Beeler-Reuter Mammalian Ventricular Model 1977
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Status
This model has been curated by Penny Noble using Flavio Fenton's Java code as a reference (See http://thevirtualheart.org/ for Java applet rendering of model - Java code is available from Dr Fenton.) An artificial stimulus component has been added this model to allow it to reproduce the action potential simulation shown in Figure 4 of the publication. The model is known to run and integrate in the PCEnv and COR CellML environments. A PCEnv session file is also associated with this model.
ValidateCellML detects unit inconsistency within this model.
Model Structure
In contrast to the earlier Purkinje fibre ionic current models of D. Noble (1962) and R.E. McAllister, D. Noble and R.W. Tsien (1975), the G.W. Beeler and H. Reuter 1977 model was developed to describe the mammalian ventricular action potential. Not all the ionic currents of the Purkinje fibre model are present in ventricular tissue; therefore, this model is simpler than the MNT model. The total ionic flux is divided into only four discrete, individual ionic currents (see below). The main additional feature of the Beeler-Reuter ionic current model is a representation of the intracellular calcium ion concentration.
The complete original paper reference is cited below:
Reconstruction of the action potential of ventricular myocardial fibres, Beeler, G.W. and Reuter, H. 1977
Journal of Physiology
, 268, 177-210. PubMed ID: 874889
cell diagram of the Beeler-Reuter model showing ionic currents across the cell surface membrane
A schematic diagram describing the current flows across the cell membrane that are captured in the BR model.
the cellml rendering of the Beeler-Reuter model
The network defined in the CellML description of the Beeler-Reuter model. A key describing the significance of the shapes of the components and the colours of the connections between them is in the notation guide. For simplicity, not all the variables are shown.
The membrane physically contains the currents as indicated by the blue arrows in . The currents act independently and are not connected to each other. Several of the channels encapsulate and contain further components which represent activation and inactivation gates. The addition of an encapsulation relationship informs modellers and processing software that the gates are important parts of the current model. It also prevents any other components that aren't also encapsulated by the parent component from connecting to its gates, effectively hiding them from the rest of the model.
The breakdown of the model into components and the definition of encapsulation and containment relationships between them is somewhat arbitrary. When considering how a model should be broken into components, modellers are encouraged to consider which parts of a model might be re-used and how the physiological elements of the system being modelled are naturally bounded. Containment relationships should be used to provide simple rendering information for processing software (ideally, this will correspond to the layout of the physical system), and encapsulation should be used to group sets of components into sub-models.
$\frac{d V}{d \mathrm{time}}=\frac{\mathrm{Istim}-\mathrm{i\_Na}+\mathrm{i\_s}+\mathrm{i\_x1}+\mathrm{i\_K1}}{C}$
$\mathrm{i\_Na}=(\mathrm{g\_Na}m^{3}hj+\mathrm{g\_Nac})(V-\mathrm{E\_Na})$
$\mathrm{alpha\_m}=\frac{-1(V+47)}{e^{-0.1(V+47)}-1}\mathrm{beta\_m}=40e^{-0.056(V+72)}\frac{d m}{d \mathrm{time}}=\mathrm{alpha\_m}(1-m)-\mathrm{beta\_m}m$
