Stoichiometric Tumor Model:
Phosphorus Flow
By: Bo Deng, Vladimir Ufimtsev
and Melissa Wilson
Funded By NSF/REU Grant to University of Nebraska-Lincoln
StoichiometryThe quantitative relationship between
elements in a system.

Mass Balance:
The concept that matter is conserved.
Whatever is put into a system either remains in
the system or is excreted from it.
OH   H 3O   2 H 2O
On each side of the equation there are:
2-Oxygens
4-Hydrogens
PHOSPHORUS!!

Metabolic Pathways

ATP –
Adenosine
Tri-Phosphate

DNA/RNA Replication
http://www.blc.arizona.edu/Molecular_Graphics/DNA_Structure/DNA_Tutorial.HTML#Components
http://www.accessexcellence.org/AB/GG/dna_replicating.html
Measuring Phosphorus
Phosphorus makes up:
1% of Healthy Cells
Kuang et al. 2004, Williams 2002
2% of Tumor Cells
Using a healthy person of average mass,70kg, excluding
bone mass, 8.9kg, with no tumor, her mass becomes:
61.1 kg of Healthy Cells * (1 kg of Phos/100 kg of Healthy Cells)
=.611 kg of Phos* (1000 g/kg)
= 611 g of Phos
Holling’s uptaking/capturing functional form (1959)
P
P
P
P
P
P
P
P
P
Total area =
P
A
,
P
Area of 1 P =
P
AP
Number of discs = P (remains constant)
Machine will tap at the board at a rate of: k taps per unit of time.
How many units of prey (discs) will the machine tap, in a
given search time; t s ?
Ptapped 
AP  P
k  ts 
A
 AP 
 k 
  ts  P  a  ts  P
A

P
P
P
P
P
P
P
P
P
P
P
P
A
Area of 1 P = AP
Total area =
Handling time for each disc = t h
How many units of prey (discs) will the machine capture, in
a given time; t ?
The total time spent searching becomes:
Pcaptured  k  (t  th  Pcaptured ) 
t s  t  t h  Pcaptured
AP  P  a  (t  t  P
h
captured )  P
A
Pcaptured  a  (t  th  Pcaptured )  P
a t  P
Pcaptured 
1  a  th  P
a1 PP
Pcaptured
t
t
1  ah  t h  P
Pcaptured
t
1
P
a  th
Pcaptured
1
th
P
0
1
P
th
1
lim
 , the max. rate of uptaking/capture
P 
1
 P th
a  th
Total number of P captured by Y many machines or cells
in a unit time is
a t  P
PcapturedY 
Y
1  a  th  P
1
P
c P
t
Y
 h
Y  Y
1
sY  P
P
a  th
Also known as the Monod-Jacob form (1961)
or the Michaelis-Menten form (1913)
http://www4.tpgi.com.au/users/amcgann/body/circulatory.html
IMPORTANT: The amounts of all quantities in the system are specified
by the amount of Phosphorus that each quantity possesses.
Quantities
H = Amount of Healthy cells that compose the body (without the organ).
O = Amount of Healthy cells that compose the organ.
P = Amount of Phosphorus in the blood stream.
2
cH P
dH

m
H
H
 s  P H  dH H
H
dt
dO
2
 cO P O  d OO  mOO
dt
sO  P
cO P
cH P
dP

O  d H  d O  m H 2 m O 2

H
  P0   P
O
H
O
H
sO  P
sH  P
dt
IMPORTANT: The amounts of all quantities in the system are specified
by the amount of Phosphorus that each quantity possesses.
Quantities
H = Amount of Healthy cells that compose the body (without the organ).
O = Amount of Healthy cells that compose the organ.
P = Amount of Phosphorus in the blood stream.
2
cH P
dH

m
H
H
 s  P H  dH H
H
dt
dO
2
 cO P O  d OO  mOO
dt
sO  P
cO P
cH P
dP

O  r (d H  d O  m H 2  m O 2 )

