Class I agents are divided into three groups (Ia, Ib, and Ic) based upon their effect on the length of the action potential. [ 10 ] [ 11 ]


Class II agents are conventional  beta blockers . They act by blocking the effects of  catecholamines  at the  β 1 -adrenergic receptors , thereby decreasing sympathetic activity on the heart, which reduces intracellular cAMP levels and hence reduces Ca 2+  influx. These agents are particularly useful in the treatment of  supraventricular tachycardias . They decrease conduction through the  AV node .


Class II agents include  atenolol ,  esmolol ,  propranolol , and  metoprolol .


Class III agents predominantly  block the potassium channels , thereby prolonging repolarization. [ 12 ]  Since these agents do not affect the sodium channel, conduction velocity is not decreased. The prolongation of the action potential duration and refractory period, combined with the maintenance of normal conduction velocity, prevent re-entrant arrhythmias. (The re-entrant rhythm is less likely to interact with tissue that has become refractory). The class III agents exhibit reverse-use dependence (their potency increases with slower heart rates, and therefore improves maintenance of sinus rhythm). Inhibiting potassium channels results in slowed atrial-ventricular myocyte repolarization. Class III agents have the potential to prolong the QT interval of the EKG, and may be proarrhythmic (more associated with development of polymorphic VT).


Class III agents include:  bretylium ,  amiodarone ,  ibutilide ,  sotalol ,  dofetilide ,  vernakalant , and  dronedarone .


Class IV agents are slow  non-dihydropyridine   calcium channel blockers . They decrease conduction through the  AV node , and shorten phase two (the plateau) of the  cardiac action potential . They thus reduce the contractility of the heart, so may be inappropriate in heart failure. However, in contrast to beta blockers, they allow the body to retain adrenergic control of heart rate and contractility. [ citation needed ]


Class IV agents include  verapamil  and  diltiazem .


Since the development of the original Vaughan Williams classification system, additional agents have been used that do not fit cleanly into categories I through IV. Such agents include:


The initial classification system had 4 classes, although their definitions different from the modern classification. Those proposed in 1970 were: [ 2 ]


Another approach, known as the "Sicilian gambit", placed a greater approach on the underlying mechanism. [ 20 ] [ 21 ] [ 22 ]


It presents the drugs on two axes, instead of one, and is presented in tabular form. On the Y axis, each drug is listed, in roughly the Singh-Vaughan Williams order. On the X axis, the channels, receptors, pumps, and clinical effects are listed for each drug, with the results listed in a grid. It is, therefore, not a true classification in that it does not aggregate drugs into categories. [ 23 ]


A recent publication (2018) has now emerged with a fully modernised drug classification. [ 24 ]  This preserves the simplicity of the original Vaughan Williams framework while capturing subsequent discoveries of sarcolemmal, sarcoplasmic reticular and cytosolic biomolecules. The result is an expanded but pragmatic classification that encompasses approved and potential anti-arrhythmic drugs. This will aid our understanding and clinical management of cardiac arrhythmias and facilitate future therapeutic developments. It starts by considering the range of pharmacological targets, and tracks these to their particular cellular electrophysiological effects. It retains but expands the original Vaughan Williams classes I to IV, respectively covering actions on Na+ current components, autonomic signalling, K +  channel subspecies, and molecular targets related to Ca 2+  homeostasis. It now introduces new classes incorporating additional targets, including:


It also allows for multiple drug targets/actions and adverse pro-arrhythmic effects. The new scheme will additionally aid development of novel drugs under development and is illustrated here.


Asian Heart Institute  is a  cardiology  and  cardiac surgery  hospital in  Mumbai , India, which was founded in 2002 by  Ramakanta Panda . Over the last two decades, Asian Heart Institute has treated more than 450,000 patients.


Some of the cases operated at the Asian Heart Institute, led by a team of doctors under the leadership of Dr.  Ramakanta Panda , include:


The hospital added a sports medicine facility in 2007 as a number of hospitals across the country expanded their services in that practice area prior to the  2010 Commonwealth Games . [ 10 ]


As of July 2012, it was one of the two private hospitals in  Mumbai ,  India  to participate in a government insurance scheme to provide coverage for the poor. [ 11 ]


In April 2013, through its Pediatric Cardiac Center, AHI committed to providing free heart surgery for  impoverished children  who have congenital heart disease. [ 12 ] [ 13 ]


"Medgate Today" honoured Dr. Ramakanta Panda, the Chief Consultant Cardiovascular Thoracic Surgery and the Vice Chairman and managing director of Asian Heart Institute, as the No1 cardiac surgeon and one of the 25 living legends in the Healthcare of India. [ 14 ]


Asian Heart Institute is also the health partner of the Mumbai Marathon. [ 15 ]  AHI has been the Official Medical Partner for this event since its inception in 2004 and maintains First Aid stations and provides emergency medical services all along the 42-kilometer route. A contingent of AHI patients always runs in the marathon, each one having undergone a bypass surgery at AHI. [ 16 ]  Many complete the entire marathon. [ 17 ] [ 18 ]


In  electrocardiography , the  atrial action potential  are  action potentials  that occur in the  heart atrium . They are similar to  ventricular action potential  with the exception of having a more narrow phase 2 (plateau phase) due to a smaller calcium influx. Also, in comparison to the ventricular action potential, atrial action potentials have a more gradual  repolarization  period. This indicates that the atria's repolarization currents are not very large and they do not undergo a large repolarization peak. [ 1 ]


Cardiac action potential


This  muscle  article is a  stub . You can help Wikipedia by  expanding it .


An  automated external defibrillator or automatic electronic defibrillator  ( AED )  is a portable electronic device that automatically diagnoses the life-threatening  cardiac   arrhythmias  of  ventricular fibrillation  (VF) and  pulseless ventricular tachycardia , [ 1 ]  and is able to treat them through  defibrillation , the application of electricity which stops the arrhythmia, allowing the heart to re-establish an effective rhythm.


With simple audio and visual commands, AEDs are designed to be simple to use for the layperson, and the use of AEDs is taught in many  first aid ,  certified first responder , and  basic life support  (BLS) level  cardiopulmonary resuscitation  (CPR) classes. [ 2 ]


The portable version of the defibrillator was invented in the mid-1960s by  Frank Pantridge  in  Belfast ,  Northern Ireland  and the first automatic, public-use defibrillator was produced by the Cardiac Resuscitation Company in the late 1970s. The unit was launched under the name Heart-Aid. [ 3 ]


An automated external  defibrillator  is used in cases of life-threatening cardiac  arrhythmias  which lead to sudden  cardiac arrest , which is not the same as a heart attack. The rhythms that the device will treat are usually limited to:


In each of these two types of shockable  cardiac arrhythmia , the heart is electrically active, but in a dysfunctional pattern that does not allow it to pump and circulate blood. In ventricular tachycardia, the heart beats too fast to effectively pump blood. Ultimately, ventricular tachycardia leads to ventricular fibrillation. In ventricular fibrillation, the electrical activity of the heart becomes chaotic, preventing the  ventricle  from effectively pumping blood. The fibrillation in the heart decreases over time, and will eventually reach  asystole .


AEDs, like all defibrillators, are not designed to shock asystole ('flat line' patterns) as this will not have a positive clinical outcome. The asystolic patient only has a chance of survival if, through a combination of CPR and  cardiac stimulant  drugs, one of the shockable rhythms can be established, which makes it imperative for CPR to be carried out prior to the arrival of a defibrillator.


Uncorrected, these cardiac conditions (ventricular tachycardia, ventricular fibrillation, asystole) rapidly lead to irreversible  brain damage  and death, once cardiac arrest takes place. After approximately three to five minutes in cardiac arrest, [ 4 ]  irreversible brain/tissue damage may begin to occur. For every minute that a person in cardiac arrest goes without being successfully treated (by defibrillation), the chance of survival decreases by 7 percent per minute in the first three minutes, and decreases by 10 percent per minute as time advances beyond ~three minutes. [ 5 ]


AEDs are designed to be used by laypersons who ideally should have received AED training. However, sixth-grade students have been reported to begin defibrillation within 90 seconds, as opposed to a trained operator beginning within 67 seconds. [ 6 ]  This is in contrast to more sophisticated manual and semi-automatic defibrillators used by health professionals, which can act as a  pacemaker  if the heart rate is too slow ( bradycardia ) and perform other functions which require a skilled operator able to read  electrocardiograms .


Bras with a metal  underwire  and piercings on the torso must be removed before using the AED on someone to avoid interference. [ 7 ] [ 8 ]  The American television show  MythBusters  found evidence that use of a defibrillator on a woman wearing an underwire bra can lead to arcing or fire but only in unusual and unlikely circumstances. [ 9 ]


In a study analyzing the effects of having AEDs immediately present during Chicago's Heart Start program over a two-year period, of 22 individuals, 18 were in a cardiac arrhythmia which AEDs can treat. Of these 18, 11 survived. Of these 11 patients, 6 were treated by bystanders with absolutely no previous training in AED use. [ 10 ] [ 11 ]


Automated external defibrillators are generally either kept where health professionals and first responders can use them (health facilities and ambulances) as well as public access units which can be found in public places including corporate and government offices, shopping centres, restaurants, public transport, and any other location where people may congregate.


In order to make them highly visible, public access AEDs are often brightly coloured and are mounted in protective cases near the entrance of a building. When these protective cases are opened or the defibrillator is removed, some will sound a buzzer to alert nearby staff to their removal, though this does not necessarily summon emergency services; trained AED operators should know to phone for an ambulance when sending for or using an AED.  In September 2008, the  International Liaison Committee on Resuscitation  issued a 'universal AED sign' to be adopted throughout the world to indicate the presence of an AED, and this is shown on the right. [ 12 ]


A trend that is developing is the purchase of AEDs to be used in the home, particularly by those with known existing heart conditions. [ 13 ]  The number of devices in the community has grown as prices have fallen to affordable levels. There has been some concern among medical professionals that these home users do not necessarily have appropriate training, [ 14 ]  and many advocate the more widespread use of community responders, who can be appropriately trained and managed.


Typically, an AED kit will contain a  face shield  for providing a barrier between patient and first aider during rescue breathing; a pair of  nitrile rubber  gloves; a pair of  trauma shears  for cutting through a patient's clothing to expose the chest; a small towel for wiping away any moisture on the chest, and a razor for shaving those with very hairy chests. [ 15 ]


Most manufacturers recommend checking the AED before every period of duty or on a regular basis for fixed units. Some units need to be switched on in order to perform a self check; other models have a self check system built in with a visible indicator.


All manufacturers mark their electrode pads with an expiration date, and it is important to ensure that the pads are in date. The typical life expectancy of AED pads are between 18 and 30 months. [ 16 ]  This is usually marked on the outside of the pads. Some models are designed to make this date visible through a 'window', although others will require the opening of the case to find the date stamp. [ citation needed ]


It is also important to ensure that the AED unit's batteries have not expired.  The AED manufacturer will specify how often the batteries should be replaced. Each AED has a different recommended maintenance schedule outlined in the user manual. Common checkpoints on every checklist, however, also include a monthly check of the battery power by checking the green indicator light when powered on, condition and cleanliness of all cables and the unit, and check for the adequate supplies. [ 17 ]


An AED is "automatic" because of the unit's ability to autonomously analyse the patient's condition. To assist this, the vast majority of units have spoken prompts, and some may also have visual displays to instruct the user.


"External" refers to the fact that the operator applies the electrode pads to the bare  chest  of the victim (as opposed to  internal defibrillators , which have  electrodes  surgically implanted inside the body of a patient).


When turned on or opened, the AED will instruct the user to connect the electrodes (pads) to the patient. Once the pads are attached, everyone should avoid touching the patient so as to avoid false readings by the unit. The pads allow the AED to examine the electrical output from the heart and determine if the patient is in a shockable rhythm (either ventricular fibrillation or ventricular tachycardia). If the device determines that a shock is warranted, it will use the battery to charge its internal  capacitor  in preparation to deliver the shock. The device system is not only safer - charging only when required, but also allows for a faster delivery of the electric current.


