MATHEMATICS STANDARD LEVEL INTERNAL ASSESSMENT;

A MATHEMATICAL APPROACH TO OPTIMIZING BLOOD VESSEL BRANCHING INTRODUCTION

PERSONAL INTEREST Coming from a family with a history of both hyper- and hypo-tension, I have always had a general interest in the workings of the circulatory system - particularly how the circulatory system can fail. This is an interest which I’ve expressed in the subjects I take in the IB, such as biology and physics, and has also played a role in motivating me to pursue a future career in medicine. The moment when my interest in the circulatory system intersected with mathematics was when learning about differential optimality in class. I remember being taken aback by the versatility of optimization and its seemingly endless applications in the real world. This got me thinking; “could optimality principles be applied to the circulatory system, and how significant could these applications be for human health?”. After doing some further research into the applications of optimality in biological organisms inspired me to focus on the branching of blood vessels, an overlooked yet essential aspect of the circulatory system’s functionality. Not only would this topic be an exciting application of differential optimality in biology, it would also allow me to apply knowledge from my physics and biology classes at school to view medicine from a new, mathematical perspective.

AIM OF THE EXPLORATION In 1878, German zoologist Wilhelm Roux made a set of hypotheses regarding the angle at which blood vessels branch in the circulatory system . These hypotheses were made on the basis of a set of real-life observations by Roux. This exploration 1

will aim to visualize Roux’s hypotheses by constructing a model of a branching blood vessel which mathematically optimizes the branching angle. Ultimately, this mathematical model will be tested against Roux’s hypotheses in order to determine the model’s accuracy.

BACKGROUND INFORMATION In order to understand this exploration, an adequate understanding of the circulatory system is necessary. The circulatory system consists of blood vessels (arteries, veins and capillaries) which transport blood around the body . In order to do this, 2

blood vessels such as arteries need to branch into smaller vessels, forming “arterial trees” which allow the blood to intercept body tissues . The aorta, which is the largest artery, branches into smaller arteries (the arterioles), which branch into the 3

smallest blood vessels (the capillaries) . This branching pattern will be the main focus of this exploration. 4

Blood in itself is also an important topic to explain, particularly the way in which blood flows around the body. The flow of blood is said to be “laminar” , meaning that blood flow can be imagined as consisting of many parallel layers of the fluid 5

sliding past one another, with a resistance existing between layers . This resistance is known as the “viscosity” of blood . The 6 7

opposite of laminar flow is “turbulent” flow, which is characterized by the “chaotic movement of particles in a fluid” . 8

The nature of blood as a fluid is also important to understand. Blood is known as an incompressible, non-Newtonian fluid, meaning that the density of blood (mass per unit volume) remains constant yet its viscosity changes depending on the amount of force applied to it . Blood is specifically known as a “shear-thinning” liquid, meaning that blood becomes less viscous the 9

more force is applied it . Conversely, a fluid whose viscosity remains constant independent of the forces applied to it is 10

called a Newtonian fluid.

ASSUMPTIONS MADE WHEN CONSTRUCTING THE MODEL As with most mathematical models of real-life concepts, the model which will be constructed in this exploration is accompanied by a set of simplifying assumptions. A first assumption made concerns the nature of blood as a fluid. In this exploration, blood will be modelled as a Newtonian fluid, despite being classified as a non-Newtonian fluid in real life . 11

1 Kurz, Haymo, Konrad Sandau, and Bodo Christ. 1997. “On the Bifurcation of Blood Vessels — Wilhelm Roux’s Doctoral Thesis (Jena 1878) — A Seminal Work for Biophysical Modelling in Developmental Biology.” Annals of Anatomy - Anatomischer Anzeiger 179 (1): 33–36. https://doi.org/10.1016/S0940-9602(97)80132-X. 2 Khan Academy. n.d. The circulatory system review . Accessed September 8, 2019. https://www.khanacademy.org/science/high-school-biology/hs-human-body-systems/hs-the-circulatory-and-respiratory-systems/a/hs-the-circulatory-system-review. 3 Ibid. 4 Ball, Karen. 2018. Academy of Ancient Reflexology. August 15. Accessed September 15, 2019. http://academyofancientreflexology.com/author/karen-ball/page/4/?cv=1. 5 Klabunde, Richard. 2018. Turbulent Flow. March 1. Accessed September 8, 2019. https://www.cvphysiology.com/Hemodynamics/H007. 6 What is Laminar Flow? 2018. Accessed September 8, 2019. https://www.simscale.com/docs/content/simwiki/cfd/what-is-laminar-flow.html. 7 Princeton. 2018. What is viscosity? Accessed September 8, 2019. https://www.princeton.edu/~gasdyn/Research/T-C_Research_Folder/Viscosity_def.html. 8 “What Is Turbulent Flow - Turbulent Flow Definition.” Nuclear Power. Accessed February 11, 2020. https://www.nuclear-power.net/nuclear-engineering/fluid-dynamics/turbulent-flow/. 9 Hill, Kyle. 2015. What kind of liquid is blood? October 30. Accessed September 8, 2019. https://archive.nerdist.com/what-kind-of-liquid-is-blood/. 10 Ibid. 11 Ibid.

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Given the fact that the viscosity of non-Newtonian fluids changes due to the amount of force applied to the fluid, modelling blood as a non-Newtonian fluid would be very difficult since the force applied to blood is different in different blood vessels, meaning that the viscosity of blood would be different in different blood vessels around the body. Therefore, in order to apply math which is commensurate with the maths standard level course, blood will be assumed to have a constant viscosity, as is the case with Newtonian fluids. This will, ultimately, help simplify the mathematical operations in this exploration.

Secondly, blood will be assumed to exhibit laminar flow, as is the case in real life. Not only will this assumption allow me to construct a model that is more accurate in real life, but it will also allow me to explore concepts such as resistance to flow which are important when optimizing blood flow in blood vessels.

A last assumption made is with regards to the structure of blood vessels. As is shown in Figure 1, blood vessels can have different shapes and may branch in different ways. Given the complexity of blood vessel branching, this exploration will assume that blood vessels have a constant, linear shape. This allows for an appropriate estimation of the shape of blood vessels, despite the fact that blood vessels may also be bent. Moreover, blood vessels will be considered to be rigid in this investigation - not flexible as they are in real life. This estimation will further simplify the mathematics used in the investigation given that accounting for the flexible nature of blood vessels would be very complex. Another assumption made is that blood is an incompressible fluid, as is also the case in real life. This will further contribute to the accuracy of my investigation. Additionally, the viscous flow which is characteristic of incompressible fluids is essential for this exploration, as described by Poiseuille’s Law (explained later on in this exploration).

