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**ABSTRACT**

**INTRODUCTION:** The purpose of the study was to build and test a mathematical model that allows new information about efferent neural activity for the bladder and sphincter to be extracted from urodynamic recordings during the micturition cycle.

**METHODS:** The main features of the VBN micturition model were reviewed. An extended VBN model that includes a more detailed model of nervous control was then described. The mean **firing rate F(t)** of efferent neurons was linked first to an * effective calcium excitation* E(t) in the muscle cells and then to the detrusor and urethral pressures. Finally, this model was used to retrospectively analyze urodynamic recordings of 166 male and female patients with various voiding characteristics.

**RESULTS:** The main result was the striking simplicity of the computed **F(t)** curves for both the detrusor and sphincter during the 3 phases of the micturition cycle (storage-continence, micturition, and return to continence). This result occurred in most of the micturitions: (1) normal with or without break of detrusor excitation, (2) abnormal with large residual volume, (3) abnormal with effect of a urethral catheter in situ. These curves **F(t)** were found as a sequence of constant values during the onset of flow and during the plateau phase, and then as a decreasing exponential during the return to continence phase. The transition between the 2 phases was always brisk, suggesting ON/OFF switching of reflexes.

**CONCLUSION:** The extended VBN model for analyzing urodynamic recordings facilitates comparison and discussion of successive micturitions from a given patient and is a valuable complement to animal studies and to functional imaging of the human brain.

Françoise A Valentini,^{1,2} Leonor Mazières,^{2} Pierre P Nelson^{2}

^{1} Department of Physical Medicine and Rehabilitation, Charles Foix Hospital, Ivry-sur-Seine, France

^{2} Research Team #6 (Physiology and Pathophysiology of Motility in Humans), Pierre and Marie Curie University, Paris, France

**Submitted** April 15, 2010 - **Accepted for Publication** June 13, 2010

**KEYWORDS:** Mathematical modeling; Nervous control; Micturition; Urodynamics.

**CORRESPONDENCE:** Dr. FranÃ§oise A. Valentini, Department of Physical Medicine and Rehabilitation, HÃ´pital Charles Foix, 7, avenue de la RÃ©publique, 94200 â€“ Ivry-sur-Seine, France (; ).

**CITATION**: Urotoday Int J. 2010 Aug;3(4). doi:10.3834/uij.1944-5784.2010.08.10

**ABBREVIATIONS AND ACRONYMS**: BPE, benign prostatic enlargement; D, index of voiding dysfunction (VBN parameter); DO, detrusor overactivity; E(t), Â« Effective calcium excitation Â»; F, firing rate; FF, free uroflow; G, Griffithsâ€™ function; k, detrusor contractility (VBN parameter); LUT, LUTS, lower urinary tract, lower urinary tract symptom; NIDC, noninhibited detrusor contraction; N(t), Â« modulating factor Â» of pressure; p_{abd}, p_{abd.eff}, abdominal pressure, effective abdominal pressure; p_{contractile}, detrusor pressure due to contractile component; p_{det}, p_{det.Qmax}, detusor pressure, detrusor pressure at maximum flow rate; p_{elastic}, detrusor pressure due to elastic component; PF, pressure-flow; p_{rec}, rectal presure; p_{ves}, bladder pressure; pucp, prostatic urethra counter pressure (VBN parameter); PVR, postvoid residual volume; Q, Q_{max}, flow rate, maximum flow rate; T, time constant; V, V_{u}, V_{ini}, bladder volume, voided volume, initial bladder volume; VBN, Valentini, Besson, Nelson; W, ratio of active regulatory protein.

**INTRODUCTION**

Many lower urinary tract symptoms (LUTS) are due to a combination of mechanical and nervous causes, many of which are poorly understood. For instance, benign prostatic enlargement (BPE), a mechanical phenomenon, often generates the nervous phenomenon of urgency. Any method that would facilitate studying and quantifying both types of disorders *in vivo* would be helpful. The purpose of the present investigation was to determine the usefulness of a modeled analysis of urodynamic traces.