When charged, the device instructs the user to ensure no one is touching the patient and then to press a button to deliver the shock; human intervention is usually required to deliver the shock to the patient in order to avoid the possibility of accidental injury to another person (which can result from a responder or bystander touching the patient at the time of the shock). Depending on the manufacturer and particular model, after the shock is delivered most devices will analyze the patient and either instruct CPR to be performed, or prepare to administer another shock.


Many AED units have an 'event memory' which store the ECG of the patient along with details of the time the unit was activated and the number and strength of any shocks delivered. Some units also have voice recording abilities [ 18 ]  to monitor the actions taken by the personnel in order to ascertain if these had any impact on the survival outcome. All this recorded data can be either downloaded to a computer or printed out so that the providing organisation or responsible body is able to see the effectiveness of both CPR and defibrillation. Some AED units even provide feedback on the quality of the compressions provided by the rescuer. [ 19 ] [ 20 ]


The first commercially available AEDs were all of a monophasic type, which gave a high-energy shock, up to 360 to 400  joules  depending on the model. This caused increased cardiac injury and in some cases second and third-degree burns around the shock pad sites.  Newer AEDs (manufactured after late 2003) have tended to utilise biphasic algorithms which give two sequential lower-energy shocks of 120–200 joules, with each shock moving in an opposite polarity between the pads.  Others may give a stepped approach to energy delivery, usually in a 200J, a second 200J, then 300J, and finally 360J shock, with any further shocks also being 360 Joules. This lower-energy waveform has proven more effective in clinical tests, as well as offering a reduced rate of complications and reduced recovery time. [ 21 ]


Unlike regular  defibrillators , an automated external defibrillator (AED) requires minimal training to be used (or even no training). That is possible because all AEDs approved for use in the  United States  and many other countries use an electronic voice to prompt users through each step. Many AEDs now include visual prompts in case of a  hearing impaired  user. Most units are designed for use by non-medical operators. Their ease of use has given rise to the notion of public access defibrillation (PAD).


An AED automatically diagnoses the heart rhythm and determines if a shock is needed. Automatic models will administer the shock without the user's command. Semi-automatic models will tell the user that a shock is needed, but the user must tell the machine to do so, usually by pressing a button. In most circumstances, the user cannot override a "no shock" advisory by an AED. Some AEDs may be used on children – those under 55 lbs (25 kg) in weight or under age 8. If a particular model of AED is approved for pediatric use,  all that is required is the use of more appropriate pads [ failed verification ] . [ 22 ]


Observational studies have shown that in out of hospital cardiac arrest, public access defibrillators when used were associated with 40% median survival. When operated by non-dispatched lay first responders they have the highest likelihood of leading to survival. [ 23 ]


Automated external defibrillators are now easy enough to use that most states in the United States include the "good faith" use of an AED by any person under  Good Samaritan laws . [ 24 ]  "Good faith" protection under a Good Samaritan law means that a volunteer responder (not acting as a part of one's occupation) cannot be held civilly liable for the harm or death of a victim by providing improper or inadequate care, given that the harm or death was not intentional and the responder was acting within the limits of their training and in good faith. In the United States, Good Samaritan laws provide some protection for the use of AEDs by trained and untrained responders. [ 25 ]  AEDs create little liability if used correctly; [ 26 ]  NREMT-B and many state  Emergency Medical Technician  (EMT) training and many CPR classes incorporate or offer AED education as a part of their program.


In addition to Good Samaritan laws, Ontario, Canada also has the " Chase McEachern  Act (Heart Defibrillator Civil Liability), 2007 (Bill 171 – Subsection N)", passed in June, 2007, [ 27 ]  which protects individuals from liability for damages that may occur from their use of an AED to save someone's life at the immediate scene of an emergency unless damages are caused by gross negligence.


Legislation in Australia varies by state, with separate liability issues relating to providing and using AED equipment.
Each state and territory has enacted "Good Samaritan" laws that offer legal protection to a person who gives assistance in a medical emergency - the standard of care expected corresponds to their training (or lack of training). [ 28 ]
In New South Wales, the Work Health and Safety Regulation (2011) requires an employer to use a risk assessment to ensure that there is adequate provision for first aid; when there is a sufficient risk it warrants providing a defibrillator. [ 29 ]


In 2012, AED's (automated external defibrillators) were under scrutiny by the U.S.  Food and Drug Administration  (FDA) which considered reclassifying AEDs as class III  premarket approval  devices. Technical malfunctions likely contributed to more than 750 deaths in the 5-year period between 2004 and 2009, in most cases by component failures or design errors. During the same period, up to 70 types of AEDs were recalled, including recalls from every AED manufacturer in the world.
 [ 30 ]


In January and February 2015, the FDA issued this news release: "The FDA issued a final order that will require AED manufacturers to submit premarket approval applications (PMAs), which undergo a more rigorous review than what was required to market these devices in the past. The agency's strengthened review will focus on the critical requirements needed to ensure the safety and reliability of AEDs and their necessary accessories, including batteries, pad electrodes, adapters and hardware keys for pediatric use." [ 31 ] [ 32 ]


In the United Kingdom there is concern that poor maintenance may make public defibrillators unreliable.  The  Henley Standard  reported on 21 July 2017 that more than half the defibrillators in  Henley-on-Thames  and the surrounding area were at risk of failing, either because of low battery power or because adhesive pads had deteriorated. [ 33 ]


The first use of an external defibrillator on a human was in 1947 by Claude Beck. [ 34 ]  The portable version of the external defibrillator was invented in 1957 by  Frank Pantridge  in  Belfast ,  Northern Ireland , a pioneer in emergency medical treatment. [ 35 ] [ 36 ]  Pantridge's defibrillator required a trained operator to perform the shock procedure and charted a course for many new innovations in external defibrillation. [ 37 ]


In the late 1970s the Heart-Aid was developed as the first truly automated external defibrillator that was designed for the public. The principles of ABC assessment and a human voice relaying instructions helped bystanders respond to a sudden cardiac event while waiting for the first responders to get to scene. [ 3 ]  Many of the early innovations in the Heart-Aid model are still part of the current generation of AEDs, although some innovations, like the airway electrode have fallen from use.


In a study published in 2017, researchers in Poland selected the main entrances of buildings which had AEDs, although the researchers themselves did not know the exact locations of the devices.  In drills of pretend heart attack, the average time to bring the AED to the patient was 96 seconds, with a time that ranged from 52 to 144 seconds.  This met the three minute goal.  In some cases, the use of the AED required the continuous presence of building personnel.  Future improvements include more obvious signage and public-access AEDs which do not require a staff member of the building to either retrieve or use the device. [ 38 ]
 [ 39 ]


The  ballistocardiograph  ( BCG ) is a measure of  ballistic  forces generated by the heart. [ 1 ]  The downward movement of blood through the  descending aorta  produces an  upward recoil , moving the body upward with each heartbeat. [ 2 ]  As different parts of the aorta expand and contract, the body continues to move downward and upward in a repeating pattern. [ 3 ]  Ballistocardiography is a technique for producing a graphical representation of repetitive motions of the human body arising from the sudden ejection of blood into the great vessels with each heart beat. [ 4 ]    It is a  vital sign  in the 1–20  Hz   frequency  range which is caused by the mechanical movement of the  heart  and can be recorded by  noninvasive  methods from the surface of the  body . It was shown for the first time, after an extensive research work by Dr.  Isaac Starr , that the effect of main heart malfunctions can be identified by observing and analyzing the BCG signal. [ 5 ]  Recent [ when? ]  work also validates BCG could be monitored using camera in a non-contact manner. [ 6 ]


One example of the use of a BCG is a ballistocardiographic scale, which measures the recoil of the persons body who is on the scale. A BCG scale is able to show a person's heart rate as well as their weight. [ citation needed ]


The term ballistocardiograph originated from the Roman  ballista , which is derived from the  Greek  word  ballein  (to throw), a machine for launching missiles, plus the Greek words for heart and writing. [ citation needed ]


The  bidomain model  is a  mathematical model  to define the electrical activity of the  heart . It consists in a continuum (volume-average) approach in which the cardiac microstructure is defined in terms of muscle fibers grouped in sheets, creating a complex  three-dimensional  structure with anisotropical properties. Then, to define the electrical activity, two interpenetrating domains are considered, which are the  intracellular  and  extracellular  domains, representing respectively the space inside the cells and the region between them. [ 1 ]


The bidomain model was first proposed by  Schmitt  in 1969 [ 2 ]  before being formulated mathematically in the late 1970s. [ 3 ] [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ]


Since it is a continuum model, rather than describing each cell individually, it represents the average properties and behaviour of group of cells organized in complex structure. Thus, the model results to be a complex one and can be seen as a generalization of the  cable theory  to higher dimensions and, going to define the so-called  bidomain equations . [ 11 ] [ 12 ]


Many of the interesting properties of the bidomain model arise from the condition of unequal anisotropy ratios. The  electrical conductivity  in anisotropic tissues is not unique in all directions, but it is different in parallel and perpendicular direction with respect to the fiber one.
Moreover, in tissues with unequal anisotropy ratios, the ratio of conductivities parallel and perpendicular to the fibers are different in the intracellular and extracellular spaces. For instance, in cardiac tissue, the anisotropy ratio in the  intracellular space  is about 10:1, while in the  extracellular space  it is about 5:2. [ 13 ]
Mathematically, unequal anisotropy ratios means that the effect of anisotropy cannot be removed by a change in the distance scale in one direction. [ 14 ]
Instead, the anisotropy has a more profound influence on the electrical behavior. [ 15 ]


Three examples of the impact of unequal anisotropy ratios are


The bidomain domain is principally represented by two main regions: the cardiac cells, called intracellular domain, and the space surrounding them, called extracellular domain. Moreover, usually another region is considered, called extramyocardial region.
The intracellular and extracellular domains, which are separate by the  cellular membrane , are considered to be a unique physical space representing the heart (

 H

 {\displaystyle \mathbb {H} }

 ), while the extramyocardial domain is a unique physical space adjacent of them (

 T

 {\displaystyle \mathbb {T} }

 ). The extramyocardial region can be considered as a fluid bath, especially when one wants to simulate experimental conditions, or as a  human torso  to simulate physiological conditions. [ 12 ]
The boundary of the two principal physical domains defined are important to solve the bidomain model. Here the heart boundary is denoted as

 ∂

 H

 {\displaystyle \partial \mathbb {H} }

  while the torso domain boundary is

 ∂

 T

 .

