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# gentex

GENTEX is a whole organ model of the vascular network providing intraorgan flow heterogeneity and accounts for substrate transmembrane transport, binding, and metabolism in erythrocytes, plasma, endothelial cells, interstitial space, and cardiomyocytes.

Model number: 0084

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## Description

  GENTEX is a GENeric Tissue EXchange model characterizing the flow and transformation of metabolites.
GENTEX is a whole organ model of the vascular network providing intraorgan flow heterogeneity and
accounts for substrate transmembrane transport, binding, and metabolism in erythrocytes, plasma,
endothelial cells, interstitial space, and cardiomyocytes.



## Equations

• Flow velocities inside the capillary:

• In a large vessel, the blood flow can be partitioned between plasma (p) flow and red blood cell (rbc) flow. The flows are given by:

Fp = Fblood*(1-hLV) and Frbc = Fblood*hLV

and the corresponding volumes are given by:
Vp = Vblood*(1-hLV) and Vrbc = Vblood*hLV,
where hLV stands for the hematocrit in the large vessel.

It is easily seen that the velocities of plasma and rbcs are the same in the large vessel. (Velocity=Flow*Length/Volume).

It has been noted that hematocrit inside capillaries, hc, is different then the hematocrit inside the large vessel supplying them. Since the volumes of plasma and rbc in the capillaries depend on hc, the volumes are different in the capillaries than in the large vessel. Hence the ratio of velocities must be different.

Using:
Vpcap=Vblood*(1-hc) and Vrbccap=Vblood*hc,

the ratio of velocities is given by:
RatioVel=(Frbc*L/Vrbccap)/(Fp*L/Vpcap) = hLV*(1-hc)/( (1-hLV)*hc ).

If the Ratio of velocities is known, the capillary plasma hematocrit is given by:
hc = hLV/(hLV-RatioVel*HLV +RatioVel).

• Permeant Equations:

• $\large\frac{\partial pC_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial pC_p}{\partial x}- \frac{G_p}{V_p} \cdot pC_p -\frac{PS_{rbc1}}{V_p}\cdot pC_{p} +\frac{PS_{rbc2}}{V_p}\cdot pC_{rbc} -\frac{PS_{g1}}{V_p} \cdot pC_p +\frac{PS_{g2}}{V_p}\cdot pC_{isf} -\frac{PS_{ecl1}}{V_p}\cdot pC_{p} +\frac{PS_{ecl2}}{V_p}\cdot pC_{ec} +D_p \cdot \frac{\partial^2 pC_p}{\partial x^2}$

$\large \frac{\partial pC_{ec}}{\partial t} = \frac{-G_{ec}}{V'_{ec}} \cdot pC_{ec} -\frac{PS_{ecl}}{V'_{ec}} \cdot (pC_{ec}-pC_{p}) -\frac{PS_{eca}}{V'_{ec}} \cdot (pC_{ec}-pC_{isf}) +D_{ec} \cdot \frac{\partial^2 pC_{ec}}{\partial x^2}$ $\large \frac{\partial pC_{isf}}{\partial t} = \frac{-G_{isf}}{V'_{isf}} \cdot pC_{isf} - \frac{PS_g}{V'_{ist}}*(pC_{isf}-pC_p) - \frac{PS_{pc}}{V'_{isf}} \cdot (pC_{isf}-pC_{pc}) -\frac{PS_{eca}}{V'_{isf}} \cdot (pC_{isf}-pC_{ec}) +D_{isf} \cdot \frac{\partial^2 pC_{isf}}{\partial x^2}$
$\large \frac{\partial pC_{pc}}{\partial t} = \frac{-G_{pc}}{V'_{pc}} \cdot pC_{pc} -\frac{PS_pc}{V'_{pc}} \cdot (pC_{pc}-pC_{isf}) +D_{pc} \cdot \frac{\partial^2 pC_{pc}}{\partial x^2}$
Initial Conditions:
$\large pC_p(t=0,x)=0$ , $\large pC_{ec}(t=0,x)=0$ , $\large pC_{isf}(t=0,x)=0$ , $\large pC_{pc}(t=0,x)=0$ .

