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What's inside a Black Hole


This is the third writeup where I'll be documenting my research with Dr. Joyce Lin.
If you want to read the my introductory article regarding the research series you can check it out [here](https://www.srirag.dev/article/research-with-dr-joyce-lin) 


### Introduction

The bidomain model is a mathematical model like the cable equation model which attempts to define the propagation of action potentials through the heart. The model itself was proposed by Tung and Geselowitz in the late 1970s. [[1]](#1)

The model presents a more realistic approach when compared to the cable equation model due to it consideration of both the intracellular and extracellular spaces. It can also be considered as a higher order version of the cable equation.

A thing to keep note of is that this model is a first order approximation to a full detailed model thus the realism gets watered down when compared to newer models(like the microdomain model) due to its homogenization of the fine micro-structure that exists within both the intracellular and extracellar spaces.


### Equation 

<u>The governing equations are as follows :</u>

For intracellular spaces : 
$$
\chi ( C_m \frac{\partial V}{\partial t} + I_{ion} ) = \nabla \cdot ( \sigma_i \nabla V_i )
$$

For extracellular spaces : 
$$
\chi(  C_m \frac{\partial V}{\partial t} + I_{ion} ) = - \nabla \cdot ( \sigma_e \nabla V_e )
$$


Here, 
- The $$ \chi $$ is the membrane surface-to-volume ratio, it is needed to convert the transmembrane current per unit area into transmembrane current per unit volume. 
- The $$ C_m $$ represents the membrane capacitance.
- The $$ \frac{\partial V}{\partial t} $$ represents the small minute change in potential accross time.
- The $$ I_{ion} $$ represents the ionic current.
  	-  For our simulation we use the Hodgkin Huxley Equations.
- The $$ \nabla $$ represents the gradient/laplacian
- The $$ \sigma_i $$ and $$ \sigma_e $$ represent the conductivity tensors for the intracellular and extracellular spaces respectively.


![Bidomain schema diagram](https://res.cloudinary.com/dtbzgpkps/image/upload/v1618293474/bidomain_e3054625c4.png)
*Representation of intracellular and extracellular space in bidomain model * [[2]](#2)


### Assumptions required for the bidomain model 

1. The $$ \nabla \cdot i_{T} = 0 $$ where $$ i_T = i_i + i_e $$ 
	1. Another assumption here is that the total current is conserved.
2. Homogenizing the fine microstructure of both the intracellular and extracellular spaces will not bring forward adverse effects in regards to the accuracy of the model.

3. The membrane parameter $$ C_m $$ is constant and does not depend on the transmembrane potential.

### Simulation 

Considering all of the above assumptions I was able to model the behavior of the action potential as it traverses through the cardiac tissue. The simulation below is animation of multiple snapshots in time (in $$ s $$) of the transmembrane potential (in $$ mV $$) vs points space of the tissue(in $$ cm $$). The simulation itself was coded up using Python and related libraries (Numpy, Pandas, Matplotlib, seaborn), to view the code behind the simulation click [here](https://github.com/srirag-vuppala/Bidomain-model)

![bidomain_eqn.gif](https://res.cloudinary.com/dtbzgpkps/image/upload/v1622684519/bidomain_tissue_4014b56caf.gif)

Here the tissue starts out with a resting potential of -70 mV and is given a stimulus by holding the the opposite ends of the tissue at 40mV and -40mV respectively to create a potential difference. This stimulus is given for only 5 time steps (1 time step = ~0.01 sec). With this short stimulus, the progression of the action potential begins. 

Another important note is that each frame of this simulation is a snapshot for every 600 time steps, thus giving us an **sped up progression** of the action potential.


**The entire source code is available here** [Github](https://github.com/srirag-vuppala/Bidomain-Model)

***
 
## Citations 


<a id="1">[1]</a> Keener, Sneyd, and Sneyd, James. Mathematical Physiology: I: Cellular Physiology. 2nd Ed. 2009. ed. New York, NY: Springer New York : Imprint: Springer, 2009. Interdisciplinary Applied Mathematics, 8/1. Web.