 {\displaystyle \partial \mathbb {T} .}

 [ 12 ]


The unknowns in the bidomain model are three, the intracellular potential

 v

 i

 {\displaystyle v_{i}}

 , the extracellular potential

 v

 e

 {\displaystyle v_{e}}

  and the  transmembrane potential

 v

 {\displaystyle v}

 , which is defined as the difference of the potential across the cell membrane

 v
 =

 v

 i

 −

 v

 e

 {\displaystyle v=v_{i}-v_{e}}

 . [ 12 ]


Moreover, some important parameters need to be taken in account, especially the intracellular conductivity tensor matrix

 Σ

 i

 {\displaystyle \mathbf {\Sigma } _{i}}

 ,  the extracellular conductivity tensor matrix

 Σ

 e

 {\displaystyle \mathbf {\Sigma } _{e}}

 .  The transmembrane current flows between the intracellular and extracellular regions and it is in part described by the corresponding ionic current over the membrane per unit area

 I

 ion

 {\displaystyle I_{\text{ion}}}

 .
Moreover, the membrane capacitance per unit area

 C

 m

 {\displaystyle C_{m}}

  and the surface to volume ratio of the cell membrane

 χ

 {\displaystyle \chi }

  need to be considered to derive the bidomain model formulation, which is done in the following  section . [ 12 ]


The bidomain model is defined through two  partial differential equations  (PDE) the first of which is a  reaction diffusion equation  in terms of the  transmembrane potential , while the second one computes the extracellular potential starting from a given transmembran potential distribution. [ 12 ]


Thus, the bidomain model can be formulated as follows:

 ∇
 ⋅

 (

 Σ

 i

 ∇
 v

 )

 +
 ∇
 ⋅

 (

 Σ

 i

 ∇

 v

 e

 )

 =
 χ

 (

 C

 m

 ∂
 v

 ∂
 t

 +

 I

 i
 o
 n

 )

 −

 I

 s

 1

 ∇
 ⋅

 (

 (

 Σ

 i

 +

 Σ

 e

 )

 ∇

 v

 e

 )

 =
 −
 ∇
 ⋅

 (

 Σ

 i

 ∇
 v

 )

 +

 I

 s

 2

 {\displaystyle {\begin{alignedat}{2}&\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s_{1}}\\[1ex]&\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+I_{s_{2}}\end{alignedat}}}

where

 I

 s

 1

 {\displaystyle I_{s_{1}}}

  and

 I

 s

 2

 {\displaystyle I_{s_{2}}}

  can be defined as applied external stimulus currents. [ 12 ]


The ionic current is usually represented by an  ionic model  through a system of  ordinary differential equations  (ODEs). Mathematically, one can write

 I

 ion

 =

 I

 ion

 (
 v
 ,

 w

 )

 {\displaystyle I_{\text{ion}}=I_{\text{ion}}(v,\mathbf {w} )}

  where

 w

 {\displaystyle \mathbf {w} }

  is called ionic variable. Then, in general, for all

 t
 >
 0

 {\displaystyle t>0}

 , the system reads [ 19 ]

 {

 ∂

 w

 ∂
 t

 =

 F

 (
 v
 ,

 w

 )

 in

 H

 w

 (
 t
 =
 0
 )
 =

 w

 0

 in

 H

 {\displaystyle {\begin{cases}{\dfrac {\partial \mathbf {w} }{\partial t}}=\mathbf {F} (v,\mathbf {w} )&{\text{in }}\mathbb {H} \\\mathbf {w} (t=0)=\mathbf {w} _{0}&{\text{in }}\mathbb {H} \end{cases}}}


Different ionic models have been proposed: [ 19 ]


In some cases, an extramyocardial region is considered. This implies the addition to the bidomain model of an equation describing the potential propagation inside the extramyocardial domain. [ 12 ]


Usually, this equation is a simple generalized  Laplace equation  of type [ 12 ]

 −
 ∇
 ⋅
 (

 Σ

 0

 ∇

 v

 0

 )
 =
 0

 x

 ∈

 T

 {\displaystyle -\nabla \cdot (\mathbf {\Sigma } _{0}\nabla v_{0})=0\quad \mathbf {x} \in \mathbb {T} }

where

 v

 0

 {\displaystyle v_{0}}

  is the potential in the extramyocardial region and

 Σ

 0

 {\displaystyle \mathbf {\Sigma } _{0}}

  is the corresponding conductivity tensor.


Moreover, an isolated domain assumption is considered, which means that the following boundary conditions are added

 (

 Σ

 0

 ∇

 v

 0

 )
 ⋅

 n

 0

 =
 0

 x

 ∈
 ∂

 T

 ,

 {\displaystyle (\mathbf {\Sigma } _{0}\nabla v_{0})\cdot \mathbf {n} _{0}=0\quad \mathbf {x} \in \partial \mathbb {T} ,}

 n

 0

 {\displaystyle \mathbf {n} _{0}}

  being the unit normal directed outside of the extramyocardial domain. [ 12 ]


If the extramyocardial region is the human torso, this model gives rise to the  forward problem of electrocardiology . [ 12 ]


The bidomain equations are derived from the  Maxwell's equations  of the electromagnetism, considering some simplifications. [ 12 ]


The first assumption is that the intracellular current can flow only between the intracellular and extracellular regions, while the intracellular and extramyocardial regions can comunicate between them, so that the current can flow into and from the extramyocardial regions but only in the extracellular space. [ 12 ]


Using  Ohm's law  and a quasi-static assumption, the gradient of a scalar potential field

 φ

 {\displaystyle \varphi }

  can describe an electrical field

 E

 {\displaystyle \mathbf {E} }

 , which means that [ 12 ]

 E

 =
 −
 ∇
 φ
 .

 {\displaystyle \mathbf {E} =-\nabla \varphi .}


Then, if

 J

 {\displaystyle J}

  represent the current density of the electric field

 E

 {\displaystyle \mathbf {E} }

 , two equations can be obtained [ 12 ]

 J

 i

 =
 −

 Σ

 i

 ∇

 v

 i

 J

 e

 =
 −

 Σ

 e

 ∇

 v

 e

 .

 {\displaystyle {\begin{alignedat}{2}J_{i}&=-\mathbf {\Sigma } _{i}\nabla v_{i}\\J_{e}&=-\mathbf {\Sigma } _{e}\nabla v_{e}.\end{alignedat}}}

where the subscript

 i

 {\displaystyle i}

  and

 e

 {\displaystyle e}

  represent the intracellular and extracellular quantities respectively. [ 12 ]


The second assumption is that the heart is isolated so that the current that leaves one region need to flow into the other. Then, the current density in each of the intracellular and extracellular domain must be equal in magnitude but opposite in sign, and can be defined as the product of the surface to volume ratio of the cell membrane and the transmembrane ionic current density

 I

 m

 {\displaystyle I_{m}}

  per unit area, which means that [ 12 ]

 −
 ∇
 ⋅

 J

 i

 =
 ∇
 ⋅

 J

 e

 =
 χ

 I

 m

 .

 {\displaystyle -\nabla \cdot J_{i}=\nabla \cdot J_{e}=\chi I_{m}.}


By combining the previous assumptions, the conservation of current densities is obtained, namely [ 12 ]


from which, summing the two equations [ 12 ]


This equation states exactly that all currents exiting one domain must enter the other. [ 12 ]


From here, it is easy to find the second equation of the bidomain model subtracting

 ∇
 ⋅
 (

 Σ

 i

 ∇

 v

 e

 )

 {\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}

  from both sides. In fact, [ 12 ]

 ∇
 ⋅
 (

 Σ

 i

 ∇

 v

 i

 )
 −
 ∇
 ⋅
 (

 Σ

 i

 ∇

 v

 e

 )
 =
 −
 ∇
 ⋅
 (

 Σ

 e

 ∇

 v

 e

 )
 −
 ∇
 ⋅
 (

 Σ

 i

 ∇

 v

 e

 )

 {\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{i})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})=-\nabla \cdot (\mathbf {\Sigma } _{e}\nabla v_{e})-\nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}

and knowing that the transmembral potential is defined as

 v
 =

 v

 i

 −

 v

 e

 {\displaystyle v=v_{i}-v_{e}}

 [ 12 ]

 ∇
 ⋅
 (

 Σ

 i

 ∇
 v
 )
 =
 −
 ∇
 ⋅
 (
 (

 Σ

 i

 +

 Σ

 e

 )
 ∇

 v

 e

 )
 .

 {\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v)=-\nabla \cdot ((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}).}

Then, knowing the transmembral potential, one can recover the extracellular potential.


Then, the current that flows across the cell membrane can be modelled with the  cable equation , [ 12 ]


Combining equations ( 1 ) and ( 2 ) gives [ 12 ]

 ∇
 ⋅

 (

 Σ

 i

 ∇

 v

 i

 )

 =
 χ

 (

 C

 m

 ∂
 v

 ∂
 t

 +

 I

 i
 o
 n

 )

 .

 {\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{i}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{ion}\right).}

Finally, adding and subtracting

 ∇
 ⋅
 (

 Σ

 i

 ∇

 v

 e

 )

 {\displaystyle \nabla \cdot (\mathbf {\Sigma } _{i}\nabla v_{e})}

  on the left and rearranging

 v
 =

 v

 i

 −

 v

 e

 {\displaystyle v=v_{i}-v_{e}}

 , one can get the first equation of the bidomain model [ 12 ]

 ∇
 ⋅

 (

 Σ

 i

 ∇
 v

 )

 +
 ∇
 ⋅

 (

 Σ

 i

 ∇

 v

 e

 )

 =
 χ

 (

 C

 m

 ∂
 v

 ∂
 t

 +

 I

 i
 o
 n

 )

 ,

 {\displaystyle \nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right),}

which describes the evolution of the transmembrane potential in time.


The final formulation described in the  standard formulation  section is obtained through a generalization, considering possible external stimulus which can be given through the external applied currents

 I

 s

 1

 {\displaystyle I_{s_{1}}}

  and

 I

 s

 2

 {\displaystyle I_{s_{2}}}

 . [ 12 ]


In order to solve the model, boundary conditions are needed. The more classical boundary conditions are the following ones, formulated by Tung. [ 6 ]


First of all, as state before in the  derive  section, there ca not been any flow of current between the intracellular and extramyocardial domains. This can be mathematically described as [ 12 ]

 (

 Σ

 i

 ∇

 v

 i

 )
 ⋅

 n

 =
 0

 x

 ∈
 ∂

 H

 {\displaystyle (\mathbf {\Sigma } _{i}\nabla v_{i})\cdot \mathbf {n} =0\quad \mathbf {x} \in \partial \mathbb {H} }

where

 n

 {\displaystyle \mathbf {n} }

  is the vector that represents the outwardly unit normal to the myocardial surface of the heart.
Since the intracellular potential is not explicitily presented in the bidomain formulation, this condition is usually described in terms of the transmembrane and extracellular potential, knowing that

 v
 =

 v

 i

 −

 v

 e

 {\displaystyle v=v_{i}-v_{e}}

 , namely [ 12 ]

 (

 Σ

 i

 ∇
 v
 )
 ⋅

 n

 =
 −
 (

 Σ

 i

 ∇

 v

 e

 )
 ⋅

 n

 x

 ∈
 ∂

 H

 .

 {\displaystyle (\mathbf {\Sigma } _{i}\nabla v)\cdot \mathbf {n} =-(\mathbf {\Sigma } _{i}\nabla v_{e})\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}


For the extracellular potential, if the myocardial region is presented, a balance in the flow between the extracellular and the extramyocardial regions is considered [ 12 ]

 (

 Σ

 e

 ∇

 v

 e

 )

 ⋅

 n

 e

 =
 −

 (

 Σ

 0

 ∇

 v

 0

 )

 ⋅

 n

 0

 x

 ∈
 ∂

 H

 .

 {\displaystyle \left(\mathbf {\Sigma } _{e}\nabla v_{e}\right)\cdot \mathbf {n} _{e}=-\left(\mathbf {\Sigma } _{0}\nabla v_{0}\right)\cdot \mathbf {n} _{0}\quad \mathbf {x} \in \partial \mathbb {H} .}

Here the normal vectors from the perspective of both domains are considered, thus the negative sign are necessary. Moreover, a perfect transmission of the potential on the cardiac boundary is necessary, which gives [ 12 ]

 v

 e

 =

 v

 0

 x

 ∈
 ∂

 H

 .

 {\displaystyle v_{e}=v_{0}\quad \mathbf {x} \in \partial \mathbb {H} .}


Instead, if the heart is considered as isolated, which means that no myocardial region is presented, a possible boundary condition for the extracellular problem is [ 12 ]

 (

 Σ

 i

 ∇
 v

 )

 ⋅

 n

 =
 −

 (

 (

 Σ

 i

 +

 Σ

 e

 )
 ∇

 v

 e

 )

 ⋅

 n

 x

 ∈
 ∂

 H

 .

 {\displaystyle \left(\mathbf {\Sigma } _{i}\nabla v\right)\cdot \mathbf {n} =-\left((\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e})\nabla v_{e}\right)\cdot \mathbf {n} \quad \mathbf {x} \in \partial \mathbb {H} .}


By assuming equal anisotropy ratios for the intra- and extracellular domains, i.e.

