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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. IEEE-TBME 2011 LETTERS: SPECIAL ISSUE ON MULTI-SCALE MODELING

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A Preliminary Model of Gastrointestinal Electromechanical Coupling Peng Du, Yong Cheng Poh, Jee Lean Lim, Viveka Gajendiran, Greg O’Grady, Martin L. Buist, Andrew J. Pullan, and Leo K. Cheng

Abstract—Gastrointestinal (GI) motility is coordinated by several cooperating mechanisms, including electrical slow wave activity, the enteric nervous system (ENS), and other factors. Slow waves generated in ICC depolarize SMC, generating basic GI contractions. This unique electrical coupling presents an added layer of complexity to GI electromechanical models, and a current barrier to further progress is the lack of a framework for ICCSMC-contraction coupling. In this study, an initial framework for the electromechanical coupling was developed in a 2D model. At each solution step, the slow wave propagation was solved first and [Ca2+ ]i in the SMC model was related to a Ca2+ tension-extension relationship to simulate active contraction. With identification of more GI-specific constitutive laws and material parameters, the ICC-SMC-contraction approach may underpin future GI electromechanical models of health and disease states. Index Terms—bidomain, ICC, SMC, slow waves, motility

I. I NTRODUCTION Gastrointestinal (GI) motility is coordinated by several cooperating mechanisms, including electrical slow waves, the enteric nervous system (ENS), autonomic nerves, hormones, and other factors [1]–[3]. The relative contributions of these mechanisms to GI motility patterns remain an area of research. GI slow waves occur at ∼3 cycles per minute (cpm) in the human stomach and ∼8-12 cpm in the small intestine [2]. Slow waves are generated by pacemaker cells known as the interstitial cells of Cajal (ICC), and depolarize the surrounding gastric smooth muscle cells (SMC). This is one of the mechanisms that affects coordinated GI motility, i.e., SMC contractions [4]. Dysfunction in either the ICC and/or SMC is increasingly being linked to a number of functional GI disorders, such as gastroparesis and dyspespia [5]. Existing GI mechanical models have typically modeled the movements of the GI luminal contents by imposing the This study was supported by a University of Auckland Doctoral Scholarship, A Riddet Institute Scholarship, NUS Engineering Doctoral Scholarship, NUS NGS Doctoral Scholarship, grants from the New Zealand Health Research Council, and the U.S. National Institutes of Health (No. R01 DK64775). P. Du, J.L. Lim, G. O’Grady (Department of Surgery) and L.K. Cheng are with the Auckland Bioengineering Institute, The University of Auckland, NZ. A.J. Pullan is with the Department of Engineering Science, Auckland Bioengineering Institute, The University of Auckland; The Riddet Institute, NZ; Department of Surgery, Vanderbilt University, Nashville, TN, USA. Y.C. Poh, V. Gajendiran and M.L. Buist are with the Division of Bioengineering; Y.C. Poh and M.L. Buist are also with the National University of Singapore Graduate School of Integrative Sciences and Engineering. Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

mechanical deformation of the GI wall from imaging evidence [6]. The ability to link slow wave activity to motlilty would offer the opportunity to translate the established multiscale continuum modeling techniques to the GI field [7]–[9] - in order to obtain a better understanding of the interplay of slow wave and GI motility across multiple spatiotemporal scales. However, as outlined above, the unique and complex physiology of the GI tract also necessitates novel adaptations and approaches. This study presents a proof-of-concept framework and perspectives for: (i) cellular electromechanical coupling of ICC-SMC-contraction; (ii) simulation of slow wave propagation in a 1D model; (iii) electromechanical activity in a 2D model representing a thin strip of GI tissue. II. C ELL E LECTRICAL AND E LECTROMECHANICAL C OUPLING In recent studies, mathematical ICC and SMC slow wave models [10], [11], were coupled via a gap junction conductance, through which the self-excitatory slow wave activity in the ICC model initiates the slow wave activity in the SMC model (Fig. 1a) [12], [13]. Equation 1 describes the gap junction current (Icouple ) that was used to couple the slow wave activity in the ICC model to drive the SMC model: Icouple = Gcouple ηICC (Vm(ICC) − Vm(SM C) ),

