Background: An intraurethral pressure-time profile as urodynamic information was obtained in a non-invasive manner using an equivalent equation as a voiding model.
Methods: The reasonability of the voiding model was confirmed by applying it to an experimental flow curve likened to urinary flow. The flow curve was approximated and the pressure profile was estimated. From the uroflowmetric curves obtained in a normal subject and a patient with bladder outlet obstruction, the respective intraurethral pressure profiles were estimated.
Results: The pressure profile estimated from the approximated flow curve was found consistent with the profile of the difference between the pressure actually measured at two different portions in the experimental system.
Conclusion: Non-invasive estimation of intraurethral pressure profile from uroflowmetric curves may be very useful to grasp intraurethral urodynamic information in clinical practice.
Because of its simplicity and non-invasiveness, uroflowmetry (UFM) has been routinely used to obtain intraurethral information. The question is what to read from the uroflowmetric curve. Pressure flow (P/F) studies are common examinations, especially in patients suspected of having benign prostatic hyperplasia (BPH) or bladder outlet obstruction (BOO). Strictly speaking, the estimation of intraurethral events should be based on intraurethral information. For example, the calculation of intraurethral resistance should not be based on detrusor pressure, but on intraurethral pressure. However, it is difficult to clinically measure intraurethral total pressure. Thus, we applied an equivalent equation, which has been commonly used in the field of engineering, to obtain intraurethral urodynamic information. For this theoretical calculation, the whole urethra was supposed as one component and an equation approximating UFM curve was set.1 Then, based on this equation, the fluid pressure-time profile was estimated.
From the standpoint of dynamics, voiding phenomenon consists of intravesical pressure as a driving force and resistance system, including inertial resistance, frictional resistance and elastic resistance of the urethra. The changes in the interaction of these dynamic factors in a time course (during voiding) are expressed as a UFM curve. Based on the premise that the pressure against the inertial resistance is proportional to the change in urinary flow rate in time course (dQ(t)/dt), pressure against the friction resistance is proportional to urinary flow rate (Q(t)) and pressure against the elastic resistance is proportional to the voided volume at the moment (∫Qdt) the intraurethral pressure-time profile was estimated.
Materials and methods
First, an experimental system was prepared (Fig. 1). The total pressure of the fluid was measured at two points of this system: portion 1 compared to the internal urethral orifice and portion 2 was likened to the external urethral meatus. The total pressure at the portion 1 was changed by changing the height (H) of the water container in order to compare to the changes in intravesical pressure.
The pressures at the portion 1 and 2 and the water flow at the portion 2 were measured with Menuet Compact (Dantec Inc. Denmark). The water discharge from portion 2 was compared to urinary voiding, which is expressed as a time–profile of interaction between vesical pressure, as a driving force and as intraurethral resistance. The pressure-time profile P(t) was supposed as a cubic function of time (t) and expressed as follows with a premise mentioned before:
where L, R, C are constants.
By differentiating this infinitesimal equation, it is converted into the second-order differential equation in order to calculate Q and P in the following equations:
Where λ, A, B, α1, α2 are constants and t represents time
where and ψ = CL.
On the other hand, if the level of the portion 1 is equal to that of the portion 2, there is no energy loss between two portions and since the total pressure can be defined as the sum of static pressure and dynamic pressure, the following equation can be obtained from Bernoulli's theorem for the total pressure at portion 1 (P1) and at portion 2 (P2, Fig. 1):
where p1 and p2 are static pressures at portions 1 and 2, respectively, ρ is the density of fluid, v1 and v2 are the velocities at the portions 1 and 2, respectively, and (1/2ρv2) is the dynamic pressure.
Supposing that there is an energy loss between the two portions, there must be a loss of total pressure corresponding to this energy loss between the two portions that can be expressed as follows:
P1 − P2 = loss of total pressure (Ploss)(5)
The energy corresponding to this pressure loss is equal to the whole energy used to make the fluid stream from portion 1 to portion 2. That is, it is equal to the sum of energy spent against the inertial resistance of the fluid, energy spent for frictional resistance and energy used to dilate the urethra against the elastic resistance of the urethra. The pressure loss obtained from Equation 5 is equal to P(t) shown in Equation 1.
Therefore, the following equation can be set from Equations 1 and 5:
P1 − P2 = Ploss = P(t)(6)
Approximation of flow-time profile
One of the solutions to Equation 1 can be expressed as follows:1
To determine the five unknowns in Equation 2 (λ, φ, ψ, A and B), the least square method was used. Selecting five points which express the characteristics of a UFM curve and reading the coordinates of these points (tn, Qn; n = 1∼5) and substituting t = tn (n = 1∼5) in Equation 2, is a simulated value including the five unknowns (λ, φ, ψ, A and B). Consequently, the sum (S) of can be expressed by the following equation:
Then, the five unknowns (λ, φ, ψ, A and B) were determined so that S should be the least.
Estimation of intraurethral pressure
Supposing that the intraurethral pressure, P(t), can be expressed as a cubic equation, the Equation 3 can be given as:
From Equation 1 as a voiding model expressing the urodynamic composition in the urethra, Equation 2 which approximates UFM curve and Equation 3 which shows intraurethral pressure can be obtained. By approximating a UFM pattern using Equation 2, the coefficients are calculated and from the calculated coefficients, the coefficients in Equation 3 can be determined automatically.