<a id="2">[2]</a> Ibrahim, A. , Adediji, A. and Olufemi, D. (2013) Dynamical modelling of cardiac electrical activity using bidomain approach: The effects of variation of ionic model parameters. Journal of Biomedical Science and Engineering, 6, 598-608. doi: 10.4236/jbise.2013.66076. 


Program flow of the bidomain model. Using Python3 and other data visualization libraries.

thumbnail_bidomain.png

bidomain.png

Bidomain Model

# Hello !

This is a quick introductory post to provide some context for the next few blog posts that will come out soon (hopefully?). First things first I have to credit [Dr. Joyce Lin](https://math.calpoly.edu/joyce-lin) and [BEACoN Research](https://beaconmentors.calpoly.edu/about-us) for bringing me in and providing this opportunity to me. Also, huge shout out to [LSAMP Research ](https://lsamp.calpoly.edu/) for providing us funding over the summer!

Now that we have that out of the way, I'll start off by laying out the 4W's .

### What are we researching ?  

**Short Answer:** The heart tissue. 

**Long Answer:**

 The propagation of an Cardiac Action potential through the heart tissue. 

Well, what are Action potentials you may ask? Here's a what [Wikipedia](https://en.wikipedia.org/wiki/Cardiac_action_potential) had to say
> The cardiac action potential is a brief change in voltage (membrane potential) across the cell membrane of heart cells. This is caused by the movement of charged atoms (called ions) between the inside and outside of the cell, through proteins called ion channels. 

At its core all they really are is the electrical impulses through cardiac tissue that allows for a heart to beat and pump blood throughout the body. 

### What do you do?

My work lies in bridging the gap between the mathematical models and programmatic simulations to see the behavior of these models with empirical evidence.

### Why are we researching this ?

While the above explanation of cardiac action potentials may seem simple enough we still don't have a working model for how they propagate at a cellular/tissue level. This is concerning as action potential failure has been strongly linked to sudden cardiac death, thus we seek a better theoretical model to study parameters that can cause arrhythmia.

Also, classically, the propagation of action potentials were considered to heavily rely on the intracellular to intracellular transfer using gap junctions, and often neglected the extracellular space. This understanding did not entirely hold up in experimental studies when the percentage of gap junctions decreased. Thus our research uses this as motivation for why we cannot rely solely on the intracellular transfer using gap junctions but also consider the extracellular space.


### Where is this happening ?

Right now, everything is online but is hosted through Cal Poly SLO.


***

**Here are the links for the rest of the blogs if you want to continue to read:**
1. [Cable Equation Model](https://www.srirag.dev/article/cable-equation)
2. [Bidomain Model](https://www.srirag.dev/article/bidomain-model)
3. Microdomain Model (TBA)











A quick intro post to the next series of posts related to my research with Dr. Joyce Lin.

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medium_BEACoNLogo.jpg

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thumbnail_BEACoNLogo.jpg

BEACoNLogo.jpg

Research with Dr. Joyce Lin



This is the second write-up where I'll be documenting my research with Dr. Joyce Lin.
If you want to read the my introductory article regarding the research series you can check it out [here](https://www.srirag.dev/article/research-with-dr-joyce-lin) 


### Introduction

The cable equation was first synthesized by Lord Kelvin (formerly William Thomson) in 1855; he derived the cable equation to study the signal decay in the transatlantic telegraph cable(then under construction). His foundational work would become extremely valuable in realm of neuronal behavior mainly due to Hodgkin and Ruston in 1946 which was then followed by a series of papers by Rall. [[1]](#1)

### Equation 

<u>The equation itself is as follows :</u>
$$
I_t = p(C_m \frac{\partial V}{\partial t} + I_{ion} ) = \frac{\partial}{\partial x} (\frac{1}{r_i + r_e} \frac{\partial V}{\partial x} )
$$
After nondimensionalizing it we reach to a more simpler equation like this,
$$
\frac{\partial V}{\partial T} = \frac{\partial^2 V}{\partial X^2} + f(V, T)
$$