 Σ

 i

 =
 λ

 Σ

 e

 {\displaystyle \mathbf {\Sigma } _{i}=\lambda \mathbf {\Sigma } _{e}}

  for some scalar

 λ

 {\displaystyle \lambda }

 , the model can be reduced to one single equation, called  monodomain equation

 ∇
 ⋅
 (

 Σ

 ∇
 v
 )
 =
 χ

 (

 C

 m

 ∂
 v

 ∂
 t

 +

 I

 i
 o
 n

 )

 −

 I

 s

 {\displaystyle \nabla \cdot (\mathbf {\Sigma } \nabla v)=\chi \left(C_{m}{\frac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s}}

where the only variable is now the transmembrane potential, and the conductivity tensor

 Σ

 {\displaystyle \mathbf {\Sigma } }

  is a combination of

 Σ

 i

 {\displaystyle \mathbf {\Sigma } _{i}}

  and

 Σ

 e

 .

 {\displaystyle \mathbf {\Sigma } _{e}.}

 [ 12 ]


If the heart is considered as an isolated tissue, which means that no current can flow outside of it, the final formulation with boundary conditions reads [ 12 ]

 {

 ∇
 ⋅

 (

 Σ

 i

 ∇
 v

 )

 +
 ∇
 ⋅

 (

 Σ

 i

 ∇

 v

 e

 )

 =
 χ

 (

 C

 m

 ∂
 v

 ∂
 t

 +

 I

 i
 o
 n

 )

 −

 I

 s

 1

 x

 ∈

 H

 ∇
 ⋅

 (

 (

 Σ

 i

 +

 Σ

 e

 )

 ∇

 v

 e

 )

 =
 −
 ∇
 ⋅

 (

 Σ

 i

 ∇
 v

 )

 +

 I

 s

 2

 x

 ∈

 H

 Σ

 i

 (
 ∇
 v
 +
 ∇

 v

 e

 )
 ⋅

 n

 =
 0

 x

 ∈
 ∂

 H

 [

 Σ

 i

 (
 ∇
 v
 +
 ∇

 v

 e

 )
 +

 Σ

 e

 ∇

 v

 e

 ]

 ⋅

 n

 =
 0

 x

 ∈
 ∂

 H

 {\displaystyle {\begin{cases}\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v_{e}\right)=\chi \left(C_{m}{\dfrac {\partial v}{\partial t}}+I_{\mathrm {ion} }\right)-I_{s_{1}}&\mathbf {x} \in \mathbb {H} \\[1ex]\nabla \cdot \left(\left(\mathbf {\Sigma } _{i}+\mathbf {\Sigma } _{e}\right)\nabla v_{e}\right)=-\nabla \cdot \left(\mathbf {\Sigma } _{i}\nabla v\right)+I_{s_{2}}&\mathbf {x} \in \mathbb {H} \\[1ex]\mathbf {\Sigma } _{i}(\nabla v+\nabla v_{e})\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \\[1ex]\left[\mathbf {\Sigma } _{i}(\nabla v+\nabla v_{e})+\mathbf {\Sigma } _{e}\nabla v_{e}\right]\cdot \mathbf {n} =0&\mathbf {x} \in \partial \mathbb {H} \end{cases}}}


There are various possible techniques to solve the bidomain equations. Between them, one can find  finite difference schemes ,  finite element schemes  and also  finite volume schemes . Special considerations can be made for the numerical solution of these equations, due to the high time and space resolution needed for numerical convergence. [ 20 ] [ 21 ]


A  biological pacemaker  is one or more types of cellular components that, when "implanted or injected into certain regions of the heart," produce specific electrical stimuli that mimic that of the body's natural  pacemaker cells . [ 1 ]  Biological pacemakers are indicated for issues such as  heart block ,  slow heart rate , and  asynchronous heart ventricle contractions . [ 2 ] [ 3 ]


The biological pacemaker is intended as an alternative to the  artificial cardiac pacemaker  that has been in human use since the late 1950s. Despite their success, several limitations and problems with artificial pacemakers have emerged during the past decades such as  electrode  fracture or damage to  insulation ,  infection , re-operations for  battery  exchange, and  venous thrombosis . The need for an alternative is most obvious in children, including premature  newborn babies , where size mismatch and the fact that pacemaker leads do not grow with children are a problem. [ 1 ]   A more biological approach has been taken in order to mitigate many of these issues. However, the implanted biological pacemaker cells still typically need to be supplemented with an artificial pacemaker while the cells form the necessary electrical connections with cardiac tissue. [ 1 ]


The first successful experiment with biological pacemakers was carried out by  Arjang Ruhparwar  's group at  Hannover Medical School  in Germany using transplanted  fetal  heart  muscle cells . The process was first introduced at the scientific sessions of the  American Heart Association  in  Anaheim  in 2001, and the results were published in 2002. [ 4 ]  A few months later, Eduardo Marban's group from  Johns Hopkins University  published the first successful gene-therapeutic approach towards the generation of pacemaking activity in otherwise non-pacemaking adult  cardiomyocytes  using a  guinea pig  model. [ 5 ]  The investigators postulated latent pacemaker capability in normal  heart muscle  cells. This potential ability is suppressed by the  inward-rectifier potassium current  Ik1 encoded by the  gene  Kir2 which is not expressed in  pacemaker cells . By specific inhibition of Ik1 below a certain level, spontaneous activity of cardiomyocytes was observed with resemblance to the action potential pattern of genuine pacemaker cells.


Meanwhile, other genes and cells have been discovered, including heart muscle cells derived from  embryonic stem cells , "HCN" genes which encode the  wild type  pacemaker current I(f). Michael Rosen's group demonstrated that transplantation of HCN2-transfected human mesenchymal  stem cells  (hMSCs) leads to expression of functional HCN2 channels  in vitro  and  in vivo , mimicking overexpression of HCN2 genes in cardiac  myocytes . [ 6 ]  In 2010, Ruhparwar's group again demonstrated a type of biological pacemaker, this time showing that by injection of the "Adenylate Cyclase" gene into the heart muscle a biological cardiac pacemaker can be created. [ 7 ]


In 2014, a gene called  TBX18  has been non-invasively applied to speed up heart rates caused by heart block. [ 2 ]  More recent studies in 2015, has been experimented  optogenetic  approach in the rats heart, where a light sensitive  transgene  ( Channelrhodopsin -2) injected to several sites of rat's  ventricular , which, furthermore, can simultaneously stimulate the injection sites by a blue light irradiation. [ 3 ]


Biotronik  (BIOTRONIK  SE  & Co.  KG ; [ 1 ] [ 2 ]  Biotronik Worldwide) is a  limited partnership [ 1 ]  multi-national  cardiovascular   biomedical  research and technology company, headquartered in  Berlin , Germany.


The company offers equipment for diagnosis, treatment, and therapy support in the areas of  cardiac rhythm  management,  electrophysiology , and  vascular  intervention. [ 3 ]   In the area of cardiac rhythm management, Biotronik Home Monitoring uses tele-monitoring technology to provide doctors with up-to-date information for implant patients. [ 4 ]


Biotronik employs more than 9,000 people worldwide in over 100 countries, with  research and development  activities in Europe, North America, and Singapore.  It produces all critical components of its products in-house.  One in every five employees at its Berlin headquarters works in research and development (R&D). [ citation needed ]


BIOTRONIK began with the development of the first German implantable  pacemaker  (Biotronik IP-03) in 1963.  The pacemaker was developed at  Technische Universität Berlin  by physicist Professor. Dr.  Max Schaldach  (1936–2001 [ 5 ] ), [ 6 ] [ 7 ]  a professor of biomedical technology at the Friedrich-Alexander- University of Erlangen-Nuremberg  (FAU), [ 8 ]  and electrical engineer Otto Franke.  In the early years, the company worked to improve pacemakers' capacity and battery life, and secure the connections among electrodes, pacemakers, and the heart.  Since its start, more than 19 million BIOTRONIK devices in over 100 countries have been implanted. [ citation needed ]


In 1976, the company moved to Sieversufer 8 in  Berlin-Neukölln .  In 1979, a US production site in  Lake Oswego, Oregon  was built.  This subsidiary resulted from the acquisition of the American pacemaker producer Stimulation Technology, Inc.  At the same time, with the development and production of advanced hybrid circuitry and structural components for the medical technology industry, the company also began to develop circuits for pacemakers.  In the 1980s, the  dual-chamber stimulation method  (DDD) was developed, leading to the manufacturing of a pacemaker that could read and react to spontaneous contractions of the atrium, and better respond to them of its own accord.  To this end, BIOTRONIK developed the Diplos 03, a multi-programme DDD pacemaker with bilateral telemetry, which made it a European market leader and increased its presence in South America and Asia.


In 1987, the firm moved its headquarters to Woermannkehre 1, next door to its previous location.  In 1993, BIOTRONIK produced the first German implantable cardioverter defibrillator (ICD), among them the Phylax 06.  Closed loop stimulation (CLS), which integrates the pacemaker into the body's own regulatory system, thereby allowing it to react to patients’ changing physical and emotional activity, was introduced in the 1990s.  Also in 1993, BIOTRONIK developed fractal coating for implantable electrodes.  This coating optimises the electrically active surface of the electrode, thereby improving its perception and stimulation properties.  BIOTRONIK remains the only manufacturer of fractal-coated electrodes.


In 1994 and 1995 respectively, BIOTRONIK began offering a full spectrum of  electrophysiology  products and vascular intervention products.  BIOTRONIK also develops and produces  balloon catheters  and  stents  for the treatment of  coronary artery disease .  With Philos, the company has offered a complete pacemaker family with telemetry since 2000, when it received  CE Mark  approval for the product, and also successfully implanted the first pacemaker with Home Monitoring ( remote patient monitoring ).  Home Monitoring has shown significant clinical benefits, including over a 50% reduction in mortality of heart failure patients. [ 9 ]


With the Lumax 540 VR-T DX in 2010, BIOTRONIK launched the first and only single-chamber defibrillator with comprehensive atrial diagnostics worldwide.  Additionally, the company entered into an exclusive international distribution partnership with the Swiss medical technology company  Endosense  to distribute their ablation catheter TactiCath with optical contact force.  The following year, BIOTRONIK released Orsiro to the market, the world's first hybrid drug-eluting stent with a bio-absorbable coating, adding to innovative treatment options combatting coronary artery disease. [ citation needed ]


In 2011, BIOTRONIK was the subject of an investigation by the  United States Department of Justice  (US DoJ) into payments made to doctors in  Nevada , United States, who use the company's products in their practices. [ 10 ]   The case was settled in 2014. [ 11 ]   In 2013, a similar investigation began and was settled involving payments to physicians in  Oregon . [ 12 ] [ 13 ]


In 2012, the company acquired the old  Postfuhramt , a historical brick postal building on Berlin's  Oranienburger Strasse  in the sub-neighbourhood of  Spandauer Vorstadt , in the district of  Mitte . [ 14 ]


The following year, BIOTRONIK launched BioMonitor, a type of mini  ECG  device that offers continuous monitoring and daily remote data collection.  In addition, BIOTRONIK also developed the world's first series of implantable defibrillators that enable patients, including those suffering heart failure, to undergo  MRI scans  under certain conditions.  BIOTRONIK's ProMRI® technology includes systems approved for 1.5 T and 3.0 T MR scanning, as well as full-body scanning.  The company offers the broadest portfolio of pacemakers, defibrillators, and therapies approved to undergo MRIs, with more than 1,000,000 ProMRI® devices and leads implanted worldwide. [ 15 ]   To navigate through this portfolio, BIOTRONIK invented two online tools in 2016 and 2017; the  ProMRI®SystemCheck  and  ProMRI®Configurator , correspondingly.