Figure 1: visual model illustrating the branching path of a blood vessels

The last assumption made in this exploration is that smaller blood vessels branch from larger, primary blood vessels.

Therefore, the radius of the branching blood vessel is assumed to be smaller than the radius of the primary blood vessel. RELEVANCE OF THE EXPLORATION

The application of optimization to blood vessel branching will allow me to gain a better understanding of the circulatory system, but also has relevance in the real world, particularly in the field of organ engineering. Organ engineering is a very fast-advancing field, with the first organ containing blood vessels having been 3D printed in April 2019 . Since organ 12

engineering focuses on producing biological substitutes to injured or diseased organs, the ability to apply the mathematical approach of optimality to blood vessel branching may be valuable in modelling blood vessels that minimize the energy expended by the body for blood circulation, thus making artificial organs more efficient and thus more successful. While this exploration won’t provide an absolute model of a perfectly optimized blood vessel (due to the many factors involved in blood flow), it may shed light on some of the approaches and difficulties in developing, for example, artificial organs.

UNDERSTANDING POISEUILLE’S LAW In order to construct a first blood vessel model, it’s necessary to begin by mathematically describing the flow of a fluid. One way to do this is by utilizing Poiseuille’s Law, a physical law which was experimentally derived by French physicist Jean Poiseuille in order to explain the laminar flow of incompressible fluids such as blood through a tube. Poiseuille’s law is 13

shown in equation (1) below: 14

(1)

When applying equation (1) in the context of blood, Q is the volumetric flow rate of blood (m3s-1), ΔP is the change in pressure of blood (Pa), r is the radius of a blood vessel (m), μ is the viscosity of blood ( ) and L is the length of blood aP · s vessel (m). As is evident from the equation, the volumetric flow rate Q of blood is proportional to the radius of the blood vessel to the fourth power (r4) and inversely proportional to the length of the blood vessel (L).

12 Jee, Charlotte. 2019. A 3D-printed heart with blood vessels has been made using human tissue. April 16. Accessed September 8, 2019. https://www.technologyreview.com/f/613316/a-3d-printed-heart-with-blood-vessels-has-been-made-using-human-tissue/. 13 Brittanica. 2019. Jean-Louis-Marie Poiseuille . Accessed September 8, 2019. https://www.britannica.com/biography/Jean-Louis-Marie-Poiseuille. 14 OpenStaxCollege. 2018. Viscosity and Laminar Flow; Poiseuille’s Law . Accessed September 8, 2019. https://opentextbc.ca/physicstestbook2/chapter/viscosity-and-laminar-flow-poiseuilles-law/.

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MODEL 1: MINIMIZING HYDRAULIC RESISTANCE

DERIVING HYDRAULIC RESISTANCE From my physics classes, I know that a certain amount of work must be done to push a certain volume of a fluid through a tube. The amount of work which must be done can be expressed as being equivalent to the amount of resistance in the tube; the greater the resistance, the greater the amount of work which must be done. Ultimately, when applying this physical law to the circulatory system, it seems that the lower the resistance to the flow of blood in a blood vessel, the smaller the amount of work which must be done by the heart to pump that blood around the body. This, essentially, means that less energy will need to be expended by the body in order to sustain the circulatory system, improving the system’s efficiency and thus biologically “optimizing” it. Critically, the first mathematical model which I can construct could optimize a branching blood vessel by minimizing the hydraulic resistance to blood flow in blood vessels.

In order to derive hydraulic resistance, it’s helpful to refer to another aspect of physics; electricity. As a physics student, I am familiar with Ohm’s Law, an equation illustrating the relationship between voltage, current and resistance in an electrical circuit . Ohm’s Law states that voltage V is equal to current I multiplied by resistance R ( ). Importantly, it seems IRV = 15

that Ohm’s Law can be seen as a hydraulic equivalent to Poiseuille’s Law, following the theory of the electronic-hydraulic analogy . In this sense, voltage V is comparable to the change in pressure ΔP, current I is comparable to the volumetric flow 16

rate Q, and electric resistance Re is comparable to hydraulic resistance Rh . Using this analogy, I am able to rewrite Ohm’s 17

Law in a hydraulic form, as seen in equation (2):

R V = I e P R = Δ = Q h

(2)= Q = R hΔP

By rewriting the equation in this form, I saw that I am able to equate the value of Q in Poiseuille’s Law [equation (1)] and the value of Q in Ohm’s Law [equation (2)], giving me an expression for hydraulic resistance, as seen in equation (3):

RΔP = 8μL

πΔP r 4

= RΔP = 8μL

πΔP r 4

(3)R= = πr 48μL

By examining equation (3), several comments regarding hydraulic resistance to blood flow can be made. Firstly, it can be seen that only 3 variables; blood vessel length (L), viscosity (μ) and blood vessel radius (r), affect hydraulic resistance, as 8 and π are both constants. Secondly, the hydraulic resistance is inversely proportional to the blood vessel radius to the fourth power (r4), meaning that the hydraulic resistance increases as the radius of the blood vessel decreases. This makes sense, as a smaller radius means that the same volume of blood needs to be pushed through a smaller volume, which increases the resistance to flow. Additionally, the hydraulic resistance is proportional to the length of the blood vessel (L), meaning that hydraulic resistance increases as the length increases. This also makes sense as longer vessels have longer walls which provide more friction, increasing the resistance to flow. Lastly, it’s evident that hydraulic resistance is proportional to the blood viscosity. This makes sense since viscosity is the resistive force between the many “layers” of blood in laminar flow.

CONSTRUCTING A VISUAL MODEL While my understanding of hydraulic resistance was strengthened by deriving equation (3), I wanted to find a way to better contextualize how this equation would affect the flow of blood in a blood vessel. Consequently, I decided that making a visual model of a branching blood vessel (applying the previously made assumptions in this investigation) would best allow me to proceed in this exploration. The visual model which I created is shown in Figure 2.

