Modeling and Simulation of Bladder Artificial Control - IEEE Xplore

Proceedings of the 2010 3rd IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics, The University of Tokyo, Tokyo, Japan, September 26-29, 2010

Modeling and simulation of bladder artificial control J´er´emy Laforˆet, Christine Azevedo-Coste, David Andreu, David Guiraud, IEEE EMBS member

Abstract— This paper presents a bladder model including detrusor and sphincter dynamics. The model focuses on artificially controlled bladder contractions under Electrical Stimulation. We developed a smooth muscle model linked to a geometrical description of the bladder. In order to illustrate the model performances, we simulate a well known example: the behavior of the bladder under electrical stimulation using a Brindley/Finetch implant. This approach allows us to compare our qualitative results with experimental data available in the literature. Simulated outputs (pressure, volume and urine flux) show good consistency both in shape and time course. Model sensitivity to parameter errors is evaluated. We also show how duty cycle of intermittent stimulation influences the efficiency of the bladder voiding and how simulation can help to select a stimulation pattern in order to optimize voiding while maintaining a low pressure and minimizing contraction duration.

I. INTRODUCTION The bladder is the organ that collects urine excreted by the kidneys prior to disposal by micturition. Urine enters the bladder via the ureters and exits via the urethra (Figure 1). The bladder system is composed of two muscle types: 1) the detrusor muscle which is a layer of the bladder wall made of smooth muscle fibers and 2) the sphincter muscles (smooth and striated ones). When the bladder is stretched, it signals the parasympathetic nervous system so that the detrusor muscle may contract in order to expel urine from the bladder through the urethra. For the urine to exit the bladder, both the autonomically controlled internal sphincter (smooth muscle) and the voluntarily controlled external sphincter (striated muscle) must be opened. In normal conditions, the bladder is supposed to empty by a synergistic contraction of the detrusor and relaxation of both sphincters during micturition. By contrast, supra-sacral spinal cord injured (SCI) patients may loss this synergy, as well as detrusor contraction efficiency. Bladder modeling is most often considered under two aspects: 1) in terms of the biochemistry involved in smooth muscle contraction [1], [2], [3] or 2) in terms of the neural mechanisms of urination [4], [5], [6]. Our approach is different: it offers a compromise between including detailed activation as well as contraction mechanisms and keeping the model complexity relatively low. Furthermore, it focuses on bladder contractions elicited by Electrical Stimulation (ES). J. Laforˆet is with DEMAR Universite Montpellier 2/LIRMM, Montpellier, France [email protected] D. Guiraud is with DEMAR INRIA/LIRMM, Montpellier, France

[email protected] C. Azevedo-Coste is with DEMAR INRIA/LIRMM, Montpellier, France

[email protected] D. Andreu is with DEMAR Universite Montpellier 2/LIRMM, Montpellier, France [email protected]

978-1-4244-7709-8/10/$26.00 ©2010 IEEE

Fig. 1.

Lower urinary tract description.

ES has been used for approximately 30 years to restore bladder function especially in SCI individuals [7]. Classically, stimulation is applied to sacral roots to induce detrusor contraction. One main drawback of this approach is that both the detrusor and the striated sphincter are stimulated at the same time. Sphincters’ contraction mechanically occludes the urethra. Nevertheless, voiding is possible due to the fact that detrusor has very slow dynamics compared to striated sphincter. Thus intermittent contraction is performed to allow micturition. It is important to note that, if ES is not correctly managed, intravesical pressure may increase and provoke urine feedback in the kidneys [8] while ES is ON. Individual modeling and estimation of the parameters for each patient may help to tune accurately the stimulators to limit such problems. To do so, we developed a smooth muscle model linked to a geometrical description of the bladder. We used this model to simulate different tuning cases of the stimulator parameters that can influence the internal pressure and the energy needed to void the bladder. A simplified sphincter model including fluids dynamics accounts for urine outflow. In the following, the next section describes the model, the third section presents simulation results, and finally, the fourth section discusses the outcomes of this study. II. ES CONTROLLED BLADDER MODELLING The model has a chain structure: the output of one stage is the input of the next stage. The input of the model is the ES signal vi and the output is the pressure Pves and volume of the bladder Vint (Figure 2). In the following we describe the main equations of the model, for more details see [9]. In order to study micturition function, this model, initially developed for isometric conditions, was extended to the following new features:

265

TABLE I M AIN VARIABLES OF THE BLADDER MODEL .