H
  P0   P
H
O
H
O
sO  P
sH  P
dt
Organ and Tumor
Liver Cancer
Angiogenesis
http://www.maths.dundee.ac.uk/~sanderso/tumour.htm
http://www.robertsreview.com/cancer_pictures.html
Quantities
H = Amount of Healthy Cells that compose the body (without the organ)
O = Amount of Healthy Cells that compose the organ
T = Amount of Cancerous Cell that compose the tumor
V = Amount of cells that compose the vessels in the tumor
P = Amount of Phosphorus in the blood stream
dH
c P
 H
H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2
dt sO  P
V
cT P(
)
dT
H  O  V
2

T  dT T  m T
T
V
dt
sT  P(
)
H  O  V
dV
 cV P T  d V
V
dt
sV  P
V
c
P
(
)
T
cO P
dP
cH P
cV P
H  O  V
T 
 [ P0  P ] 
H
O 
T
V
sV  P
sT  P(
)
dt
sH  P
sO  P
H  O  V
 r (d H H  dOO  dT T  dV V  mH H 2 mOO 2  m T 2)
T
Quantities
H = Amount of Healthy Cells that compose the body (without the organ)
O = Amount of Healthy Cells that compose the organ
T = Amount of Cancerous Cell that compose the tumor
V = Amount of cells that compose the vessels in the tumor
P = Amount of Phosphorus in the blood stream
dH
c P
 H
H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2 m (T  V )O
C
dt sO  P
V
cT P(
)
dT
H  O  V
2

T  dT T  m T
T
V
dt
sT  P(
)
H  O  V
dV
 cV P T  d V
V
dt
sV  P
V
c
P
(
)
T
cO P
dP
cH P
cV P
H  O  V
T 
 [ P0  P ] 
H
O 
T
V
sV  P
sT  P(
)
dt
sH  P
sO  P
H  O  V
 r (d H H  dOO  dT T  dV V  mH H 2 mOO 2 m (T  V )O m T 2)
C
T
The term for Tumor growth:


V

cT P
 H   O   V  T


V

sT  P
 H  O  V 
V
The H  O  V
term is used to explain the
distribution of Phosphorus throughout the circulatory
system.
P
H  O
V
Immune Response

Effector Cells


NK Cells
(Natural
Killers)
Attack like
T-Cells
Natural Killer Cells
Natural Killer Cell
(NK Cell) cell
destroying a
targeted tumor
cell.
NK Cell
Tumor cell
http://www.cancerfoundation.com/NKcells.html, 2004
dH
cH P

H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2  mC (T  V )O
dt sO  P
dT
dt
dV
dt
dE
dt
dP
dt

V
)
H  O  V
T
V
sT  P(
)
H  O  V
cT P(
 dT T  mT T 2
 cV P T  dV V
sV  P

c E1 P
E  d E E  mE E 2
s E1  P
cO P
cH P
 [ P0  P ] 
H
O
sH  P
sO  P
V
)
H  O  V

T
V
sT  P(
)
H  O V
cT P(

cV P
c P
T  E1 E
sV  P
s E1  P
2
 r (d H H  dOO dT T  dV V d E E mH H 2  mOO
mC (T  V )O mT T 2 mE E 2)
Antigenicity
Antigenicity
Tdestroyed  (t  t hTdestroyed )aT
a
t
s ET  t
Tdestroyed  (t  t hTdestroyed ) 
t
s ET  t
T
 t 

c ET T 
Tdestroyed
s ET  t 
c ET T  t



t
 t  a ET ( s ET  T )  T  t

a ET  T 
 s ET  t 
dH
cH P

H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2  mC (T  V )O
dt sO  P
dT
dt
dV
dt
dE
dt
dP
dt

V
)
H  O  V
T
V
sT  P(
)
H  O  V
cT P(
 dT T  mT T
2

c ET T  t
E
a ET ( s ET  T )  T  t
 cV P T  dV V
sV  P



c P
c ET T  t
c E1 P
E 
E  d E E  mE E 2  E 2  ind  
sE 2  P
s E1  P
 a ET ( s ET  T )  T  t 
V
cT P(
)
cO P
cH P
c P
c E1 P