15 AspenCore. n.d. Ohms Law and Power. Accessed September 8, 2019. https://www.electronics-tutorials.ws/dccircuits/dcp_2.html. 16 Johnson, Jack. 2014. Hydraulic-Electric Analogies, Part 1 . April 18. Accessed September 8, 2019. https://www.hydraulicspneumatics.com/technologies/hydraulic-electric-analogies-part-1. 17 Ibid.

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Figure 2: diagram of branching blood vessel (self-made diagram)

When looking at Figure 2, it was necessary for me to consider where a branch would arise from the “primary vessel” AOC. As mentioned earlier, the radius of the primary blood vessel r1 is assumed to be larger than the radius of the branching vessel r2. Referring back to equation (3), I need to consider minimizing the length and maximizing the radius of the branching path in order to minimize the hydraulic resistance to blood flow. This can be done by considering the right-angled triangle ABC in Figure 2.

If a branch were to originate from point A (Branch 1), the total length of the branching path AB would be minimized. This can be deduced using Pythagoras’ theorem, which states that “ ” where a is one side of the triangle (AC), b a2 + b2 = c2 is another side of the triangle (BC) and c is the hypotenuse of the triangle (AB). Ultimately, Pythagroas’s theorem shows that the length of Branch 1 (which is the length of the hypotenuse) will be smaller than any other combination of length which join point A to point B. However, a branch originating from point A will also have the effect of decreasing the average radius of the branching path AB, given that a larger proportion of the branching path will be in Branch 1 which has a smaller radius (r2) than the primary vessel.

On the other hand, if a branch were to originate at point C (Branch 2) the average radius of the branching path ACB would be maximized, given that the larger proportion of the branching path will be in the primary vessel which has a larger radius (r1) than the vessel branch. However, a branch originating at point C will also have the effect of increasing the total length of the branching path, as is also evident from applying Pythagoras’ theorem.

When considering these two fictional branches, it became evident to me that the branch from the primary vessel should originate somewhere in between points A and C at a point O where the branching path has the optimal ratio of average radius to length in order to optimally minimize the hydraulic resistance against blood flow along the path AOB. Given this, it becomes necessary to find at what angle θ this optimal branch should originate.

CREATING AN OPTIMAL EQUATION In order to minimize the hydraulic resistance to blood flow, I need to employ optimality principles using differential equations. In order to do this, I must first alter equation (3) in order to express the total hydraulic resistance in the branching path AOB. In this model, the blood’s viscosity is constant, as blood is modelled as a Newtonian fluid, therefore making the product of equal to a constant This is highlighted in the equation below:π

8μ .k

R = k Lr 4

Utilizing this equation, the total hydraulic resistance RT in the branching path AOB will incorporate the lengths and radii of sections AO and OB : 18

(4)( )RT = kr 41

L 1 +r 42

L 2

In equation (4), L1 represents the length of the blood vessel segment AO and L2 represents the length of the blood vessel segment OB. I decided to further develop equation (4) by trigonometrically deriving the lengths L1 and L2. I did this by visualizing triangle OBC in a self-made diagram, shown in Diagram 1, and utilizing trigonometric principles, particularly sine, cosine, cotangent (which is defined as ) and cosecant (which is defined as ). My derivations of side L1 and L2 1

tanθ 1sinθ

are illustrated below.

18 Adam, John. “Blood Vessel Branching: Beyond the Standard Calculus Problem.” Mathematic Magazine, June 2011. 4

AOL 1 =

Diagram 1: trigonometric analysis of blood vessel branching (self-made diagram)AC C = − O AC BC )= − ( · 1

tanθ AC Ccot(θ)= − B L cot(θ)= 4 − L 3

BL 2 = O ypotenuse= h pposite= o · opposite

hypotenuse C= B · 1

sin(θ) Ccsc(θ)= B csc(θ)= L 3 In these formulae, L3 is the length of the blood vessel segment BC and L4 is the length of the blood vessel segment AC. Substituting the above values of L1 and L2 into equation (4) creates the following equation:

RT (5)( )= kr 41

L −L cot(θ)4 3 +r 42

L csc(θ)3

In order to find the value of angle θ which minimizes the hydraulic resistance, I need to find the minimum value of the function RT, also referred to as the optimum solution to the function. In order to do this, I can use differential calculus to identify the turning points of this function, which may be the maximum or minimum points. First, I must find the first derivative of equation (5). Equation (5) includes two terms which aren’t included in the maths standard level syllabus, namely cosecant and cotangent, whose derivative are unknown to me. However, I am able to mathematically prove that “the first derivative of is and the first derivative of is ” , as is shown below:sc(θ)c sc(θ)cot(θ)− c ot(θ) c sc (θ) − c 2 19

sc(θ)c = 1sin(θ)

Therefore, [csc(θ)] [ ]ddθ = d

dθ1

sin(θ)

Using the quotient rule: [ ]d

dθ1

sin(θ) = sin (θ)2(sin(θ))(0) − (1)(cos(θ)) = − cos(θ)

sin (θ)2

Given that and :sc(θ)c = 1sin(θ) ot(θ)c = sin(θ)

cos(θ) [ ] sc(θ)cot(θ)d

dθ1

sin(θ) = − cos(θ)sin (θ)2 = − 1

sin(θ) · sin(θ)cos(θ) = − c

ot(θ)c = sin(θ)cos(θ) = 1

tan(θ) Therefore, [cot(θ)] [ ]d

dθ = ddθ

1tan(θ)

Using the quotient rule:

[ ]ddθ

1tan(θ) = tan (θ)2

(tan(θ))(0) − (1)( )1cos (θ)2

= −1

cos (θ)2

tan (θ)2 = −1

cos (θ)2

( )sin (θ)2

cos (θ)2

= − 1sin (θ)2

Given that :sc(θ)c = 1

sinθ [ ] sc(θ) sc(θ) sc (θ)d

dθ1

tan(θ) = − 1sin (θ) · 1

sin (θ) = − c · c = − c 2

Ultimately, this knowledge allows me to find the first derivative of equation (5), as is shown below:

{ [ (L ) (L cot(θ))] [ (csc(θ))]}dθdR T = k 1

r 41d

dθ 4 − ddθ 3 +

r 42

L 3 ddθ

{ [0 [ (cot(θ))]] [− sc(θ)cot(θ)]}= dθdR T = k 1

r 41− L 3

ddθ +

r 42

L 3 c

{ [(L csc (θ))] [csc(θ)cot(θ)]}= dθdR T = k 1

r 41 32 −

r 42

L 3

L { [(csc (θ))] [csc(θ)cot(θ)]}= dθdR T = k 3

1r 41

2 − 1r 42

L { }= dθdR T = k 3 r 41

csc (θ)2

−r 42

csc(θ)cot(θ)

