Team:SupBiotech-Paris/Treatement modeling

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== Context ==
== Context ==
-
Non-small cell lung carcinoma, or NSCLC, is an aggressive cancer, with a relatively high speed growth. Treatments are often ineficient, because the tumour growth is faster than  the elimination by the drug<br>
+
Non-small cell lung carcinoma, or NSCLC, is an aggressive cancer, with a relatively high speed growth. Treatments are often ineficient, because the tumor growth is faster than  the elimination by the drug.<br>
== Objective ==
== Objective ==
-
We have decided to model our treatment efficacy for this kind of tumour. Therefore we have modelled the tumour progression, our treatment evolution and efficacy.
+
We have decided to model our treatment efficacy for this kind of tumor. Therefore we have modelled the tumour progression, our treatment evolution and efficacy.
-
The objective of the modelling is to verify if our treatment is able to eliminate the entire tumour.<br>
+
The objective of the modelling is to verify if our treatment is able to eliminate the entire tumor.<br>
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
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== Model segmentation ==
== Model segmentation ==
-
First, we had to recreate [[Team:SupBiotech-Paris/Concept#DVS|DVS]] complete mechanism and the tumour evolution. Then, for each step of the treatment, we have identified all the paramters that intervene, their actions and their interactions, in order to determine the model equations.<br>
+
First, we had to recreate [[Team:SupBiotech-Paris/Concept#DVS|DVS]] complete mechanism and the tumor evolution. Then, for each step of the treatment, we have identified all the parameters that intervene, their actions and their interactions, in order to determine the model equations.<br>
To simplifly the equation we have devided the mechanism and we have modelled each step separately.<br>
To simplifly the equation we have devided the mechanism and we have modelled each step separately.<br>
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=== Tumor and DVS evolution versus time ===
=== Tumor and DVS evolution versus time ===
-
==== First step : Tumour development according to time ====
+
==== First step : Tumor development according to time ====
-
We consider the '''tumour is non métastatic and its growth is exponential'''.<br>
+
We consider the '''tumor is non metastatic and its growth is exponential'''.<br>
-
Let the tumour have a volume V1 in cm3 at an instant t1.<br>
+
Let the tumor have a volume V1 in cm3 at an instant t1.<br>
-
Let the same tumour, at an instant t2, have a volume V2.<br>
+
Let the same tumor, at an instant t2, have a volume V2.<br>
-
The tumour is considered in exponetial growth phase and without metastasis therefore its development equation,  '''Tumor Growth Rate (TGR)''', is equal to :<br>
+
The tumor is considered in exponential growth phase and without metastasis therefore its development equation,  '''Tumor Growth Rate (TGR)''', is equal to :<br>
[[Image :TGR.jpg|center|200px]]  
[[Image :TGR.jpg|center|200px]]  
-
Thus, the'''tumour volume according to the time (V(t))''' is equal to :<br>
+
Thus, the '''tumor volume according to the time (V(t))''' is equal to :<br>
[[Image :V(t).jpg|center|300px]]  
[[Image :V(t).jpg|center|300px]]  
-
Finally, knowing the ''' Average volume of a cancerous cell (Vcc)''' (experimental data), if we regard the tumour as fraught (without cavity or blood vessel),  we can determine that the '''Number of cancerous cells according to time (Nc(t))''', without treatment effect, is equal to :<br>
+
Finally, knowing the ''' Average volume of a cancerous cell (Vcc)''' (experimental data), if we regard the tumor as fraught (without cavity or blood vessel),  we can determine that the '''Number of cancerous cells according to time (Nc(t))''', without treatment effect, is equal to :<br>
[[Image :N(c).jpg|center|150px]] <br>
[[Image :N(c).jpg|center|150px]] <br>
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The number of [[Team:SupBiotech-Paris/Concept1#drapeau|Tissue vectors]] increases until injection of  doxycycline. F rom then, tissue vectors lysis releases the [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] in the lung.<br>
