Project of RBC

THE PROJECT OF THE RED BLOOD CELL

APPLYING THE RECENT STUDY TO DERIVE AN ANCHOR-RING RBC MODEL

A PROJECT

Presented to the Department of Mechanical and Aerospace Engineering,

California State University, Long Beach

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science in Mechanical Engineering

By Hung-Wen Huang

B.S., 2002, Northern Taiwan Institute of Science and Technology,

Taipei, Taiwan

December 2008

iv

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ……………………………………………………….. iii

LIST OF TABLES …………………………………….……………………….…… v

LIST OF FIGURES …………………………………………………………..……. vi

Introduction .......................……..…..............................................……………......... 1

Previous Experimental Researches for the RBC ..….............................….………… 1

Previous Mathematical Models and Analytical Solutions .…........….…………....... 6

Recent Study of the Anchor-Ring Vesicle Red Blood Cell .....…………….............. 10

Finding the Compatibility Equations ..…………............…………........................... 13

Procedure of Solving the Equations ..………….................………............................ 16

MATLAB Codes....……………………..…......... ..………….…............................... 18

Running an Example ……………......…………………............................................. 19

Inspection of the Result ..........………………………………..........…....................... 23

Final Results ........……….....................….............................…………….................. 23

Possible Improvement in the Future ........…………….............................……….….. 29

Summary ......................................................…………………….………..……….... 33

Reference ...................................…............………………………………..……….... 35

Appendix A .............................….……………....…………………..……….……… 37

Appendix B ..........................…….….…………………………………..…………… 38

Appendix C .............................……………..………………..……….……………… 39

v

LIST OF TABLES

TABLE Page

1. Valid Ranges for λ , p∆ , ck , and ρ ………….…………………….…… 19

2. The First result after running RK4 ………….…………………...…….…… 22

3. Final values for λ , p∆ , ck , ρ , a and c0 ………….…………………..… 24

vi

LIST OF FIGURES

FIGURE Page

1. Cross-Sectional Shape of the Average RBC and other

Geometrical Data ……………………………………………………...……. 3

2. Three fundamental 3D RBC Models ………….……….………...…….…… 4

3. Different Shapes of RBC ……………………………..…………………..… 5

4. Human Red Blood Cell Membrane Structure ……………………………… 7

5. The cross section of a ring vesicle with generating radii …………………... 9

6. The coordinate system of RBC’s quarter cross-section ……………………. 12

7. The flow chart for solving the differential equation ……………………….. 18

8. The first result for testing the codes …………………………………….…. 22

9. The quarter cross-section of the final result ……………………………….. 24

10. The full cross-section of the final result …………………………………. 25

11. The full 3D model of the final result of the RBC shape equation

(i.e. Eq.5 and Eq.6) ………………………………………………………... 26

12. The cross-section of the 3D model of the RBC shape equation

(i.e. Eq.5 and Eq.6) ………………………………………………………... 27

13. The cross-section and coordinate system of a biconcave-

discoid model ………………………………………………………………. 31

14. The full cross-section of the biconcave- discoid model which is

derived from Zhou’s shape equation ……………………………………….. 32

1

Introduction The Red Blood Cell (RBC) is one of the most common blood cells in our body. From

many previous research results, several RBC proprieties such as shape, dimension, and

osmotic pressure can now be determined by experiments with precise measurements. In

addition, there are several physical and mathematical models of the RBC that were

proposed by some biophysics pioneers. However, since finding the analytical solutions of

these RBC models involves a high complexity and nonlinearity, it requires further study

and research. The purpose of this project is to find the geometrical solutions for one of

the most recent study of the RBC, namely, solutions of the shape equation proposed by

Zhou [1]. The project focuses on the anchor-ring RBC [2]. By accessing several

mechanical properties such as osmotic pressure, bending module, and tensile stress, the

close form solutions were obtained, and the results are examined and verified with

previous experimental studies.

Previous Experimental Researches for the RBC Fung and Evan are two of the most important scholars in bioengineering. In their

previous work about the measurement of the RBC [3], they proposed an improved

method of measurements for RBC. According to Fung and Evan, dimensions of an object

in the living state such as a RBC are the keys of solving problems in biophysics. Also,

dimensions are the most important factors to determine what an object’s shape would be.

Thus, the measurements accuracy is very important before one can determine the shapes

of RBCs. However, measuring dimensions of the RBC with traditional or normal micro

measurement would present some limitations since the dimensions of the RBCs are too

2

small to be comparable with the light wavelength. Nevertheless, observing a RBC is such

a challenge due to its physical weakness. For example, when researchers wish to study

the elasticity of the RBC, they must observe the specimen under stress and strain,

whereas there will be a tremendous chance to spoil RBC’s biophysical properties because

the stress-and-strain relationship of RBC can be easily altered by fixation or radiation.

Thus, Fung and Evan processed the optical image through an interference microscope to

obtain a higher resolution of the image which could mitigate the problem and obtain the

further accuracy from the measurements of RBC. The results was recorded and produced

based on three different statistics conditions (RBC at 300, 217, and 131 milliosmoles

(mOsm), a unit of osmotic pressure that could be converted to other common pressure

unit. One osm is the osmotic pressure of a one molar solution, i.e. 1 mole/L of solvent).

Then, the three average shapes of RBC’s cross sections from the tonicity can be described

as shown in Figure 1 [3]. From Figure 1, several parameters of RBC’s geometry are

revealed. The diameter of RBC from tonicity is approximately between 6.78 and 7.82 �

m. The volume is from 94 to 164 �m3, and the surface area is from 135 to 145�m2.