The minimum hydraulic resistance would occur when or, in other words, when the gradient of the function of dθ

dR T = 0 equation (5) is equal to 0. Ultimately, when , I can solve to find θmin, the optimal branching angle which minimizes dθ

dR T = 0 hydraulic resistance:

19 Table of Derivatives. 2005. Accessed September 15, 2019. http://www.math.com/tables/derivatives/tableof.htm. 5

L { }= k 3 r 41

csc (θ)2

−r 42

csc(θ)cot(θ) = 0

=r 41

csc (θ)2

=r 42

csc(θ)cot(θ)

=r 41

r 42 = csc (θ)2csc(θ)cot(θ)

) = ( r 1r 2 4 = csc(θ)

cot(θ)

) = ( r 1r 2 4 = 1

sin(θ)

1tan(θ)

) = ( r 1r 2 4 = sin(θ)

tan(θ)

) os(θ)= ( r 1r 2 4 = c

(6) os (( ) )= θ min = c −1r 1

r 2 4

While equation (6) showed me a function for the optimal branching angle which minimizes the hydraulic resistance, it didn’t give me a visual indication of how the optimal branching angle changes when the ratio of r2 to r1 changes. For this reason, I decided to graph equation (6) using the online program Desmos. In this graph (Graph 1), the ratio was considered as the r 1

r 2 x-value and the value of θ in radians was considered as the y-value. This creates a graph of the function .(x) os (x )f = c −1 4

Graph 1: graph of equation (6) with adjusted domain Defining the domain and range of Graph 1 is important. Since the range of is { }, the domain (x) os (x )f = c 4 os(x)− 1 ≤ c ≤ 1 of must be { }. However, when considering this domain in the context of the blood vessel model, I os (x)c −1 − 1 ≤ x ≤ 1 came to the realization that it isn’t suitable. Given that this model assumes that a smaller vessel branches from a larger vessel, the value of must be larger than 0 and smaller than 1 (since 0 < r2 < r1). Therefore, the suitable domain for this r 1

r 2 graph is { }, as is illustrated in Graph 1. Moreover, while the range of is { }, the 0 < x < 1 (x) os (x )f = c −1 4 os (x )0 ≤ c −1 4 ≤ 2

π suitable range for this graph is { }, given that the branching angle must be greater than 0º (0 radians) but os (x )0 < c −1 4 < 2

π smaller than 90º ( radians). This is also seen in Graph 1.2

π

The value of θmin expressed in equation (6) could either represent a local maximum or a local minimum value, given that the first derivative of a graph can be 0 at both of these points. Given this, it is necessary to confirm that the value for θmin shown in equation (6) is, in fact, the minimum value. To do this, I can figure out the second derivative of RT at the point where . If > 0, then equation (6) is a minimum value. Conversely, if < 0, then equation (6) is a maximum x )( = θ

dθ 2d R 2

T dθ 2

d R 2T

value. I will determine the second derivative of RT by finding the derivative of the first derivative, as is shown below:

(kL ( )dθ 2d R 2

T = dθdR T

3 r 41

csc (θ)2

−r 42

csc(θ)cot(θ)

6

L (( [ [csc (θ)]]) [ [csc(θ)cot(θ)]])= k 3

1r 41

ddθ

2 − ( 1r 42

ddθ

Using the chain rule and the product rule I was subsequently able to find the derivatives for the terms and sc (θ)c 2

respectively, using their derivatives which I mathematically proved previously:sc(θ)cot(θ)c

L (( (− csc (θ)cot(θ))) [(− sc (θ)) csc(θ)) (cot(θ) (− ot(θ)csc(θ))]])= k 31

r 412 2 − ( 1

r 42c 2 × + × c

L (− )= k 3 r 41

2csc (θ)cot(θ)2

−r 42

−csc (θ) − cot (θ)csc(θ)3 2

Plugging in the value of θ ( ) from equation (6) and solving gives: os (( ) )θ min = c −1

r 1

r 2 4 The above equation represents the second derivative of RT, but due to its complexity and the many variables involved in it I was still unable to deduce whether its value was positive (denoting a minimum point on the graph) or negative (denoting a maximum point on the graph). To overcome this problem, I decided to choose random values for the radii r1 and r2 and plugging them in to the equation, keeping in mind that this model assumes that r1 > r2. For example, if r1 = 1 and r2 = 0.5:

Given that k and L3 are always positive, the value of the second derivative is positive (16.03kL3 > 0), indicating that the value for θ in equation (6) is, in fact, a minimum value. MODEL 2: MINIMIZING THE ENERGY REQUIRED TO MAINTAIN THE BLOOD VESSEL STRUCTURE After constructing my first model [equation (6)], which minimized the hydraulic resistance to flow, I couldn’t help but wonder if there were other factors which could be optimized in order to maximize the efficiency of the circulatory system. After doing some research I came across one such factor; the energy required to maintain the blood vessel structure. In deriving equation (5), I assumed that the only energy which an organism must expend is in the pumping of blood around the body. However, it is also suggested that there is a certain amount of energy which an organism must expend to maintain the structure of a blood vessel (i.e. keeping the blood vessel dilated). This energy expenditure must also be minimized in order to maximize the efficiency of the circulatory system. Theoretical biologist Robert Rosen proposed this form of energy expenditure by the body, stating that organic structures carry with them a “metabolic cost which roughly represents the energy expenditure required by an organism to maintain said structure” . Ultimately, if this energy expenditure is 20

minimized, then an organism would need to expend less energy to maintain the structure of blood vessels, thus optimizing the circulatory system. Rosen suggested that the energy required to maintain a given anatomical structure would be proportional to the volume of that structure. Given this, “it can be assumed that the energy required to maintain a blood vessel structure, S, would be proportional to the volume of the blood vessel, which can be assumed to be a cylinder. In this

20 Rosen, Robert. 1967. Optimality Principles in Biology. http://link.springer.com/openurl?genre=book&isbn=978-1-4899-6207-2. 7

equation, represents the volume of a blood vessel, with L being the length of the blood vessel and r being the radius of πr L 2 the blood vessel” : 21

