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Stoichiometric Tumor Model: Phosphorus Flow. By: Bo Deng, Vladimir Ufimtsev and Melissa Wilson. Funded By NSF/REU Grant to University of Nebraska-Lincoln. Stoichiometry-. The quantitative relationship between elements in a system. Mass Balance: The concept that matter is conserved. - PowerPoint PPT Presentation
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Stoichiometric Tumor Model:Phosphorus Flow
By: Bo Deng, Vladimir Ufimtsev and Melissa Wilson
Funded By NSF/REU Grant to University of Nebraska-Lincoln
Stoichiometry-
Mass Balance:
The concept that matter is conserved.
Whatever is put into a system either remains in the system or is excreted from it.
OHOHOH 23 2
On each side of the equation there are:
2-Oxygens
4-Hydrogens
The quantitative relationship between elements in a system.
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
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
Kuang et al. 2004, Williams 2002
Holling’s uptaking/capturing functional form (1959)
P
P
P
P
P
P
P
Total area = ,A Area of 1 P = PA
Machine will tap at the board at a rate of: k taps per unit of time.
PP P
P
P
How many units of prey (discs) will the machine tap, in a given search time; ?st
tappedP stk A
PAP Pt
A
Ak s
P
Pta s
Number of discs = P (remains constant)
P
P
P
P
P
P
P
Total area = AArea of 1 P =
PA
PP P
P
P
How many units of prey (discs) will the machine capture, in a given time; ?t
capturedP )( capturedh Pttk
The total time spent searching becomes:
Handling time for each disc = ht
capturedhs Pttt
A
PAP PPtta capturedh )(
PPttaP capturedhcaptured )(
Pta
PtaP
h
captured
1
Pta
Pa
t
P
h
captured
1
Pta
Pt
t
P
h
hcaptured
1
1
aptureuptaking/c of rate max. the ,1
1
1
limh
h
h
P tPta
Pt
P
capturedP
ht
1
0
YP
ta
Pt
h
h
1
1
YPta
PtaYP
hcaptured
1
Total number of P captured by Y many machines or cells in a unit time is
YPs
Pc
Y
Y
Also known as the Monod-Jacob form (1961) or the Michaelis-Menten form (1913)
http://www4.tpgi.com.au/users/amcgann/body/circulatory.html
QuantitiesH = 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.
IMPORTANT: The amounts of all quantities in the system are specified
by the amount of Phosphorus that each quantity possesses.
dH
dt
dO
dt
dP
dt
H
H
c PH
s P
O
O
c PO
s P
0P P H
H
c PH
s P
O
O
c PO
s P
HdH
OdO
HdH OdO
2HmH
2OmO
2HmH 2OmO
QuantitiesH = 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.
IMPORTANT: The amounts of all quantities in the system are specified
by the amount of Phosphorus that each quantity possesses.
dH
dt
dO
dt
dP
dt
H
H
c PH
s P
O
O
c PO
s P
0P P H
H
c PH
s P
O
O
c PO
s P
HdH
OdO
2HmH
2OmO
)( 22 OmHmOdHdr OHOH
http://www.robertsreview.com/cancer_pictures.html
http://www.maths.dundee.ac.uk/~sanderso/tumour.htm
Liver Cancer Angiogenesis
Organ and Tumor
QuantitiesH = Amount of Healthy Cells that compose the body (without the organ)O = Amount of Healthy Cells that compose the organT = Amount of Cancerous Cell that compose the tumor V = Amount of cells that compose the vessels in the tumorP = Amount of Phosphorus in the blood stream
OPs
PcH
Ps
PcPP
dt
dPdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
T
VOH
VPs
VOH
VPc
T
T
)(
)(
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
TPs
Pc
V
V
OdHdr OH ( )
TdT
VdV
TdT VdV
2HmH
2OmO
2TmT
2HmH 2OmO 2TmT
QuantitiesH = Amount of Healthy Cells that compose the body (without the organ)O = Amount of Healthy Cells that compose the organT = Amount of Cancerous Cell that compose the tumor V = Amount of cells that compose the vessels in the tumorP = Amount of Phosphorus in the blood stream
OPs
PcH
Ps
PcPP
dt
dPdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
T
VOH
VPs
VOH
VPc
T
T
)(
)(
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
TPs
Pc
V
V
OdHdr OH ( )
TdT
VdV
TdT VdV
2HmH
2OmO
2TmT
OVTmC )(
2HmH 2OmO OVTmC )( 2TmT
T
VOH
VPs
VOH
VPc
T
T
The term for Tumor growth:
The term is used to explain the
distribution of Phosphorus throughout the circulatory system.