Urodynamic traces contain a wealth of information, but usually only a part of this information is extracted and analyzed. To overcome this deficiency, the mathematical micturition model VBN (abbreviated from the names of its authors: Valentini, Besson, and Nelson) and its associated software VBN (Laborie Med Tech, Inc; Mississauga, Canada) were developed [1]. The first version of the VBN model allowed the user to study the mechanical phenomena with reasonably reliable results [1,2]. However, when nervous control was introduced in an empirical way, use of the model was limited.

The present study is mainly devoted to building an extended VBN model. It includes a proposed model of the nervous excitation that is closely related to the sequence of phenomena beginning with the firing rate of the efferent neurons and ending with muscular contraction. The extended model was then tested on a large data bank of urodynamic recordings.

**METHODS**

**Leading Ideas of VBN Models**

The goal of VBN models is to link the observed effects of a lower urinary tract (LUT) dysfunction (eg, abnormal pressure-flow [PF] studies or free uroflow [FF] studies) to their causes (eg, low maximum flow rate due to urethral compression from an enlarged prostate) for each patient. Building of the initial model began with a mechanical description of each LUT component (bladder, urethra, and sphincter) and of uroflow (hydrodynamics) for a healthy (standard) participant [3]. Then, for each studied dysfunction, the nature of the physiological changes was considered. In BPE (which induces urethral compression) changes of detrusor contractility are well known, but changes of nervous control leading to urgency are poorly understood. Each change in LUT function is characterized by a specific parameter. Applying the model to the analysis of urodynamic recordings allows evaluation of the magnitude of change in the value of each parameter(s).

*Displaying the results*. The inputs are gender, voided volume (V_{u}), and postvoid residual (PVR) volume. Specific parameters have to be tried until a satisfactory evaluation is found. For each set of these parameters, the VBN software computes detrusor pressure (p_{det}) and flow rate (Q) vs time. Recorded and computed curves are compared on a screen. A sophisticated quadratic evaluation of the distance between these pairs of curves (called *error*) is computed (Appendix 1). Minimum error defines the *best fitting* of computed curves. For the present study, it was assumed that the identification of the parameters was reasonably reliable if the error was < 0.03. The software allows an automatic search for the set of parameters that minimizes the error.

**Initial VBN Model**

The initial VBN model consists of 5 submodels: hydrodynamics, bladder, urethra and sphincter, nervous control of the detrusor and sphincter.

The first 3 submodels have a robust experimental and theoretical basis (Appendix 2). In this initial model, an empirical nervous excitation has been introduced that modifies Griffithsâ€™ law for the contractile pressure of the detrusor. Griffithsâ€™ law [4] is a mathematical relation giving the detrusor contractile pressure as a function of actual bladder volume (V) and flow rate (Q) for a fully excited detrusor: p_{contractile} = G(V,Q). The isometric case obeys this law with Q = 0. The VBN parameter **k** (standard value = 1) characterizes the contractile state of the detrusor.

In a first step, the nervous excitation of the detrusor is described by an empirical *modulating factor* N(t) [range 0-1]; then: p_{contractile} = **k**.N(t).G(V,Q) and can be deduced from the recorded curves.

*Urethra* [1]. A description of the male or female standard urethra leads to a relation, giving the local fluid cross section as a function of the inside hydrodynamic pressure. Compressive obstruction is described by a pressure exerted on the outside of the urethra; gaping or constrictive obstruction is described by multiplying the standard cross section by a urethral parameter (equal to 1 for standard, > 1 for gaping, and < 1 for constriction). Nspg

*Sphincter nervous control*. A sphincter causes a compressive force that is exerted on the outside of urethra. The maximal sphincter pressure is assumed to be the maximal urethral closure pressure p_{closure}, which is known to be age-dependant [5]. A modulating force factor **N _{sph}(t)** for a sphincter allows it to describe all conditions: p

_{sph}=

**N**.p

_{sph}_{closure}. The standard sphincter excitation is

**N**= 1 during continence or 0 during voiding.

_{sph}When applied to the analysis of uroflow studies from healthy volunteers, this initial model allows restoration of the recorded curves; it also allows verification that the influence of voided volume is adequately taken into account [6]. Figure 1a demonstrates this action in successive voidings of a healthy male volunteer.

