3.4.    Blood and BTEX Operators

3.4.1.   Introduction

The configuration of the circulatory operators in GENTEX is shown in Fig. 3.1.1. Of the operators shown, all large vessel operators including input tubing, artery, arterioles, venules, vein and output tubing, are modeled with blood operators. The exchange units are modeled with BTEX operators. Parameters related to the circulatory operators are listed in Table 3.4.5. Note that the labels of the variables shown on the XSIM model graphs may be different from the full XSIM parameter name, which must be unique.

Blood flow through vascular and BTEX operators

The total blood flow and the large vessele hematocrit are set in the GENTEX model layout window ( Fig. 3.4.1). The total blood flow is delivered to the input tubing, artery, vein, and output tubing. The fraction of this flow that is delivered to each pathway is controlled by the number of paths and the flow heterogeneity. All the flow delivered to a path goes to the arteriole, BTEX unit, and venule.

3.4.2.   Blood operators

Introduction

In GENTEX, a nonexchanging vessel is defined as a large vessel that does not exchange with extravascular regions. However, exchange between the RBC's and the plasma is allowed in these "nonexchanging" vessels modeled by blood operators. Each operator is characterized by two parameters: volume, V, and relative dispersion, RD. Other parameters governing exchange and chemical reactions in the blood are from the capillary-tissue unit. Fig. 3.4.2 shows a XSIM control window of such an operator.

The maximum value permitted for relative dispersion is 0.48, and the minimum value is 0. Note that the blood operator approximates the desired RD with axial diffusion coeffcients.

The volume should be the physical volume of that component of the circulation in ml g^-1. If the volume of an operator is set to 0, it is effectively removed from the system.

Arterial operators

The vascular operators on the arterial side are the inlet tubing, artery, and arterioles. For each tracer or nontracer substance, there is a single inlet tube and artery. Separate operators are provided for these two components for use when the tissue content of the tracer and nontracer (residue) is being observed. The inlet tube can provide delay and dispersion of the injection before the marterial reaches the tissue that is included in the residue signal.

There is one arteriole ( Fig. 3.4.2) for each of the flow paths. The volume specified for this component is divided equally amongst the paths.

Venous operators

The vascular operators on the venous side are the venules, vein, and outlet tubing. As with arterioles, there is one venule per path, and the volume is distributed equally. The comments made above regarding the inlet tubing and artery also apply to the vein and outlet tubing.

3.4.3.   BTEX operators

Introduction

The capillary-tissue units of each path are modeled with BTEX operators. A different type of BTEX operator is used for each tracer depending on the number of regions required to model the behavior of that tracer. Detailed information about the BTEX operators is contained in Bassingthwaighte, Chan and Wang (1992); information about the computer implementation of a typical three region model is available on-line (man btex30).

The generic parameters of a BTEX model are shown in the table below. The names used in GENTEX are prefixed with the tracer label (r: RBC tracer, v: vascular or plasma tracer, e: extracellular tracer, p: permeant tracer and nontracer substance, no prefix: all) and postfixed by the region label (cap: capillary, rbc: red blood cells, p: plasma, ec: endothelial cell, isf: interstitial fluid, and pc: parenchymal cell).

For all extravascular regions (ec, isf, and pc) the volume is a volume of distribution rather than a physical volume. Note that erythrocyte mean transit time through the capillary system is less than plasma transit time, since the velocity of red blood cells is faster than that of plasma. The ratio of RBC to plasma velocity, r_vel, is reflected in a difference between large vessel hematocrit (Hct) and capillary hematocrit (Hct_cap). In this model, the total blood flow (F_b), capillary volume (V_cap), large vessel hematocrit (Hct), and RBC to plasma velocity ratio (r_vel) are user-defined parameters. The intracapillary RBC flow, F_rbc, and plasma flow, F_p, are calculated and identical to the large vessel flows, i.e., F_rbc = Hct F_b and F_p = (1 - Hct) F_b. The volumes of RBC's and plasma (V_rbc and V_p) are calculated as V_rbc = Hct_cap V_cap and V_p = (1 - Hct_cap) V_cap where Hct_cap is the capillary hematocrit and is derived as follows: because

so

Because the species can move to all regions, it has an apparent volume of distribution (V') in all regions. In the RBC and plasma region, the volumes of distribution are

where W_rbc and W_p are the water fractions of the RBC's and the plasma, respectively. Note that W's are the contants that convert the physical volumes to volumes of distribution. They are referred as water fractions in GENTEX for convenience. See Fig. 3.4.4 for details of these model parameters.

RBC reference tracer

The RBC reference tracer is modeled with a one region BTEX model. The tracer is constrained to the RBC region.

Vascular reference tracer

The vascular tracer is modeled with a two region BTEX model ( Fig. 3.4.3). While the vascular tracer is nominally constrained to the plasma region, the second region is used to account for any binding of the tracer to the endothelial cell. An example of this effect is binding of albumin to an albumin-fatty acid receptor on the endothelial cell membrane. If appropriate, this second region can also be used to account for slow leakage of the vascular tracer into the ISF over long times. See Table 3.4.6 for details of these model parameters.

Extracellular tracer

The extracellular tracer also has access to two regions, plasma and ISF ( Fig. 3.4.4). For this tracer, however, the ISF is partitioned into two parts. This partitioning can be used to account for radial diffusion in the ISF or physical division of the ISF into two distinct regions (e.g., T-tubules in skeletal muscle). Apportionment of the total ISF volume between these two partitions is controlled by the parameter fVisf2 and the diffusion of tracer between them by the parameter PSisf. If partitioning of the ISF is not desired, set both fVisf2 and PSisf to 0. See Table 3.4.7 for details of these model parameters.