For radiation protection, BIOTRONIK distributes the Zero-Gravity Radiation Protection System, which consists of a movable suspended body and head shield.  The shield material has a significantly higher lead equivalency than traditional radiation apparel, thereby reducing radiation exposure by 87–100%. [ 16 ]


Indicated to treat peripheral artery disease in the lower limbs, BIOTRONIK released the Passeo-18 Lux in 2014 as the first peripheral drug-coated balloon.  The year following, CardioMessenger Smart was launched, its new patient device for Home Monitoring, [ 17 ]  and BioMonitor 2, the second-generation insertable cardiac monitor. [ 18 ]


In February 2016, BIOTRONIK Inc. opened an Education and Innovation Center, a training facility and meeting location in New York City. [ 19 ]


In 2007, BIOTRONIK was given the EuroPCR 2007 Novelty Award for its innovative absorbable metal stents (AMS) by the European Association of Percutaneous Cardiovascular Interventions (EAPCI).  In 2009, it was nominated for the German Future Prize for its Home Monitoring system by the German Federal Ministry of Education and Research.  In 2010, BIOTRONIK endowed the  Berlin-Brandenburg Academy of Sciences and Humanities ' Technical Science Prize, first awarded to Till Schlösser.  The CARDIOSTIM Innovation Award for Practice Improvement was granted to BIOTRONIK for its MRI AutoDetect. [ 20 ]


BIOTRONIK collaborates with the EPIC Alliance, a global network of female electrophysiologists. The alliance aims to enhance collaboration and support for women in the field of electrophysiology.


In 2014, Biotronik faced allegations regarding kickback payments to cardiologists, leading to a settlement of nearly five million dollars. [ 21 ] [ 22 ] [ 23 ] [ 24 ]  Currently, another case is ongoing in the California Central District Court. Additionally,


In a landmark case from New Mexico in 2016, a jury found Biotronik negligent for unnecessarily implanting a pacemaker in Tommy Sowards due to an alleged conspiracy involving the company, a salesman, a hospital, and a cardiologist. The lawsuit exposed a scheme where Biotronik paid kickbacks to the cardiologist, significantly inflating his earnings per procedure. Despite later advice from medical professionals that the pacemaker was not needed, it could not be removed without risking heart damage. As a result, the jury awarded Sowards $2.3 million in compensatory damages and $65 million in punitive damages. [ 25 ]


Biotronik is under investigation in the Netherlands, with inquiries initiated in autumn 2020. This investigation, reported by Der Spiegel and the Dutch newspaper NRC Handelsblad, focuses on suspicions that medical specialists received millions of euros to promote Biotronik products. [ 26 ]  Over ten residences have been searched, and assets worth 3.1 million euros belonging to medical specialists have been confiscated. Biotronik has not provided comments on these allegations.


In July 2022, Biotronik resolved a prolonged legal dispute in the USA by agreeing to a multimillion dollar settlement. The case involved allegations from two whistleblowers, who were former independent sales representatives, concerning a kick-back scheme. According to the whistleblowers the scheme entailed treating cardiologists to lavish dinners, winery tours, baseball games, strip club visits, and golf outings, sometimes including their spouses or employees. [ 27 ]  Additionally, cardiologists were allegedly compensated for making brief appearances at international conferences or provided with business class flights. Another aspect of the scheme involved a trainee program where Biotronik employees observed cardiologists during the implantation of ICDs, for which the cardiologists were paid $400. [ 28 ] [ 27 ]  Despite concerns raised by the compliance department, the payments continued. The alleged kick-back system, intended to encouraged the use of Biotronik products, occurred between 2013 and 2021. Although Biotronik denied the allegations, they settled with the US authorities without admitting guilt, paying nearly $13 million. [ 29 ] [ 30 ] [ 31 ] [ 32 ]


The  Bueno-Orovio–Cherry–Fenton model , also simply called  Bueno-Orovio model , is a minimal  ionic model  for human  ventricular   cells . [ 1 ]  It belongs to the category of  phenomenological models , because of its characteristic of describing the  electrophysiological  behaviour of  cardiac muscle cells  without taking into account in a detailed way the underlying  physiology  and the specific mechanisms occurring inside the cells. [ 2 ] [ 3 ]


This  mathematical model  reproduces both single  cell  and important  tissue -level properties, accounting for physiological  action potential  development and  conduction velocity  estimations. [ 1 ]
It also provides specific parameters choices, derived from parameter-fitting algorithms of the  MATLAB   Optimization Toolbox , for the modeling of epicardial,  endocardial  and  myd-myocardial  tissues. [ 1 ]
In this way it is possible to match the action potential morphologies, observed from experimental data, in the three different regions of the human ventricles. [ 1 ]
The Bueno-Orovio–Cherry–Fenton model is also able to describe reentrant and  spiral wave  dynamics, which occurs for instance during  tachycardia  or other types of  arrhythmias . [ 1 ]


From the mathematical perspective, it consists of a system of four  differential equations . [ 1 ]  One  PDE , similar to the  monodomain model , for an adimensional version of the  transmembrane potential , and three  ODEs  that define the evolution of the so-called  gating variables , i.e.  probability density functions  whose aim is to model the fraction of open  ion channels  across a  cell membrane . [ 1 ] [ 4 ] [ 2 ]


The  system  of four differential equations reads as follows: [ 1 ]


where

 Ω

 {\displaystyle \Omega }

  is the spatial domain and

 T

 {\displaystyle T}

  is the final time. The initial conditions are

 u

 0

 =
 0

 {\displaystyle u_{0}=0}

 ,

 v

 0

 =
 1

 {\displaystyle v_{0}=1}

 ,

 w

 0

 =
 1

 {\displaystyle w_{0}=1}

 ,

 s

 0

 =
 0

 {\displaystyle s_{0}=0}

 . [ 1 ]


H
 (
 x
 −

 x

 0

 )

 {\displaystyle H(x-x_{0})}

  refers to the  Heaviside function  centered in

 x

 0

 {\displaystyle x_{0}}

 . The adimensional transmembrane potential

 u

 {\displaystyle u}

  can be rescaled in  mV  by means of the  affine transformation

 V

 m
 V

 =
 85.7
 u
 −
 84

 {\displaystyle V_{mV}=85.7u-84}

 . [ 1 ]

 v

 {\displaystyle v}

 ,

 w

 {\displaystyle w}

  and

 s

 {\displaystyle s}

  refer to gating variables, where in particular

 s

 {\displaystyle s}

  can be also used as an indication of  intracellular calcium

 C
 a

 i

 2
 +

 {\displaystyle {{Ca}_{i}}^{2+}}

  concentration (given in the adimensional range [0, 1] instead of  molar concentration ). [ 5 ]


All the above-mentioned ionic  density currents  are partially adimensional and are expressed in

 1
 seconds

 {\displaystyle {\dfrac {1}{\text{seconds}}}}

 . [ 1 ]


Different parameters sets, as shown in Table 1, can be used to reproduce the action potential development of epicardial, endocardial and mid-myocardial human ventricular cells. There are some constants of the model, which are not located in Table 1, that can be deduced with the following formulas: [ 1 ]


where the temporal constants, i.e.

 τ

 v

 −

 ,

 τ

 w

 −

 ,

 τ

 s
 o

 ,

 τ

 s

 ,

 τ

 o

 {\displaystyle \tau _{v}^{-},\tau _{w}^{-},\tau _{so},\tau _{s},\tau _{o}}

  are expressed in seconds, whereas

 v

 ∞

 {\displaystyle v_{\infty }}

  and

 w

 ∞

 {\displaystyle w_{\infty }}

  are adimensional. [ 1 ]


The diffusion coefficient

 D

 {\displaystyle D}

  results in a value of

 1.171
 ±
 0.221

 cm

 2

 seconds

 {\displaystyle 1.171\pm 0.221{\dfrac {{\text{cm}}^{2}}{\text{seconds}}}}

 , which comes from experimental tests on human ventricular tissues. [ 1 ]


In order to trigger the action potential development in a certain position of the domain

 Ω

 {\displaystyle \Omega }

 , a forcing term

 J

 app

 (

 x

 ,
 t
 )

 {\displaystyle J_{\text{app}}({\boldsymbol {x}},t)}

 , which represents an externally applied density current, is usually added at the right hand side of the PDE and acts for a short time interval only. [ 5 ]


Assume that

 z

 {\displaystyle {\boldsymbol {z}}}

  refers to the vector containing all the gating variables, i.e.

 z

 =
 [
 v
 ,
 w
 ,
 s

 ]

 T

 {\displaystyle {\boldsymbol {z}}=[v,w,s]^{T}}

 , and

 F

 :

 R

 4

 →

 R

 3

 {\displaystyle {\boldsymbol {F}}:\mathbb {R} ^{4}\rightarrow \mathbb {R} ^{3}}

  contains the corresponding three right hand sides of the ionic model. The Bueno-Orovio–Cherry–Fenton model can be rewritten in the compact form: [ 6 ]


Let

 p
 ∈
 U
 =

 H

 1

 (
 Ω
 )

 {\displaystyle p\in U=H^{1}(\Omega )}

  and

 q

 ∈

 W

 =
 [

 L

 2

 (
 Ω
 )

 ]

 3

 {\displaystyle {\boldsymbol {q}}\in {\boldsymbol {W}}=[L^{2}(\Omega )]^{3}}

  be two generic test functions. [ 6 ]


To obtain the  weak formulation : [ 6 ]


Finally the weak formulation reads:


There are several methods to discretize in space this system of equations, such as the  finite element method  (FEM) or  isogeometric analysis  (IGA). [ 7 ] [ 8 ] [ 5 ] [ 6 ]


Time discretization  can be performed in several ways as well, such as using a  backward differentiation formula  (BDF) of order

 σ

 {\displaystyle \sigma }

  or a  Runge–Kutta method  (RK). [ 7 ] [ 5 ]


Let

 T

 h

 {\displaystyle {\mathcal {T}}_{h}}

  be a  tessellation  of the computational domain

 Ω

 {\displaystyle \Omega }

  by means of a certain type of elements (such as  tetrahedrons  or  hexahedra ), with

 h

 {\displaystyle h}

  representing a chosen measure of the size of a single element

 K
 ∈

 T

 h

 {\displaystyle K\in {\mathcal {T}}_{h}}

 .
Consider the set

 P

 r

 (
 K
 )

 {\displaystyle \mathbb {P} ^{r}(K)}

  of  polynomials  with degree smaller or equal than

 r

 {\displaystyle r}

  over an element

 K

 {\displaystyle K}

 . Define

 X

 h

 r

 =
 {
 f
 ∈

 C

 0

 (

 Ω
 ¯

 )
 :
 f

 |

 K

 ∈

 P

 r

 (
 K
 )

 ∀
 K
 ∈

 T

 h

 }

 {\displaystyle {\mathcal {X}}_{h}^{r}=\{f\in C^{0}({\bar {\Omega }}):f|_{K}\in \mathbb {P} ^{r}(K)\,\,\forall K\in {\mathcal {T}}_{h}\}}

  as the finite dimensional space, whose dimension is

 N

 h

 =
 dim

 (

 X

 h

 r

 )

 {\displaystyle N_{h}=\dim({\mathcal {X}}_{h}^{r})}

 .
The set of basis functions of

 X

 h

 r

 {\displaystyle {\mathcal {X}}_{h}^{r}}

  is referred to as

 {

 ϕ

 i

 }

 i
 =
 1

 N

 h

 {\displaystyle \{\phi _{i}\}_{i=1}^{N_{h}}}

 . [ 5 ]


The semidiscretized formulation of the first equation of the model reads: find

 u

 h

 =

 u

 h

 (
 t
 )
 =

 ∑

 j
 =
 1

 N

 h

 u
 ¯

 j

 (
 t
 )

 ϕ

 j

 {\displaystyle u_{h}=u_{h}(t)=\sum _{j=1}^{N_{h}}{\bar {u}}_{j}(t)\phi _{j}}

  projection of the solution

 u
 (
 t
 )