H


O


V
 [ P0  P ] 
H
O
T V
T

E
V
sH  P
sO  P
s

P
sT  P(
)
s E1  P
V
H  O  V
 r (d H H  dOO dT T  dV V d E E mH H 2  mOO 2  mC (T  V )O mT T 2  mE E 2)


c ET T  t
 r 
E 
 a ET ( s ET  T )  T  t 



cE 2 P
c ET T  t
 ind  
E 
sE 2  P
 a ET ( s ET  T )  T  t 
dH
cH P

H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2  mC (T  V )O
dt sO  P
dT
dt
dV
dt
dE
dt
dP
dt

V
)
H  O  V
T
V
sT  P(
)
H  O V
cT P(
 dT T  mT T
2

c ET T  t
E
a ET ( s ET  T )  T  t
 cV P T  dV V
sV  P

c E1 P
E  d E E  mE E 2
s E1  P



cE 2 P
c ET T  t
 ind  
E 
sE 2  P
 a ET ( s ET  T )  T  t 
 I (t )
V
cT P(
)
cO P
cH P
c P
c E1 P

H


O


V
 [ P0  P ] 
H
O
T V
T

E
V
sH  P
sO  P
s

P
sT  P(
)
s E1  P
V
H  O V
 r (d H H  dOO dT T  dV V d E E mH H 2  mOO 2  mC (T  V )O mT T 2  mE E 2)


c ET T  t
 r 
E 
 a ET ( s ET  T )  T  t 



cE 2 P
c ET T  t
 ind  
E 
sE 2  P
 a ET ( s ET  T )  T  t 
dH
cH P

H  d H H  mH H 2
dt
sH  P
cO P
dO

O  d O O  mO O 2  mC (T  V )O
dt sO  P
dT
dt
dV
dt
dE
dt
dP
dt

V
)
H  O  V
T
V
sT  P(
)
H  O  V
cT P(
 dT T  mT T
2

c ET T  t
E
a ET ( s ET  T )  T  t
 cV P T  dV V
sV  P

c E1 P
E  d E E  mE E 2
s E1  P



cE 2 P
c ET T  t
 ind  
E 
sE 2  P
 a ET ( s ET  T )  T  t 
 I (t )
V
cT P(
)
cO P
cH P
c P
c E1 P

H


O


V
 [ P0  P ] 
H
O
T V
T

E
V
sH  P
sO  P
s

P
sT  P(
)
s E1  P
V
H  O  V
 r (d H H  dOO dT T  dV V d E E mH H 2  mOO 2  mC (T  V )O mT T 2  mE E 2)


c ET T  t
 r 
E 
 a ET ( s ET  T )  T  t 



cE 2 P
c ET T  t
 ind  
E 
sE 2  P
 a ET ( s ET  T )  T  t 
Days until tumor detected
Tumor Dynamics without Therapy
Chemotherapy and Tumor Death
cH P
dH
H

s

P
dt
H
 d H H  mH H 2
cO P
dO
O

s

P
dt
O
 d OO  mO O 2  mC (T  V )O
dT

dt


V

cT P
 H  O  V  T


V

sT  P
 H  O  V 
cV P
dV
T

s

P
dt
V

c ET T  t
E
a ET ( s ET  T )  T  t
 dV V
c E1 P
dE
E

s

P
dt
E1
dP
 P0  P
dt
 dT T  mT T 2
 d E E  mE E 2
c P
c P
 H H O O
sO  P
sH  P



cE 2 P
c ET T  t
ind 
E 
sE 2  P
 a ET ( s ET  T )  T  t 


V

cT P

H


O


V

 T



V

sT  P

H


O


V



 I (t )
c P
cV P
T  E1 E
s E1  P
sV  P
 r ( d H H d OO dT T dV V  d E E  mH H 2  mOO 2 mC (T  V )O mT T 2  mE E 2 



cE 2 P
c ET T  t
ind 
E 
sE 2  P
 a ET ( s ET  T )  T  t 
c ET T  t
E
a ET ( s ET  T )  T  t
)