(Lπr )S = B 2

The above equation, suggested by Rosen, included a constant B, whose significance wasn’t specified. In order to determine what this constant B represents, I once again turned to my knowledge in physics, particularly to pressure-volume work; “the work done by a fluid against a certain amount of external pressure” . The equation for pressure-volume work is shown 22

below, where P is pressure in Pascals and V is volume in m3;

pressure olume" work V " − v = − P In physics, “work” is the transfer of energy from one point to another, meaning that the equation for pressure-volume could express the energy required to maintain the blood vessel structure in the body. Ultimately, this would suggest that the constant B in Rosen’s equation is equal to -P (the external pressure on a blood vessel), given that “ ” represents the πr L 2 volume of a blood vessel V. Ultimately, applying this knowledge allows me to alter Rosen’s equation to express the total energy, ST, required to maintain the structure of the branching path AOB :

(L πr πr )S T = B 1

21 + L 2

22

I then substituted the values of L1 and L2 which I trigonometrically derived earlier into this equation:

(7) {[(L csc(θ))πr ] (L cot(θ))πr ]} S T = B 3

21 + [ 4 − L 3

22

Proceeding just as in the previous model, I proceeded to find the optimum solution of equation (7) by determining its first derivative, as shown below.

(L πr csc(θ)) πr ( (− cot(θ)) L ) dθdS T = B 3

21 · d

dθ + ( 22

ddθ L 3 + d

dθ 4 (L πr (− ot(θ)csc(θ))) πr ((− cot(θ)) ) = B 3

21 · c + ( 2

2 L 3 · ddθ + 0

L πr (cot(θ)csc(θ))) − πr ((− sc (θ)))= B − ( 321 · + ( L 3

22 c 2

(− πr cot(θ)csc(θ) πr csc (θ)) = B L 321 + L 3

22

2 L πcsc(θ)[− cot(θ) csc(θ)]= B 3 r 2

1 + r 22

The critical point of this function would occur when . I can therefore solve to find θmin, the optimal branching angle dθ

dS T = 0 which minimizes the energy required to maintain the blood vessel branching structure:

L πcsc(θ)[− cot(θ) csc(θ)]B 3 r 21 + r 2

2 = 0 r csc(θ)) r cot(θ))= ( 2

2 = ( 21

= csc(θ)cot(θ) =

r 21

r 22

)= 1sin(θ)

sin(θ)cos(θ)

= (r 1

r 2 2

os(θ) ) = c = (r 1

r 2 2

(8) os (( ) )= θ min = c −1r 1

r 2 2

I also decided to graph equation (8) using Desmos in order to better visualize it. In this graph (Graph 2), I considered the ratio as the x-value and the value of θ in radians as the y-value. This creates a graph of the function .r 1

r 2 (x) os (x )f = c −1 2

21Rosen, Robert. 1967. Optimality Principles in Biology. http://link.springer.com/openurl?genre=book&isbn=978-1-4899-6207-2. 22 Khan Academy. n.d. Pressure-volume work . Accessed October 6, 2019. https://www.khanacademy.org/science/chemistry/thermodynamics-chemistry/internal-energy-sal/a/pressure-volume-work.

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Graph 2: graph of equation (8) with adjusted domain

Evidently, Graph 2 is different than Graph 1. Although both graphs incorporate the same ratio on the x-axis, in Graph 1 r 1

r 2 this ratio is graphed to the fourth power while in Graph 2 it is graphed to the second power. Despite this difference, the appropriate domain and range of Graphs 1 and 2 remains the same (the domain is { } and the range is 0 < x < 1 { }), as both equations are inverse cosine equations and both abide to the same assumptions made in thisos (x )0 < c −1 2 < 2

π exploration. Again, in order to confirm that the value for θ expressed in equation (8) is, in fact, the minimum value, I will figure out its second derivative.

{BL πcsc(θ)[r csc(θ) cot(θ)}dθ 2d S 2

T = dθdS T

321 − r 2

2 L π[((− ot(θ)csc(θ))(r csc(θ) cot(θ))) csc(θ))(r (csc(θ)) (cot(θ)))]= B 3 c 2

1 − r 22 + ( 2

1 · ddθ − r 2

2 · ddθ

L π[((− ot(θ)csc(θ))(r csc(θ) cot(θ))) csc(θ))(r − ot(θ)csc(θ)) − sc (θ)))]= B 3 c 21 − r 2

2 + ( 21 · ( c − r 2

2 · ( c 2

KL πcsc(θ)[(r csc (θ)) 2r cot(θ)csc(θ)) r cot (θ))]= B 322

2 + ( 21 + ( 2

22

Plugging in the value of θ ( ) from equation (8) and solving gives: os (( ) )θ min = c −1r 1

r 2 2

L πcsc(cos (( ) )[(r csc (cos (( ) )) 2r cot(cos (( ) )csc(cos (( ) )) r cot (cos (( ) ))]dθ 2

d S 2T = B 3

−1r 1

r 2 2 22

2 −1r 1

r 2 2 + ( 21

−1r 1

r 2 2 −1r 1

r 2 2 + ( 22

2 −1r 1

r 2 2

Once again, I used the same random values for radii r1 and r2, r1 = 1 and r2 = 0.5, to determine whether equation (8) optimizes function ST and is a minimum solution to it:

L πcsc(cos ((0.5) )[((0.5) csc (cos ((0.5) )) 2cot(cos ((0.5) )csc(cos ((0.5) )) (0.5) cot (cos ((0.5) ))]dθ 2

d S 2T = B 3

−1 2 2 2 −1 2 + ( −1 2 −1 2 + ( 2 2 −1 2

L πcsc(1.318)[((0.25)csc (1.318)) 2cot(1.318)csc(1.318)) (0.25)cot (1.318))] = B 32 + ( + ( 2

L π(1.033)(0.267) (0.534) 0.017)) = B 3 + ( + (

.276BL π .551 = 0 3 + 0

Given that B, L3 and π are always positive, the value for the second derivative is greater than zero . 0.276BL π .551 ) ( 3 + 0 > 0 This means that the value for θ in equation (8) does, in fact, express a minimum value. MODEL 3: FINAL OPTIMAL MODEL After constructing models 1 and 2, I wanted to develop a final optimal equation which considered minimizing the hydraulic resistance and the energy required to maintain the blood vessel structure. I was able to do this by combining the equation for RT and the equation for ST which I had developed earlier. My newly formed equation, OT, is shown below:

OT = R T + S T

9

( ) (L r L r )= k

r 41

L 1 +r 42

L 2 + B 121 + 2

22

I further simplified this equation by removing the common term L1 and rearranging. For simplicity, the terms andr )( kr 41

+ B 21

can be named W1 and W2 respectively: Br )( kr 42

+ 22

( r ) ( Br )= L 1

kr 41

+ B 21 + L 2

kr 42

+ 22

W W = L 1 1 + L 2 2

I then proceeded to substitute the values of L1 and L2 derived earlier into the above equation:

L cot(θ))(W ) L csc(θ))(W ) = ( 4 − L 3 1 + ( 3 2 (L cot(θ)) L csc(θ) = W 1 4 − L 3 + W 2 3

Proceeding just as in the previous models, I found the optimum solution of this equation by determining its first derivative:

[ (L ) (cot(θ))] L [csc(θ)]dθdO T = W 1

ddθ 4 − L 3

ddθ + W 2 3

ddθ

[0 (− sc (θ))] L cot(θ)csc(θ)= W 1 − L 3 c 2 − W 2 3 L csc (θ) L cot(θ)csc(θ)= W 1 3

2 − W 2 3 csc(θ)[W csc(θ) cot(θ)] = L 3 1 − W 2

csc(θ)[W ( csc(θ) ot(θ))]= L 3 2 W 2W 1 − c

Here again, I further simplified the equation by removing the term csc(θ) from the parentheses:

L csc(θ)[csc(θ)( )]= W 2 3 W 2W 1 − csc(θ)

cot(θ)

L csc (θ)[ ]= W 2 32

W 2W 1 − 1

sin(θ)

sin(θ)cos(θ)

L csc (θ)[ os(θ)]= W 2 32

W 2W 1 − c

The critical point of this function would occur when . At this point θ would be the optimal branching angle:dθ

dO T = 0

L csc (θ)[ os(θ)]= W 2 32

W 2W 1 − c = 0

os(θ)]= [ W 2W 1 − c = 0

os(θ)= c = W 2W 1

os ( )= θ = c −1W 2W 1

After finding this equation, I was able to substitute the values for W1 and W2 from before : 23

os ( )= θ = c −1+Br k

r 41

21

+ Br kr 42

22

(9)os [(( ) )( )]= θ = c −1r 1

r 2 4k+Br 62

k+Br 61

The value of θ in equation (9) depends on the relative values of k and B. There exist two limiting situations where the value of θ is identical to the value of θ in the previous models: the first limiting situation is when the value of r2 approaches the value of r1 (r2 → r1). In this situation, the value of θ in equation (9) becomes identical with the value of θ in equation (6) as:

, thus making limr → r 2 1

k+Br 62

k+Br 61 = 1 os (( ) )θ = c −1r 1

r 2 4

23 Adam, John. “Blood Vessel Branching: Beyond the Standard Calculus Problem.” Mathematic Magazine, June 2011. 10

The second limiting situation is when the value of r2 approaches zero (r2 → 0). In this situation, the value of θ in equation (9) becomes identical with the value of θ in equations (6) and (8) as:

, thus making limr → 02

r 1

r 2 = 0 os (0)θ = c −1 = 2π

However, between these two limiting situations equations (6), (8) and (9) do not agree. One way in which I attempted to better understand equation (9) was by graphing it but, unlike equations (6) and (8), equation (9) has several different variables and is difficult to graph. Instead, in order to better understand equation (9) I decided to better understand the significance of in the equation. Given that all four values in this expression; k, B, r1 and r2, are positive, the value of k+Br 61

k+Br 62

will also be positive. Essentially, the effect of the expression is to horizontally stretch the graph ofk+Br 62

k+Br 61 k+Br 62

k+Br 61

by the scale factor . If the value of is greater than 1, it moves the points ofos (( ) )θ = c −1r 1

r 2 4 k+Br 61

k+Br 62 k+Br 62

k+Br 61 os (( ) )θ = c −1r 1

r 2 4

closer to the y-axis. Conversely, if the value of is greater than zero but smaller than 1 , it moves the k+Br 62

k+Br 61 0 )( < k+Br 62

k+Br 61 < 1

points of further away from the y-axis.os (( ) )θ = c −1r 1

r 2 4

After understanding the significance in the equation, I decided to try graph equation (9) by calculating values of θ k+Br 61k+Br 6

2

for different values of r1 and r2 and plotting these on a graph with x-axis “ ” and y-axis “θ”, as I had done with equations r 1

r 2

(6) and (8). In the expression , the value of constant k is , where µ is the viscosity of blood in pascal-seconds ( k+Br 62

k+Br 61 π8μ

). Since this value is constant it can be calculated, taking the viscosity of blood as at bodyaP · s .78 0 P a 2 × 1 −3 · s temperature (37ºC) : 24

.00708 P ak = π8μ = 3.1415926535

8 · (2.78×10 )−3

= 0 · s I was also able to deduce the value of B in equation (9) as it represents the external pressure on blood vessels in the body, as was explained when deriving model 2. This external pressure is 13,000 Pascals (Pa) . When graphing equation (6) and (8), 25

the graph produced illustrated the values of θ at every possible ratio of . However, to graph equation (9), I decided to r 1

r 2 apply the equation to real values of blood vessel radii, in order to make this optimal model more applicable to real life. I was able to do this by utilizing data collected by Zamir and Phipps which showed the diameter of arteries in the arterial tree of dogs , as illustrated in Table 1. These values are in metres given that Poiseuille’s Law also uses values in metres. 26

Table 1: radii of different types of blood vessels in a dog’s arterial tree

Blood vessel type Diameter of blood vessel (metres/m) Radius of blood vessel (metres/m)

Aorta 10.000 5.000

Large artery 3.000 1.500

Main branch 1.000 0.500

Secondary branch 0.600 0.300

Tertiary branch 0.140 0.070

Terminal artery 0.050 0.025

Terminal branch 0.030 0.015

Arterioles 0.020 0.010

Using these blood vessel radii values, I was able to find the corresponding values of θ in radians using equation (9) in different branching scenarios (different ratios of . These calculations can be seen in Table 2 below:)r 1

r 2

24 Anton Paar GmbH. n.d. Viscosity of Whole Blood . Accessed September 8, 2019. https://wiki.anton-paar.com/en/whole-blood/. 25 Jaliman, Debra. 2017. What is "normal" blood pressure? July 11. Accessed October 6, 2019. https://www.webmd.com/hypertension-high-blood-pressure/qa/what-is-normal-blood-pressure. 26 Zamir, M., and S. Phipps. 1988. “Network Analysis of an Arterial Tree.” Journal of Biomechanics 21 (1): 25–34. https://doi.org/10.1016/0021-9290(88)90188-1.