Fig. 2.

•

•

•

Variable vi [Ca2+ ]i k1 [M ] [M ∗ ] [AM ∗ ] [AM ] kc σc εp εc Pves Vint

Simplified block diagram of the model.

A more complex mechanical model is necessary to take into consideration the bladder geometric changes and their impact on contraction dynamics in a non isometric case. All the lengths are related to the macroscopic lengths of a Hill-Maxwell structure. The stress and the active stiffness are also related to the macroscopic behavior. The integration between the sarcomere scale and the global muscle is not described in this paper. Sphincters’ action is considered in order to include the two modes: isometric contraction when sphincters are closed and micturition when sphincters are open. Fluid mechanics influenced by sphicnter’s pressure is introduced in order to simulate urine flow properties. The influence of urethra itself is neglected to reduce the model complexity.

The model comprises a set of nonlinear differential equations that can be summarized via the following state equation, given
X = [Ca2+ ]i ; [M ]; [M ∗ ]; [AM ∗ ]; [AM ]; kc ; σc ; εp

       X˙ =       

J − Jvocc − Jextrusion k7 [AM ] − k1 [M ] + k2 [M ∗ ] k1 [M ] − (k2 + k3 )[M ∗ ] + k4 [AM ∗ ] k3 [M ∗ ] − (k4 + k5 )[AM ∗ ] + k6 [AM ] k5 [AM ∗ ] − (k6 + k7 )[AM ] kcm f − (f + g) kc σcm f − (f + g) 2 pσc + kc Lc0 ε˙c 2π 2 su (1−Esph )

2

Pves ρ

             

(1)

L3p0 ε2p

The main variables of the model’s equation are described in table I. The first stage of the model computes the calcium dynam˙ ics induced by the ES signal (eq. X((1)). The evolution of calcium concentration in the cell is expressed as the sum of all calcium currents through the membrane and with the sarcoplasmic reticulum ( Koeninsberg et al. [2]). We report only the currents that are directly influenced by ES and thus we consider the input to be the membrane potential vi , J is

Definition Electrical membrane potential induced by ES Intracellular calcium concentration reaction rate depending on [Ca2+ ]i (= k6 ) Non-activated myosin fraction Activated myosin fraction Crossbridges fraction Latchbridges fraction Active stiffness of the contractile element Stress of the contractile element Relative deformation of the parallel element Relative deformation of the contractile element Intravesical pressure Internal volume of the bladder

the contribution of all the other currents: vi − vCa1 (2) Jvocc = GCa 1 + e(vi −vCa2 )/RCa vi − vd Jextrusion = D [Ca2+ ]i 1 + (3) Rd Once the calcium concentration is computed, it is used as the input command of our modified Hai & Murphy model [1] ˙ ˙ described via the equations X(2) to X(5). The rate constants k1 and k6 are calcium dependent via a sigmoidal relationship. k1,6 ∝

[Ca2+ ]2i [Ca2+ ]2i + x20

(4)

k4 and k7 depend on the relative shortening speed |ε˙c | and then stand for the stress-velocity relationship at the level of the crossbridges forming process. This reflects the higher probability for a bridge to break if the actin and myosin filaments are moving relatively to each other. k4 (ε˙c ) = k4 + |ε˙c | k7 (ε˙c ) = k7 + |ε˙c | (5) ˙ ˙ The next part from X(6) to X(7) of the model links the kinetic equations from Hai & Murphy to Huxley’s crossbridge model.  [M ∗ ] 0 < ξ < 1,  kf k3 [M ]+[M ∗ ] , f (ξ, t)= (6)  0, ξ 6∈ [0; 1] n ∗ ]+k7 /k4 [AM ] − f (ξ, t), ∀ξ g(ξ, t)= kg k4 [AM [AM ]+[AM ∗ ] Considering our definition of f and g the derivative of the first and second order moments of n, portion of attached bridges, can be obtained [9]. They are equivalent to the stiffness and force of the contractile element Ec . At the macroscopic level, we use a three compartment model: Ec , the contractile element; Es , the serial linear elasticity element with a stiffness ks ; and Ep the exponential elasticity elements. Besides, the bladder is considered to be spherical. Thus, the link between geometrical variables, pressure and ˙ stress is described by the X(8) equation with:

266

Pves

=

Vint

=

4πσc e Lp0 εp (Lp0 εp )3 6π 2

(7)

TABLE II S UMMARY OF THE MODEL PARAMETERS AND THEIR VALUES .