Figure 1 shows the cross-sections of all three basic types of RBC shapes. By revolving

each cross-section shape with each central axis, three RBC shapes can be generated by

3D plotting as shown in Figure 2 [1]. Beside these three basic shapes, there are still

several mutations. As shown in Figure 2, the type of 300 and 217 mO are described as a

biconcave-discoid model. In fact, several variations of RBC shapes could exist due to

different environments and conditions.

3

Figure 1. Cross-Sectional Shape of the Average RBC and other Geometrical Data.

7.82 ��

Volume = 94 ��

Surface Area = 135 ��

300 mO

7.59 ��

Volume = 116 ��


217 mO

6.78 ��

Volume = 164 ��


131 mO

4

Figure 2. Three fundamental 3D RBC Models.

According to Gov, N. [4], altering the difference in osmotic pressure between the outside

and inside environment could cause the changes in RBC shape. This could result in

RBC’s geometric shape change into a swollen or crenated type. Moreover, altering the

structure of the underlying membranous cytoskeleton could also change RBC’s shape.

Figure 3a [5] shows the shapes of RBC inconsistently exist compared to the three

standard types while they are enduring different physical and biological conditions in our

bodies. In Figure 3b [5], both closed standard shape (left one) and the swollen shape

(right one) are shown. Figure 3c [5] indicates the existence of the nearly spherical shape.

In addition, a serious variation is shown in Figure 3d [5]. It is called an anchor-ring

vesicle without the membrane in the middle.

300 mO 217 mO 131 mO

5

Figure 3. Different Shapes of RBC.

6

Previous Mathematical Models and Analytical Solutions

For several decades, researchers have attempted to find out the analytical solutions to

describe the RBC’s physical and geometrical properties. In 1973, Helfrich [6] proposed

an energy equation that could determine the equilibrium shape of a vesicle by minimized

the free curvature energy, namely shape energy [2, 6]. The equation is shown below:

( )� � �+∆+−+= dAdVpdAccckF c λ20212

1 , (1)

where,

ck is the bending module (elastic constant).

1c and 2c are two principal curvatures.

0c is the spontaneous curvature of the vesicle membrane surface.

10 ppp −=∆ is the osmotic pressure difference between outer and inner media.

λ is a tensile stress acting on the membrane.

The Helfrich’s energy equation becomes the most significant analytical foundation of the

recent studies for RBC shapes. As a result, four of the above parameters ( ck , 0c , p∆ , λ )

become key factor in the BC studies. Therefore, understanding those parameters would be

the first step to investigate analytical solutions of RBC shapes. First of all, the parameter

ck represents the bending rigidity of the vesicle membrane. It is used as an elastic

constant to indicate the stiffness of the vesicle. The range of RBC’s bending module ( ck )

is between 9107.1 −× to 19107 −× (N.m) [7]. p∆ is the difference of the osmotic

pressure between RBC’s inner and outer media with the values between 65 to 140 N/m2

7

[7]. λ can be dynamic or static. This project studies the static case only, so the range of

λ is between 6102 −× to 6106 −× [8]. The RBC’s membrane structure is very complex.

It is constructed by lipid bilayers, spectrin network, and transmembrane proteins [4]. As

shown in Figure 4 [4], there could be many vacancies and uniform

Figure 4. Human Red Blood Cell Membrane Structure.

structures in RBC’s membrane, so it is very difficult to describe the detail of this

phenomenon. Thus, Helfrich [6] defined the spontaneous curvature ( 0c ) in the

mathematical model as a parameter to adjust the practical difference. Although 0c

usually appears as an unknown constant in the equation, it could be determined when the

boundary conditions and experimental data can be found though the process of solving

the differential equation. Therefore, finding 0c became the first important work of this

project. In 1987, Ou-Yang Zhong-Can, a theoretical physicist in crystal solid physics and

8

optics in China, and Helfrich derived the general shape equation (OH General Shape

Equation) based on Helfrich’s spontaneous curvature theory as follows [2, 9].

02))((42 202

1202

1 =∇+−−++−∆ HkHcKHcHkHp ccλ , (2)

where 2∇ denotes the Laplacian operator. H is the mean curvature, and K is Gauss

curvature. Unlike Helfrich’s energy equation, the OH general shape equation further

improves the use of ∆ p and λ without just considering those two parameters as

Lagrange multipliers. According to Ou-Yang [2], the various solutions of Eq.2 are

calculated by numerical methods under the assumption of the azimuthal symmetry and

topological spheres. He attempted to introduce the ring functions or toroidal functions

into certain aforementioned parameters [10]. Ou-Yang defined the name of the vesicle’s

shape for a further genus (besides those with spherical topology) as an anchor ring. Then,

transferred the mean curvature H, the Gauss curvature K, and the term 2∇ H into the

hyperbolic and trigonometric functions. After plugging those hyperbolic and

trigonometric functions of the mean curvature, Gauss curvature, and 2∇ H back into Eq.2,

certain coefficients are obtained, and it shows that the generating circles have the radii

with ratio 2/1 in the anchor-ring vesicles of RBC. Also, these rings are stable for

negative values of the spontaneous curvature 0c which satisfies

9.3)4/2()2( 232/1

00 −≈+−< πrc . (3)

According to Ou-Yang, since the mechanical equilibrium equation of vesicle membranes

9

(Eq.2) is highly nonlinear, it is not an easy task to find exact and analytic solutions. So far,

besides the case of a sphere, no analytical solution is known yet. Based on the result of

what Ou-Yang had found, Mutz and Bensimon made an observation of RBC’s toroidal

vesicles and found an agreement with Ou-Yang’s theory [11]. They tried to measure the

dimension of the outer diameter (D) and the width (d) of the ring vesicle as shown in

Figure 5 [11].

Figure 5. The cross section of a ring vesicle with generating radii. D represents the outer

diameter and d is the width of the ring.