 {\displaystyle u(t)}

  on

 X

 h

 r

 {\displaystyle {\mathcal {X}}_{h}^{r}}

 ,

 ∀
 t
 ∈
 (
 0
 ,
 T
 )

 {\displaystyle \forall t\in (0,T)}

 , such that [ 5 ]


with

 u

 h

 (
 0
 )
 =

 ∑

 j
 =
 1

 N

 h

 (

 ∫

 Ω

 u

 0

 ϕ

 j

 d
 Ω

 )

 ϕ

 j

 {\displaystyle u_{h}(0)=\sum _{j=1}^{N_{h}}\left(\int _{\Omega }u_{0}\phi _{j}\,d\Omega \right)\phi _{j}}

 ,

 z

 h

 =

 z

 h

 (
 t
 )
 =
 [

 v

 h

 (
 t
 )
 ,

 w

 h

 (
 t
 )
 ,

 s

 h

 (
 t
 )

 ]

 T

 {\displaystyle {\boldsymbol {z}}_{h}={\boldsymbol {z}}_{h}(t)=[v_{h}(t),w_{h}(t),s_{h}(t)]^{T}}

  semidiscretized version of the three gating variables, and

 J

 ion

 (

 u

 h

 ,

 z

 h

 )
 =

 J

 f
 i

 (

 u

 h

 ,

 z

 h

 )
 +

 J

 s
 o

 (

 u

 h

 ,

 z

 h

 )
 +

 J

 s
 i

 (

 u

 h

 ,

 z

 h

 )

 {\displaystyle J_{\text{ion}}(u_{h},{\boldsymbol {z}}_{h})=J_{fi}(u_{h},{\boldsymbol {z}}_{h})+J_{so}(u_{h},{\boldsymbol {z}}_{h})+J_{si}(u_{h},{\boldsymbol {z}}_{h})}

  is the total ionic density current. [ 5 ]


The space discretized version of the first equation can be rewritten as a system of non-linear ODEs by setting

 U

 h

 (
 t
 )
 =
 {

 u
 ¯

 j

 (
 t
 )

 }

 j
 =
 1

 N

 h

 {\displaystyle {\boldsymbol {U}}_{h}(t)=\{{\bar {u}}_{j}(t)\}_{j=1}^{N_{h}}}

  and

 Z

 h

 (
 t
 )
 =
 {

 z
 ¯

 j

 (
 t
 )

 }

 j
 =
 1

 N

 h

 {\displaystyle {\boldsymbol {Z}}_{h}(t)=\{{\bar {\boldsymbol {z}}}_{j}(t)\}_{j=1}^{N_{h}}}

 : [ 5 ]


where

 M

 i
 j

 =

 ∫

 Ω

 ϕ

 j

 ϕ

 i

 d
 Ω

 {\displaystyle \mathbb {M} _{ij}=\int _{\Omega }\phi _{j}\phi _{i}\,d\Omega }

 ,

 K

 i
 j

 =

 ∫

 Ω

 D
 ∇

 ϕ

 j

 ⋅
 ∇

 ϕ

 i

 d
 Ω

 {\displaystyle \mathbb {K} _{ij}=\int _{\Omega }D\nabla \phi _{j}\cdot \nabla \phi _{i}\,d\Omega }

  and

 (

 J

 ion

 (

 U

 h

 (
 t
 )
 ,

 z

 h

 (
 t
 )
 )

 )

 i

 =

 ∫

 Ω

 J

 ion

 (

 u

 h

 ,

 z

 h

 )

 ϕ

 i

 d
 Ω

 {\displaystyle \left({\boldsymbol {J}}_{\text{ion}}({\boldsymbol {U}}_{h}(t),{\boldsymbol {z}}_{h}(t))\right)_{i}=\int _{\Omega }J_{\text{ion}}(u_{h},{\boldsymbol {z}}_{h})\phi _{i}\,d\Omega }

 . [ 5 ]


The non-linear term

 J

 ion

 (

 U

 h

 (
 t
 )
 ,

 Z

 h

 (
 t
 )
 )

 {\displaystyle {\boldsymbol {J}}_{\text{ion}}({\boldsymbol {U}}_{h}(t),{\boldsymbol {Z}}_{h}(t))}

  can be treated in different ways, such as using state variable interpolation (SVI) or ionic currents interpolation (ICI). [ 9 ] [ 10 ]
In the framework of SVI, by denoting with

 {

 x

 s

 K

 }

 s
 =
 1

 N

 q

 {\displaystyle \{{\boldsymbol {x}}_{s}^{K}\}_{s=1}^{N_{q}}}

  and

 {

 ω

 s

 K

 }

 s
 =
 1

 N

 q

 {\displaystyle \{\omega _{s}^{K}\}_{s=1}^{N_{q}}}

  the  quadrature  nodes and weights of a generic element of the mesh

 K
 ∈

 T

 h

 {\displaystyle K\in {\mathcal {T}}_{h}}

 , both

 u

 h

 {\displaystyle u_{h}}

  and

 z

 h

 {\displaystyle {\boldsymbol {z}}_{h}}

  are evaluated at the quadrature nodes: [ 5 ]


The equations for the three gating variables, which are ODEs, are directly solved in all the  degrees of freedom  (DOF) of the tessellation

 T

 h

 {\displaystyle {\mathcal {T}}_{h}}

  separately, leading to the following semidiscrete form: [ 5 ]


With reference to the time interval

 (
 0
 ,
 T
 ]

 {\displaystyle (0,T]}

 , let

 Δ
 t
 =

 T
 N

 {\displaystyle \Delta t={\dfrac {T}{N}}}

  be the chosen time step, with

 N

 {\displaystyle N}

  number of subintervals. A uniform partition in time

 [

 t

 0

 =
 0
 ,

 t

 1

 =
 Δ
 t
 ,
 …
 ,

 t

 k

 ,
 …
 ,

 t

 N
 −
 1

 ,

 t

 N

 =
 T
 ]

 {\displaystyle [t_{0}=0,t_{1}=\Delta t,\ldots ,t_{k},\ldots ,t_{N-1},t_{N}=T]}

  is finally obtained. [ 7 ]


At this stage, the full discretization of the Bueno-Orovio ionic model can be performed both in a monolithic and segregated fashion. [ 11 ]
With respect to the first methodology (the monolithic one), at time

 t
 =

 t

 k

 {\displaystyle t=t^{k}}

 , the full problem is entirely solved in one step in order to get

 (

 U

 h

 k
 +
 1

 ,

 Z

 h

 k
 +
 1

 )

 {\displaystyle ({\boldsymbol {U}}_{h}^{k+1},{\boldsymbol {Z}}_{h}^{k+1})}

  by means of either  Newton method  or  Fixed-point iterations : [ 11 ]


where

 U

 ext

 ,
 h

 k

 {\displaystyle {\boldsymbol {U}}_{{\text{ext}},h}^{k}}

  and

 Z

 ext

 ,
 h

 k

 {\displaystyle {\boldsymbol {Z}}_{{\text{ext}},h}^{k}}

  are extrapolations of transmembrane potential and gating variables at previous timesteps with respect to

 t

 k
 +
 1

 {\displaystyle t^{k+1}}

 , considering as many time instants as the order

 σ

 {\displaystyle \sigma }

  of the BDF scheme.

 α

 {\displaystyle \alpha }

  is a coefficient which depends on the BDF order

 σ

 {\displaystyle \sigma }

 . [ 11 ]


If a segregated method is employed, the equation for the evolution in time of the transmembrane potential and the ones of the gating variables are numerically solved separately: [ 11 ]


Another possible segregated scheme would be the one in which

 U

 h

 k
 +
 1

 {\displaystyle {\boldsymbol {U}}_{h}^{k+1}}

  is calculated first, and then it is used in the equations for

 Z

 h

 k
 +
 1

 {\displaystyle {\boldsymbol {Z}}_{h}^{k+1}}

 . [ 11 ]


A  bundle branch block  is a partial or complete interruption in the flow of electrical impulses in either of the  bundle branches  of the   heart's electrical system . [ 1 ]


The  heart's electrical activity  begins in the  sinoatrial node  (the heart's natural  pacemaker ), which is situated on the upper right  atrium . The impulse travels next through the left and right atria and summates at the  atrioventricular node . From the AV node the electrical impulse travels down the  bundle of His  and divides into the right and left  bundle branches . The right bundle branch contains one  fascicle . The left bundle branch subdivides into two fascicles: the  left anterior fascicle , and the  left posterior fascicle . Other sources divide the left bundle branch into three fascicles: the left anterior, the left posterior, and the left septal fascicle. The thicker left posterior fascicle bifurcates, with one fascicle being in the septal aspect.  Ultimately, the fascicles divide into millions of  Purkinje fibres , which in turn interdigitate with individual cardiac myocytes, allowing for rapid, coordinated, and synchronous physiologic depolarization of the ventricles. [ citation needed ]


When a bundle branch or fascicle becomes injured (by underlying  heart disease ,  myocardial infarction , or cardiac surgery), it may cease to conduct electrical impulses appropriately. This results in altered pathways for ventricular depolarization. Since the electrical impulse can no longer use the preferred pathway across the bundle branch, it may move instead through muscle fibers in a way that both slows the electrical movement and changes the directional propagation of the impulses. As a result, there is a loss of ventricular synchrony, ventricular depolarization is prolonged, and there may be a corresponding drop in cardiac output. When  heart failure  is present, a specialized  pacemaker  may be used to resynchronize the ventricles. In theory a pacemaker like this will shorten the  QRS  interval, thus bringing the timing of contraction of the left and right ventricles closer together and slightly increasing the  ejection fraction . [ citation needed ]


A bundle branch block can be diagnosed when the duration of the  QRS complex  on the  ECG  exceeds 120 ms. A right bundle branch block typically causes prolongation of the last part of the QRS complex and may shift the heart's electrical axis slightly to the right. The ECG will show a terminal R wave in lead V1 and a slurred S wave in lead I.
Left bundle branch block widens the entire QRS, and in most cases shifts the heart's electrical axis to the left. The ECG will show a QS or rS complex in lead V1 and a monophasic R wave in lead I. Another normal finding with bundle branch block is appropriate T wave discordance. In other words, the T wave will be deflected opposite the terminal deflection of the QRS complex.
Bundle branch block, especially left bundle branch block, can lead to cardiac dyssynchrony. The simultaneous occurrence of left and right bundle branch block leads to total AV block.


Depending on the anatomical location of the defect which leads to a bundle branch block, the blocks are further classified into:


The left bundle branch block can be further sub classified into:


Other classifications of bundle branch blocks are;


A simple way to quickly differentiate between the two types is to note the deflection of the QRS complex in the V1 lead. A (V1) QRS segment deflected down indicates left bundle branch block, while a deflection up indicates right bundle branch block. In both types, the QRS is wide (> 0.12 seconds).


Some people with bundle branch blocks are born with this condition. Many others acquire it as a consequence of heart disease. People with bundle branch blocks may still be quite active, and may have nothing more remarkable than an abnormal appearance to their ECG.  However, when bundle blocks are complex and diffuse in the bundle systems, or are associated with additional and significant ventricular muscle damage, they may be a sign of serious underlying heart disease. In more severe cases, a  pacemaker  may be required to restore an optimal electrical supply to the heart muscle. [ citation needed ]


The  bundle of His  ( BH ) [ 1 ] : 58   or  His bundle  ( HB ) [ 1 ] : 232   ( / h ɪ s /  "hiss" [ 2 ] ) is a collection of  heart muscle cells  specialized for  electrical conduction . As part of the  electrical conduction system of the heart , it transmits the  electrical impulses  from the  atrioventricular node  (located between the  atria  and the  ventricles ) to the point of the apex of the fascicular branches via the  bundle branches . The fascicular branches then lead to the  Purkinje fibers , which provide electrical conduction to the ventricles, causing the  cardiac muscle  of the ventricles to  contract  at a  paced  interval.


