Thesis Update I
Geoffrey Hunter
Last Updated:
October 16, 2006
Contents
1 Thesis Progress Report I
1
1.1
Timeline: May ’06 - mid Sept ’06 . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Future Timelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Research Summary
2.1
2
Active Membrane Changes During Ischemia . . . . . . . . . . . . . . . . . . . . .
2
2.1.1
Anoxia & Acidosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1.2
Hyperkalemia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Passive Membrane Changes During Ischemia . . . . . . . . . . . . . . . . . . . .
5
2.3
Border Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Recommended Review Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.5
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3 Opinion
7
3.1
Thesis Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2
Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
References
11
1
1
Thesis Progress Report I
With this progress report I will summarize the preliminary reading on border zone (BZ)
arrhythmias and highlight what I will work on in the coming months. I intend to send out
progress reports like this when warranted to those on my supervisory committee and others
who are interested to keep everyone in the loop even if I don’t talk to all of you all the time.
Prospective Supervisory Committee
Chairperson: Prof J. Keener
Committee Members: Prof A. Fogelson, Prof P. Bressloff.
Possible Committee Members: Prof A. Zaitsev∗ , Prof F. Sachse∗
Additional Correspondence: Dr.Luca Pavarino (U of Milano, Italy)
* Faculty at the CVRTI (http://www.cvrti.utah.edu/).
1.1
Timeline: May ’06 - mid Sept ’06
• May ’06 - mid July ’06: General background reading on BZ arrhythmias, ischemia, and
cardiac physiology.
• July 28 ’06: Attended a talk at the CVRTI by Dr.Luca Pavarino on his research into
designing accurate computational models of myocardium. He and his collaborator, Prof
Colli Franzone, think that a more detailed model of the passive membrane properties in
normal myocardium is required before disease cases can be considered. I e-mailed him
after his talk and asked him whether he knew of any math models that showed: healthy
tissue + ischemia → BZ arrythmia. He wasn’t aware of any models showing this, but he
referred me to some papers that might help ([8], [17], and [19]). He has asked to be
updated on my work.
• mid July ’06 - late Aug ’06: Minimal reading done. Took the rest of the summer to relax
and prepare for last qualifying exam (DE’s).
• late Aug ’06 - beg Oct ’06: Passed all qualifying exams (whew!) and am taking currently
taking Math Biology, thesis research, a Bioengineering Journal Club, and a
computational cardiology reading course with Prof Frank Sachse of the CVRTI. I’m also
working on this summary and am trying to formulate possible thesis problems.
1.2
Future Timelines
• Dec ’06: Formulate a thesis problem and have the foundation for an oral defense ready.
• April ’07: Write and present oral defense. Finalize supervisory committee.
• May ’07 - Aug ’07: Cellular or macroscopic wet lab work at the CVRTI.
• Oct ’08: Assemble and write thesis.
• April ’09: Final defense.
2
2
Research Summary
The following is a summary of my preliminary reading on myocardial ischemia as well as border
zone and reperfusion arrhythmias. This summary focusses on the time course of ischemia with
a particular emphasis on arrhythmogenic mechanisms. I will also summarize some information
on border zones and intermittently mention reperfusion arrhythmias. Following the summary,
I will make some personal comments and share some thesis problem ideas.
2.1
Active Membrane Changes During Ischemia
Active membrane properties are the factors that affect excitation and propagation during the
initial stages of acute ischemia and are broken down into three categories: anoxia, acidosis, and
hyperkalemia ([3], [22]).
2.1.1
Anoxia & Acidosis
Anoxia is an absence or reduction in O2 and acidosis is an abnormal rise in acidity. Ischemic
myocardium becomes anoxic and acidic on occlusion, and this adversely affects the function of
the myocardium thereafter. Anoxia is first to affect the myocardium. Immediately after
occlusion ATP and creatine phosphate concentrations decrease, oxidative phosphorylation is
inhibited, and anaerobic respiration begins [3]. This causes a rise in the intracellular
concentration of lactic acid and [pH]i [3]. The active pumps, such as the Na+ –K+ pump and
SR Ca2+ cycling, are inhibited by reduced ATP levels, so [Na+ ]i and [Ca2+ ]i rise and [pH]i
falls ([2], [3], [26]). Meanwhile, [pH]o decreases from the Na+ –H+ exchange and Na+ –HCO−
3
cotransport of H+ to the extracellular space [3]. [pH]o also decreases from lactate transport
and diffusion of lactic acid and of CO2 out of the cell ([2], [3]). Lastly, some glycolytic enzymes
are inhibited by acidosis, so H+ is a negative feedback mechanism to limit acidification of the
cell [3].