Some results have been obtained: (1) correlation of the VBN parameter pucp (prostatic urethral counter-pressure) with the Abrams-Griffiths number, obstructive coefficient (OCO) and urethral resistance factor (URA) [2]; (2) creation of the VBN parameter k for detrusor contractility with both a modified projected isometric value (PIP) and the Watt factor [2]; (3) definition of an index of voiding dysfunction (D) for patients with BPE [7]; (4) demonstration of the effect of a urethral catheter on voiding [8]; (5) assessment of an alpha-blocker treatment for BPE and of surgery for incontinence with suburethral tapes [9].

The empirical definition of the detrusor excitation implied a slowly increasing function toward an asymptotic value that can be fitted, without any consequence, by an exponential or a hyperbola. Despite that rough definition, the function was found identical for male and female participants. Another result was that fading of detrusor excitation implied a second function deduced from the basic one by an affinity.

**An Extended VBN Model**

*Need for an Extended Model*

With the empirical model of excitation, it was impossible to accurately restore the last phase of the micturition process (ie, return to continence). Additionally, if the properties of nervous control were observed, they were not explained. The present authors assumed that a knowledge model of excitation would bring some answers. The first step was the choice of a well-defined and measurable variable to quantify nervous excitation. The firing rate F(t) of efferent nerves (or, more conveniently, the normalized ratio F/F_{max}) was chosen. The second step was to link F(t) to the detrusor pressure starting from N(t) and E(t).

*Skeletal Muscles, Guts, and the Detrusor*

The detrusor is a smooth muscle, like those in the gut. It does not own the same regulatory proteins and myosin light chain as skeletal muscle [10]. In the gut, contraction is mainly due to self-generated excitations. Spontaneous contractile activity (ie, localized contractions) is observed in the bladder wall [11,12] at rest or during filling, but the muscle is unable to induce the synchronous activity that leads to voiding [13] and no tetanic contraction can be observed. In the human, the vicinity of smooth muscle cells [14] and nervous fiber and the ratio of axons to detrusor cells [15] could explain a possible synchronous activation of the detrusor by direct nerve stimulation of each cell [16]. Then, it seems reasonable to consider that the detrusor contraction mainly results from simultaneous and direct stimulation, similar to the activation of skeletal muscle. Skeletal muscle contraction is governed by the following sequence of phenomena: (1) firing of the motor neurons, (2) release (and reuptake) of calcium ions from the sarcoplasmic reticulum, and (3) activation of the regulatory proteins that activate the sliding filaments. Each of these phenomena has been intensively studied. Using experimental data from Miledi et al [17], Huxley [18] and Hill [19], the whole chain has been modeled [20]. In their model, Valentini and Nelson [20] called *effective calcium excitation* **E** the concentration [Ca^{2+}] in the sarcoplasm, multiplied by the chemical equilibrium constant K_{2} between calcium and troponin [21]. In the initial model, that definition of E was used because the same behavior for detrusor and skeletal muscle was assumed. Equations (Eq2) and (Eq3) in Appendix 3 describe the relationship between the *modulating factor* N(t) and E(t).

*New Model*

It has been shown that the sequence of events leading to the contraction of skeletal and smooth muscles is qualitatively the same [22]. Therefore, the 2 models must have the same number of equations. However, because the regulatory proteins are different, equations are expected to be different. Specific biochemical studies on the detrusor have been published [23,24]. Unfortunately, the results are too specific and fragmentary to be used. The solution comes from the following observation: the curve N(E) plotted by combining the 2 nonlinear equations (Eq2) and (Eq3) is very close to a straight line. That property expresses an optimal fitting between the contractile proteins in skeletal muscle. An analogy exists for driving a vehicle, where the angular position of the wheels must be proportional to the steering wheel angle; a delay or too much amplification of the control would be damaging. Assuming a similar optimization for detrusor contraction, the equations in Appendix 3 remain usable; the detrusor behaves like a skeletal muscle but with a longer delay of response. The same equations apply to the sphincter, but with other values for the parameters.