Their observation reveals the mean ratio between the outer diameter D and the width d is

02.041.0 ± which is in good agreement with the theoretical prediction proposed by

Ou-Yang [2] (i.e. 12 − , the appropriate value for the ratio of the diameters).

10

Recent Study of the Anchor-Ring Vesicle Red Blood Cell In the studying of mathematical model for RBC shapes, three important shape equations

were developed. They are Deuling and Helfrich (DH) Shape Equation, Seifert, Berndl

and Lipowsky (SBL) Shape Equation, and Hu and Ou-Yang (HO) Shape Equation [1].

The first two were developed based on Helfrich’s spontaneous curvature theory and

energy, and the last one was derived based on OH equation (Eq.2) [12]. According to a

recent study [1], general analytical solutions of these shape equations are still unavailable

so far. Therefore, Zhou proposed a new method to obtain the solutions for anchor

ring-vesicle membranes of RBC [1]. One of the examples for the nth order (highest order),

variable coefficients, inhomogeneous and nonlinear ODE (NVIN-ODE). is the HO Shape

Equation, which was derived from anchor-ring vesicles’ geometry as shown in Eq.4 [12].

2

23322

2

22

3

33 cos2

cos21

sincoscossin4cosρϕ

ρϕ

ρϕϕϕϕ

ρϕ

ρϕϕϕ

ρϕϕ

dd

dd

dd

dd

dd +��

�

��

��

��

� −+−

ρϕλ

ρϕ

ρϕϕϕ

ρϕ

ρϕϕ

ddc

kc

dd

c��

��

��

��

�+−−−+��

�

��

�−2

sin22

cos2sincos

2cossin7 2

002

222

2

2

( )0

sin22

cos1sin 20

3

2

=∆+��

��

�+−++

cc kpc

k ρϕλ

ρϕϕ

(4)

where, ρ is a spatial variable and )(ρϕϕ = is an angular function of ρ , and

0c , ck , p∆ , and λ are same as defined in Eq.1. This class of NVIN-ODEs possess a very

high nonlinearity. Zhou indicated that it is not possible to obtain the analytical solutions

for this type of NVIN-ODEs by using the common and traditional techniques for several

11

reasons. First of all, the individual methods in the differential equation handbooks cannot

be used to solve the Eq.4. In addition, the approximate analytical methods can only be

valid when the equations have less complexity and nonlinearity. Furthermore,

semi-analytic methods can be applied if the nonlinear or inhomogeneous terms in an

NVIN-ODE are transformable to polynomials or variable coefficients. Finally, using

numerical techniques can only obtain a specific solution with certain conditions. Hence,

Zhou proposed a new and specific method that could find the analytical solution for the

HO Shape Equation (Eq.4) as follows [13]:

Let

)(tan)( ρϕ

ρρ =

ddz

(5)

where, )(ρϕ contains three solutions shown as follows [13]:

( )��

�

�

��

�

�

+

��

��

�−+−��

�

��

�−+∆+

= −

ρρλ

ρρλρλ

ρϕ0

2

30

222

030

11 2

223

sinca

k

aaca

kca

kpc

c

cc (6)

( )��

�

�

��

�

�

−+

+−��

��

�−+∆+

= −

ρρρλ

ρρλρλ

ρϕ23

05

320

242

060

12

22

223

sinbcb

k

bcbk

cbk

pc

c

cc (7)

12

( )( )

��

�

�

��

�

�

−−++

++−+��

��

�−−+

��

��

�−−+��

�

��

�−+∆+

= −

ρρρρρλ

ρρρ

ρλρλρλ

ρϕ

2230

40

5

3220

223

0

3

402

20

520

60

13

22

223

23

32

23

sin

babcbcak

babcbbaacab

a

kbca

cbk

cak

pc

c

ccc

(8)

The coordinate system of Zhou’s shape equation solution is shown in Figure 6. Variable z

in Eq.5 is the central axis of the spherical symmetric body in the cross-section shown in

Figure 6. ρ axis is used to represent the inner radius ( iρ ) and the outer radius ( fρ ) of

RBC in the cross-section. ϕ indicates the angle between ρ axis and the tangential

lines of RBC’s outline, and y is the axis of the depth in 3D space.

Figure 6. The coordinate system of RBC’s quarter cross-section.

z

ρ

ϕ

iρ fρ

y

13

Finding Boundary Conditions

Before starting to solve those equations, the preliminary targets and procedures must be

clarified by our observation due to the complexity of the equations. First of all, it is

important to clarify what are known and what are unknown in those equations. From

Eq.5, it is known that )(ρϕ represents the angle of the curve as shown in Figure 6, and

there are three solutions for )(ρϕ represented by three inverse trigonometric functions

(sin-1) from Eq.6 to Eq.8. In those three sin-1 functions, there are several common

parameters defined from the previous studies as mentioned in Eq.1. Those parameters are

the bending module ( ck ), osmotic pressure difference ( p∆ ), and tensile stress ( λ ). The

values of these parameters were determined by researches mentioned in previous sections

[7, 8]. Hence, those parameters can be treated as given constants. Moreover, although the

geometry of the anchor ring-vesicle RBC is slightly different from the typical shapes that

were presented by Fung [3], it is still acceptable to use the size of the original shapes to

approximate the anchor ring-vesicle RBC. Therefore, the variable ρ could also be

considered as a known quantity. As a result, the unknown constants in Eq.6 are a and c0;

whereas b and c0 are unknowns in Eq.7; and a, b, c0 are unknowns in Eq.8. The unknown

constants a and b are integral constants when the solution equations were derived, and c0

is the spontaneous curvature of the vesicle membrane surface mentioned in Helfrich’s

energy equation (Eq.1). There is no sufficient information from previous studies to help

finding what values those constants should be. Therefore, the only way to determine

those constants is to use the boundary conditions for Eq.6, Eq.7, and Eq.8 mathematically.