3
2.1.2
Hyperkalemia
K+ is the primary determinant of depressed conduction and conduction block [27]. During
anoxia and acidosis the active Na+ –K+ pump slows and [K+ ]o accumulates in three phases as
illustrated in figure 1 ([2], [18]).
Phase I
Phase II
Phase III
[K+]o
5
20
30
Time (min)
Figure 1: The three phases of [K+ ]o accumulation
Phase I: This phase is characterized by a fast increase in [K+ ]o to 8-12mM within 5-10
minutes after occlusion [27] and is correlated to the rise in phase 1a VT and VF ([12],
[18], [27]). Higher [K+ ]o levels depolarize the resting membrane potential leading to an
inhibition of INa , the excitatory current ([3], [19]). INa is responsible for the initial
depolarization of the membrane and its inhibition decreases the magnitude of the inward
current and speed of impulse propagation ([3], [15], [22]). The action potential (AP)
duration (APD) decreases ([19], [27]) from activation of the IKATP and IKAAA channels
[3]. Sufficient ATP remains to sustain important ionic homeostatic processes even though
ATP levels are significantly below normal [3].
Higher resting membrane potentials lead to TQ segment depression (-10 mV to -15 mV)
after 1.5 minutes and decreasing action potential amplitude (APA) leads to ST segment
elevation (+15 mV to +25 mV) after 4 minutes ([8], [9], [11]). Monophasic AP’s are seen
after 8 minutes and the effective refractory period (ERP) increases ([3], [9]). The ST
segment, however, will become more depressed as ischemia becomes more transmural [8].
Phase II: This phase is characterized by a plateau or even, perhaps, a slight decrease in
[K+ ]o ∼ 8-20 minutes after occlusion as free energy from hydrolysis of ATP stabilizes [3].
During the plateau phase, catecholamines are massively released from nerve
endings and cells, glycolysis is stimulated, and lactate production is increased.
The plateau level of [K+ ]o is dependent on glycolytic activity. When the heart
4
is depleted of glycogen during successive ischemia periods, lactate production
and acidification are less, and [K+ ]o rises faster. [3]
Interestingly, impulse propagation might improve from catecholamine stimulation of the
Na+ –K+ pump and/or a loss of interaction from depolarized ischemic myocardium [3].
If the tissue is reperfused at a time up to and including this phase, then metabolic and
biochemical levels return to normal within 15 minutes of reperfusion ([2], [9], [12]). The
highest incidence of reperfusion arrhythmias occurs after 20-30 minutes of ischemia when
there is a heterogeneous distribution of reversibly and irreversibly damaged cells [12].
Immediately on reperfusion the AP’s are heterogeneous, out of phase, and may shorten
by as much as 60ms [12]. A reperfusion arrhythmia might be triggered by a delayed after
depolarization (DAD) and maintained by Ca2+ –entry blockers and by α−adrenergic
blocking agents [12].
Phase III: This final phase is characterized by a very fast rise in [K+ ]o [3] and is
correlated with rapid conduction failure from gap junction uncoupling and, consequently,
with the onset of phase 1b arrhythmias ([12], [27]). Conduction failure accompanies
diminished [Na+ ]i and [Ca2+ ]i load and contributes to the substrate for reentry ([3],
[27]), while catecholamines can trigger an arrhythmia ([2], [27]). In particular, reversal of
neuronal uptake leads to enormous extracellular norepinephrine that induces early after
depolarizations (EAD), DAD’s, and triggered activity facilitated by [Ca2+ ]o ([12], [27]).
Cell-to-cell uncoupling might be important in the BZ where changes in [K+ ]o and [pH]o
are less severe and active membrane properties are less depressed than in the ischemic
zone (IZ) [3].
Only partial recovery of myocardium is expected if it is reperfused during or after this
phase because of irreversible cell damage (i.e. permanent cell-to-cell uncoupling, muscle
tension, and increase membrane permeability) ([2],[9]). The reperfused myocardium can
take several weeks to reach a final healed state [12].
The three main currents sensitive to hyperkalemia are: IKo (transient outward current), IKr
(delayed rectifying current), and IK1 (inward rectifying current) [2].