**The Database**

This study was conducted in accordance with the declaration of Helsinki. According to the local practice of the ethics committee at the authors' institution, no formal institutional review board approval is required for retrospective studies.

A large data bank of urodynamic tracings from FF and PF studies of patients referred for urodynamics between 2003 and 2008 was reviewed. Urodynamics were performed using either Aquarius or Dorado (Laborie Med Tech, Inc; Mississauga, Canada) or Menuet (Dantec Medical Ltd, Royal Portbury, UK) units. For both FF and PF recordings, men voided in a standing position and women voided from a comfortable seated position. Cystometry was performed with a double-lumen catheter 6F or 7F in men or a 3-lumen catheter 7F or 10F for women, which also allowed for urethral pressure recording. Rectal pressure (p_{rec}) was recorded using a punctured intrarectal balloon catheter that was filled with 2 mL of saline. The transducers were placed at the level of the upper edge of the symphisis pubis and the pressures were zeroed to atmosphere. For urethral pressure recording, the catheter eye was positioned at the level of the maximum urethral closure pressure (located from a urethral profilometry with the bladder empty).

Participant exclusion criteria were: (1) all neurological diseases, (2) diabetes mellitus, or (3) pelvic organ prolapse ≥ grade 2. These criteria allowed the authors to focus on male and female patients with no sign of impaired nervous control of the bladder. All participants had a continuous flow of more than 100 mL of voided volume.

Urodynamic files of 63 men and 130 women were screened; 166 participants (56 men and 110 women) met the criteria for inclusion. Thus, a total of 218 PF studies and 248 FF studies with a large variety of Q_{max}, p_{det.Qmax}, V_{ini}, and shape of the tracings could be analyzed. Table 1 contains the number of PF and FF studies available for the participants.

The mean age of the men was 66.3 years (range, 45-86 years); the mean age of the women was 54.5 years (range, 24-86 years). The primary diagnosis for the men was LUTS due to BPE. The primary diagnosis for the women was incontinence (stress 30.6%, urge 29.7%, mixed 39.6%); 26% of the women had undergone previous pelvic surgery for incontinence or pelvic organ prolapse.

**RESULTS**

At this stage of the study, the authors had invented a tool that enabled them to deduce N(t) from the recorded curves, E(t) from N(t) using equations (Eq3) and then (Eq2), and then F(t) using (Eq1). At the initiation of micturition, E(t) had an exponential shape; during voiding, the E(t) curve was equal to or lower than the initial curve (never higher). These findings agreed with F(t)=F_{max} at onset; that property allowed the authors to deduce the time constant T. However, they did not know if this function was useful or meaningful.

**General Results**

The authors proceeded to the VBN analysis of all tracings of the participant files. The main result was the striking simplicity of the computed F(t) curves for the detrusor and sphincter during the 3 phases: (1) storage-continence, (2) micturition, and (3) return to continence. The result is demonstrated for all these files in Figure 1a; Figure 1b; Figure 2; Figure 3; Figure 4; Figure 5.

In the sequel, * voiding* will refer to the phase of actual flow (Q > 0), and

*will refer to the phase starting with the central nervous system order to*

**micturition***induce*voiding (whether resulting in voiding or not). The authors called t

_{0}the time of the onset of flow.

*Identification of the parameters*. From a preliminary study on a sample of 50 PF studies devoted to excitation at the onset of voiding, the parameters T and E_{max} in the equation (Eq1) were identified for both the detrusor and sphincter: T_{det} = 6 ± 1 s, T_{sph} = 3 ± 2 s, E_{max} = 5 ± 0.2 for both. The mean values T_{det} = 6 s, T_{sph} = 3 s (much greater than the values for skeletal muscles) and E_{max} = 5 were used for all of the computations.