In Figure 6, the curve is a quarter cross-section of anchor ring-vesicle RBC. Thus, it

shows clearly in Figure 6 that the angle between the tangent line of the curve and the ρ

14

axis, at the points iρ and fρ are 2π and 2

π− . In addition, )(ρϕ is represented by

sin-1 functions from Eq.6 to Eq.8, so two boundary conditions for )(ρϕ can be written

as follows:

)1(sin2

)( 1−== πρϕ i (9)

)1(sin2

)( 1 −=−= −πρϕ f (10)

Therefore, using Eq.9 and Eq.10 to revise Eq.6, Eq.7 and Eq.8, and the boundary

conditions now can be established in following manner [13]:

��

��

�

−=+

��

��

�−+−��

�

��

�−+∆+

=+

��

��

�−+−��

�

��

�−+∆+

12

223

12

223

02

30

222

030

02

30

222

030

ffc

ffc

fc

iic

iic

ic

cak

aaca

kca

kpc

cak

aaca

kca

kpc

ρρλ

ρρλρλ

ρρλ

ρρλρλ

��

��

�

−=−+

+−��

��

�−+∆+

=−+

+−��

��

�−+∆+

12

222

3

12

222

3

230

5

320

242

060

230

5

320

242

060

fffc

ffc

fc

iiic

iic

ic

bcbk

bcbk

cbk

pc

bcbk

bcbk

cbk

pc

ρρρλ

ρρλρλ

ρρρλ

ρρλρλ

(12)

(11a)

(11b)

15

( )

( )

��

��

�

−=−−++

++−+��

��

�−−+

��

��

�−−+��

�

��

�−+∆+

=−−++

++−+��

��

�−−+

��

��

�−−+��

�

��

�−+∆+

12

222

32

33

2

23

12

222

32

33

2

23

2230

40

5

3220

223

0

3

402

20

520

60

2230

40

5

3220

223

0

3

402

20

520

60

fffffc

fff

fc

fc

fc

iiiiic

iii

ic

ic

ic

babcbcak

babcbbaacab

a

kbca

cbk

cak

pc

babcbcak

babcbbaacab

a

kbca

cbk

cak

pc

ρρρρρλ

ρρρ

ρλρλρλ

ρρρρρλ

ρρρ

ρλρλρλ

(13)

It is obvious that there are three unknowns in Eq.8 (a, b, and c0), but there are only two

boundary conditions in Eq.13. Thus, one more boundary condition is required. Since

there is a peak (the slope equals zero) of the curve for the anchor-ring vesicle between iρ

and fρ , and for this smooth and continuous curve, )(3 ρϕ must be zero. By defining the

peak point as mρ , )(3 mρϕ can be described below:

)0sin(0)(3 ==mρϕ (14)

And, the third boundary condition for Eq.8 can be derived by substituting Eq.14 into

Eq.8, thus:

16

( )0

22

223

23

32

23

2230

40

5

3220

223

0

3

402

20

520

60

=−−++

++−+��

��

�−−+

��

��

�−−+��

�

��

�−+∆+

mmmmmc

mmm

mc

mc

mc

babcbcak

babcbbaacab

a

kbca

cbk

cak

pc

ρρρρρλ

ρρρ

ρλρλρλ

(15)

With all of those boundary conditions available, now it is possible to determine the

solutions of unknown constants a, b, and c0. and further obtain the solutions for Eq.5 ~

Eq.8.

Procedure of Solving the Equations

Although all boundary conditions for three types of )(ρϕ are found, there are still

several difficulties for solving the final differential equation (Eq.5 and Eq.6). These

difficulties are listed as follows:

1. ck , p∆ , λ , and ρ in the equations of )(ρϕ are treated as known constants.

However, the current data for those parameters are presented by various ranges, so it

cannot assign specific values to each parameter immediately. Hence, it is necessary to

seek the best combination of those parameters by assigning values from those various

ranges one by one, and that could be hundreds or thousands of combinations, which

could spend tremendous time in trial and error even if run these calculations by using

computers.

2. Because all boundary conditions contain a lot of unknowns, there will be more than

one solution for those variables. For example, there will be 7 solutions for the

unknown constants a and c0 in the Eq.11. Each solution could be used to solve the

17

differential equation (Eq.5), so the multi-solutions enhance the difficult task of solving

the equation.

3. The boundary conditions are available, but all of them are very complex. Therefore,

finding the solutions could be quite a challenge work.

4. The differential equation (Eq.5) is highly nonlinear. It is not possible to apply

traditional methods to solve this type of differential equation.

Since the first and second problem as shown in Eq.5 and Eq.6 indicate that seeking

proper values for ck , p∆ , λ , and ρ is truly a difficult task, especially when there are

more than one solution, it is necessary to spread out tasks for finding the solutions of

those equations. Thus, this report targets primarily on the Eq.11 and solve for the

unknown constants (a and c0). Another team member for the RBC project will deal

with Eq.12. For the 3rd problem listed above in Eq.11, the modern computing software

such as MATLAB could solve most of the algebraic problems. However, so far, it is still

unable to find the algebraic solution for Eq.13 and Eq.15 by using MATLAB due to their

high complexity, so the task for solving )(3 ρϕ is still an ongoing job and can be

considered as the future research topic. After finding solutions for Eq.11, it is possible to

plug all parameters including ck , p∆ , λ , ρ , a, and c0 back to the Eq.6 and solve Eq.5.

