

$\overline{b}=\sum _{i=1}^N\frac{b_i}{N}=2,413\times {10}^7 N/m^2$;

$\overline{с}=\sum _{i=1}^N\frac{c_i}{N}=12,691$,

where N=30 – the number of studied samples.

 

Table 1. Results of Mathematical Processing of Experimental Data

№ of sample	Age, years	Coefficients			R-square	№ of sample	Age, years	Coefficient			R-square
		a, N/m2	b, N/m2	c				a, N/m2	b, N/m2	c
1	44	7,942∙107	2,373∙107	10,891	0,9999	16	48	8,132∙107	2,308∙107	11,274	0,9998
2	50	8,113∙107	2,391∙107	12,063	0,9999	17	52	7,655∙107	2,380∙107	11,846	0,9997
3	48	7,845∙107	2,550∙107	11,812	0,9999	18	80	8,968∙107	2,568∙107	15,074	0,9998
4	79	8,711∙107	2,672∙107	15,121	0,9996	19	76	8,891∙107	2,581∙107	14,353	0,9997
5	43	7,824∙107	2,252∙107	10,731	0,9998	20	43	7,673∙107	2,312∙107	10,986	0,9993
6	53	8,216∙107	2,442∙107	12,345	0,9992	21	39	7,424∙107	2,278∙107	10,763	0,9994
7	81	9,053∙107	2,680∙107	15,447	1,0000	22	41	7,846∙107	2,231∙107	10,954	0,9998
8	81	8,600∙107	2,420∙107	14,981	0,9998	23	45	7,653∙107	2,274∙107	11,428	0,9999
9	66	8,350∙107	2,280∙107	13,322	0,9995	24	65	8,521∙107	2,455∙107	13,427	0,9999
10	75	8,841∙107	2,615∙107	14,915	0,9998	25	49	7,623∙107	2,269∙107	11,711	0,9997
11	83	8,632∙107	2,640∙107	15,552	0,9998	26	77	8,210∙107	2,450∙107	14,583	0,9998
12	36	7,621∙107	2,278∙107	10,514	0,9998	27	76	8,411∙107	2,548∙107	15,017	0,9999
13	56	7,830∙107	2,310∙107	12,315	0,9985	28	76	8,773∙107	2,663∙107	14,871	0,9998
14	61	8,415∙107	2,420∙107	12,942	0,9997	29	26	7,156∙107	2,135∙107	9,377	0,9987
15	67	8,621∙107	2,519∙107	13,907	0,9984	30	18	6,952∙107	2,097∙107	8,195	0,9994

 

Fig. 1. Dependence of initial elasticity of DDV on age

 

 As it follows from the given data, the initial (maximal) value of the average elasticity ᾱ₀ at F →0 is of the order of 5.4∙10^-8 m²/N.

With increase in the force load the ave-rage value of the elasticity rapidly declines and asymptotically approaches the stabilized parameter of the order of 1.4∙10^-8 m²/N. Here, the most considerable change is seen with force loads less than 3∙10⁶ N/m².

 

Fig. 2. Dependence of the average value of elasticity of DDV on force load

 

 It should be noticed that the results of the calculations well agree with the results of the direct experiments. In particular, they demonstrate a rather rapid transition of relative increments of length Δl/l at low force loads and their subsequent decline with a smooth transition to practically linear dependence on the mechanical tension F/S.

Veins are referred to capacitance vessels that accommodate the most part of circulating blood and provide its return to the heart. This explains a high elasticity of a venous vessel at the beginning of its filling and smooth reduction in the elasticity with increase in the load. Nevertheless, with high loads, a vein can withstand high pressure in it [3, 4]. The data obtained by us also confirm this regularity.

Mathematical analysis of the results of direct measurements of the elastic properties of DDV revealed a considerable – about 20% – reduction in the elasticity of vein in the studied age range 18-83 years from α₀ = 6,2·10^-8 m²/N to α₀ = 5,0·10^-8 m²/N. With increase in the functional load on the vein the elasticity rapidly decreases at the initial stage and then practically does not change having achieved a certain level of saturation. With this, the age-related tendency to reduction in the elasticity with different extent of functional load is preserved.

The revealed regularities of reduction in the elasticity reflect changes in the venous walls with age that may play a certain role in age-related increase in erectile dysfunction. The obtained data enrich our knowledge about peculiarities of elastic properties of veins, and also about age-related changes in their functional properties.

The used method of quantitative evaluation of the elasticity of vein can be used in study of other problems of scientific and clinical medicine.

Conclusion

1. A direct measurement of the elasticity of the deep dorsal vein of human penis (elasticity with low force loads) revealed a decline of the initial elasticity of the vein in the age range of 18-83 years from α₀ = 6,2·10^-8 m²/N to α₀ = 5,0·10^-8 m²/N. With increase in the functional load on the vein elasticity rapidly decreases in the initial stage, then it reaches a certain level and after that practically does not change.
2. The proposed method can be used for determination of elasticity of vessels of other localization, and of other biological tissues in norm and pathology.

About the authors

Alexey N. Strelkov

Ryazan Regional Clinical Hospital

Author for correspondence.
Email: anstrel11@yandex.ru
ORCID iD: 0000-0003-1761-0529
SPIN-code: 3944-3061

MD, PhD, Urologist

Russian Federation, Ryazan

Alexandr I. Ulitenko

Ryazan State Radioengineering University

Email: anstrel11@yandex.ru
ORCID iD: 0000-0002-9334-7489
SPIN-code: 3004-9258

Grand PhD in Engineering sciences, Professor of the Industrial Electronics Department

Russian Federation, 59/1, Gagarina street, Ryazan, 390005

References

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