**Detailed Results**

*Normal Voiding*

*Normal storage-continence*: Q = 0; p_{det} = p_{elastic}; p_{sph} = p_{closure}; F_{det} = 0; F_{sph} = F_{max}; *transition from storage-continence to micturition*. The PF and FF studies of 36 men and 60 women were available for the analysis of normal voidings Table 2. Change of F_{det} was simultaneous with the beginning of p_{det} increase. The onset of Q appeared later. Brisk change of F_{sph} from F_{max} to 0 was not exactly synchronous with the brisk increase of F_{det}: usually, the decrease of p_{sph} began before the increase in F_{det} (range, -5 s to -1 s). However, in 6 of the 96 files, the onset of the decrease in sphincter excitation was delayed up to 10 s.

*Micturition*

The general course of the *firing rate* F_{det}(t) fell into one of 2 types:

*Type 1: The simplest type*. This type was demonstrated by 14 men and 23 women. F_{det} = F_{max} until the onset of the return to continence process (Figure 1a; Figure 1b). For this type, if 1 voiding was normal, all of the voidings of the same patient were normal.

*Type 2: Early break in detrusor excitation*. This type was demonstrated by 22 men and 27 women. Micturition began as in the simplest type, by F_{det} = F_{max}. Then, at a critical time, t_{break}, F_{det} took a new value F_{1}< F_{max}. This was kept until the onset of the return to the continence process (Figure 2), with F_{1} values in the range [0.3*F_{max} - 0.9*F_{max}]. The F_{1} value was the same for all voidings of a given patient. The break occurred 3.0 ± 2.0 s after the onset of flow in 57 out of 79 PF studies (72%) and 33 out of 89 FF studies (37%). In the remaining cases, the break occurred approximately 10 s later, when the flow rate approached its maximum value.

*Return to Continence*

This active phenomenon allowed the return to continence in the absence of significant PVR volume (PVR/V_{ini} < 20%).

*Time of the end of micturition phase*. Changes of F_{det} and F_{sph} were simultaneous. They occurred when the decreasing bladder volume reached a critical volume V_{end} = 30 Â± 20 mL.

*Mechanism of return to continence*. When V < V_{end}, the flow abruptly dropped to 0 while p_{det} was slowly decreasing; F_{sph} = F_{max} and p_{sph} increased toward p_{closure}, while F_{det} decreased as an exponential with time constant T = 5 s (Figure 1a; Figure 1b; Figure 2; Figure 3; Figure 4).

**Abnormalities During Voiding**

*Effect of a Urethral Catheter in Situ*

The PF and FF studies of 10 men and 26 women were available for analysis of the effect of a urethral catheter in situ (Table 2; Figure 3). Q_{max} during the PF studies was dramatically smaller than Q_{max} during the FF studies, even though they were obtained on the same day and the patients had similar initial bladder volumes. VBN analysis showed that the sphincter, which relaxed normally (F_{sph} = 0) during the FF studies, kept its continence value F_{sph} = F_{max} during some of the PF studies.

*Voidings With* PVR/V_{ini} > 20% *for at Least 1 Voiding*

The PF and FF studies of 10 men and 24 women were available for analysis of at least 1 voiding with large PVR volume (Table 2; Figure 4). Break of detrusor excitation occurred as in normal voidings. However, although computations predicted large flow times for normal voidings, return to continence began 60 Â± 20 s after the onset of flow, with V >> V_{end}.

*Abnormal Continence: Preliminary Study of Phasic Detrusor Overactivity *

The PF and FF studies of 4 men and 6 women were available for analysis of phasic detrusor overactivity (DO) and return to continence (Table 2; Figure 5). Phasic DO is the occurrence of noninhibited detrusor contraction (NIDC) during bladder filling before the voluntary decision to void. Although it decreased during each NIDC, F_{sph} kept a large value; hence, Q = 0. The ascending limb of the p_{det} during each NIDC was fitted by F_{det} = F_{max}, and the decreasing limb was fitted by F_{det} = 0. After NIDCs, micturition phases were of the simplest type described above (F_{det} = 1, F_{sph} = 0).