(1)

where Gcouple is the gap junction conductance between the membrane potentials of ICC (Vm(ICC) ) and SMC (Vm(SM C) ). The ratio of ICC-to-SMC (ηICC ) specifies the physiological load i.e., how many SMC an ICC depolarizes [13]. Although many details of the intracellular processes involved in GI motility are still debatable, the consensus is that contraction of SMC is initiated by myosin light chain (MLC) phosphorylation via the activation of calcium-calmodulindependent MLC kinase (MLCK) [4]. Complexity lies in the additional physiological factors that co-regulate the sensitivity of the contractile apparatus to calcium (Ca2+ ), such as spike activity in intestines, hormones, ENS etc [4]. For this initial framework, we assumed that each slow wave would initiate a corresponding SMC contraction [4]. A bimodular active force cellular mechanical model has been developed recently to relate Ca2+ to contraction [14]. In the first module of the mechanical model, given an input of [Ca2+ ]i from the SMC, Ca2+ interacts with calmodulin, which activates MLCK. The second module describes the pathway of actin-myosin bridging formation, which involves MLCK phosphorylation of myosin. Contractile force (F ) generated is proportional to the total number of bridges formed,

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i.e., the sum of cross-bridges ([AMp ]) and latch-bridges ([AM ]), in a given total myosin concentration ([Mtotal ]) [14], as shown in Eqn. 2: F = Fmax

[AMp ] + [AM ] , [Mtotal ]

(2)

where Fmax is the maximum achievable force. The active force was simulated by using the [Ca2+ ]i from the SMC model as an input to the mechanical model (Fig. 1b) [10], [14].

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of Vm -dependent IP3 synthesis, K4 and Kv are the halfsaturation constants for non-linear IP3 degradation and Vm dependent IP3 synthesis, respectively. A full description of the parameters in Eqn. 5 can be found in [12]. An alternative model of entrainment is a modified Vm dependent dihydropyridine-resistant conductance (IV DDR ) model in ICC, here denoted as IP U for short [13]. Equation 6 describes the formulation of IP U : IP U = GP U dr dP U fP U (Vm − ECaP U ),

III. C ONTINUUM M ODELS OF E NTRAINMENT A system of two equations (3 and 4) constitute the basic bidomain model, which is used to simulate propagation of slow waves at higher spatial scales [7], [9], [15]: ∇ · (σi ∇Vm ) = −∇ · ((σi + σe )∇ϕe ), (3) ∂Vm Am (Cm + Iion ) − ∇ · (σi ∇ϕe ) = ∇ · (σi ∇Vm ), (4) ∂t where Vm and ϕe are membrane and extracellular potentials, respectively. The ionic current (Iion ) term from the ICC and/or SMC model relates the slow wave activity from the cell model to the bidomain equations. The tissue conductivity tensors (σ), with subscript i denotes the intracellular domain and subscript e denotes the extracellular domain [7], [9], [15]. Slow wave entrainment is a well established process by which slow waves in a group of ICC with a gradient of intrinsic frequencies phase-lock to form a wavefront [2], [3]. Two entrainment models that can be incorporated into the bidomain equations have been proposed [12], [13]. First, an IP3 -dependent Ca2+ -release model (Eqn. 5) was incorporated into the ICC model [12]:

(5)

where β represents IP3 production in response to chemical stimulus agent (such as Acetylcholine), η is linear rate constant for IP3 degradation, Vm4 is the maximum non-linear rate constant for IP3 degradation, PM V is the maximum rate

(6)

where GP U , dP U , and fP U are the maximum channel conductance, activation gate, and inactivation gate, respectively [13]. The fraction of the IP U in IV DDR conductance is denoted by dr . ECaP U is the Nernst potential of Ca2+ in the pacemaker unit in the ICC model. A Ca2+ -extrusion mechanism was also added to the ICC cytoplasm to maintain long-term Ca2+ homeostasis in the ICC model (Eqn. 7): ICaEXT =

Camax 1+e

Fig. 1. Simulated gastric slow waves and mechanical response. (a) Simulated slow waves in gastric smooth muscle cell (SMC) (solid line) and interstitial cell of Cajal (dashed line). (b) Simulated SMC [Ca2+ ]i (solid line) and the resultant normlized active force (dashed line).

