Stoichiometric Model - University of Nebraska-Lincoln

Stoichiometric Model - University of Nebraska-Lincoln

Stoichiometric Tumor Model: Phosphorus Flow By: Bo Deng, Vladimir Ufimtsev and Melissa Wilson Funded By NSF/REU Grant to University of NebraskaLincoln 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 * (1 kg of Phos/100 kg of Cells Healthy Cells) =.611 kg of * (1000 Phos g/kg) = 611 g of Phos Hollings uptaking/capturing functional form (1959) P P P P P P P Total area = A , P P

P Area of 1 P = 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 AP k t s P a t s P k t s A 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 = th 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 ) AP t s t t h Pcaptured P a (t t P h captured ) P A Pcaptured a (t th Pcaptured ) P Pcaptured a t P 1 a t h P a1 PP Pcaptured t h t 1 a t h P

Pcaptured t 1 P a t h Pcaptured 1 th P 0 1 P th 1 lim , the max. rate of uptaking/c apture P 1 P th a t h Total number of P captured by Y many machines or cells in a unit time is a t P Pcaptured Y Y 1 a t h P 1 P c P t

Y h Y Y 1 sY P P a t h Also known as the Monod-Jacob form (1961) or the Michaelis-Menten form (1913) http://www4.tpgi.com.au/users/amcgann/body/ 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 d H H H s P H H dt dO 2 cO P O d O O mO O dt

sO P cO P cH P dP O d H H d O m H 2 mO O 2 H P0 P 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 d H H H s P H H dt dO 2

cO P O d O O mO O dt sO P cO P cH P dP O r (d H H d O O mH H 2 mO O 2 ) H P0 P 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 c P dP c P [ P0 P ] H H O O dt sH P sO P V ) H O V

T V sT P ( ) H O V cT P ( cV P T sV P r (d H H d O O dT T dV V mH H 2 mO O 2 mT T 2 ) 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 c P dP c P [ P0 P ] H H O O dt sH P sO P V ) H O V T V sT P ( ) H O V cT P (

cV P T sV P r (d H H d O O dT T dV V mH H 2 mO O 2 mC (T V )O mT T 2 ) 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 c P dO O 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 s P E d E E mE E 2 E1 cH P cO 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 d O O dT T dV V d E E mH H 2 mO O 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 c P dO O 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 ( c ET T t dT T mT T a ( s T ) T t E ET ET 2 cV P T dV V sV P cE 2 P c ET T t c E1 P 2 ind E E d E mE E s P E sE 2 P a ET ( s ET T ) T t E1 V cT P( ) cH P

cO P H O V [ P0 P] H O T V sH P sO P sT P( ) H O V cV P c P T E1 E sV P s E1 P r (d H H d O O dT T dV V d E E mH H 2 mO O 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 c P dO O 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 s P E d E E mE E 2 E1 cE 2 P c ET T t ind E sE 2 P a ET ( s ET T ) T t cH P cO P [ P0 P] H O sH P sO P V ) H O V T V sT P ( ) H O V

cT P ( I (t ) cV P c P T E1 E sV P s E1 P r (d H H d O O dT T dV V d E E mH H 2 mO O 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 c P dO O 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 s P E d E E mE E 2

E1 cE 2 P c ET T t ind E sE 2 P a ET ( s ET T ) T t V cT P ( ) cH P cO P H O V [ P0 P] H O T V sH P sO P sT P ( ) H O V I (t ) cV P c P T E1 E sV P s E1 P

r (d H H d O O dT T dV V d E E mH H 2 mO O 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 O O mOO 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

dT T mT T 2 c ET T t E a ET ( s ET T ) T t dV V c E1 P dE E d E m E2 E E s P dt E1 dP P0 P dt 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 O O dT T dV V d E E mH H 2 mO O 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 )

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