**DISCUSSION**

*Why is a Model Usable?*

Any model is a simplification; therefore, no one can prove that it is exact. A micturition model is usable if: (1) most of the recorded curves are well fitted by the computed curves; (2) several successive voidings of a patient lead to the same values of the standard or specific model parameters. Then, the discrepancies observed in some cases have to be imputed to some secondary phenomena (eg, nonstandard sphincter behavior). Modeled analysis allows the user to separate secondary phenomena from the other concomitant phenomena.

The extended VBN model allows detailed analysis of a large number of urodynamic tracings with simple and coherent results. This is an *a posteriori* justification of the present model. Two questions need to be answered: (1) why is the model working, and (2) what is the physiological meaning of the surprisingly simple computed behavior of F(t).

*Why is the Model Working?*

The main weakness of the model is the lack of a direct biomolecular justification of the equations. As explained in the Methods section, this difficulty was circumvented by using the analogy between the contractile mechanism of skeletal muscle and detrusor muscle; ie, linearity of the relationship N(E).

*What is the Physiological Meaning of the F(t) Behavior?*

The most striking result is that F(t)/F_{max} is a sequence of constant values except during return to continence, where F(t)/F_{max} is a decreasing exponential. Finding a sequence of constant values for F(t)/F_{max} appears to be such a simple and general result that it must be meaningful.

The time constants T_{det} and T_{sph} in (Eq1), which give E(t), were the same for every participant. This is not surprising, because the same property was observed in skeletal muscles by Hill [19]. Different muscles have different time constants, but a given muscle has the same time constant for everyone.

In 1977, Malony et al [25] reported a total of 12 reflexes for the release of urine or urine storage. Each year, attempts are made to independently study each reflex in animals or humans. However, a global description of the nervous control must encompass all of these partial descriptions. In humans, the study of all reflexes involved in nervous control of micturition is not possible. Van Duin et al [26] tried to simulate the effect of various reflexes on lower urinary tract function without obtaining satisfactory responses to disturbance. Today, most studies are related to the identification and mechanism of action of the transmitters. Analysis of the F(t) curves could help to detect the occurrence of some reflexes and to provide information about their connection. The main strength of the model is to enlighten some phenomena such as fading of detrusor excitation during voiding; each curve is the representation of many phenomena. Each difference from the standard curve must have a physiological explanation. Thus, a break (fading) of the detrusor excitation could result from the following sequence of events: at the onset of micturition, a saturating order from the pontine micturition center leads to F = F_{max}, while the F value after the break of excitation would be due to the bladder-to-bladder reflex [27]. In the same way, a urethral catheter in situ could modify the sensory level by an additional urethral reflex; thus, the time of break would not be the same with or without a urethral catheter.

A question is: what is the variability of the characteristics of the F(t) curves? When a break of detrusor excitation is observed for successive voidings of a given patient, the value of F/F_{max} after the break is always the same, but this value depends on the patient. These findings seem to open the way for further specific studies. Already, the present authors have analyzed voidings with large PVR [8]; occurrence of large PVR can be related to recurrent inhibition [28] or to fatigue. VBN analysis shows that the onset of the return to continence in normal voidings depends on bladder volume (and so is governed by a bladder sensor), while it depends on flow duration (probably measured by a urethral sensor) for large PVR. A hypothesis is that afferent input from the urethra may facilitate the onset of bladder emptying (urethral afferent mediated reflex) while, in case of large flow-time, an inhibitory reflex would occur. A preliminary study of detrusor overactivity is given in this paper; characterization of both phasic and terminal detrusor overactivity is in process. Other studies are needed to determine the impact on micturition of diseases such as diabetes mellitus or multiple sclerosis, and to examine the effects of surgery for incontinence.

**CONCLUSIONS**

Modeled analysis of urodynamic PF and FF recordings with the extended VBN model gives some insight into the signal processing that leads to the excitation of the bladder and urethra. This method is a complement to animal studies and, although it is not concerned with brain localizations, to the functional imaging of the brain. This method could be the basis for a modeled analysis of neurological LUT dysfunction in humans.