The bundle of His is an important part of the  electrical conduction system of the heart , as it transmits  impulses  from the atrioventricular node, located at the anterior-inferior end of the  interatrial septum , to the ventricles of the heart. The bundle of His branches into the left and the right  bundle branches , which run along the  interventricular septum . The left bundle branch further divides into the  left anterior fascicle  and the  left posterior fascicle . These bundles and fascicles give rise to thin filaments known as  Purkinje fibers . These fibers distribute the impulse to the ventricular muscle. The ventricular conduction system comprises the bundle branches and the Purkinje networks. It takes about 0.03–0.04 seconds for the impulse to travel from the bundle of His to the  ventricular muscle .


Disorders affecting the  cardiomyocytes  that make up the electrical conduction system of the heart are called  heart blocks . Heart blocks are separated into different categories based on the location of the cellular damage. Damage to any of the conducting cells in or below the bundle of His is collectively referred to as "infra-Hisian blocks". To be specific, blocks that occur in the right or left bundle branches are called " bundle branch blocks ", and those that occur in either the left anterior or the left posterior fascicles are called "fascicular blocks", or "hemiblocks". The conditions in which both the right bundle branch and either the left anterior fascicle or the left posterior fascicle are blocked are collectively referred to as  bifascicular blocks , and the condition in which the right bundle branch, the left anterior fascicle, and the left posterior fascicle are blocked is called  trifascicular block . Infra-hisian blocks limit the heart's ability to coordinate the activities of the atria and ventricles, which usually results in a decrease in its efficiency in pumping blood.


A 2000 study found that direct His bundle pacing is more effective in producing synchronized ventricular contraction—and therefore in improving cardiac function—than  apical  pacing. [ 3 ]


These specialized muscle fibers in the heart were named after the  Swiss   cardiologist   Wilhelm His Jr. , who discovered them in 1893. [ 4 ] [ 5 ]


In  neuroscience , classical  cable theory  uses  mathematical models  to calculate the  electric current  (and accompanying  voltage ) along passive [ a ]   neurites , particularly the  dendrites  that receive  synaptic  inputs at different sites and times. Estimates are made by modeling dendrites and  axons  as cylinders composed of segments with  capacitances

 c

 m

 {\displaystyle c_{m}}

  and  resistances

 r

 m

 {\displaystyle r_{m}}

  combined in parallel (see Fig. 1). The capacitance of a neuronal fiber comes about because  electrostatic  forces are acting through the very thin lipid bilayer (see Figure 2). The resistance in series along the fiber

 r

 l

 {\displaystyle r_{l}}

  is due to the  axoplasm 's significant resistance to movement of  electric charge .


Cable theory in computational  neuroscience  has roots leading back to the 1850s, when Professor  William Thomson  (later known as Lord Kelvin) began developing mathematical models of signal decay in submarine (underwater) telegraphic cables. The models resembled the partial differential equations used by  Fourier  to describe heat conduction in a wire.


The 1870s saw the first attempts by Hermann to model  neuronal   electrotonic potentials  also by focusing on analogies with heat conduction. However, it was Hoorweg who first discovered the analogies with Kelvin's undersea cables in 1898 and then Hermann and Cremer who independently developed the cable theory for neuronal fibers in the early 20th century. Further mathematical theories of nerve fiber conduction based on cable theory were developed by Cole and  Hodgkin  (1920s–1930s), Offner et al. (1940), and Rushton (1951).


Experimental evidence for the importance of cable theory in modelling the behavior of  axons  began surfacing in the 1930s from work done by Cole, Curtis, Hodgkin,  Sir Bernard Katz , Rushton, Tasaki and others. Two key papers from this era are those of Davis and  Lorente de Nó  (1947) and Hodgkin and Rushton (1946).


The 1950s saw improvements in techniques for measuring the electric activity of individual  neurons . Thus cable theory became important for analyzing data collected from intracellular microelectrode recordings and for analyzing the electrical properties of neuronal  dendrites . Scientists like Coombs,  Eccles , Fatt, Frank, Fuortes and others now relied heavily on cable theory to obtain functional insights of neurons and for guiding them in the design of new experiments.


Later, cable theory with its mathematical derivatives allowed ever more sophisticated  neuron models  to be explored by workers such as Jack,  Rall , Redman, Rinzel, Idan Segev, Tuckwell, Bell, and Iannella. More recently, cable theory has been applied to model electrical activity in bundled neurons in the white matter of the brain. [ 1 ]


Note, various conventions of  r m  exist.
Here  r m  and  c m , as introduced above, are measured per membrane-length unit (per meter (m)). Thus  r m  is measured in  ohm ·meters (Ω·m) and  c m  in  farads  per meter (F/m). This is in contrast to  R m  (in Ω·m 2 ) and  C m  (in F/m 2 ), which represent the specific resistance and capacitance respectively of one unit area of membrane (in m 2 ). Thus, if the radius,  a , of the axon is known, [ b ]  then its circumference is 2 πa , and its  r m , and its  c m  values can be calculated as:


These relationships make sense intuitively, because the greater the circumference of the axon, the greater the area for charge to escape through its membrane, and therefore the lower the membrane resistance (dividing  R m  by 2 πa ); and the more membrane available to store charge (multiplying  C m  by 2 πa ).
The  specific electrical resistance ,  ρ l , of the axoplasm allows one to calculate the longitudinal intracellular resistance per unit length,  r l , (in Ω·m −1 ) by the equation:


The greater the cross sectional area of the axon,  πa 2 , the greater the number of paths for the charge to flow through its axoplasm, and the lower the axoplasmic resistance.


Several important avenues of extending classical cable theory have recently seen the introduction of endogenous structures in order to analyze the effects of protein polarization within dendrites and different synaptic input distributions over the dendritic surface of a neuron.


To better understand how the cable equation is derived, first simplify the theoretical neuron even further and pretend it has a perfectly sealed membrane ( r m =∞) with no loss of current to the outside, and no capacitance ( c m  = 0). A current injected into the fiber  [ c ]  at position  x  = 0 would move along the inside of the fiber unchanged. Moving away from the point of injection and by using  Ohm's law  ( V  =  IR ) we can calculate the voltage change as:


where the negative is because current flows down the potential gradient.


Letting Δ x  go towards zero and having infinitely small increments of  x , one can write ( 4 ) as:


or


Bringing  r m  back into the picture is like making holes in a garden hose. The more holes, the faster the water will escape from the hose, and the less water will travel all the way from the beginning of the hose to the end. Similarly, in an axon, some of the current traveling longitudinally through the axoplasm will escape through the membrane.


If  i m  is the current escaping through the membrane per length unit, m, then the total current escaping along  y  units must be  y·i m . Thus, the change of current in the axoplasm, Δ i l , at distance, Δ x , from position  x =0 can be written as:


or, using continuous, infinitesimally small increments:


i

 m

 {\displaystyle i_{m}}

  can be expressed with yet another formula, by including the capacitance. The capacitance will cause a flow of charge (a current) towards the membrane on the side of the cytoplasm. This current is usually referred to as displacement current (here denoted

 i

 c

 {\displaystyle i_{c}}

 .) The flow will only take place as long as the membrane's storage capacity has not been reached.

 i

 c

 {\displaystyle i_{c}}

  can then be expressed as:


where

 c

 m

 {\displaystyle c_{m}}

  is the membrane's capacitance and

 ∂
 V

 /

 ∂
 t

 {\displaystyle {\partial V}/{\partial t}}

  is the change in voltage over time.
The current that passes the membrane (

 i

 r

 {\displaystyle i_{r}}

 ) can be expressed as:


and because

 i

 m

 =

 i

 r

 +

 i

 c

 {\displaystyle i_{m}=i_{r}+i_{c}}

  the following equation for

 i

 m

 {\displaystyle i_{m}}

  can be derived if no additional current is added from an electrode:


where

 ∂

 i

 l

 /

 ∂
 x

 {\displaystyle {\partial i_{l}}/{\partial x}}

  represents the change per unit length of the longitudinal current.


Combining equations ( 6 ) and ( 11 ) gives a first version of a cable equation:


which is a second-order  partial differential equation  (PDE).


By a simple rearrangement of equation ( 12 ) (see later) it is possible to make two important terms appear, namely the length constant (sometimes referred to as the space constant) denoted

 λ

 {\displaystyle \lambda }

  and the time constant denoted

 τ

 {\displaystyle \tau }

 . The following sections focus on these terms.


The length constant,

 λ

 {\displaystyle \lambda }

  (lambda), is a parameter that indicates how far a stationary current will influence the voltage along the cable. The larger the value of

 λ

 {\displaystyle \lambda }

 , the farther the charge will flow. The length constant can be expressed as:


The larger the membrane resistance,  r m , the greater the value of

 λ

 {\displaystyle \lambda }

 , and the more current will remain inside the axoplasm to travel longitudinally through the axon. The higher the axoplasmic resistance,

 r

 l

 {\displaystyle r_{l}}

 , the smaller the value of

 λ

 {\displaystyle \lambda }

 , the harder it will be for current to travel through the axoplasm, and the shorter the current will be able to travel.
It is possible to solve equation ( 12 ) and arrive at the following equation (which is valid in steady-state conditions, i.e. when time approaches infinity):


Where

 V

 0

 {\displaystyle V_{0}}

  is the depolarization at

 x
 =
 0

 {\displaystyle x=0}

  (point of current injection),  e  is the exponential constant (approximate value 2.71828) and

 V

 x

 {\displaystyle V_{x}}

  is the voltage at a given distance  x  from  x =0. When

 x
 =
 λ

 {\displaystyle x=\lambda }

  then


and


which means that when we measure

 V

 {\displaystyle V}

  at distance

 λ

 {\displaystyle \lambda }

  from

 x
 =
 0

 {\displaystyle x=0}

  we get


Thus

 V

 λ

 {\displaystyle V_{\lambda }}

  is always 36.8 percent of

 V

 0

 {\displaystyle V_{0}}

 .


Neuroscientists are often interested in knowing how fast the membrane potential,

 V

 m

 {\displaystyle V_{m}}

 , of an axon changes in response to changes in the current injected into the axoplasm. The time constant,

 τ

 {\displaystyle \tau }

 , is an index that provides information about that value.

 τ

 {\displaystyle \tau }

  can be calculated as:


The larger the membrane capacitance,

 c

 m

 {\displaystyle c_{m}}

 , the more current it takes to charge and discharge a patch of membrane and the longer this process will take. The larger the membrane resistance

 r

 m

 {\displaystyle r_{m}}

 , the harder it is for a current to induce a change in membrane potential. So the higher the

 τ

 {\displaystyle \tau }

  the slower the nerve impulse can travel. That means, membrane potential (voltage across the membrane) lags more behind current injections. Response times vary from 1–2 milliseconds in neurons that are processing information that needs high temporal precision to 100 milliseconds or longer. A typical response time is around 20 milliseconds.


If one multiplies equation ( 12 ) by

 r

 m

 {\displaystyle r_{m}}

  on both sides of the equal sign we get:


and recognize

 λ

 2

 =

 r

 m

 /

 r

 l

 {\displaystyle \lambda ^{2}={r_{m}}/{r_{l}}}

  on the left side and

 τ
 =

 c

 m

 r

 m

 {\displaystyle \tau =c_{m}r_{m}}

  on the right side. The cable equation can now be written in its perhaps best known form:


This is a 1D  heat equation  or  diffusion equation  for which many solution methods, such as Green's functions and Fourier methods, have been developed.


It is also a special degenerate case of the  Telegrapher's equation , where the inductance

 L

 {\displaystyle L}

  vanishes and the signal propagation speed

 1

 /

 L
 C

 {\displaystyle 1/{\sqrt {LC}}}

  is infinite.