Even though the 4th problem listed in Eq.6 indicates that it is not possible to solve the

differential equation (Eq.5) directly, it may use the Forth-Order Runge-Kutta (RK4) as

the alternative route to find the solution for final results [14]. In conclusion, the entire

procedure for solving Eq.5 with the boundary conditions for )(1 ρϕ is listed in the flow

chart below (it also indicates what MATHLAB code is used in running a code if the

process is necessary).

18

Figure 7. The flow chart for solving the differential equation.

MATLAB Codes

As shown in Figure 7, there are 3 MATLAB codes used to solve Eq.11 (solutions for a

and c0), run RK4, and graph the solution, which may describe the quarter cross-section of

the anchor-ring RBC. The 3 codes are following:

1. c0a.m is a valid solver for 0c and a . It can be run independently or called by other

program such c0a_RK4 which will be mentioned in the next code description. This

code is shown in Appendix A.

Start Pick Values

for ck , p∆ , λ , ρ Soving Eq.11

(Run c0a.m in MATLAB)

Solve Eq.5 by RK4 (Run RK4Method.m)

i = 1 Select the ith Solution of Eq.11

If the result satisfies our expectation

No, and needs new solutions for Eq.11.

No, i+1

Final Results Yes

19

2. RK4Method.m is a code of the Fourth-Order Rung-Kutta Method [14]. It is used to

solve the differential equation (Eq.5 and Eq.6) that describes the shape of the quarter

cross-section of the anchor-ring RBC numerically. This code is shown in Appendix B.

3. c0a_RK4.m is a coordinate code which mainly combines the above two codes and

achieves the tasks below:

a. Using c0a.m to obtain the solutions of a and c0.

b. Using RK4Method.m to solve the differential Equation.

c. Plot the solutions and show the quarter cross-section for the anchor-ring RBC.

This code is shown in Appendix C.

Running an Example

Now pick values for λ , p∆ , ck , iρ , and fρ form the valid various ranges provided by

Fang, Evan, Daoa, Limb, and Suresh [3, 7, 8]. The valid ranges for λ , p∆ , ck , iρ ,

and fρ are obtained in the literatures [3, 7, 8], and listed in Table 1 [3, 7, 8].

Table 1 Valid Ranges for λ , p∆ , ck , and ρ

Parameter Valid Range

λ (Static) 2 610−× ~ 6 610−× (N/m)

p∆ 65 ~ 140 (N/m2)

ck 1.7 1910−× ~ 7 1910−× (N�m)

ρ 3.5 ~ 4.25 (�m)

Before running the codes, there are two mathematical and numerical concepts that need

20

be notified in order to obtain converged solution.

a. Due to the mathematical property of the equation, it will obtain a zero value in the

denominator when iρ is the first value of ρ in RK4. Therefore, picked the initial

value hi += ρρ , where h is the distance in ρ axis between each node in RK4.

b. To obtain the correct and proper geometric shape of anchor-ring, the initial value of z,

)( iz ρ , should be zero. However, since iρ cannot be used as the first value of ρ , we

use )( hz i +ρ instead as the initial condition. In this step, the value of

[ ( )ii zhz ρρ −+ )( ] is actually extremely small, so it is assumed that

( ) 0)( ==+ ii zhz ρρ as the initial condition.

Now codes can be tested in MATLAB by picking up a set of values from Table 1

randomly. The following executed lines and results indicate how the codes can be run in

MATLAB:

>> L = 4e-6; %λ = 4e-6

>> k = 4.35e-19; % ck = 4.35e-19

>> p = 102.5; % p∆ = 102.5

>> rowi = 1e-7 % iρ = 1e-7, which is very close to z-axis.

>> rowf = 3.875e-6; % fρ = 3.875e-6

>> n = 1000; %Number of the steps for RK4 >>[f, a, c0, row ,z] = c0a_RK4([L k p], [rowi rowf n]);

>> f

f =

Inline function:

f(r,z) =

(-8682305152453676218.3569555080501.*r.^3+15401138462824.196266346940557774.*r.^2+

19473.772350985706009785227108636.*r-.22099581656634766103189420321293e-1)./(23540

21

22988505747./128.*r.^2-587311.70922759403309972010759134.*r)./(1-(-868230515245367

6218.3569555080501.*r.^3+15401138462824.196266346940557774.*r.^2+19473.77235098570

6009785227108636.*r-.22099581656634766103189420321293e-1).^2./(2354022988505747./1

28.*r.^2-587311.70922759403309972010759134.*r).^2).^(1./2)

>> a

a = (solutions of the integral constant a)

.22104982238020007329205431289300e-1 (1st Solution)

2.4616811451618729848649739665309+3.5587725830356032674413153925189*i (2nd Solution)

-.20890334653782119959493640774216+2.1809967962511503510305030279674*i (3rd Solution)

-2.2638302897430617889577062896969+.40301909052246416082061811282175*i (4th Solution)

-2.2638302897430617889577062896969-.40301909052246416082061811282175*i (5th Solution)

-.20890334653782119959493640774216-2.1809967962511503510305030279674*i (6th Solution)

2.4616811451618729848649739665309-3.5587725830356032674413153925189*i (7th Solution)

>> c0

c0 = (solutions of the spontaneousnes curvature c0)

-26569200.685329338483356421364725 (1st Solution)

4248583.280266107248254593678458+16903749.17458962984855662460197*i (2nd Solution)

14956931.255616129098041258346095+23621786.310599776672610785107495*i (3rd Solution)

-14392247.775785395070239077161247-692797.622609872547821476254073*i (4th Solution)

-14392247.775785395070239077161247+692797.622609872547821476254073*i (5th Solution)

14956931.255616129098041258346095-23621786.310599776672610785107495*i (6th Solution)

4248583.280266107248254593678458-16903749.17458962984855662460197*i (7th Solution)