The number of [[Team:SupBiotech-Paris/Concept1#drapeau|Tissue vectors]] increases until injection of  doxycycline. F rom then, tissue vectors lysis releases the [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] in the lung.<br>
-
This injection time is not insignificant. Indeed, si if we wait long enough, [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors]] number is sufficient to eliminate the tumour or at least to significantly reduce it. On the other hand, if we wait too long, a higher dose of doxycycline (and so potentially toxic) is necessary for [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]] release.<br>
+
This injection time is not insignificant. Indeed, si if we wait long enough, [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors]] number is sufficient to eliminate the tumor or at least to significantly reduce it. On the other hand, if we wait too long, a higher dose of doxycycline (and so potentially toxic) is necessary for [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]] release.<br>
Thus we can use modelling to determine the ''' optimal injection time of doxycyline (Tdox)'''.<br>
Thus we can use modelling to determine the ''' optimal injection time of doxycyline (Tdox)'''.<br>
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[[Image :Np(t)2.jpg|center|300px]]  
[[Image :Np(t)2.jpg|center|300px]]  
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The phage vector dispersion steps  in the tumour and for cell penetration are the steps below  '''Fourth''' and '''Fifth''') because of their complexity.<br>
+
The phage vector dispersion steps  in the tumor and for cell penetration are the steps below  '''Fourth''' and '''Fifth''') because of their complexity.<br>
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:* The size of the tumor versus time (initial volume + growth)
:* The size of the tumor versus time (initial volume + growth)
:* The amount of tissue vector versus time
:* The amount of tissue vector versus time
-
:* The amount of released cellular vectors for a tissue vector  
+
:* The amount of released [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] for a tissue vector  
Now, we're going to determine the efficiency of our vectors for penetring cancer cells. <br>
Now, we're going to determine the efficiency of our vectors for penetring cancer cells. <br>
For that we are studying:
For that we are studying:
-
:* The area of dispersal vector cell
+
:* The area of dispersal [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]]
:* The importance of the cellular internalization of the vector in cancer cells.<br>
:* The importance of the cellular internalization of the vector in cancer cells.<br>
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This requires knowing:
This requires knowing:
-
        * The spread of phages in the bloodstream
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:* The spread of phages in the bloodstream
-
        * Their diffusion through the walls of blood vessels
+
:* Their diffusion through the walls of blood vessels
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        * The surface of a cancer cell
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:* The surface of a cancer cell
For our modeling, we consider the blood as a '''Newtonian fluid with a constant velocity Vmax'''. Here, We neglect the heart-related jolts and turbulent flows caused by the cavities of the blood epithelium.<br>
For our modeling, we consider the blood as a '''Newtonian fluid with a constant velocity Vmax'''. Here, We neglect the heart-related jolts and turbulent flows caused by the cavities of the blood epithelium.<br>
-
The cell vector moves along two axes. An X axis in the direction of blood flow and a Y axis orthogonal to the axis X.<br>
+
The [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]] moves along two axes. An X axis in the direction of blood flow and a Y axis orthogonal to the axis X.<br>
-
[[Image : Repère mouvement mécanique en.png|center|170px]]
 