Cameron Health  was a  medical device  developer based in  San Clemente, California , US. Cameron Health had its European office, Cameron Health BV, in  Arnhem ,  The Netherlands . The privately held company's focus was on a new generation of minimally invasive  implantable cardioverter-defibrillator  (ICD) which they called a Subcutaneous Implantable Defibrillator (S-ICD). Cameron Health's approach avoided implanting transvenous leads into the  heart , which had been the usual procedure for cardiac devices. Instead, the Cameron ICD was entirely implanted outside the thoracic wall.


In June 2012,  Boston Scientific  acquired Cameron Health for a total sum of $1.3 Billion, paid out incrementally as various revenue milestones were achieved. [ 1 ]  As of February 2016 [update] , Boston Scientific still markets the S-ICD system. [ 2 ]


Every ICD is designed to detect heart rhythms consistent with a catastrophic failure of the body's natural regulation of the heartbeat, which, untreated, could result in death. When an ICD detects a serious  arrhythmia , it issues an electrical impulse to the heart muscle, of a magnitude sufficient to cause the heart to revert to a normal rhythm.  ICDs with transvenous leads administer this shock to the interior of the heart muscle; the Cameron Health device generated a more powerful shock which can be effective from outside the heart.  In the view of Cameron Health, transvenous leads into the heart needlessly complicated the process of implanting a device, and raised other issues and risks which their less invasive approach avoids. [ 3 ]


The Cameron Health subcutaneous ICD sat outside the ribcage and has no connection to the interior of the heart.  The surgical procedure for implantation was minimally invasive as opposed to the traditional procedure of threading leads into the subclavian venous system, through the  superior vena cava  and into one or more endocardial areas of the heart, a procedure often requiring a  cardiologist  with specialized training in  electrophysiology . [ 4 ]  In addition to the risks inherent in cardiac surgery, the leads have themselves proved to be a weakness in some ICD designs. [ 5 ]  According to one estimate, patients with ICDs have a 20 percent chance of lead failure within 10 years, and replacing the leads carries a risk of death of between two and five percent. [ 6 ]  Some device manufacturers have had to replace defective leads which exposed implanted individuals to unnecessary shocks or other malfunctions, in some cases possibly resulting in fatalities. [ 7 ] [ 8 ]


The Cameron Health S-ICD had the disadvantage of being somewhat bulkier than existing ICDs. [ 6 ]  Also, this kind of ICD did not include a pacemaker, which narrowed the range of patients for whom it would be appropriate; it was estimated that a majority of patients receiving combination pacemaker/ICD implants would qualify for a pure ICD. These patients tended to have genetic or other conditions predisposing them to sudden cardiac death due to a failure of the heart of maintain a normal rhythm. [ 9 ]


A trial involving 53 patients, who were temporarily implanted with S-ICDs, was reported in 2005 at the  European Society of Cardiology  Congress. [ 10 ]
A second series of 55 trial patients was conducted in 2008 and 2009 in 10 centers in  Europe  and  New Zealand .  Of the 55 patients, 53 had two instances of  fibrillation  and in 52 these were successfully converted.  These findings were reported to European Union authorities in 2009, and resulted in approval for marketing the device.  A study of 300 patients is in progress for US approvals. [ 6 ] [ 11 ]   Small nonrandomized early-phase studies primarily intended to show the feasibility of an entirely subcutaneous ICD were updated, combined and published in May 2010.  In this report,  the system successfully and consistently detected and converted episodes of ventricular fibrillation that were induced during electrophysiological testing. In the European trial of 55 patients, after 46 patient-years of follow-up, 54 of 55 patients were alive, and the single death was due to renal failure. In this trial the system successfully detected and treated 12 episodes (100%)of spontaneous, sustained ventricular tachyarrhythmia in three patients,  prior to the onset of syncope, and with no adverse events. One of the three patients was successfully treated for seven successive episodes of  ventricular tachycardia , a condition known as a  "VT storm".


Boston Scientific acquired an exclusive option to purchase Cameron Health in 2004, and made an undisclosed equity investment in the company at that time. [ 12 ]
In 2008, several additional investors organized by the investment company  Piper Jaffray [ 13 ]
and including PTV Healthcare Capital, Delphi Ventures, Sorrento Ventures, Three Arch Partners and  Versant Ventures  provided just over $50 million to finance the continuing operation of the company. [ 14 ]


In June 2012, Boston Scientific officially acquired Cameron Health for a total sum of $1.3 Billion, paid out incrementally as various revenue milestones were achieved. [ 1 ]


Cardiac aberrancy  is a type of disruption in the shape of the  electrocardiogram  signal, representing abnormal activation of the ventricular heart muscle via the  electrical conduction system of the heart .
Normal activation utilizes the  bundle of His  and  Purkinje fibers  to produce a narrow (QRS) electrical signal.
Aberration occurs when the electrical activation of the heart, which is caused by a series of  action potentials , is conducting improperly which can result in temporary changes in the morphology that looks like:


This is in contrast to a permanent dysfunction of the electrical pathways that produces wide QRS complexes in one of the above patterns or combinations of patterns (ie,  bifascicular block ).


In the context of  atrial fibrillation , the  Ashman phenomenon  is a form of aberrancy.


Aberrancy is due to prematurity in which part of the conduction system is still  refractory  and cannot conduct the premature depolarization. This effect can sometimes be seen in the setting of a faster heart rate ( tachycardia ) and so is termed "rate-related aberrancy." After the first aberrant complex, subsequent complexes may be wide due to  concealed conduction  rather than aberrancy.


Unlike the  action potential  in  skeletal muscle cells , the cardiac action potential is not initiated by nervous activity. Instead, it arises from a group of specialized cells known as  pacemaker cells , that have automatic action potential generation capability. In healthy hearts, these cells form the  cardiac pacemaker  and are found in the  sinoatrial node  in the right  atrium . They produce roughly 60–100 action potentials every minute. The action potential passes along the cell membrane causing the cell to contract, therefore the activity of the sinoatrial node results in a resting heart rate of roughly 60–100 beats per minute. All cardiac muscle cells are electrically linked to one another, by  intercalated discs  which allow the action potential to pass from one cell to the next. [ 1 ] [ 2 ]  This means that all atrial cells can contract together, and then all ventricular cells.


Rate dependence of the action potential is a fundamental property of cardiac cells and alterations can lead to severe cardiac diseases including  cardiac arrhythmia  and sometimes sudden death. [ 3 ]
Action potential activity within the heart can be recorded to produce an  electrocardiogram  (ECG). This is a series of upward and downward spikes (labelled P, Q, R, S and T) that represent the depolarization (voltage becoming more positive) and repolarization (voltage becoming more negative) of the action potential in the  atria  and  ventricles . [ 4 ]


Similar to skeletal muscle, the  resting membrane potential  (voltage when the cell is not electrically excited) of  ventricular cells  is around −90 millivolts (mV; 1 mV = 0.001 V), i.e. the inside of the membrane is more negative than the outside. The main ions found outside the cell at rest are sodium (Na + ), and chloride (Cl − ), whereas inside the cell it is mainly potassium (K + ). [ 6 ]


The action potential begins with the voltage becoming more positive; this is known as  depolarization  and is mainly due to the opening of  sodium channels  that allow  Na +  to flow into the cell. After a delay (known as the  absolute refractory period ), the action potential terminates as potassium channels open, allowing K +  to leave the cell and causing the membrane potential to return to negative, this is known as  repolarization . Another important ion is  calcium (Ca 2+ ) , which can be found inside the cell in the  sarcoplasmic reticulum  (SR) where calcium is stored, and is also found outside of the cell. Release of Ca 2+  from the SR, via a process called  calcium-induced calcium release , is vital for the plateau phase of the action potential (see phase 2, below) and is a fundamental step in  cardiac excitation-contraction coupling . [ 7 ]


There are important physiological differences between the  pacemaker cells  of the  sinoatrial node , that spontaneously generate the cardiac action potential and those non-pacemaker cells that simply conduct it, such as  ventricular myocytes ). The specific differences in the types of  ion channels  expressed and mechanisms by which they are activated results in differences in the configuration of the action potential waveform, as shown in figure 2.


Cardiac automaticity  also known as  autorhythmicity , is the property of the specialized  conductive   muscle cells  of the heart to generate spontaneous cardiac action potentials. [ 8 ] [ 9 ]  Automaticity can be normal or abnormal, caused by temporary  ion channel  characteristic changes such as certain medication usage, or in the case of abnormal automaticity the changes are in  electrotonic environment , caused, for example, by  myocardial infarction . [ 10 ]


The standard model used to understand the cardiac action potential is that of the ventricular myocyte. Outlined below are the five phases of the ventricular myocyte action potential, with reference also to the SAN action potential.


In the ventricular myocyte, phase 4 occurs when the cell is at rest, in a period known as  diastole . In the standard non-pacemaker cell the voltage during this phase is more or less constant, at roughly -90 mV. [ 11 ]  The  resting membrane potential  results from the flux of ions having flowed into the cell (e.g. sodium and calcium), the flux of ions having flowed out of the cell (e.g. potassium, chloride and bicarbonate), as well as the flux of ions generated by the different membrane pumps, being perfectly balanced.


The activity of these  pumps  serve two purposes. The first is to maintain the existence of the resting membrane potential by countering the depolarisation due to the leakage of ions not at the electrochemical equilibrium (e.g. sodium and calcium). These ions not being at the equilibrium is the reason for the existence of an electrical gradient, for they represent a net displacement of charges across the membrane, which are unable to immediately re-enter the cell to restore the electrical equilibrium. Therefore, their slow re-entrance in the cell needs to be counterbalanced or the cell would slowly lose its membrane potential.


The second purpose, intricately linked to the first, is to keep the intracellular concentration more or less constant, and in this case to re-establish the original chemical gradients, that is to force the sodium and calcium which previously flowed into the cell out of it, and the potassium which previously flowed out of the cell back into it (though as the potassium is mostly at the electrochemical equilibrium, its chemical gradient will naturally reequilibrate itself opposite to the electrical gradient, without the need for an active transport mechanism).


For example, the  sodium (Na + )  and  potassium (K + )  ions are maintained by the  sodium-potassium pump  which uses energy (in the form of  adenosine triphosphate (ATP) ) to move three Na +  out of the cell and two K +  into the cell. Another example is the  sodium-calcium exchanger  which removes one Ca 2+  from the cell for three Na +  into the cell. [ 12 ]


During this phase the membrane is most permeable to K + , which can travel into or out of cell through leak channels, including the inwardly rectifying potassium channel. [ 13 ]  Therefore, the resting membrane potential is mostly equal to K +   equilibrium potential  and can be calculated using the  Goldman-Hodgkin-Katz voltage equation .


However,  pacemaker cells  are never at rest. In these cells, phase 4 is also known as the  pacemaker potential . During this phase, the membrane potential slowly becomes more positive, until it reaches a set value (around -40 mV; known as the threshold potential) or until it is depolarized by another action potential, coming from a neighboring cell.


The pacemaker potential is thought to be due to a group of channels, referred to as  HCN channels (Hyperpolarization-activated cyclic nucleotide-gated) . These channels open at very negative voltages (i.e. immediately after phase 3 of the previous action potential; see below) and allow the passage of both K +  and Na +  into the cell. Due to their unusual property of being activated by very negative membrane potentials, the movement of ions through the HCN channels is referred to as the  funny current  (see below). [ 14 ]


Another hypothesis regarding the pacemaker potential is the 'calcium clock'. Calcium is released from the  sarcoplasmic reticulum  within the cell. This calcium then increases activation of the  sodium-calcium exchanger  resulting in the increase in membrane potential (as a +3 charge is being brought into the cell (by the 3Na + ) but only a +2 charge is leaving the cell (by the Ca 2+ ) therefore there is a net charge of +1 entering the cell). This calcium is then pumped back into the cell and back into the SR via calcium pumps (including the  SERCA ). [ 15 ]


This phase consists of a rapid, positive change in voltage across the cell membrane ( depolarization ) lasting less than 2 ms in ventricular cells and 10–20 ms in  SAN  cells. [ 16 ]  This occurs due to a net flow of positive charge into the cell.