[IP3 ]4 d[IP3 ] = β − η[IP3 ] − Vm4 4 dt K4 + [IP3 ]4 V8 + PM V (1 − 8 m 8 ), Kv + Vm

2

[Ca2+ ]−Ca50 k

,

(7)

where Camax was set to be 0.0315 µM ms−1 ; Ca50 and k were set to be 100 nM and 15, respectively. A full description of the parameters in Eqns. 6 and 7 can be found in [13]. Both mechanisms of entrainment were combined separately with the ICC model [12], [13], and slow waves were simulated in a 1D linear model (8 mm in length). A total of 101 gridpoints, i.e., solution points, were distributed at equal spacing along the geometric element of the 1D model. Zero-flux boundary conditions were imposed. To simulate propagation of slow waves, a linear gradient of ICC intrinsic frequencies (3-2.68 cpm) was prescribed to the 1D model. The slow wave activity at each grid point was self-excitatory at the prescribed frequency and occurs independently of each other in a decoupled setting, i.e., no entrainment. No stimulus current term was used to invoke an electrical propagation [9]. IV. 2D M ODEL OF E LECTROMECHANICAL ACTIVITY A 2D model was used to demonstrate electromechanical activity. The 2D model represented a thin strip of GI tissue (8×8 mm) and consisted of 8×8 bi-linear elements, at the same solution resolution as the 1D model. The x and y coordinates of the 2D model represented the longitudinal and circular direction in the GI tract, respectively. A linear gradient of ICC slow wave intrinsic frequencies (3-2.68 cpm) was prescribed horizontally across the 2D model. Here, as a proofof-concept step to demonstrate coupled electromechanical activity, an established steady-state Ca2+ -tension (T )-extension relationship (Eqn. 8) was used to relate the slow wave to contraction [8]: T = Camax T0 (1 + β(λ − 1))

[Ca2+ ]hi , + Cah50

[Ca2+ ]hi

(8)

where Camax was set to be 0.6 µM and T0 is the maximum isometric tension. The values of the non-dimensional slope parameter (β), Ca50 , and Hill coefficient (h) were set to be 1.45, 1.95, and 2.5, respectively; these values were estimated by minimizing the root-mean-square error between a set of



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λ = 1.3 λ = 1.2 λ = 1.1 λ = 1.0 λ = 0.9 λ = 0.8

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waves through coupled ICC slow waves in the 1D model; while the decoupled slow waves may contribute to ectopic sources of entrainment in disease states. V (mV)

isometric tension-[Ca2+ ]i recording from guinea-pig taenia coli [16], and the simulated isometric tension using Eqn. 8 (Fig. 2a). A full description of Eqn. 8 can be found in [8]. The isometric tensions were then simulated over a range of extension-ratios (λ) (Fig. 2b). The pole-zero constitutive law

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log([Ca 2+]i) Fig. 2. Steady-state Ca2+ -tension-extension relationship. (a) Simulated normalized isometric tension-[Ca2+ ]i relationship. (circle: experimental data [16].) The [Ca2+ ]i was presented in a logarithmic scale. (b) Simulated isometric tension-[Ca2+ ]i relationship at different extension-ratios (λ).

was used to describe the passive mechanical behavior of the tissue [8]. The limiting strains in the fiber and sheet directions were specified as the strains corresponding to maximum stress obtained during uniaxial tensile testing on human small intestine [17]. The 2nd Piola-Kirchhoff stress tensor was used to relate the tension in the reference configuration to areas in the reference configuration. Zero displacement boundary conditions were imposed. The LSODA and Euler solvers were used to solve for the intracellular and extracellular domains of slow wave activity, respectively, as previously described [12]. The cell models were encoded using the CellML standard [18]. The mechanical deformation was solved using the Newton-Raphson method. At each solution step, the electrical propagation was solved first and then the mechanical deformation was solved to update the mechanical parameters in the geometric elements. The local deformation was quantified by calculating the area of each column of elements in the 2D model at each solution step. The average tension generated by the 2D model was also calculated at each solution step. V. R ESULTS AND D ISCUSSION A. Entrainment Models Decoupled slow waves activity is shown in Fig. 3a where entrainment does not occur. Simulations of the entrainment of slow waves demonstrated coupling of ICC with different intrinsic frequencies to a single frequency of 3 cpm. The IP3 -dependent Ca2+ -release model (Fig. 3b) produced a steeper rate-of-upstroke than the IP U model (Fig. 3c) (73 vs 23 mV s−1 ). In both cases, the entrainment models produced a constant velocity (∼12 mm s−1 ), in accordance with physiological data [2]. Rather than a Vm -threshold based propagation mechanism, the entrainment models actively propagate slow