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Table 2: radii of different types of blood vessels in a dog’s arterial tree

Branching scenario r2/r1 θ in radians

Large artery branching from the aorta 0.300 1.542

Main branch branching from the larger artery 0.333 1.534

Secondary branch branching from the main branch 0.600 1.378

Tertiary branch branching from the secondary branch 0.233 1.555

Terminal artery branching from the tertiary branch 0.835 0.991

Arteriole branching from the terminal branch 0.667 1.293

Secondary branch branching from the large artery 0.200 1.569

Tertiary branch branching from the large artery 0.047 1.571

Ultimately, I was able to plot the values calculated in Table 2 to help me visualize what the graph of equation (9) would look like as well as how its relative shape compares to the graphs of equations (6) and (8). This is shown in Graph 3 below, where the blue dotted line is the graph of equation (6), the red dotted line is the graph of equation (8) and the black dots are the plotted values from equation (9):

Graph 3: comparison of equations (6), (8) and (9) as graphs

COMPARING THE MODEL TO ROUX’S HYPOTHESES To verify whether the values of θ from equation (9) are applicable to the real world, I referred to the hypotheses which Roux made regarding the branching of blood vessels in the circulatory system. These hypotheses are: “if one of the two branches is smaller than the other, the primary vessel will make a larger angle with the smaller branch than it does with the larger branch” and “branches which are so small that they don’t seem to weaken or diminish the blood flow from the primary vessel come off from it at a large angle from about 70º to 90º” . Values for the ratio from Table 2 can be used to test these r 1

r 2 27

hypotheses. This requires me to convert the radian values for θ from radians to degrees, which is done by multiplying the radian value by . As is seen, if = 0.200, then θ = 1.569 radians (≈ 89.50º); if = 0.600, then θ = 1.378 radians (≈ π

180 r 1

r 2 r 1

r 2

78.95º); and if = 0.835, then θ = 0.991 (≈ 56.78º). Evidently, as the ratio of increases (and thus the size of the r 1

r 2 r 1

r 2 branching vessel increases), the branching angle θ decreases. This adheres to Roux’s hypotheses, which state that smaller branching vessels come off from the primary vessel at larger angles, thus showing how my model’s results are in line with Roux’s hypotheses.

27 Passariello, Alessandra. "Wilhelm Roux (1850-1924) and blood vessel branching." Journal of Theoretical and Applied Vascular Research 1, no. 1 (2016): 25-40.

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EVALUATION OF THE MODELS In order to evaluate the models I developed, it's necessary to consider the assumptions made in this exploration and how these assumptions either improve or weaken the model’s application to the real world. One of the more prominent assumptions made was in the modelling of blood as a Newtonian fluid. Although this assumption greatly simplified the mathematics required to construct the models, it does hinder the real-life applications of these models to the circulatory system, such as in capillaries. The diameter of capillaries is roughly that of singular red blood cells , meaning red blood cells 28

can only travel through capillaries in a single layer. Given this, it is unreasonable to assume that blood travels as layers of fluid sliding past each other with viscosity in capillaries, as the present models assume. This presents the model’s limitations in applications in, for example, a medical context. Another limiting assumption made is that the blood vessels are perfectly linear, cylindrical and rigid. Although this assumption simplified the mathematics used, it is contrary to observations in real life, where blood vessels are often curved and made from flexible muscle fibers. This is said to have an impact on blood dynamics, a factor which was critically overlooked in this exploration which ultimately also limits the applications of these models to real life. Ultimately, the assumption made in this exploration allowed me to explore this topic with a level of mathematics that is commensurate with the standard level course. If this topic were to be explored more seriously, more advanced mathematics could be used to make the models more applicable to reality - an aspect which limited the applicability of my own exploration.

One more aspect of these models to consider is their scope and thus the extent to which they are able to fulfill the aim of this exploration. This exploration has been successful in producing an optimal equation which maximizes the efficiency of the circulatory system by optimizing the vessel branching angle whilst considering two factors to be minimized; hydraulic resistance and the metabolic cost of maintaining the blood vessel structure. Despite this success, deducing the optimal branching angle is only a small portion of the investigation into maximizing the efficiency of the circulatory system. For instance, while blood vessel branching was covered, blood vessel bifurcations, which arguably occur as frequently as blood vessel branching , were not considered at all. The models focused on optimizing blood flow in the path AOB yet overlooked 29

the significance of section C; the continuation of the primary vessel. This is an example of an extension to this exploration. Apart from this, further extensions which could be made to this investigation would be to minimize other factors which may increase resistance to blood flow. Returning to equation (1), it might be worth investigating the impact which different blood viscosities have on resistance to blood flow. This would allow the model to cover a larger scope, thus maximizing the efficiency of the circulatory system to a greater extent.

While the limitations of the models might seem overpowering, it’s also important to consider the successes of the models created. While they aren’t perfect, the models present a commensurate and mathematically accurate way to optimize the branching angle in blood vessels. The final optimal model was shown to adhere to Roux’s observations, thus somewhat confirming the credibility of equation (9). Furthermore, it’s also important to consider the inherent difficulties of applying optimality principles to biological organisms. This fact became increasingly apparent to me as I worked on this exploration and understood the complexity of biological organisms. This brought me back to one of the first questions I asked myself when beginning this exploration; “could optimality principles be applied to the circulatory system?”. It also sparked further questions which I asked myself, such as “why should I expect optimality principles to manifest in the natural world?”. This exploration has allowed me to discover that it is difficult to pragmatically justify the application of mathematical principles in biology, particularly given the many factors which affect biological systems such as the circulatory system. Ultimately, this has not only given me a better understanding of some of the fields in which mathematics still needs to grow but has also given me hope that mathematics will one day allow us to optimize functions of the human body; positively impacting human health and well-being. CONCLUSION AND EXTENSIONS TO THE EXPLORATION Although this mathematical exploration has limitations which prevent it from becoming directly applicable to real life, the results of the exploration should not be overlooked. The many successes of this model, alongside the difficulty of mathematically modelling biological systems, are still a step forward in understanding the applications of mathematical principles in the natural world. I have also come to realize that the significance of this exploration stems farther than the circulatory system. With certain modifications, e.g. to viscosity, this equation can also be applied to the transport of other bodily fluids in vessels through the body, such as lymphatic fluid in lymph vessels.