$\mathrm{alpha\_h}=0.126e^{-0.25(V+77)}\mathrm{beta\_h}=\frac{1.7}{e^{-0.082(V+22.5)}+1}\frac{d h}{d \mathrm{time}}=\mathrm{alpha\_h}(1-h)-\mathrm{beta\_h}h$
$\mathrm{alpha\_j}=\frac{0.055e^{-0.25(V+78)}}{e^{-0.2(V+78)}+1}\mathrm{beta\_j}=\frac{0.3}{e^{-0.1(V+32)}+1}\frac{d j}{d \mathrm{time}}=\mathrm{alpha\_j}(1-j)-\mathrm{beta\_j}j$
$\mathrm{E\_s}=-82.3-13.0287\ln (\mathrm{Cai}\times 0.001)\mathrm{i\_s}=\mathrm{g\_s}df(V-\mathrm{E\_s})\frac{d \mathrm{Cai}}{d \mathrm{time}}=\frac{-0.01\mathrm{i\_s}}{1}+0.07(0.0001-\mathrm{Cai})$
$\mathrm{alpha\_d}=\frac{0.095e^{\frac{-(V-5)}{100}}}{1+e^{\frac{-(V-5)}{13.89}}}\mathrm{beta\_d}=\frac{0.07e^{\frac{-(V+44)}{59}}}{1+e^{\frac{V+44}{20}}}\frac{d d}{d \mathrm{time}}=\mathrm{alpha\_d}(1-d)-\mathrm{beta\_d}d$
$\mathrm{alpha\_f}=\frac{0.012e^{\frac{-(V+28)}{125}}}{1+e^{\frac{V+28}{6.67}}}\mathrm{beta\_f}=\frac{0.0065e^{\frac{-(V+30)}{50}}}{1+e^{\frac{-(V+30)}{5}}}\frac{d f}{d \mathrm{time}}=\mathrm{alpha\_f}(1-f)-\mathrm{beta\_f}f$
$\mathrm{i\_x1}=\frac{\mathrm{x1}\times 8E-3(e^{0.04(V+77)}-1)}{e^{0.04(V+35)}}$
$\mathrm{alpha\_x1}=\frac{5E-4e^{\frac{V+50}{12.1}}}{1+e^{\frac{V+50}{17.5}}}\mathrm{beta\_x1}=\frac{0.0013e^{\frac{-(V+20)}{16.67}}}{1+e^{\frac{-(V+20)}{25}}}\frac{d \mathrm{x1}}{d \mathrm{time}}=\mathrm{alpha\_x1}(1-\mathrm{x1})-\mathrm{beta\_x1}\mathrm{x1}$
$\mathrm{i\_K1}=0.0035(\frac{4(e^{0.04(V+85)}-1)}{e^{0.08(V+53)}+e^{0.04(V+53)}}+\frac{0.2(V+23)}{1-e^{-0.04(V+23)}})$
$\mathrm{Istim}=\begin{cases}\mathrm{IstimAmplitude} & \text{if $(\mathrm{time}\ge \mathrm{IstimStart})\land (\mathrm{time}\le \mathrm{IstimEnd})\land (\mathrm{time}-\mathrm{IstimStart}-\lfloor \frac{\mathrm{time}-\mathrm{IstimStart}}{\mathrm{IstimPeriod}}\rfloor \mathrm{IstimPeriod}\le \mathrm{IstimPulseDuration})$}\\ 0 & \text{otherwise}\end{cases}$
cardiac
cardiac electrophysiology
electrophysiology
ventricular myocyte
electrophysiological
Changed model cmeta:id from beeler_reuter_1977_version06 to beeler_reuter_1977
keyword
James
Lawson
Richard
Penny
Noble
Added an intial value for X1 to enable the model to run.
Catherine
Lloyd
May
874889
Re-added cmeta:id's for 4 major currents that had been deleted by COR
Reconstruction of the action potential of ventricular myocardial fibres
268(1)
177
210
2008-05-20T11:16:23+12:00
G
Beeler
2008-05-08T00:00:00+00:00
This model has been curated and is known to run and reproduce the published results in PCEnv and COR. A PCEnv session file is also associated with this model.
Penny has curated this model from Flavio Fenton's model code. See http://thevirtualheart.org/ for Java applet rendering of model. Code available from Dr Fenton
Updated cmeta:id's for reference by PCEnv sessions.
Added simulation metadata to allow simulation for 10,000 ms
2008-05-08T03:15:26+12:00
Journal of Physiology
James Lawson
Catherine Lloyd
In contrast to the earlier Purkinje fibre ionic current models of D. Noble (1962) and R.E. McAllister, D. Noble and R.W. Tsien (1975) (MNT model), the G.W. Beeler and H. Reuter 1977 model was developed to describe the mammalian ventricular action potential. Not all the ionic currents of the Purkinje fibre model are present in ventricular tissue; therefore, this model is simpler than the MNT model. The total ionic flux is divided into only four discrete, individual ionic currents. The main additional feature of the Beeler-Reuter ionic current model is a representation of the intracellular calcium ion concentration.
University of Auckland
Auckland Bioengineering Institute
2008-05-20T10:56:34+12:00
10000
10000
0.1
H
Reuter
James
Lawson
Richard
2008-05-20T11:41:27+12:00
c.lloyd@auckland.ac.nz
1977-06-00 00:00
James
Lawson
Richard