VOH
V
P
VOH
Immune Response
Effector Cells NK Cells
(NaturalKillers)
Attack likeT-Cells
Natural Killer Cells
http://www.cancerfoundation.com/NKcells.html, 2004
Natural Killer Cell (NK Cell) cell destroying a targeted tumor cell.
Tumor cell
NK Cell
OPs
PcH
Ps
PcPP
dt
dPdt
dEdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
T
VOH
VPs
VOH
VPc
T
T
)(
)(
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
TPs
Pc
V
V
OdHdr OH ( )
TdT
VdV
TdT VdV
2HmH
2OmO
2TmT
OVTmC )(
2HmH 2OmO OVTmC )( 2TmT
EPs
Pc
E
E
11
EPs
Pc
E
E
1
1
EdE
EdE
2EmE
2EmE
Antigenicity
AntigenicityaTTttT destroyedhdestroyed )(
ts
ta
ET
Tts
tTttT
ETdestroyedhdestroyed
)(
tTTsa
tTc
ts
tTa
ts
tTc
t
T
ETET
ET
ETET
ETET
destroyed
)(
OPs
PcH
Ps
PcPP
dt
dPdt
dEdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TdT 2TmT
TPs
Pc
V
V
VdV
EPs
Pc
E
E
11 EdE 2EmE
EdE
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
E
Ps
Pc
E
E
1
1
OdHdr OH ( TdT 2HmH 2OmO OVTmC )( 2TmT 2EmE )
2HmH
2OmO OVTmC )(
VdV
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
EtTTsa
tTc
ETET
ET
)(
E
tTTsa
tTcr
ETET
ET
)(
OPs
PcH
Ps
PcPP
dt
dPdt
dEdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
EtTTsa
tTc
ETET
ET
)(
E
tTTsa
tTcr
ETET
ET
)(
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TdT 2TmT
TPs
Pc
V
V
VdV
EPs
Pc
E
E
11 EdE 2EmE
EdE
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
E
Ps
Pc
E
E
1
1
OdHdr OH ( TdT 2HmH 2OmO OVTmC )( 2TmT 2EmE )
2HmH
2OmO OVTmC )(
VdV
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
)(tI
OPs
PcH
Ps
PcPP
dt
dPdt
dEdt
dVdt
dT
OdOPs
Pc
dt
dO
HdHPs
Pc
dt
dH
O
O
H
H
OO
O
HH
H
][ 0
EtTTsa
tTc
ETET
ET
)(
E
tTTsa
tTcr
ETET
ET
)(
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TdT 2TmT
TPs
Pc
V
V
VdV
EPs
Pc
E
E
11 EdE 2EmE
EdE
T
VOH
VPs
VOH
VPc
T
T
)(
)(
TPs
Pc
V
V
E
Ps
Pc
E
E
1
1
OdHdr OH ( TdT 2HmH 2OmO OVTmC )( 2TmT 2EmE )
2HmH
2OmO OVTmC )(
VdV
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
E
tTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2
)(tI
Days until tumor detected
Tumor Dynamics without Therapy
Chemotherapy and Tumor Death
dt
dH HPs
Pc
H
H
HdH 2HmH
OVTmC )( 2OmO
dt
dV
dt
dT
OdOOPs
Pc
O
O
dt
dO
dt
dP
dt
dE
TdT 2TmTT
VOH
VPs
VOH
VPc
T
T
EtTTsa
tTc
ETET
ET
)(
EtTTsa
tTc
ETET
ET
)(
VdV
EPs
Pc
E
E
11
TPs
Pc
V
V
EdE 2EmE
EtTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2 )(tI
PP 0
EtTTsa
tTcind
Ps
Pc
ETET
ET
E
E
)(2
2HPs
Pc
H
H
TdTOdOHdH
T
VOH
VPs
VOH
VPc
T
T
EPs
Pc
E
E
1
1TPs
Pc
V
V
O
Ps
Pc
O
O
(r VdV EdE )2HmH 2TmT2OmO OVTmC )( 2EmE