Boundary Conditions:
$\large pC_{out}=pC_p(t,L).$
$\large pC_p(t>0,0)=pC_{in}(t)$ , $\large \frac{\partial pC_p(t,L)}{\partial x} =0$

$\large \frac{\partial pC_{ec}(t,0)}{\partial x} =0$ , $\large \frac{\partial pC_{ec}(t,L)}{\partial x} =0$

$\large \frac{\partial pC_{isf}(t,0)}{\partial x} =0$ , $\large \frac{\partial pC_{isf}(t,L)}{\partial x} =0$

$\large \frac{\partial pC_{pc}(t,0)}{\partial x} =0$ , $\large \frac{\partial pC_{pc}(t,L)}{\partial x} =0$ .

• Extracellular Equations:

• $\large \frac{\partial eC_p}{\partial t} = \frac{-F_p \cdot L}{V_p} \cdot \frac{\partial eC_p}{\partial x}- \frac{G_p}{V_p} \cdot eC_p -\frac{PS_g}{V_{p}}\cdot (eC_{isf}-eC_p)+D_p \cdot \frac{\partial^2 eC_p}{\partial x^2}$

$\large \frac{\partial eC_{isf}}{\partial t} = \frac{-G_{isf}}{V_{isf}} \cdot eC_{isf} -PS_g \cdot (eC_p-eC_{isf}) +D_{isf} \cdot \frac{\partial^2 eC_{isf}}{\partial x^2}$
Initial Conditions:

$\large eC_p(t=0,x)=0$

$\large eC_{isf}(t=0,x)=0$
Boundary Conditions:

$\large \frac{\partial eC_p(t,L)}{\partial x} =0$

$\large eC_p(t>0,0)=C_{in}(t)$

$\large \frac{\partial eC_{isf}(t,0)}{\partial x} =0$

$\large \frac{\partial eC_isf(t,L)}{\partial x} =0$

• Vascular Equations:

• $\large \frac{\partial vC_p}{\partial t} =\frac{-F_p \cdot L}{V_p} \cdot \frac{\partial vC_p}{\partial x}- \frac{G_p}{V_p} \cdot vC_p +D_p \cdot \frac{\partial^2 vC_p}{\partial x^2}$

$\large vC_{out}=vC_p(t,x=L)$

$\large vC_p(t=0,x=0)=0$

$\large vC_p(t>0,0)=C_{in}(t)$

$\large \frac{\partial vC_p(t,L)}{\partial x} =0$

* Note: This JSim model project will run ONLY with JSim version 1.6.94 and above.
** Note: This JSim model project may not run in a Microsoft Windows XP or Vista environment.

The sample solution plots the input function in black with the concentration outflow curves for the vascular, extracellular, and permeant tracers (red, green, and blue, respectively). In the right panel, the extraction is plotted along with an estimated PS for the permeant tracer which is approximately the sum of the transfer rate between the capillary and the isf regions and the permeabiity-surface product governing the flux of material between the capillary and endothelial cell, i.e.,

$\large PS \approx PS_g+PS_{ecl}$ .

## References

  Poulain CA, Finlayson BA, Bassingthwaighte JB.,Efficient numerical methods for nonlinear-facilitated
transport and exchange in a blood-tissue exchange unit, Ann Biomed Eng. 1997 May-Jun;25(3):547-64.

Bassingthwaighte JB, Raymond GR, Ploger JD, Schwartz LM, and Bukowski TR. GENTEX, a general multiscale model
for in vivo tissue exchanges and intraorgan metabolism. Phil Trans Roy Soc : Mathematical, Physical and Engineering Sciences 2006.



## Key Terms

purine nucleoside metabolism, convection-diffusion-reaction model, GENTEX, transporter, red blood cell, axially distributed blood-tissue exchange processes, constrained parameter estimation, simultaneous optimization.