-
 
-
In our model, we regard the blood as a '''newtonian fluid with a constant speed Vmax  over time'''. We neglect the cardiac jolts and the turbulent flows related to the blood epithelium cavities.<br>
 
-
The [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]] moves according to two axes. A X axe following the blood flow and an Y axe, orthogonal to the X axe.<br>
 
[[Image:RepèremouvementmécaniqueEn.png|center|400px]]<br>
[[Image:RepèremouvementmécaniqueEn.png|center|400px]]<br>
-
The move '''in  Y''' represents the '''phage vector diffusion in the blood'''. It depends on the equation of a particle  (phage) diffusion in a fluid (blood). <br>
 
-
[[Image : EqDif.jpg|center|170px]]
+
=====The phage propagation in the bloodstream=====
-
With n, the number of particles, and '''D, the diffusion ratio'''.<br>
+
The movement '''in X''' depends solely on the '''propagation of phages''' the vessel due to blood flow. Indeed, we neglect the diffusion which takes place also along the X axis because it is 1000 times less than the propagation of particles in the blood (due to the importance of blood flow). The cellular carriers are moving at '''speeds spread on a dish''' from, Vmax in the center of the vessel at V0 against the vessel wall.<br>
-
 
+
-
The move '''in X''' only depends on '''blood flow propagation''' in the vessel. Indeed, we neglect the diffusion that also occures according to the X axe because it is 1000 times inferior to the particles propagation in the blood (linked to the importance of the blood flow). Phages move with different  '''speeds distributed in a parabolic way''' from Vmax in the vessel center, to V0 at the vessel edge.<br>
+
[[Image:répartitionvitesseparaboliqueEN.png|center|500px]]<br>
[[Image:répartitionvitesseparaboliqueEN.png|center|500px]]<br>
-
Closing to the vessel edge, the phages speeds decrease because of  friction forces.<br>
+
The speed of phages decrease in approaching the vessel walls due to the friction forces which are opposing to the movement.<br>
-
We can therefore determine how long it takes for a particle with a Vmax speed to atteint l'extrémité. On obtient ainsi le temps nécessaire à l’internalisation de tous les [[Team:SupBiotech-Paris/Concept2Fr#drapeau|phages]] d’une [[Team:SupBiotech-Paris/Concept1Fr#drapeau|bactérie]].<br>
+
We can determine how long the particle (with a Vmax velocity), ie the particles in the center of the ship, reached the end. This gives the time necessary to internalize all the phages of the bacteria.<br>
-
Lorsque l'on associe le déplacement en Y (vitesse de diffusion) et le déplacement en X (vitesse du flux sanguin), on obtient, après intégration sur le périmètre d'un vaisseau sanguin, la surface d’action des [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] issus d’un [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]]. On est alors capable connaître le nombre de cellules cancéreuses détruites pour 100 [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] ou 1 [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]]<br>
+
=====The diffusion through the walls of blood vessels=====
-
La vitesse de diffusion du [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteur cellulaire]], REDUITE A D (ADPRES SIMPLIFICATION ???)est égale à 0,5µm.s-1 or la taille d’un capillaire sanguin est de 10µm de diamètre. La particule la plus éloignée met donc 10s à atteindre la paroi du vaisseau.(CE QUI EST NEGLIGEABLE ETANT DONNE L'ECHELLE DE TEMPS OBSERVEE AVANT LA SYNTHESE DE P53).<br>
+
The movement '''in Y''' is the distribution of phages in the blood (j(n)). It depends on the equation of diffusion of a particle (n) in a fluid (Fick's Law).<br>
-
Grâce à cette durée de diffusion (10s) et à la vitesse du flux sanguin (1x10^3µm.s-1) dans les capillaires, on peut déterminer :
+
[[Image:Equation diffusion du phage.png|center|170px]]<br>
-
:*La longueur (L) couverte par les [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] libérés par un [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]].
+
-
:*La surface (S) occupée par les phages dans un vaisseau sanguin de diamètre 2r.<br>
+
 +
With n the number of particles (phages), grad n the difference between the concentrations and D the diffusion coefficient.
 +
The cellular distribution of vectors within the blood vessel and then through the wall is a phenomenon of diffusion with output. So, there will always be a strong gradient of concentration of phage in the blood. We can therefore say that the gradient is constant (equal to 1) over time. Thus the diffusion rate (j(n)) is equal to D.
-
L = 1 x 10^4 µm <br>
 
-
2r = 10 µm <br>
 
-
S = 2 π x L x r = 31,4x10^4µm² <br>
 
-
Ainsi, un [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]] peut potentiellement cibler plus de 31 000 cellules cancéreuses, or, il ne possède que 100 [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]]. On peut effectuer une simplification en disant que 100 [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] détruisent 100 cellules cancéreuses et donc réduire l’équation de dispersion à une constante (NOM + VALEUR OU MOYEN DE DETERMINER LA VALEUR ??).<br>
+
=====Dispersal area of phage=====
-
Pour le phage, une fois la paroi atteinte, entre en jeu l’internalisation cellulaire. Ce modèle répond à deux schéma d’action.<br>
+
When we combine moving '''Y''' ('''diffusion rate''') and moving '''in X''' ('''blood flow velocity'''), we obtain, after integration on the '''perimeter of a blood vessel''', the action surface of [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]]. Then, we are able to determine the number of cancer cells per 100 [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] destroyed or 1 [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vector]].
 +
 