One normal micturition case and 12 cases of six BPH subjects were also investigated. The normal case had no subjective voiding symptoms and the shape of uroflowmetric curve of the case met the criteria of normal uroflowmetrogram.3 Six BPH cases had symptoms of dysuria, demonstrated as abnormal uroflowmetrograms,3 subjective symptoms and a small residual urine volume. Uroflowmetrograms of six cases were taken both before and after transurethral resection of the prostate (TURP). Age-related differences were not investigated.
A flow curve obtained with the experimental system was taken as an example (Fig. 2). Five coordinates (t, Q) which characterized the flow profile well, were taken on this curve: (0.44, 7.3); (3.0, 2.0); (7.0, 4.3); (12, 11.7); and (16, 6.9). The coefficients were calculated by use of a self-made program in order to determine the pattern of Equation 2 passing all these coordinates.
α1 = −0.000929
α2 = −3.59
λ = 8.38
A = 1.53
B = 0.101
L = 0.690
R = 2.48
C = 435
and, from equation 3,
Figure 3 shows the curves of Q(t) and P(t). It was found that the pressure-time profile in Figure 3 was nearly equal to the profile of (P1 − P2) shown in Figure 2.
Second, UFM curves obtained in a normal subject and a patient suspected of having BOO are shown in Figures 4 and 5 with respective estimated pressure-time profiles.
We applied an equivalent equation,1 which is commonly used in the field of engineering, to non-invasively obtain intraurethral hydraulic information in form of an intraurethral pressure-time profile.
Yasuda et al. have reported a series of studies using a transducer catheter.2,4 Their method is characterized by the measurement of pressure at the maximum urinary flow rate (Qmax) at different levels of the urethra. Namely, their method measures ‘cross-sectional’ urinary parameters at Qmax or in the stable phase of urinary flow. Strictly speaking, their data are lateral or static pressures, but their method permits assuming at what level of the urethra a lesion lies, if any. On the other hand, the present estimation of an intraurethral pressure-time profile totally depends on the whole duration of voiding. The present method permits grasping the changes in intraurethral pressure during voiding as a ‘longitudinal’ profile of total pressure in the urethra.
Figure 6 shows a pressure profile obtained in a condition where resistance was increased by pressing the tube with a roller cramp in the experimental system shown in Figure 1. As the resistance increases, P1 increases and, in contrast, P2 decreases. Therefore, the difference (ΔP = Ploss = P(t)) between P1 and P2 increases and the flow rate decreases. This phenomenon may be called ‘high-pressure low-flow’ from the standpoint of the conventional P/F study and the resistance must be great.5 If both P1 and the flow rate are high, the residual energy must be great at the portion 2 because of high dynamic pressure (1/2ρv22), or ΔP must be small. This condition may be called ‘high-pressure high-flow’ from the standpoint of the conventional P/F study. However, from our point of view, it is a ‘low-pressure high-flow’ phenomenon and resistance must be small. The pressure that determines the urinary flow in the urethra is not P1, but the difference between P1 and P2 (ΔP). P1 may reflect the intravesical pressure, but it does not determine the urinary flow rate. That is, detrusor pressure or intravesical pressure is the driving force for voiding, but it is ΔP that directly makes urine run in the urethra.
Figure 7shows the profiles of the three pressure components: in Equation 1 which expresses a pressure against the inertial resistance (PL), RQ expressing a pressure against the frictional resistance (PR) and as a pressure against the elastic resistance (PC). As expressed by Equation 1, P(t) is expressed as the sum of these three components (PL + PR + PC) and it explains that the pressure is mostly used to drive the fluid at the initial phase of voiding and is spent against the frictional resistance at the mid-phase of voiding, the state of equilibrium, and at the late phase of voiding the remaining pressure is used against the elastic resistance. However, it is impossible to measure these component pressures separately from each other clinically or physically and, consequently, our voiding model (Eqn 1) is useful to simulate urodynamic phenomena in the urethra during voiding.
In the present study, the profile of intraurethral pressure could be estimated from a UFM curve as a model of voiding in a non-contact, non-invasive manner and this voiding model was concluded to express hydraulic phenomena in the urethra.
For example, once each pressure is calculated (Fig. 7), using the pressure and the flow, the energy can be calculated as consumed against the inertial resistance (WL) in the urethra. The average ratio of the energy for inertial resistance against the total energy (W) can also be calculated. This mean energy ratio (WL/W) became normalized to be 11.0% of post TURP operation from 3.6% of pre-TURP operation (Table 1). The normal range of this ratio is considered to be 4.5% or larger. This shows that more energy can be used to accelerate the urine in the urethra after the resection of prostate. The degree of the improvement of the voiding condition after the operation can be quantitatively expressed.
Table 1. Energy ratio of pre and post transurethral resection of the prostate (TURP) cases. There is a significant difference between pre and post TURP results (paired t-test, P = 0.014)
This simulation method is convenient as non-invasive clinical application for voiding phenomena.