Further research and alternative approaches to this investigation would be valuable in elevating the success of the models created. Apart from considering other parts of the circulatory system, such as blood vessel bifurcation, it might also be sensible to further apply the models to real-life data. For example, integral calculus can be used to find the exact length of real-life blood vessels from a photo, which could be applied to the models to more closely mimic the true structure of the circulatory system (instead of the original assumption that blood vessels are linear structures).

28 Bailey, Regina. 2019. Understanding Capillary Fluid Exchange . August 19. Accessed September 14, 2019. https://www.thoughtco.com/capillary-anatomy-373239. 29 National Cancer Institute. n.d. Classification & Structure of Blood Vessels . Accessed September 14, 2019. https://training.seer.cancer.gov/anatomy/cardiovascular/blood/classification.html.

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LITERATURE

Adam, John. “Blood Vessel Branching: Beyond the Standard Calculus Problem.” Mathematic Magazine, June 2011. Anton Paar GmbH. n.d. Viscosity of Whole Blood . Accessed September 8, 2019.

https://wiki.anton-paar.com/en/whole-blood/. AspenCore. n.d. Ohms Law and Power. Accessed September 8, 2019.

https://www.electronics-tutorials.ws/dccircuits/dcp_2.html. Bailey, Regina. 2019. Understanding Capillary Fluid Exchange . August 19. Accessed September 14, 2019.

https://www.thoughtco.com/capillary-anatomy-373239. Ball, Karen. 2018. Academy of Ancient Reflexology. August 15. Accessed September 15, 2019.

http://academyofancientreflexology.com/author/karen-ball/page/4/?cv=1. Britannica. 2019. Jean-Louis-Marie Poiseuille . Accessed September 8, 2019.

https://www.britannica.com/biography/Jean-Louis-Marie-Poiseuille. Hill, Kyle. 2015. What kind of liquid is blood? October 30. Accessed September 8, 2019.

https://archive.nerdist.com/what-kind-of-liquid-is-blood/. Jaliman, Debra. 2017. What is "normal" blood pressure? July 11. Accessed October 6, 2019.

https://www.webmd.com/hypertension-high-blood-pressure/qa/what-is-normal-blood-pressure. Jayanti, Sreenivas. 2018. Computational Fluid Dynamics for Engineers and Scientists. Jee, Charlotte. 2019. A 3D-printed heart with blood vessels has been made using human tissue. April 16. Accessed

September 8, 2019. https://www.technologyreview.com/f/613316/a-3d-printed-heart-with-blood-vessels-has-been-made-using-human-tissue/.

Johnson, Jack. 2014. Hydraulic-Electric Analogies, Part 1 . April 18. Accessed September 8, 2019. https://www.hydraulicspneumatics.com/technologies/hydraulic-electric-analogies-part-1.

Khan Academy. n.d. Pressure-volume work . Accessed October 6, 2019. https://www.khanacademy.org/science/chemistry/thermodynamics-chemistry/internal-energy-sal/a/pressure-volume-work.

Khan Academy. n.d. The circulatory system review . Accessed September 8, 2019. https://www.khanacademy.org/science/high-school-biology/hs-human-body-systems/hs-the-circulatory-and-respiratory-systems/a/hs-the-circulatory-system-review.

Klabunde, Richard. 2018. Turbulent Flow. March 1. Accessed September 8, 2019. https://www.cvphysiology.com/Hemodynamics/H007.

Kurz, Haymo, Konrad Sandau, and Bodo Christ. 1997. “On the Bifurcation of Blood Vessels — Wilhelm Roux’s Doctoral Thesis (Jena 1878) — A Seminal Work for Biophysical Modelling in Developmental Biology.” Annals of Anatomy - Anatomischer Anzeiger 179 (1): 33–36. https://doi.org/10.1016/S0940-9602(97)80132-X.

National Cancer Institute. n.d. Classification & Structure of Blood Vessels . Accessed September 14, 2019. https://training.seer.cancer.gov/anatomy/cardiovascular/blood/classification.html.

OpenStaxCollege. 2018. Viscosity and Laminar Flow; Poiseuille’s Law . Accessed September 8, 2019. https://opentextbc.ca/physicstestbook2/chapter/viscosity-and-laminar-flow-poiseuilles-law/.

Passariello, Alessandra. "Wilhelm Roux (1850-1924) and blood vessel branching." Journal of Theoretical and Applied Vascular Research 1, no. 1 (2016): 25-40.

Princeton. 2018. What is viscosity? Accessed September 8, 2019. https://www.princeton.edu/~gasdyn/Research/T-C_Research_Folder/Viscosity_def.html.

Rosen, Robert. 1967. Optimality Principles in Biology. http://link.springer.com/openurl?genre=book&isbn=978-1-4899-6207-2.

Table of Derivatives. 2005. Accessed September 15, 2019. http://www.math.com/tables/derivatives/tableof.htm. What is Laminar Flow? 2018. Accessed September 8, 2019.

https://www.simscale.com/docs/content/simwiki/cfd/what-is-laminar-flow.html. “What Is Turbulent Flow - Turbulent Flow Definition.” Nuclear Power. Accessed February 11, 2020.

https://www.nuclear-power.net/nuclear-engineering/fluid-dynamics/turbulent-flow/. Zamir, M., and S. Phipps. 1988. “Network Analysis of an Arterial Tree.” Journal of Biomechanics 21 (1): 25–34.

https://doi.org/10.1016/0021-9290(88)90188-1. Images: Figure 1: Schreiner, Karch, Neumann, Neumann, Rodler, and End. Visual Representation of 3-Dimensional Staged

Growth Modeling a Coronary Arterial Bed. . Photograph. Vienna: International Journal of Bioelectromagnetism. Accessed February 13, 2020. http://www.ijbem.org/volume2/number2/139-143.htm.

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