Project presentation work

The topic I’ll be exploring in my project is using

mathematical methods to optimise ‘in

vitro’ spheroid models to best recapitulate ‘in vivo’ tissue for drug testing.

Culturing cells in 3d is much more representative of the in

vivo environment than traditional 2d cultures. The multicellular arrangement

allows cells to interact with each other. Spheroids have huge potential in

cancer and stem cell research. (https://www.thermofisher.com/blog/cellculture/a-brief-history-of-spheroids-and-3d-cell-culture/)

(18/01/18)

Spheroids have been shown to accurately mimic natural cell

responses and interactions. Usually, a monolayer of cells on the surface of a

cell culture dish are used in labs (classic cell culture); but these cells only

interact with their direct neighbours. In a real-life tumour a cell mass is

growing uncontrolled in all three dimensions of the space producing these

typical “knots” of tumours. As long as these are not getting supplied

by own blood vessels supply of oxygen, nutrients and growth factors and

hormones differ very much between the cells in the outer layers of the tumour

and cell in the inside. This causes different levels of stress and also

physiological reactions of the cells.

Spheroids are being used as an intermediate between

monolayer culture and in vivo studies for the screening of drugs. This method

of testing is more accurate than 2D cultures, but safer than ‘in vivo’ testing.

They are grown in a 3D cell culture, which is an

artificially created environment that allows cells in vitro to grow in 3

dimensions, similar to how a cell would grow in vivo. A 2D environment

(for example a petri dish) is not able to do this.

Even though this seems like a biology based problem, we can

use maths to try to optimise the 3D spheroid model to best mimic the in vivo

environment. Fickian diffusion is used to model the transport of oxygen and

drugs through the 3D spheroids.

The first bit of maths used in my project is Fickian

diffusion. Fickian diffusion is used to model the transport of oxygen and drugs

throughout the 3D spheroid. Fickian diffusion is used to solve for the

Diffusion coefficient D.

Fick’s first law =>

J= diffusion flux (mol m?2 s?1)

D=diffusion coefficient (m2/s)

u= is the concentration (mol/m3)

x= position, the dimension of which is length (m)

This law describes a net movement down the gradient of u.

The overall movement will be away from high concentrations and towards low

concentrations.

Fick’s second law Predicts how diffusion causes the

concentration to change with respect to space and time. It is defined using a

PDE which in one dimension reads:

T= time (s)

All of these units are dependent of the amount of substance.

The General form of the Reaction diffusion equations =>

(Also called

Turing’s instability).

The model assumes chemicals react and diffuse in the

tissue/spheroid:

u(x,t)=

concentration of the chemical

()= reaction term (e.g.

oxygen consumption)

If we assume the equation has a positive uniform steady

state ()

()=0The reaction diffusion equations (on an infinite domain) can

be solved using Fourier transforms. Fourier transforms are closely related to

the complex form of the Fourier series. Here I have applied the Fourier

transform: = , – , t > 0, u(x, t) 0 at x ± , t > 0 u(x, 0) = , Applying Fourier Transforms (F) w.r.t x gives: = D =With this initial condition = Let (, t) = such that (, 0) = We have 2 Linear ODE in t, with

solution )This leaves us

with a linear ordinary differential equation in t, with a solution in the form:(,t) = ( = Inverse Fourier

transformNow we can apply the inverse Fourier Transform x to give us:u(x, t) == Let = = = f*g (By

convolution theory)Where: g(x,t) == = u(x,t) = = = Radius length and

02 profile (http://rsif.royalsocietypublishing.org/content/11/92/20131124)

(15/01/18)Consider the diffusion equation: (1) Oxygen supply is assured to be in steady state. Tissue

consumes oxygen at rate a(r).Rewritten in terms of spherical laplacian=> (2) At the edge of a spheroid, radius r0, the volume

of oxygen per unit mass is C0. At a distance of rn from

the centre, C=0 and (3) (4) (5)where: D= diffusion coefficient C= volume of oxygen per unit mass a(r)= oxygen consumption rate at point r Initially, it is assumed that a is constant.Cross

section of a tumour spheroid of radius r0. Oxygen partial pressure

is non-zero in the region rc. This region comprises all viable cells

both hypoxic and normoxic. Hypoxia is a condition of low oxygen tension,

typically in the range 1–5% O2, and is often found in the central

region of tumours due to poor vascularisation. Normoxia is used to describe oxygen tensions between 10–21%.Oxygen cannot penetrate into the region rn, which

is anoxic.’necrosis may set in at partial pressures slightly above

zero. This does not affect the generality of the model as it is close to zero’. (6)The general solution to equation (5) can be solved with trig

identities. Define X as a function of r0:X= (7)Then the thickness of the viable region, rc and

the anoxic region, rn are given by:rc = r0

( +cos(x)) (8)rn = r0

( -cos(x)) (9) =These must also be a certain spheroid size where the partial

pressure of oxygen just reaches zero at the centre, which is the greatest

radius the spheroid can obtain and still be viable. Define this radius as the

‘diffusion limit’ r. r1 does not depend on the size of the spheroid.

The implication of this is that the viable rim thickness decreases as the

spheroid grows in size, tending towards this limit if the consumption stays

constant. The diffusion limit , is inversely related to

the consumption rate and is independent of spheroid radius. The oxygen

consumption rate and oxygen partial pressure at any point in the spheroid can

be estimated once or spheroid boundaries are

derived. The viable rim thickness, decreases towards a theoretical minimum of , with increasing spheroid

size.Analytical methods can become hard to control or deal with

when nonlinear reactions are included. My first step in this project was to

investigate the relationship between oxygen and the size of a spheroid.

Development of hypoxia and anoxia in tumour spheroids. Hypoxia is the deficiency

in the amount of oxygen reaching the tissues. Anoxia- is the absence of oxygen.

Here is a graph I created in Matlab. It shows the difference

in oxygen concentration as spheroid radius length increases. Different lines

represent different core O2 temperatures. As we can see from the graph, the optimal

spheroid radius length increases at lower concentrations of oxygen. We can see that, at lower concentrations of oxygen, the

optimal spheroid radius length increases. Spheroid core oxygen concentration with 9% and 13% oxygen at

the spheroid boundary is given by the yellow and red curves. Oxygen concentration decreases in the well that contains the

spheroid. This simulation was created in Matlab using PDE tool box. The boundary condition at the air/liquid

interface was 0.21 (21% oxygen). Oxygen (u) consumption rate inside the

spheroid is given by IntroductionMany biological systems show considerable variations over

space e.g. the growth and spread of cancer (Grimes et al. 2014 o2 Modelling)How oxygen passes through liver cellsHow drugs pass through liver cells and how it differs

from oxygenMatlab code and graphs ConclusionDrug Transport into Spheroid Cells (mainly liver cells). We are working in 1 dimension because we can assume that the

spheroid is symmetrical and even all the way through. Fourier transforms (http://web.stanford.edu/class/ee102/lectures/fourtran)

(11/10/2017)Laplacian- Cartesian and spherical coordinates (http://skisickness.com/2009/11/20/)

(11/10/2017)Core o2 for different spherical radii (https://www.syvum.com/cgi/online/serve.cgi/eng/mass/mass1801.html)

(31/10/2017)Describe graph here Hypoxia occurs when oxygen levels are too low. Cells start to begin to die at 0.01 oxygenAssumes radial symmetry