The relationship between stress and pressure is obtained using the same approach as Back [10]. Finally, we have modeled the global action of sphincters on micturition, but only the pressure exerted by sphincters is considered. Based on literature [6], the action of sphincters is linked to the difference between fluid pressure (induced by detrusor contraction) and pressure exerted by the sphincters to maintain the urethra closed. We derived a simplified excitation Esph that is linked to the ES input through a second order Butterworth low pass filter with a time constant (0.2s) consistent with data in the literature [5]. Then, the output flow is obtained from these exerted pressures, urine viscosity and minimal urethra diameter (eq. 7): ε˙c is needed to compute the dynamics and can be obtained via the following relation between Ec and Es elements:   σ˙ c 1 Lp0 ε˙p − (8) ε˙c = Lc0 ks

Symbol GCa 1 vCa1 1 vCa2 1 RCa

D1 vd 1 Rd 1 k2 k3 k4 k5 k7 kcm ks σcm 2 e Lp0

The computation can be completed combining this equation ˙ ˙ with the equations X(7) and X(8). III. RESULTS c software. We have implemented the model on SCILAB Parameters are reported in table II. The aim of these simulations is to compare the output of the model to experimental data available in the literature. We choose to reproduce the behavior of the human bladder undergoing the stimulation of a FineTech / Brindley implant [11]. Its principle is to consider the different relaxation speeds of both kinds of muscles involved in micturition. Stimulating the sacral roots induces the contraction of both detrusor (smooth) and sphincter (striated) muscles. When releasing the stimulus the sphincter relaxes faster than the detrusor, allowing the urethra to open while the pressure remains high enough in the bladder in order to empty it. The cycle contraction/relaxation is then repeated. Our simulation works in two phases: 1) closed sphincter and contracting detrusor induce an isometric contraction while the pressure increases. 2) opened sphincter while stimulation is OFF. As the sphincters have much higher dynamics (contraction and relaxation times less than 200 ms) than the detrusor (about few seconds), micturition occurs. A. Simulation results The command signal used is shown in figure 3. When its value is 1, the electrical stimulation is active, when it is 0 the stimulation is OFF. The typical pattern if 3s ON then 6s OFF. We consider a stimulation amplitude high enough to obtain

Fig. 3. Command signal complying with the behavior of a Finetech/brindley implant.

1

Lc0

Parameter Cellular conductance for vocc Reversal potential for vocc Half-point of the vocc activation sigmoid Maximum slope of the vocc activation sigmoid Extrusion rate for ATPase pumps Intercept of voltage dependence of extrusion ATPase Slope of voltage dependence of extrusion ATPase reaction rate reaction rate reaction rate reaction rate reaction rate maximal contractile stiffness serial stiffness maximal contractil stress thickness of the bladder wall rest length of the parallel elements (circumference) rest length of the contractil elements