**Conflict of Interest**: none declared.

**APPENDICES**

**Appendix 1. Definition of Error**

Let DQ be DQ =2 (Q_{measured} - Q_{computed}) / (Q_{measured} + Q_{computed}), DP a similar expression for the detrusor pressure, â€œAâ€ a constant, and f(t) a decreasing function of time such as:

The definition of error is:

The authors choose the constant A = 2/3 and a time constant = 40 s for f (t), thus giving more weight to Q than to p_{det} and to the beginning than to the end of voiding.

**Appendix 2. Mechanical Part of the VBN Models**

*Hydrodynamics*

Hydrodynamics obey the classical equations (ie, mass and impulse conservation). For flow in a distensible pipe, the solution was given by E. Mach (circa 1877) who introduced the *subsonic* and *supersonic* part of flow.

*Bladder*

The smooth muscle is composed of elastic and contractile components. The detrusor pressure p_{det} is: p_{det} = p_{elastic} + p_{contractile}

Elastic components are comprised of elastic and viscoelastic (delayed elasticity) elements. Experiments on pig bladder strips [29] and the theory of polymer elasticity [30] allow predicting their respective behavior. The time constant explaining the difference between slow filling and fast filling is 300 s.

The behavior of contractile components has been widely studied by Derek Griffiths. Griffithsâ€™ law [4] is a mathematical relation giving the detrusor contractile pressure as a function of the actual bladder volume V and the flow rate Q for a fully excited detrusor, in isovolumetric conditions, Q = 0: p_{contractile} = G(V,Q)

Generalization of Griffithsâ€™ law needs to introduce in its right part 2 terms: one is a *contractility factor* k, which characterizes the detrusor contractility, and the other is a *modulating factor* N(t), which characterizes the nervous excitation. Then: p_{contractile} = **k**.N(t).G(V,Q)

*Abdominal Pressure *

Abdominal pressure is from the Enhorning hypothesis [31] (p_{det} = p_{ves} â€“ p_{rec}) with (p_{rec} = p_{abd}). In some cases, a part p_{abd.eff} of the abdominal pressure is transmitted to the bladder, but not to the urethra. Then: p_{abd.eff} + p_{det} = p_{ves} â€“ p_{rec}

Effective abdominal pressure is introduced in an empirical way, but it is not predicted by the model.

*Urethra*

Urethra [1] : A description of a male or female standard urethra leads to a relation, giving the local fluid cross section as a function of the inside hydrodynamic pressure; p_{sph} = p_{urethral} when flow rate Q = 0 and is weaker than the local p_{urethral} when Q > 0.

**Appendix 3. Detrusor Pressure Versus Firing Rate Model**

The * nerve activity* (range: [0-1]) is F/F

_{max}where F is the mean firing rate of the efferent neurons. After each spike, calcium ions Ca

^{2+}enter into the cell sarcoplasmic reticulum where they induce muscular contraction; relaxation implies reuptake of Ca

^{2+}by the longitudinal tubules. The authors called

*E the concentration [Ca*

**effective calcium excitation**^{2+}] in the sarcoplasm multiplied by the constant K

_{2}of the calcium troponin equilibrium (K

_{2}= 3.2 *10

^{6}) [7,21]. Equation (Eq1) describes the delayed link between F/F

_{max}(t) and E(t), where the constants E

_{max}and T have to be identified for detrusor and for sphincter.

Eq1

Remark: if F/F_{max} jumps from 0 to 1, E increases exponentially with the time constant T (Eq1); if F/F_{max} increases exponentially with a time constant T_{F}, E increases exponentially with the time constant (T+T_{F}).

In skeletal muscle, Ca^{2+} ions bind with troponin C, a regulatory protein owning 6 cationic sites in chemical equilibrium with Ca^{2+} and/or Mg^{2+}. The constant K_{2} governs two binding sites with only Ca^{2+}. The ratio **W(t)** of fully bound troponin proteins is easily evaluated from chemical equilibrium balance.

Eq2

A regulatory protein can fasten to a free myosin head only when it is fully bound. From the sliding filaments theory with parameters fitted to obtain Hillâ€™s law, the authors express the ** modulating factor N(t)** (range: [0, 1]) as the force correction due to the actual value of

**W(t)**(Eq3).

Eq3

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