 +
The diffusion rate of the [[Team:SupBiotech-Paris/Concept2#drapeau|cell vector]] is equal to 0.5 μm.s-1 and the size of a capillary blood is 10μm in diameter. The particle farthest places so 10s to reach the vessel wall.
 +
 
 +
With this '''dissemination length''' (10s), the '''blood flow velocity''' (1x10 ^ 3μm.s-1) in the capillaries, and the surface of one cancer cell (1 micron square), we can determine:
 +
 
 +
:* The '''length (L)''' covered by the [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] released by one [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vector]].
 +
:* The '''surface (S)''' occupied by phages in blood vessel diameter of 2r.
 +
:* The amount of cancer cells available.
 +
 
 +
L = 1 x 10^4 µm
 +
2r = 10 µm
 +
S = 2 x π x L r = 31.4 x 10^4 µm²
 +
 
 +
Thus, a [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vector]] can potentially target more than 31 000 cancer cells, yet it has that 100 [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]]. We can make a simplification to say that 100 [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] destroy 100 cancer cells. The efficiency of the dispersion is complete.
 +
 
 +
For phage, once reached the wall, comes in the cellular internalization. This model responds to two courses of action.
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
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==== Cinquième étape : L’internalisation du vecteur cellulaire ====
+
==== Step Five: The cell vector internalization ====
-
+
-
Une fois au contact de la cellule, le [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteur cellulaire]] a deux schémas d’action possibles.
+
-
:* Le vecteur se fixe puis il se détache de la cellule.
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-
:*Le vecteur se fixe puis il se fait internaliser au sein de la cellule.
+
-
On peut modéliser cela en fonction du temps et des '''Constantes d’Association (kon) ''', de '''Dissociation (koff) ''' et d’'''Internalisation (kint) '''.<br>
+
-
On obtient ainsi: [[Image : EqInt.jpg|center|280px]]
+
-
Les étapes les plus courtes, en échelle de temps, sont certainement les étapes concernant le phage. L’internalisation est la plus courte d’entre elle, après avoir déterminé les constantes, on sait que plus de 320 [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] sont internalisés par seconde au contact d’une paroi.<br> En raison des échelle de temps, en heure, on peut réduire cette équation en fonction du temps à une simple constante.(UNE CONSTANTE NE DEPEND PAS DU TEMPS donc : kon, koff et kint forment une seule constante égale à 320 ? Ou c'est IDP qui devient une constante ?)<br>
+
Once in contact with the cell, the cell vector has two possible ways of action.
 +
:* The vector fix then it detaches from the cell.
 +
:* The vector fix then it is internalized within the cell.
-
Une fois internalisé, le  [[Team:SupBiotech-Paris/Concept3Fr#drapeau|plasmide thérapeutique]] engendre l’apoptose de la cellule en 1h, diminuant le nombre de cellule cancéreuse, Nc(t), et le volume tumoral, Vc.
+
We can model this according time and the association constant (kon) and dissociation constant (koff) and internalization constant(Kint).
 +
 
 +
This gives:
 +
[[Image : EqInt.jpg|center|280px]] <br>
 +
 
 +
With '''kon''' = 5.10^3 M-1s-1, '''koff''' = 8.10^-3 s-1 and '''Kint''' = 5,78.10^-4 s-1. If we calculate the '''global constant K'''', such as IDP = K' xt, we obtain '''K''''= 361.5 s-1. Thus more than 360 phages are internalized per second in contact with a wall.
 +
 
 +
If we compare the time required to internalize a phage compared to waiting times before the cell enters apoptosis in response to the entrance of a vector cell (1 hour), it appears logical to neglect the internalization of cell vectors (IDP = constant = 360 phages/s) in the final equation.
 +
 