Value 1,29e−3 100 -24

Unit µM/(mV s) mV mV

8.5

mV

0,24

s−1

-100

mV

250

mV

0,2 0,4 0,4 0,02 0,2 100 1000 4e5 0,3 0.22

s−1 s−1 s−1 s−1 s−1

Pa mm m

0.11

m

full activation of the bladder. This level remains constant in all simulations. The model is able to have an insight in the basic phenomena that occur at the cell’s level: 1) the activation due to ES induces the rise of calcium concentration in the smooth muscle cells, enabling its contraction (fig. 4). It falls back to its rest level as the stimulation stops. 2) The actin and myosin fractions plot (fig. 4) reveals the chemical activity of the muscle cells. The calcium rise activates the myosin ([M ] → [M ∗ ]). The actin can then bound to it and generate force ([A] + [M ∗ ] → [AM ∗ ]). Some of the bounds will loose their activation but remain attached, they become latchbridges ([AM ∗ ] → [AM ]). The oscillations are due to the changes of the calcium concentration which influence the kinetics of all the biochemical reactions. 3) Stiffness and stress produced by the contractile element are then linked to the number of crossbridges formed. The intravesical pressure obtained is presented on figure 5. The dynamics is characteristic of smooth muscle with a rise time of few seconds and a fall time even longer due to the contribution of latchbridges at the microscopical scale. The figure also shows typical oscillations induced by intermittent ES. The figure 5 shows the evolution of urine volume in the bladder. The interrupted voiding pattern is easily noticeable with its plateau between voiding phases. The interruptions occur when both the detrusor and the sphincters are contracting at the same time. These two plots can be compared to the experimental data published by Brindley et al. [11]. They show a good consistency in term of shape and time course.

267

Fig. 4. Activation stages of the model: chemical scale. Left: Intracellular calcium concentration. Right: Fraction of the different species of free actin and myosin, and crossbridges.

Fig. 5. Outputs of the model in the standard case complying with Brindley/Finetch implant typical functionning. Left: Intravesical pressure (Pa). Right: Urine volume in the bladder (mL).

B. Model sensitivity to parameter errors

important to be efficient in terms of energy and so apply the minimal duration stimulation. The figure 7 presents the variations on the outputs for 13 cases plus the standard one ( [3 6] ). Detailed simulation results are presented in table III. We wish to minimize both index and pressure to obtain the most efficient voiding of the bladder. Four cases give good results, they are circled on figure 7. They lead to good outflow while requiring less stimulation and inducing less maximum pressure. The two hollow points correspond to the last line of the table. This case is particular as it is a simulation of how selective activation of the detrusor would act if achieved. It means that the detrusor is selectively stimulated without sphincter contractions using advanced stimulation strategies as described in [12]. It leads to a great improvement on outflow, but the pressure is slightly higher.

In this section we show the effect on the model output of a ±50% variation of several parameters that are not easy to estimate (k2 ,k3 ,k4 ,k5 ,k7 ,ks ,kcm ,σcm ,Lp0 ) summarized by an histogram in fig.6. The bars show the relative maximal pressure, emptied volumes and the value of an index (the Variance Accounted For or VAF) for intravesical pressure (Pves). The VAF is calculated as follows:   var(x) − var(x0) V AF (x0) = 100 1 − (9) var(x) with x the output for the initial value of the parameter and x0 the one for a modified value. Using this method, it is easy to spot which parameters have more influence on outputs. These are the ones to identify first with care in order to obtain the best accuracy. Whereas taking the example of k5 or ks , crude approximation could be enough. Doing so patient specific studies could be performed. C. Influence of the input duty cycle We simulated the effect of the stimulation pattern on the efficiency of the voiding. Three criteria were taken into account: the intermittent stimulation duty cycle, the mean urine outflow and the maximal intravesical pressure. All are normalized to their values in the standard case (ton = 3s and tof f = 6s). We wish to obtain efficient voiding at low pressure to be close to natural micturation. It is also

Fig. 6.

268

Histogram summarizing the results of preliminary sensitivity tests

TABLE III D UTY CYCLE INFLUENCE ON THE OUTPUTS . [ton tof f ] [1 8] [1 6] [2 8] [1 4] [2 6] [3 8] [2 4] [1 2] [4 8] [3 6] [4 6] [3 4] [4 4] [3 6] s

Duty Cycle 0.11 0.14 0.2 0.2 0.25 0.27 0.33 0.33 0.33 0.33 0.4 0.43 0.5 0.33

Pmax(%) 32.82 40.07 69.67 56.7 81.55 92.75 98.46 93.59 102.87 100 104.97 106.2 108.5 111.49

Qmean(%) 80.96 89.46 106 99.29 108.32 106.02 103.5 99.22 95.66 100 103.59 94.75 90.5 135.2

(a) Intravesical pressure (Pa)

(b) Bladder volume (mL)

Fig. 8. Output of the model in the case of a fully selective activation on the detrusor.