5
2.2
Passive Membrane Changes During Ischemia
Passive membrane properties are structural changes that alter extracellular, intracellular, and
intercellular resistance and conductance such as gap junction conduction failure and volume
changes [3]. These changes changes occur during the later stages of acute ischemia (i.e. during
and after phase III hyperkalemia). “Gap junction conductance is modulated by the ionic and
metabolic state of the cell, and decreases in response to increased H+ and Ca2+ and decreased
ATP.”[3] Volume changes, on the other hand, are modulated by hydrostatic and osmotic
pressure that are altered by acidosis and poor circulation ([2], [24]). Like hyperkalemia, passive
membrane changes occur in three phases [24]:
Phase I: Extracellular longitudinal resistance rapidly increases by 25%-50% because the drop
in hydrostatic pressure in the capillaries allows a small amount of interstitial fluid to enter ([3],
[24]). Furthermore, membrane resistance decreases when K+ channels are opened and results
in improved conductivity since it concentrates the excitatory current (i.e. the space constant
decreases) [3].
Phase II: The extracellular resistance increases gradually and cells begin to swell for ∼ 15
minutes after occlusion ([2], [24]). The interstitial space has high osmolarity and begins to
draw water from the non-ischemic regions of the heart ([2], [24]). In low flow ischemia there is
low metabolite washout and a low osmotic imbalance, so there is low cell swelling [24]. High
flow ischemia, on the other hand, leads to high metabolite washout and a high osmotic
imbalance between the cell and the extracellular space, so there is high cell swelling [24]. This,
in turn, can activate stretch activated channels [27].
Phase III: There is a rapid increase in intracellular resistance from gap junctions closing ∼
15-30 minutes after occlusion ([21], [24]). Gap junctions close from considerable
dephosphorylation of connexin-43 and from an increase in intracellular acidification [24]. “The
time course of cell-to-cell electrical uncoupling follows the changes of [pH]i , [Ca2+ ]i , long-chain
acylcarnitines, and dephosphorylation of connexin-43 and is associated with the onset of VT,
and VF.”[3] In rabbits, for example, intracellular Ca2+ doubles and pH drops to 6.0 before VT
and VF [24].
6
2.3
Border Zones
The BZ is the tissue in between the IZ and normal zone (NZ). Often steep gradients in
metabolic, ionic, and neurogenic concentrations are found in the BZ [3]. There are many ways
experimentalists find the BZ including looking for regions of ST segment elevation and TQ
segment depression with multielectrode arrays ([8], [9], [10], [11], [14]), for a large intrinsic
deflection in the AP [9], and for steep gradients in either [K+ ]o ([5], [27]) or dominant
frequency maps using optical imaging methods [28]. Multielectrode arrays are the most
common technique and can have high resolution at the cost of more noise. Optical techniques
emerging as a new promising alternative to multielectrode arrays, but these techniques need
further development.
There are many naturally occurring BZ’s on the heart and these BZ’s are required for normal
heart function such as the separation between the left and right ventricles [28] and large
differences in conductivity between Purkinje fibers, the bundle of His, and myocardium [20],
for example. Ischemia-induced BZ’s are of interest because these are regions of the heart where
steep gradients of refractoriness, conduction, ATP, lactate, water, [pH]i , [K+ ]o , [Na+ ]i , [Ca2+ ]i ,
and catecholamines are found [3]. These gradients are responsible for arrhythmogenic triggers
such as wave fragmentation and impulse propagation failure ([3], [5], [9], [28]).
Steep [K+ ]o gradients seem to be the most significant gradient in the BZ because of the
associated injury currents ([3], [5], [11]) and is correlated with maximum conduction delay in
the IZ, VF triggers, spontaneous occurrence of VF, and increased proliferation of wavelets in
the BZ during ischemic VF ([5], [27], [28]).
2.4
Recommended Review Papers
The following are some (long) papers are listed here for future reference.
• Ischemia: [3] (mini-review) and [4]
• Ion Channels: [2], [5]
• Acute and Chronic Arrythmia’s: [12]
• ATP, CP, and lactate BZ data: [9]
• Computational Cardiology: [21]
• Mechanisms of Cardiac Propagation and Associated Arrhythmias: [13]
7
2.5
Glossary
Activation Time Time interval between the onset of the last pacing stimulus and the time of
maximum upstroke velocity of the action potential [19].