22

Table 2 The First result after running RK4

N ρ ( 610−× ) z ( 610−× )

1 0.103779 0

2 0.107558 0.025393051

3 0.111336 0.04487378

: : :

994 3.859885 -0.812183029

995 3.863664 -0.845861928

996 3.867442 -0.885867513

997 3.871221 -0.938098541

998 3.875 -1.008098083230237 + 18901.44915858783i

999 3.878779 -1.008098083231395 + 37802.96842509027i

1000 3.882558 -1.008098083231395 + 37803.02086313524i

Figure 8. The first result for testing the codes.

z

ρ

23

Inspection of the Result

First of all, there are 7 solutions for both a and c0 in page 21. However, there are several

complex numbers among these solutions (i.e. 6 complex numbers in both a and c0 as

shown in page 21). Practically, These complex values may not be reasonable if one

attempt to apply them in RK4. Thus, get ride of these complex solutions would be the

first job before trying to solve Eq.5 and Eq.6. Since Zhou’s shape equation solution is

focused on anchor-ring vesicle RBC, the frame of the result in Figure 6 shows a perfect

quarter anchor-ring vesicle shape. Figure 8 shows that the first attempt did not quite hit

the target. Unlike Figure 6, the last point of the curve in Figure 8 is not close enough to

the ρ axis. Moreover, the result obtained from RK4 (Table2) shows complex values in

the last 3 terms which are also considered impractical and negligible. Therefore,

according to the inspection, it is necessary to either re-apply another set of a and c0 or

reassign new values for λ , p∆ , ck , iρ , and fρ to match the best result. On the other

hand, since the result obtained is a smooth and continuous curve whose tangent angle is

restricted between 2π and 2

π− , it revels that there is a high chance to achieve the goal of

this project.

Final Results

After struggling with seeking the best values for λ , p∆ , ck , iρ , and fρ , the final

solution, which could be the best results to describe the anchor-ring vesicle RBC shape,

were found. The values for all parameters and constants are listed in Table 3.

24

Table 3 Final values for λ , p∆ , ck , ρ , a and c0

Parameter & Constant Final Value

λ (Static) 2.04 610−× (N/m)

p∆ 140 (N/m2)

ck 7 1910−× (N�m)

iρ 1 (�m)

fρ 3.875 (�m)

a 0.197807 210−×

c0 -69998148.957300

By applying those values to solve Eq.5 and Eq.6 with RK4, there are 1000 points

obtained to plot the results as follows:

Figure 9. The quarter cross-section of the final result.

ρ

z

25

As shown in Figure 9, although the last point does not exactly on the ρ axis ( 0≠z ),

the z value of the last point is –0.046337�m, which is less than 4% of the peak of the

curve (1.3146�m), so this tiny overstepping can be tolerant. Finally, one can close up the

plot by establishing a mirror image of the curve as shown in Figure 10.

Figure 10. The full cross-section of the final result.

Now the 3D model of the final result can be obtained by revolving the closed curve,

which is shown in Figure 10, around the z axis and plotted in Figure 11, and Figure 12.

ρ

z

26

Figure 11. The full 3D model of the final result of the RBC shape equation

(i.e. Eq.5 and Eq.6).

ρ

z

y

610−×

610−×

610−×

27

Figure 12. The cross-section of the 3D model of the RBC shape equation

(i.e. Eq.5 and Eq.6).

ρ

z

y

610−×

610−×

610−×

28

The 3D model indicates that the final result is quite an achievement which can almost

perfectly describe a shape of anchor-ring RBC, and further improvement is possible to

obtain a solution for anchor-ring RBC from Zhou’s shape equation solution. With these

tangible solutions, now few practical properties of Zhou’s anchor-ring vesicle RBC shape

equation solution could be examined by comparing with the experimental results. They

are:

1. Referring Figure 5, the outer diameter D of the final result equals =×= 875.322 fρ

7.75�m, which is very compatible with the results in Figure 1 (D = 6.78 ~ 7.82�m).

2. According to Mutz and Bensimon [11], the ratio between the outer diameter (D) and

the width of the ring (d) in Figure 5 is 12 − . In the above final result, as shown in

Figures 11 and 12 the ratio is 0.38 which represents a high agreement with Mutz and

Bensimon’s experimental data.

3. The total length of the curve in Figure 9 is 4.3201�m. There is no experimental data

that can be compared with. However, it is able to calculate the surface area of the final

model. Since the top view (y- ρ plane) is a circular shape, the surface can be calculated

by applying circular area equation as follows:

( ) ( ) ][2 22 πρπρ iiclA −+×= (16)

where,

A is the surface area of the final model.

cl is the curve length for the quarter section of the final model.

29

Therefore, the surface area of the final model can be calculated in the following

calculation:

( ) ( ) 172]113201.4[2 22 =−+×= ππA �m2

The calculation shows that the surface area of the final model is greater than the results

in Figure 1. However, Due to the different shapes between biconcave-discoid and

anchor-ring vesicle RBC, the inconsistent surface area is expected, but the slight

difference may possibly imply that this final model could actually be a mutation

shape from the standard biconcave-discoid model shown in Figure 1.

Based on the above comparison, this final model shows a good agreement with these

important experimental results. Although there are still some properties that are not able

to confirm due to the lack of data, this final model displays a confident demonstration of

the shape equation solutions proposed by Zhou for computing the shape of anchor-ring

vesicle RBC.

Possible Improvement in the Future

Since studying in RBC is an ongoing task, more and more features, properties, and

theories could be discovered. For this project, although the final model shows a good

match with previous studies, there are still several possible improvements that can be

studied in the future.

30

1. This report only applies the )(1 ρϕ (Eq.11a) as the equation to solve the primary

differential equation (Eq.5), so there are still two more boundary conditions that need

be tried.