 +
Thus, with a total efficiency of the phage diffusion, and a neglected internalization time, we can say that the efficiency constant '''λ is equal to 1'''.  
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
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== Evolution en simultanée du DVS et d'une tumeur ==
+
== Simultaneous evolution of DVS and tumor  ==
-
L’équation d'évolution de notre modèle en fonction du temps est égale à :<br>
+
The evolution equation of our model over time is equal to:<br>
-
[[Image :EqFinale.jpg|center|700px]][[Image:Bibou3.png|float|right|150px]]
+
[[Image :EqFinaleEN.jpg|center|700px]][[Image:Bibou3.png|float|right|150px]]
-
Avec :<br>
+
With :<br>
<div style="margin-left: 100px;">
<div style="margin-left: 100px;">
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- Nc(t), le nombre de cellules cancéreuses dans le temps,<br>
+
- Nc (t), the number of cancer cells depending time,<br>
-
- V(t), le volume tumoral,<br>
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- V (t), tumor volume,<br>
-
- V1 et V2, deux volumes tumoraux à respectivement des temps t1 et t2,<br>
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- V1 and V2, two tumors volumes respectively times t1 and t2, <br>
-
- Vcc, le volume d’une cellule cancéreuse,<br>
+
- Vcc, the volume of a cancer cell,<br>
-
- Nbi, le nombre de [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteurs tissulaires]] injectés,<br>
+
- Nbi, the number  of injected [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors]],<br>
-
- Pp, le pourcentage pulmonaire de [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteurs tissulaires]] par rapport à la dose injectée,<br>
+
- Pp, the lung percentage of [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors]] relative to the injected dose,<br>
-
- DTB, le temps de doublement du [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]],<br>
+
- DTB, the doubling time of [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vector]],<br>
-
- tinj, le temps d'injection du [[Team:SupBiotech-Paris/Concept1Fr#drapeau|vecteur tissulaire]],<br>
+
- tinj, injection time of the [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors],<br>
-
- Npl, le nombre de [[Team:SupBiotech-Paris/Concept2Fr#drapeau|vecteurs cellulaires]] libérés par [[Team:SupBiotech-Paris/Concept1Fr#drapeau|bactérie]].<br>
+
- Npl, the number of [[Team:SupBiotech-Paris/Concept2#drapeau|cell vectors]] released by [[Team:SupBiotech-Paris/Concept1Fr#drapeau|bacteria]].<br>
</div>
</div>
-
On peut négliger (aux vues des différences entre les échelles de temps ou d’espace) certains facteurs :<br>
+
We can neglected (differences between the time or space scales) some factors: <br>
<div style="margin-left: 100px;">
<div style="margin-left: 100px;">
-
- Kdeg, la constante de dégradation du phage, car tous les phages sont internalisés en 10s.<br>
+
- Kdeg, the degradation constant of the phage, because all the phages were internalized in 10s.<br>
-
- D, la diffusion du phage et IDP, l’internalisation cellulaire, car on considère que 100 phages rentrent dans 100 cellules différentes (pour une valeur potentielle de 31400) donc tout cela est égal à 1.<br>
+
</div>
</div>
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<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
<span style="float: right">[[Team:SupBiotech-Paris/Treatement_modeling#drapeau|Back to top]]</span>
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== Simulation de traitement ==
+
== Treatment simulation ==
-
 
+
 +
The DVS needs two injections:
 +
:* The DVS injection,
 +
:* The activator injection: the doxycycline.
 +
You can find just below a simulator, which calculates how many time after the DVS injection, we have to wait until the doxycycline injection.<br>
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Latest revision as of 02:37, 22 October 2009

framless


Contents

Modelling of DVS efficiency on a lung tumour

Context

Non-small cell lung carcinoma, or NSCLC, is an aggressive cancer, with a relatively high speed growth. Treatments are often ineficient, because the tumor growth is faster than the elimination by the drug.

Objective

We have decided to model our treatment efficacy for this kind of tumor. Therefore we have modelled the tumour progression, our treatment evolution and efficacy. The objective of the modelling is to verify if our treatment is able to eliminate the entire tumor.