AP Action Potential
APA Action potential amplitude
APD The Action potential duration is the time difference between activation and 90%
recovery time [19]
BZ Border Zone
DAD Delayed after depolarization
EAD Early after depolarization
ESP Epicardial Surface Potential [8]
ERF Effective refractory period: Longest coupling interval that did not elicit propagation in
the slice after the seventh paced beat [19]
IZ Ischemic zone
NZ Normal zone
VF Ventricular Fibrillation
VT Ventricular Tachycardia
3
Opinion
I’ve reserved this section for some less scientifically rigorous remarks for discussion purposes.
In general the structure of the BZ (i.e. the width, edge contour, the distribution and degree of
of cell injury) and its precise role in propagation is unknown. The nature of a BZ depends on
the duration and location of ischemia and both of which are significantly affected by the
experimental conditions. There are different types of BZ’s with little distinction between each
in the literature. In order to have a clear understanding of the distribution and degree of cell
injury in the BZ, I think it is necessary to classify the different types of BZ’s based on the
duration of ischemia. The following is a summary of the different types of BZ’s but only two of
them have been defined explicitly in other papers.
8
• Metabolic BZ [11]: This is the most intuitive definition of a BZ because it relates to the
gradients in metabolic concentrations (i.e. ATP, CP, and lactate) in regional ischemia.
Janse et al. noted that metabolic BZ’s were not present at least 10 minutes after
occlusion although electrical BZ’s were always present [11].
• Electrical BZ [11]: This is the most common BZ discussed in the case of 15-20 minute
regional ischemia (i.e. before cell-to-cell uncoupling occurs) and is identified by AP
abnormalities and gradients in ion concentrations. The majority of the papers cited in
this summary have focussed on this BZ. These papers found that, on average, electrical
BZ’s are approx 8-10mm wide ([1], [5], [9], [11], [28]), however, in many cases the BZ was
defined to be this width rather than the width being a consequence of the definition.
• Phase 1b Arrhythmia BZ: This is the phase III hyperkalemia BZ that results from
electrical abnormalities as above in addition to cell-to-cell uncoupling. I don’t recall any
models with this type of BZ because of the spatial and electrophysiological complexity
involved. This BZ comprises of normal, reversibly damaged, and irreversibly damaged
cells, however, the distribution of cell damage in the BZ is unknown. Reversibly
damaged cells heal on reperfusion after a long healing phase and this has caused some
misinterpretation of the structure of BZ’s in the literature (i.e. claims that the BZ is one
cell width instead of 8mm wide).
• Chronic Arrhythmia BZ: This is the BZ that affects reperfused hearts that have survived
acute ischemia lasting 30 minutes. The tissue comprises of normal and irreversibly
damaged cells. Some of the cells are irreversibly damaged because of permanent cardiac
remodeling such as cell-to-cell uncoupling, muscle tension, and development of scar
tissue. This BZ can be 0.1mm - 1.0mm wide, or only a few cells wide. These BZ’s are
frequently modeled in numerical simulations of 2-D and 3-D myocardium and causes
reentry and wave fragmentation.
• Functional BZ: This BZ might be a special case of electrical BZ’s or deserve a
classification on its own. This BZ comprises of normal, ischemic, and abnormally
contracting myocardium [1]. The ischemic cells are surrounded by normally perfused
myocardium with abnormal contraction characteristics. I have only seen one study with
this type of BZ [1], but most of my reading has not focussed on the relationship between
ischemia on contraction.
9
The above definitions are based on the duration of ischemia and do not include natural BZ’s
that significantly influence propagation in myocardium. Many experimentalists have concluded
that the location of the BZ is constant in time regardless of the duration of the ischemia ([6],
[9], [11], [25]), although the width might change in accordance to the type of damage the tissue
has sustained. A further appeal for a precise definition of the BZ can be found in [7].
3.1
Thesis Ideas
Most thesis problems explore the relationship between the structure of the BZ and AP
propagation.
1. How does the distribution of cell injury in the BZ affect AP propagation?
• Are there some AP diagnostic tests that an experimentalist can do to deduce the
structure of the BZ?
• How do cellular differences in metabolic concentrations affect the metabolism and
AP propagation in neighboring cells?
• What is the role of injury currents across the ischemic BZ in the initiation and
maintenance of VF?
2. How is the shape of (or the distribution of cell injury in) the BZ affected by stimulus
frequency?
• Does the width of the BZ increase with stimulus frequency and/or are only the
gradients affected?
• Does the distribution of cell injury in the BZ become more uniform or
heterogeneous as the stimulus frequency changes?