Fig. 3. Simulated 1D slow wave entrainment. (a) In the decoupled model (by setting σe and σi to 0.001 mS mm−1 ), the simulated slow waves on the two boundary nodes demonstrated different intrinsic frequencies, 3.00 cpm (solid line) and 2.68 cpm (dashed line). In the entrained model, (b) IP3 dependent Ca2+ -release model and (c) IP U model, the slow wave with the lower intrinsic frequency (dashed line) was entrained to the slow wave with the higher intrinsic frequency (solid line).

B. Electromechanical Model Electromechanical coupling was achieved in a 2D multiscale model, as shown in Fig. 4. The electromechanical activity resulted in a dynamic regional deformation of the 2D model that followed the slow wave propagation. The geometric elements in column 1 of the 2D model deformed first, followed sequentially by columns 2 to 8. The deformations of the geometric elements in column 1 led to extension in the areas of the remaining elements in the 2D model. The peak deformation occurred in column 1 at 0.54 s (Fig. 5a). The tension developed monotonically in a sigmoidal profile (Fig. 5b). The development of tension in both the cell model (Fig. 1b) and the 2D model was delayed relative to the onset of slow wave activity. C. Limitations There are a number of important limitations in this initial model that warrant further discussion. One limitation is the absence of an mechanoelectrical feedback mechanism at the cellular level. For example, the mechno-sensitive Nav 1.5 sodium conductance to mechanical stretch has been quantified in a recent study [19]. Updating the cell models to include the Nav 1.5 could offer a potential mechanoelectrical mechanism in the model. Furthermore, Nav 1.5 is of clinical interest because mutation of this channel is associated with dysmotility [5]. This model offers potential to predict the functional effects of such mutations at the tissue level. Another limitation is the application of the pole-zero constitutive law in this study. A key feature described by the polezero law is the difference in limiting strains in each of the main microstructural axes in the tissue; this has also been



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VI. C ONCLUSIONS

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2 mm y x Fig. 4. Simulated electromechanical activity. The black lines highlight the boundary of each geometric element (1×1 mm). The deformation of each element followed the direction of the simulated slow wave propagation.

This study presents an initial GI electromechanical model, demonstrating potential for relating slow waves to motility in a multi-scale framework, thereby integrating a vast volume of experimental data. The model may be expanded in future to incorporate other co-regulators of GI motility, e.g., ENS [1], to help interpret and predict the relative contributions of each mechanism to integrated motility patterns and control. The model may also be applied to show how mechanical deformation affects the propagation of slow waves. We further anticipate clinical applications, to understand the effects of slow wave disorders on tissue level function, including ICC/SMC ion channelopathies. R EFERENCES

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Time (s) Fig. 5. Quantification of the mechanical activity. (a) Changes in normalized area for columns 1-8 in the finite element mesh. (b) Average normalized stress developed in the 2D model.

observed in human intestinal tissue [17], and incorporated into the 2D model. The remaining material parameters were left with cardiac-specific values. Future GI-specific strain field data from appropriate biaxial tests can be used to properly adapt the constitutive law to describe the passive mechanical properties of GI tissue. Additional parameter values in the constitutive law can be identified through optimization procedures. Similarly, the steady-state Ca2+ -tension-extension relationship used to model the active tension generation in this study could be updated with the bimodular model [14]. Furthermore, it has been postulated that a Vm -threshold may exist for more vigorous mechanical contractions in the GI tract [2]. Incorporation of complementary modeling studies is required to include the role of ENS, which acts as a major co-regulator of GI motility [1], [3]. A proposed extension to the bidomain equations could serve as a bridge between our framework and ENS models [15]. Furthermore, there are many motor patterns in different parts of the GI tract during different states, e.g., fed vs unfed, and the coordinations of these motor patterns are dependent on the complex interactions of many control mechanisms [1], [3]. Much details of these processes are still under investigations, and future GI electromechanical models will need to incorporate these mechanisms as further details become available.

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