Back to top

Model segmentation

First, we had to recreate DVS complete mechanism and the tumor evolution. Then, for each step of the treatment, we have identified all the parameters that intervene, their actions and their interactions, in order to determine the model equations.

To simplifly the equation we have devided the mechanism and we have modelled each step separately.


Back to top

Tumor and DVS evolution versus time

First step : Tumor development according to time

We consider the tumor is non metastatic and its growth is exponential.
Let the tumor have a volume V1 in cm3 at an instant t1.
Let the same tumor, at an instant t2, have a volume V2.
The tumor is considered in exponential growth phase and without metastasis therefore its development equation, Tumor Growth Rate (TGR), is equal to :

TGR.jpg

Thus, the tumor volume according to the time (V(t)) is equal to :

V(t).jpg

Finally, knowing the Average volume of a cancerous cell (Vcc) (experimental data), if we regard the tumor as fraught (without cavity or blood vessel), we can determine that the Number of cancerous cells according to time (Nc(t)), without treatment effect, is equal to :

N(c).jpg


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Second step : Development of the tissue vector

The tissue vector is injected to the patient at an instant t, near t2. The Number of injected vectors (Nbi) is 1x10^6. The pulmonary tropism of the vector isn’t perfect, only a Percentage (Pp) goes to the lung. The total number of tissue vector in the body increases, because this vector is bacterial and therefore possesses a Doubling period (DTB).
Thus we can establish that the tissue vector number in the lungs (Nb(t)) is equal to:

Nb(t).jpg

The number of Tissue vectors increases until injection of doxycycline. F rom then, tissue vectors lysis releases the cell vectors in the lung.

This injection time is not insignificant. Indeed, si if we wait long enough, tissue vectors number is sufficient to eliminate the tumor or at least to significantly reduce it. On the other hand, if we wait too long, a higher dose of doxycycline (and so potentially toxic) is necessary for cell vector release.

Thus we can use modelling to determine the optimal injection time of doxycyline (Tdox).


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Third step : Release of the cell vector

Once the doxycycline injected, the cell vector is released. The cell vectors number is proportional to the tissue vectors number in the lung. And yet, we know the average value of recombinant phage vectors released by M. avium (Npl) is equal to 100.
We can write cell vectors number at the injection instant (Np(Tdox)) is equal to :

Np(t)1.jpg

The cell vectors number does not increase such as the tissue vectors. Indeed, it decreases with time, because of the phage vector stability and of its cell penetration (to release the therapeutic plasmide).
Its stability in the blood is equal to the phage vector deterioration constant (kdeg) according to time. If we add this constant to the cell vectors number equation according to time (Np(t)) we obtain the following formula :

Np(t)2.jpg

The phage vector dispersion steps in the tumor and for cell penetration are the steps below Fourth and Fifth) because of their complexity.


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DVS Efficiency

So, we determined:

  • The size of the tumor versus time (initial volume + growth)
  • The amount of tissue vector versus time
  • The amount of released cell vectors for a tissue vector

Now, we're going to determine the efficiency of our vectors for penetring cancer cells.
For that we are studying:

  • The area of dispersal cell vector
  • The importance of the cellular internalization of the vector in cancer cells.


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Fourth step : The cell vector dispersion

Here, we are looking for determine the maximum area that a phage can cover. This requires knowing:

  • The spread of phages in the bloodstream
  • Their diffusion through the walls of blood vessels
  • The surface of a cancer cell

For our modeling, we consider the blood as a Newtonian fluid with a constant velocity Vmax. Here, We neglect the heart-related jolts and turbulent flows caused by the cavities of the blood epithelium.

The cell vector moves along two axes. An X axis in the direction of blood flow and a Y axis orthogonal to the axis X.

RepèremouvementmécaniqueEn.png

The phage propagation in the bloodstream

The movement in X depends solely on the propagation of phages the vessel due to blood flow. Indeed, we neglect the diffusion which takes place also along the X axis because it is 1000 times less than the propagation of particles in the blood (due to the importance of blood flow). The cellular carriers are moving at speeds spread on a dish from, Vmax in the center of the vessel at V0 against the vessel wall.