• Is there a “characteristic frequency” that will lead to elongation in the ERP (or
changes in the APA, EAD, DAD, etc)?
3. How does the width of the border zone affect AP propagation?
• Is there a “characteristic width” that will lead to certain kinds of abnormal AP’s
and/or VT/VF triggers?
• How does the distribution of cell injury change affect the “characteristic width”?
• What is the relative importance of K+ , Na+ , and Ca2+ currents/concentrations at
the BZ?
Each of the above questions will also focus on...
• Electrical BZ’s (i.e. up to the first 15 minutes of occlusion and no irreversible cell injury).
• Simulate a chain of (human/guina pig/canine/pig/mammal) myocardium with one end
designated as the IZ and the other end as the NZ.
10
I’m not sure how I will model the cells in the BZ or by what measure do I quantify the “degree
of cell damage” (i.e. differences in the resting membrane potentials? [Na+ ]i ? [Ca2+ ]i ? [pH]i ,
[pH]o ?). So far I’ve read ([16], [22], [23], [26]) (and have many more) studies that simulate 1-D
chains with complex ion models, but these models have not incorporated a BZ in the
simulations. Simulations incorporating BZ’s have only considered Chronic Arrhythmia BZ’s
(not electrical BZ’s) and the width of the BZ has been chosen arbitrarily (i.e. using a step
function or a linear function spanning 0.1-1.2mm). The only exception is the study done by
Peercy and Keener [18], which looks at a 1-D compartment model with reduced HH and LRdI
models but I would like to use a more detailed ion model with spatial dependence. A detailed
ion model will allow me to show the relationship between ionic mechanisms and the AP and a
1-D chain will permit analysis of a BZ, while avoid the computational complexity of 2-D and
3-D simulations.
3.2
Next Steps
In the coming months I will look into the following:
• Find and/or define the different types of border zones and what is meant by “cell injury”.
• Find other simulations and see how ischemia is modeled.
• Look at what has already been done to model single cells and (1-D) coupled cells.
• Develop a plan to see what is required to simulate a 1-D chain of cells with detailed
physiology.
• Search for more simulations incorporating BZ’s.
11
References
[1] A.J. Buda, R.J. Zotz, and K.P. Gallagher, Characterization of the functional border zone
around regionally ischemic myocardium using circumferential flow-function maps, J. Am.
Coll. Cardiol. 8 (1986), no. 1, 150–158.
[2] Edward Carmeliet, Cardiac ionic currents and acute ischemia: From channels to
arrhythmias, Phys. Rev. 79 (1999), no. 3, 917–1017.
[3] Wayne E. Cascio, Myocardial ischemia: What factors determine arrhythmogenesis?, J
Cardiovasc Electrophysiol 12 (2001), no. 6, 726–729.
[4] Wayne E. Cascio, Timothy A. Johnson, and Leonard S. Gettes, Electrophysiologic changes
in ischemic ventricular myocardium: I. influsnces of ionic, metabolic, and energetic
changes, J Cardiovasc Electrophysiol 6 (1995), 1039–1062.
[5] R. Coronel, J. W. Fiolet, F. J. Wilms-Schopman, A. F. Schaapherder, T. A. Johnson,
L. S. Gettes, and M. J. Janse, Distribution of extracellular potassium and its relation to
electrophysiologic changes dueing acute myocardial ischemia in the isolated perfused
porcine heart, Circulation 77 (1988), 1125–1138.
[6] D.H. Drake, T.B. McClanahan, X.H. Ning, R.A. Gerren, W.R. Dunham, and K.P.
Gallagher, Changes in contractility fail to alter the size of the funcitonal border zone in
anesthetized dogs, Circ. Res. 61 (1987), no. 2, 166–180.
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Controversy or confusion?, Am. J. Cardiol. 47 (1981), no. 6, 1321–1334.
[8] Bruce Hopenfeld, Jeroen G. Stinstra, and Rob MacLeod, Mechanism for st depression
associated with contiguous subendocardial ischemia, J Cardiovasc Electrophysiol 15
(2004), 1200–1206.
[9] M. J. Janse, J. Cinca, H. Morena, J. W. Fiolet, A. G. Kleber, G. P. de Vries, A. E.
Becker, and D. Durrer, The “border zone” in myocardial ischemia. an electrophysiological,
metabolic, and histological correlation in the pig heart, Circ. Res. 44 (1979), 576–588.