2. The second equation ( )(2 ρϕ ) is run by another team member of the entire RBC project,

and it represents very different results that need be examined and compared with the

model in this report.

3. So far, the 3rd equation ( )(3 ρϕ ) cannot be solved because MATLAB is not able to

process the high complexity of the algebraic problem. The function )(3 ρϕ represents

3 boundary conditions, which could be flexibly established a more realistic model for

anchor-ring RBC. Since there is no way to obtain the solution via the aforementioned

approach, looking for other algebraic or numerical method to handle the extremely

complicated equation in the future is imperative.

4. Now, several properties such as diameter and surface area can be compared with the

experimental data of biconcave-discoid model. Based on the limited studies so far, to

better verify this analytical model proposed by Zhou, more experimental data for

anchor-ring vesicle RBC are required.

5. Although Ou-Yang mentioned [2] that it is only likely to find the solutions for the

specific shape one by one. Zhou’s shape equation solution may be applied to examine

the general RBC shape. Figure 13 shows the cross-section and coordinate system of a

biconcave-discoid model. In the figure, the angle )( iρϕ in the middle of the

biconcave-discoid model is zero. Therefore, the boundary conditions listed in Eq.11a

and Eq.11b have to be modified when applying Zhou’s shape equation to find a

biconcave-discoid model.

31

Figure 13. The cross-section and coordinate system of a biconcave-discoid model.

Then, the modified boundary conditions for the biconcave-discoid model can be

written as follows:

��

��

�

−=+

��

��

�−+−��

�

��

�−+∆+

=+

��

��

�−+−��

�

��

�−+∆+

12

223

02

223

02

30

222

030

02

30

222

030

ffc

ffc

fc

iic

iic

ic

cak

aaca

kca

kpc

cak

aaca

kca

kpc

ρρλ

ρρλρλ

ρρλ

ρρλρλ

z

ρ

ϕ

iρ fρ

0)( =iρϕ

(17a)

(17b)

32

After re-applying the three MATLAB codes with Eq.17a and Eq.17b, the examination

of how compatible that Zhou’s shape equation could fit for the biconcave-discoid

model has been checked as shown in the following figure.

Figure 14. The full cross-section of the biconcave- discoid model which is derived

from Zhou’s shape equation.

As shown in Figure 14, the attempt to describe the biconcave- discoid model by

applying Zhou’s shape equation solution is compared with the real experimentally

recorded model shown in Figure 3c. The curve length of the quarter cross-section is

4.9446�m, and iρ equals 0.05�m. Therefore, by applying Eq.16, the calculated

ρ

z

33

surface area is 157�m2 which is close to the models in Figure 1 (A = 135 ~ 145�m2)

than the anchor-ring model in Figures 11 and 12 (A = 172�m2). Although the surface

area is closer than what Fang and Evan’s results [3] when applying Zhou’s shape

equation to obtain a similar shape with Figure 1 (the biconcave-discoid model), it is

not mature to declare that Zhou’s shape equation solution is capable of dealing with

biconcave- discoid model because of the insufficient experimental reference for the

both biconcave-discoid and anchor-ring models. Therefore, further research of this

subject is highly recommended.

6. Although the valid values of λ , p∆ , and ck were found in this study, there is no

proper theory to explain how these mechanical properties could alter a RBC’s shape.

Therefore, finding how tensile stress, osmotic pressure difference, and bending module

would influence a RBC’s shape becomes another important research task in the future.

After understanding these mechanical properties better, it is possible to establish a

connection between mechanical, medical, and biophysical fields. This future study

could become a precious reference to answer many human body mysteries.

Summary

Study the RBC’s shapes is quite a massive task. For several decades, several shape

theories of RBC have been proposed by many pioneers of biophysics. By combining with

several mechanical properties, to find mathematical solutions for the RBC’s shapes may

contribute valuable information to help people understanding the mystery of the RBC.

Although several shape equations were proposed for years, these equations were not

34

solved because they were presented in the form of nth order (highest order), variable

coefficients, inhomogeneous and nonlinear ODE (NVIN-ODE). In a recent study, Zhou

[1] proposed a new method to deal with these NVIN-ODEs and obtained a new shape

equation for the anchor-ring vesicle RBC. By applying the first solution from Zhou’s

shape equation solution, it shows good agreement with several experimental results. The

model also suggests the possible values for tensile stress, osmotic pressure difference, and

bending module.

35

Reference

1. Ming Zhou, Ph.D. Dissertation (2008), “Quantizing Study in Shape of Red Blood Cells,” Department of Mechanical and Aerospace Engineering, California State University, Long Beach, California 90840, USA.

2. Ou-Yang Zhong-can (1990). "Anchor ring-vesicle membranes,” Physical Review A,

The American Physical Society, Vol. 41, No. 8. 3. E. Evans and Y.C. Fung (1972). “Improved Measurements of the Erychrocyte

Geometry,” Microvascular Research4, P. 335-347. 4. Gov, N. and Zilman, A. G. and Safran, S (2003). “Cytoskeleton Confinement and

Tension of Red Blood cell Membranes. ” Physical Review Letters, Vol. 90, No. 22, Section 228101.

5. Royalty Free Healthcare Images (2008), http://www.mediaMD.com. 6. W. Helfrich (1973). "Elastic Properties of Lipid Bilayers: Theory and Possible

Experiments," Z. Naturuforsch, C28, 693. 7. Evan A. Evans (1983). "Bending Elastic Modulus of Red Blood Cell Membrane

Derived From Buckling Instability in Micropipet Aspiration Tests,” Boiphys. J., Biophysical Society, Vol. 43, P27-P30.