RépartitionvitesseparaboliqueEN.png

The speed of phages decrease in approaching the vessel walls due to the friction forces which are opposing to the movement.

We can determine how long the particle (with a Vmax velocity), ie the particles in the center of the ship, reached the end. This gives the time necessary to internalize all the phages of the bacteria.

The diffusion through the walls of blood vessels

The movement in Y is the distribution of phages in the blood (j(n)). It depends on the equation of diffusion of a particle (n) in a fluid (Fick's Law).

Equation diffusion du phage.png

With n the number of particles (phages), grad n the difference between the concentrations and D the diffusion coefficient. The cellular distribution of vectors within the blood vessel and then through the wall is a phenomenon of diffusion with output. So, there will always be a strong gradient of concentration of phage in the blood. We can therefore say that the gradient is constant (equal to 1) over time. Thus the diffusion rate (j(n)) is equal to D.


Dispersal area of phage

When we combine moving Y (diffusion rate) and moving in X (blood flow velocity), we obtain, after integration on the perimeter of a blood vessel, the action surface of cell vectors. Then, we are able to determine the number of cancer cells per 100 cell vectors destroyed or 1 tissue vector.

The diffusion rate of the cell vector is equal to 0.5 μm.s-1 and the size of a capillary blood is 10μm in diameter. The particle farthest places so 10s to reach the vessel wall.

With this dissemination length (10s), the blood flow velocity (1x10 ^ 3μm.s-1) in the capillaries, and the surface of one cancer cell (1 micron square), we can determine:

  • The length (L) covered by the cell vectors released by one tissue vector.
  • The surface (S) occupied by phages in blood vessel diameter of 2r.
  • The amount of cancer cells available.

L = 1 x 10^4 µm 2r = 10 µm S = 2 x π x L r = 31.4 x 10^4 µm²

Thus, a tissue vector can potentially target more than 31 000 cancer cells, yet it has that 100 cell vectors. We can make a simplification to say that 100 cell vectors destroy 100 cancer cells. The efficiency of the dispersion is complete.

For phage, once reached the wall, comes in the cellular internalization. This model responds to two courses of action.


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Step Five: The cell vector internalization

Once in contact with the cell, the cell vector has two possible ways of action.

  • The vector fix then it detaches from the cell.
  • The vector fix then it is internalized within the cell.

We can model this according time and the association constant (kon) and dissociation constant (koff) and internalization constant(Kint).

This gives:

EqInt.jpg

With kon = 5.10^3 M-1s-1, koff = 8.10^-3 s-1 and Kint = 5,78.10^-4 s-1. If we calculate the global constant K', such as IDP = K' xt, we obtain K'= 361.5 s-1. Thus more than 360 phages are internalized per second in contact with a wall.

If we compare the time required to internalize a phage compared to waiting times before the cell enters apoptosis in response to the entrance of a vector cell (1 hour), it appears logical to neglect the internalization of cell vectors (IDP = constant = 360 phages/s) in the final equation.

Thus, with a total efficiency of the phage diffusion, and a neglected internalization time, we can say that the efficiency constant λ is equal to 1.


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Simultaneous evolution of DVS and tumor

The evolution equation of our model over time is equal to:

EqFinaleEN.jpg
float

With :

- Nc (t), the number of cancer cells depending time,
- V (t), tumor volume,
- V1 and V2, two tumors volumes respectively times t1 and t2,
- Vcc, the volume of a cancer cell,
- Nbi, the number of injected tissue vectors,
- Pp, the lung percentage of tissue vectors relative to the injected dose,
- DTB, the doubling time of tissue vector,
- tinj, injection time of the [[Team:SupBiotech-Paris/Concept1#drapeau|tissue vectors],
- Npl, the number of cell vectors released by bacteria.

We can neglected (differences between the time or space scales) some factors:

- Kdeg, the degradation constant of the phage, because all the phages were internalized in 10s.


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Treatment simulation

The DVS needs two injections:

  • The DVS injection,
  • The activator injection: the doxycycline.

You can find just below a simulator, which calculates how many time after the DVS injection, we have to wait until the doxycycline injection.


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