Here, the $$ X $$ and $$ T $$ are non dimensional variables each of which individually represents $$ X= x/ \lambda_m $$ and $$ T = t/\tau_m $$. Also, $$ \lambda_m $$ represents the membrane space constant and $$ \tau_m $$ represents the membrane time constant.
The $$f$$ corresponds to a function that accounts for the ionic channels that exist through the cell membrane. For our simulation we use the [Hodgkin Huxley Equations](https://github.com/srirag-vuppala/HodgkinHuxley_model) to generate the ionic current. i.e $$ I_{ion} = -\frac{f(V,t)}{R_m} $$ Here $$R_m$$ is our membrane resistivity.

At its core the cable equation provides us with a framework where one can analyze the spatial distribution of electric pulses. The capacitance of the outer insulator or "membrane" leads to the the temporal and spatial differences in voltage or "transmembrane voltage". 

![Cable equation schema diagram](https://res.cloudinary.com/dtbzgpkps/image/upload/v1621910304/cable_equation_schema_diagram_66092dc06d.png)
*Representation of Cable Model at a cellular level* [[2]](#2)


### Assumptions required for the Cable Equation Model

While the model itself is our first true attempt of reasoning the conduction of the action potential, its still far from a complete explanation. Namely, due to a few key assumptions we make to derive our equation.

1. The cable is linear, constant and infinitely long cylinder.
2. The current flow is defined along the length of the cable ($$ V = V(x,t) $$)
3. We disregard radial or angular variables thus allowing us to expect the potential to be uniform when moving radially. This assumption is known as the [*core conductor assumption*](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1367927/).
3. The resistance from the extracellular and intracellular conductor/space ($$ r_e $$ and $$ r_i $$ in  $$\Omega \cdotp cm$$) is 
	$$
	r_e = \frac{1}{\Pi \cdot (rad+0.00001)^2 \cdot 20}
	$$

	$$
	r_i = \frac{1}{\Pi \cdot (rad)^2 \cdot 6.7}
	$$

	*Here $$ rad $$ is the radius of the cross-section of the cylinder.*

4. The membrane parameter $$ C_m $$ is constant and does not depend on the transmembrane potential.

### Simulation 

Considering all of the above assumptions I was able to model the behavior of the action potential as it traverses through the "strand" of cardiac tissue. The simulation below is animation of multiple snapshots in time (in $$ s $$) of the transmembrane potential (in $$ mV$$) vs length of the strand (in $$ cm $$). The simulation itself was coded up using python and related libraries, to view the code behind the simulation click [here](https://github.com/srirag-vuppala/cable-equation-model)

![cable_eqn.gif](https://res.cloudinary.com/dtbzgpkps/image/upload/v1622005156/cable_eqn_ce180b4b08.gif)


**The entire source code is available here** [Github](https://github.com/srirag-vuppala/cable-equation-model)
***
 
## Citations 

<a id= "1">[1]</a> Douglas, PK; Douglas, DB (2019). "Reconsidering Spatial Priors In EEG Source Estimation : Does White Matter Contribute to EEG Rhythms?". 7th International Winter Conference on Brain-Computer Interface (BCI), Gangwon, Korea (South): 1-12. doi:10.1109/IWW-BCI.2019.8737307.

<a id="2">[2]</a> Keener, Sneyd, and Sneyd, James. Mathematical Physiology: I: Cellular Physiology. 2nd Ed. 2009. ed. New York, NY: Springer New York : Imprint: Springer, 2009. Interdisciplinary Applied Mathematics, 8/1. Web.


 


Hello, my name is

Srirag Vuppala

Developer, Leader, and Amateur Plant Grower 🌱

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Work Experience

Software Engineering Intern at Amazon Web Services

Student Research Assistant at PathML Project

Fellowship Student Researcher at Cal Poly BEACON Research Program 2022

Student Research Assistant at IATPP

Software Engineering Intern at Project Jupyter

Full Stack Software Engineering Intern at Celebrate!

Fellowship Student Researcher at Cal Poly DSAI

Student Assistant at TRIO Achievers