This last point is due to the low volume reached in this simulation with similar contractil stress. IV. DISCUSSION AND CONCLUSION In this paper we introduce a new bladder model including sphincter and detrusor actions able to simulate electrically stimulated contraction. We used parameter values from the literature and did not aim at quantitative simulation of human bladder. Nevertheless, the model can already be used as a tool to compare different ES strategies in a stable environment. We demonstrate the interest of modeling in the fine tuning of the implant to save energy and limit as much as possible the maximum internal pressure. We also show that some parameter values difficult to estimate on a specific patient, may lead to different simulation results. This preliminary study shows that for some parameters a careful identification protocol must be performed. Another way to obtain more efficient voiding is to use more complex ES technics. Some research groups work on selective activation of the detrusor without inducing striated sphincter contraction. The simulation presented on figure 8 shows the results of such approach. We used the same command pattern as the previously shown simulation for comparison purpose (3s on, 6s off), but with a sphincter always open. The stimulation leads to similar pressure and

enhanced outflow. Moreover, the micturition is now uninterrupted and closer to natural behavior. On going works focus on identification of the model through animal experiments first. Indeed all the experimental conditions are controlled so that accurate analysis of the simulation compared to recorded data can be performed. Human experiments will follow to investigate how to use this simulation tool in a clinical environment. R EFERENCES [1] C. M. Hai and R. A. Murphy, “Cross-bridge phosphorylation and regulation of latch state in smooth muscle.” Am J Physiol, vol. 254, no. 1 Pt 1, pp. C99–106, 1988. [2] M. Koenigsberger, R. Sauser, M. Lamboley, J. Beny, and J. Meister, “Ca2+ dynamics in a population of smooth muscle cells: modeling the recruitment and synchronization.” Biophys J, vol. 87, no. 1, pp. 92–104, 2004. [3] S. Yu, P. Crago, and H. Chiel, “A nonisometric kinetic model for smooth muscle.” Am J Physiol, vol. 272, no. 3 Pt 1, pp. C1025–39, 1997. [4] F. van Duin, P. F. Rosier, N. J. Rijkhoff, P. E. van Kerrebroek, F. M. Debruyne, and H. Wijkstra, “A computer model of the neural control of the lower urinary tract.” Neurourol Urodyn, vol. 17, no. 3, pp. 175– 196, 1998. [5] E. H. Bastiaanssen, J. L. van Leeuwen, J. Vanderschoot, and P. A. Redert, “A myocybernetic model of the lower urinary tract.” J Theor Biol, vol. 178, no. 2, pp. 113–133, Jan 1996. [6] F. Valentini, G. Besson, and P. Nelson, “[Mathematical model of micturition allowing a detailed analysis of free urine flowmetry],” Prog Urol, vol. 9, no. 2, pp. 350–60; discussion 369–70, 1999. [7] G. S. Brindley, C. E. Polkey, and D. N. Rushton, “Sacral anterior root stimulators for bladder control in paraplegia.” Paraplegia, vol. 20, no. 6, pp. 365–81, 1982. [8] N. J. M. Rijkhoff, “Neuroprostheses to treat neurogenic bladder dysfunction: current status and future perspectives.” Childs Nerv Syst, vol. 20, no. 2, pp. 75–86, Feb 2004. [9] J. Laforet and D. Guiraud, “Smooth muscle model for functional electrical stimulation applications: simulation of realistic bladder behavior under fes.” in Proc. 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society EMBS 2008, 2008, p. 37023705. [10] L. H. Back, “Left ventricular wall and fluid dynamics of cardiac contraction,” Mathematical Biosciences, vol. 36, no. 3-4, pp. 257 – 297, 1977. [11] G. S. Brindley, “An implant to empty the bladder or close the urethra.” J Neurol Neurosurg Psychiatry, vol. 40, no. 4, pp. 358–69, 1977. [12] N. Rijkhoff, H. Wijkstra, P. van Kerrebroeck, and F. Debruyne, “Selective detrusor activation by sacral ventral nerve-root stimulation: results of intraoperative testing in humans during implantation of a Finetech-Brindley system.” World J Urol, vol. 16, no. 5, pp. 337–41, 1998.

Fig. 7. Relative duty cycle, maximal pressure and mean outflow for each [ton tof f ] couple.

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