8. M. Daoa, C.T. Limb, and S. Suresh (2003). "Mechanics of the human red blood cell

deformed by optical tweezers,” Journal of the Mechanics and Physics of Solids, Vol. 51, P2259-P2280.

9. Ou-Yang Zhong-can and W. Helfrich (1987). "Instability and Deformation of a

Spherical Vesicle by Pressure,” Physical Review letters, Vol. 59, No. 21, P2486-P2488.

10. E. W. Hobson (1955), “The Theory of Spherical and Ellipsoidal Harmonics,”

Cambridge, Cambridge, England.

36

11. M. Mutz and D. Bensimon (1991). "Observation of toroidal vesicles,” Physical Review A, The American Physical Society, Vol. 43, No. 8.

12. J. G. Hu and Z. C. Ou-Yang (1993), “Shape Equations of the Axisymmetric

Vesicles,” Phys. Rev, E47, p462, 1993. 13. Ming Zhou and H.Y. Yeh (2008), “Red Blood Cell Project Guide for Spring 2008,”

Department of Mechanical and Aerospace Engineering, California State University, Long Beach, California 90840, USA.

14. Ramin S. Esfandiari (2007), “Applied Mathematics for Engineers,” Fourth Edition,

ATLANTIS PUBLISHING COMPANY, P.945-P.958.

37

Appendix A c0a.m

function [ar, cra] = c0a(coeffs) %Find c0 and a % % [a, c0] = c0a([L k p r z]) where % % L is lamda. % k is kc. % p is delta p. % r is rowf. % z is rowi. % % ar is the result of a. % cra is the result of c0. % % Example: % [a, c0] = c0a([4e-6 4.35e-19 102.5 3.875e-6 1e-7]) if nargin ~= 1 || size(coeffs,2) ~= 5 L = 4e-6; %Lamda, the middle value in the range. k = 4.35e-19; %kc, middle value in the range p = 102.5; %Delta p, the middle value in the range r = 3.875e-6; %Rowf, the middle value in the range z = 1e-7 %Rowi, which is very close to zero else L = coeffs(1,1); k = coeffs(1,2); p = coeffs(1,3); r = coeffs(1,4); z = coeffs(1,5); end syms c a %symbolized c0 and a A = ((L*c+p)/k)*z^3; B = a*(c^2-(L/k))*z^2; C = ((3*(a^2)*c)/2) * z; D = ((a^3)/2)-a; E = ((2*L)/k)*z^2; F = a*c*z; f1sac = strcat(char(A+B-C+D-E-F), '=0'); f1 = (A+B-C+D)/(E+F); A = ((L*c+p)/k)*r^3; B = a*(c^2-(L/k))*r^2; C = ((3*(a^2)*c)/2) * r; D = ((a^3)/2)-a; E = ((2*L)/k)*r^2; F = a*c*r; f2sac = strcat(char(A+B-C+D+E+F), '=0'); f2 = (A+B-C+D)/(E+F); [ar,cra] = solve(f1sac,f2sac,'a,c');

38

Appendix B RK4Method.m

function [x, y] = RK4Method(x0, N, h, f, y0) %RK4 Method %given in the form y' = f(x, y). % % [x, y] = RK4Method(x0, N, h, f, y0) where % % x0 is a scalar representing the initial value of x. % N is number of nodes. % h is the distance between each node. % f is an inline function representing y' = f(x, y). % y0 is the initial condition of y. % % x is the vector to store each value of x in each node. % y is the vector to store each value of y in each node. q1 = 0; q2 = 0; q3 = 0; q4 = 0; x = zeros(N,1); y = zeros(N,1); x(1) = x0; y(1) = y0; for n = 1:N q1 = h * f(x(n), y(n)); q2 = h * f(x(n) + (h / 2), y(n) + (q1 / 2)); q3 = h * f(x(n) + (h / 2), y(n) + (q2 / 2)); q4 = h * f(x(n) + h, y(n) + q3); y(n+1) = y(n) + ((q1 + 2 * q2 + 2 * q3 + q4) / 6); x(n+1) = x(n) + h; end

39

Appendix C c0a_RK4.m

function [f, ar, cra, row, z] = c0a_RK4(coeffs, range) % Run RK4 from c0-a % % [f, a, c0] = c0a_RK4([L k p], [rowi rowf n]) where % % L is lamda. % k is kc. % p is delta p. % r is rowf. % z is rowi. % n is numbers of the running steps for RK4. % % f is the differential equation z'. % ar is the result of a. % cra is the result of c0. % % Example: % [f, a, c0, row, z] = c0a_RK4([4e-6 4.35e-19 102.5], [1e-7 3.875e-6 1000])% [f, a, c0, row ,z] = c0a_RK4([L k p], [rowi rowf n]) L = coeffs(1,1); %Lamda k = coeffs(1,2); %kc p = coeffs(1,3); %Delta p rowi = range(1,1); rowf = range(1,2); [ar, cra] = c0a([L k p rowf rowi]); %solve c0 and a syms r z; % Symbolize row and z a = ar (1,1); %a c = cra(1,1); %c0 L = coeffs(1,1); %Lamda k = coeffs(1,2); %kc p = coeffs(1,3); %Delta p A = ((L*c+p)/k)*r^3; B = a*(c^2-(L/k))*r^2; C = ((3*(a^2)*c)/2) * r; D = ((a^3)/2)-a; E = ((2*L)/k)*r^2; F = a*c*r; f = vectorize(inline(char(tan(asin((A+B-C+D)/(E+F)))),'r','z')); N = range(1,3)-2; h = (range(1,2) - range(1,1))/(range(1,3)); row0 = range(1,1) + h; z0 = 0; [row, z] = RK4Method(row0, N, h, f, z0); plot(row,z),grid,hold;

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Project of RBC