[10] M. J. Janse and F. J. van Capelle, Electrotonic interactions across an inexcitable region as
a cause of ectopic activity in acute regional myocardial ischemia. a study in intact
porcupine and canine hearts and computer models, Circ. Red. 50 (1982), 527–537.
[11] Michiel J. Janse, Frans J. L. van Capelle, Hemmy Morsink, André G. Kléber, Francien
Wilms-Schopman, René Cardinal, Christoph Naumann d’Alnoncourt, and Dirk Durrer,
Flow of “injury” current and patterns of excitation during early ventricular arrhythmias
in acute regional myocardial ischemia in isolated porcine and canine hearts, Circ. Res. 47
(1980), no. 2, 151–165.
[12] Michiel J. Janse and Andre L. Wit, Electrophysiological mechanisms of ventricular
arrythmias resulting from myocardial ischemia and infarction, Phys. Rev. 69 (1989),
no. 4, 1049–1169.
[13] André Kléber and Yoram Rudy, Basic mechanisms of cardiac impulse propagation and
associated arrhythmias, Physiol Rev 84 (2004), 431–488.
[14] Kenneth R. Laurita, Steven D. Girouard, Yoram Rudy, and David S. Rosenbaum, Role of
passive electrical properties during action potential restitution in intact heart, Am. J.
Physiol. Heart Circ. Physiol. 273 (1997), no. 42, H1205–H1214.
12
[15] C.H. Luo and Y. Rudy, A dynamic model of the cardiac ventricular action potential: I.
simulations of ionic currents and concentration changes, Circ Res 74 (1994), 1071–1096.
[16]
, A dynamic model of the cardiac ventricular action potential: Ii.
afterdepolarizations, triggered activity, and potentiation, Circ Res 74 (1994), 1097–1113.
[17] Mary C. MacLachlan, Joakim Sundnes, and Glenn Terje Lines, Simulation of st segment
changes during subendocardial ischemia using a realistic 3-d cardiac geometry, IEEE
Trans. Biomed. Eng. 52 (2005), no. 5, 799–807.
[18] Bradford E. Peercy and James P. Keener, Coupled cell model of border zone arrhythmias,
SAIM J. Applied Dynamical Systems 4 (2005), no. 3, 679–710.
[19] Blanca Rodriguez, Brock M. Tice, James C. Eason, Felipe Aguel, José M. Ferrero Jr., and
Natalia Trayanova, Effect of acute global ischemia on the upper limit of vulnerability: A
simulation study, Am J Physiol Heart Circ Physiol 286 (2004), H2078–H2088.
[20] Robert F Rushmer, Organ physiology: Structure and function of the cardiovascular
system, 2 ed., W. B. Saunders Company, 1976.
[21] Frank B. Sachse, Computational cardiology, Springer-Verlag, Berlin, Germany, 2004.
[22] Robin M. Shaw and Yoram Rudy, Electrophysiologic effects of acute myocardial ischemia:
A mechanistic investigation of action potential conduction and conduction failure, Circ
Res 80 (1997), 124–138.
[23]
, Ionic mechanisms of propagation in cardiac tissue, Circ Res 81 (1997), 727–741.
[24] J. G. Stinstra, S. Shome, B. Hopenfeld, and R. S. MacLeod, Modelling passive cardiac
conductivity during ischaemia, Med. Biol. Eng. Comput. 43 (2005), 776–782.
[25] D.M. Yellon, D.J. Hearse, R. Crome, and R.K. Wyse, Temporal and spatial characteristics
of evolving cell injury during regional myocardial ischemia in the dog: the “border zone”
controversy., J Am Coll Cardiol. 2 (1983), no. 4, 661–70, I didn’t read this.
[26] Cheung-Seon Yi, Aaron Fogelson, James P. Keener, and Charles S. Peskin, A
mathematical study of volume shifts and ionic concentration changes during ischaemia
and hypoxia, J. Theor. Biol. 220 (2003), 83–106.
[27] Alexey V. Zaitsev, Mechnisms of ischemic ventricular fibrilation: Who’s the killer?, 4 ed.,
ch. 44, pp. 399–406, Saunders, 2004.
[28] Alexey V. Zaitsev, Prabal K. Guha, Farzad Sarmast, Arum Kolli, Omer Berenfeld,
Arkady M. Pertsov, Joris R. de Groot, Ruben Coronel, and José Jalife, Wavebreak
formation during ventricular fibrillation in the isolated regionally ischemic pig heart, Circ.
Res. 92 (2003), 546–553.