Permeant species

GENTEX allows both tracer and nontracer mother substances of multiple permeant species (up to 5). GENTEX models linear and nonlinear processes of transmembrane transport and chemical reactions. Model parameters can be set up by clicking on the PS buttons for transmembrane transport, on the G buttons for consumption, and on the Bind buttons for binding ( Fig. 3.4.5). One can also access all GENTEX functionalities via the pull-down menu of Parameter in XSIM main window (See an example in Fig. 3.4.6). See Table 3.4.8 and Table 3.4.9 for details of the model parameters.

3.4.4.   Transport and chemical reactions of permeant species in BTEX units

Transporters

There are transports at the membrane of RBC's, the luminal and abluminal membranes of endothelial cells, and the membrane of the parenchymal cells. Fig. 3.4.7 shows the control window for the transport through the luminal membrane of endothelial cells (click on the PSecl button in the BTEX Unit window for any permeant species).

For each species, users can select one of the four transport types (T_type) ( Table 3.4.1). For Linear transport the constant PS values can be set in the main control window. For Facilitated and Nonlinear transport, further setup is required (clicking on the buttons at the bottom of the window).   

Both Facilitated and Nonlinear transporters in GENTEX are carrier-mediated, i.e., substrate binds to the transporter at one side of the membrane, undergoes conformational change, and then is released at the other side of the membrane. The binding is assumed to quasi-equilibrium, i.e., much faster than the conformational change. The conformational change is assumed to be first-order with rate constants P_T for free transporters and P_TS for bound transporters. The main difference in Facilitated and Nonlinear transport is that the latter is more general and also has surface receptors.

The Facilitated transporter ( Table 3.4.8) in GENTEX can be species-specific or shared by different species (competitive). Its conformational change is always symmetric. It has zero transporter capacitance (no receptor). If two species are set up to use species-specific transporter, each species has its own set of transporter parameters. In the competitive transport case, one needs to choose Facilitated (specific) of T-type for the two or more species sharing the same transporter. The total transporter concentration (e.g., T_tot in Fig. 3.4.9 and Table 3.4.10) of the first species sharing the transporter is used in the model, i.e., only the first and one T_tot is in effect. Other transporter parameters can be species specific.

A Nonlinear transporter ( Fig. 3.4.9) in GENTEX can be shared by multiple species. The conformational change can be asymmetric. The carrier also serves as a receptor. The parameters that are used by different types of transporters are listed in Table 3.4.10. The method for setting up competitive, facilitated transporter is applicable here.

Note that the effective PS values are calculated by the model. These values are averaged over all the axial segments and are shown in the control windows during a simulation run. 

Consumptions

For every region, there is a control window ( Fig. 3.4.10) for species consumption and transformation. Consumption type can be set separately for each species ( Table 3.4.2).

The parameters for these chemical reactions can be set up by choosing the G-typefirst, then provide correspoding parameter values. For linear and Michaelis-Menten reactions, the transformation of chemical species can be set up by using fractional rate in the middle part of the consumption control window. For example, a f23 of 0.4 means 40% of species #2 that is consumed will be transformed to species #3, and f22 is the sequestration fraction. GENTEX allows flexible configuration of reaction cascade but it is the user's responsibility to assure the mass balance.

Enzyme binding and reactions can be set up at the bottom of the consumption control window. The enzyme is shared if multiple species are thosen to use this reaction type.

GENTEX also allow multiple metabolic pathways of the same species by choosing G-type of M-M & Enzyme or M-M & M-M.

Table 3.4.12 and Table 3.4.13 provide the complete lists of consumption parameters for species #1 and #2 in parenchymal cells. Consumption parameters in other regions for other species are likewise.

Binding

For every region, one can use up to three binding sites for each species: B1, B2 and B3. B1 and B2 are fast binding sites (quasi-equilibrium), and B3 is a slow on-and-off binding site. For every region, there is a control window ( Fig. 3.4.11) for setting up the binding of all the species. Binding type can be set separately for each species ( Table 3.4.3). The full lists of parameters for species #1 and #2 in the ISF space are provided in Table 3.4.14 and Table 3.4.15. Binding parameters for other regions and species are likewise. Note that mobil binding sites that move with the flow are allowed in the RBC and plasma regions.

Bireactant reaction

In addition to the consumption and binding in every region, GENTEX allows bireactant reactions in one user-specified region by using the Bireactant reactions window ( Fig. 3.4.12; in the Main XSIM Window choose menu Parameters -> Consumptions -> Bireactant reactions). Users can select the region and set up individual bireactant reactions (up to 4),

where S's are the species, and "a" and "b" are fractional coefficients. K_f and K_r are the forward and reverse rate constants, s^-1. Note that this feature applies to nontracer kinetics only, and Si, S_j and Sk must be different chemical species and tracer.

Species synthesis and time-varying parameters

GENTEX allows point or line sources to simulate species synthesis. From the XSIM menu (Parameters -> Species synthesis/sources in BTEX unit), one can get the control window ( Fig. 3.4.13). Users can have two sources.

Users also need select two source functions for the two sources (two buttons at the bottom). For the Additive Source, the function values represent the accumulative changes over time, i.e., the area under the souce function curves are the total changes. For other sources, the function values represent the absolute concentrations over time.

Time-varying parameters

GENTEX allows allows time-varying model parameters. From the XSIM menu (Parameters -> Time-varying Paremeters in BTEX unit), one can get the control window (). The function values represent the fractional change of the parameter over time. Users can have two time-varying parameters simultaneously.

3.4.5.   Model switches

There are three switches in the GENTEX model layout window: (1) initial condition (I.C.) switch, (2) switch for computing tracer or nontracer or both, and (3) ODE solver switch.

For the ODE solver, GENTEX provides four methods: (1) a fifth-order Taylor method for linear problems; (2) LSODES, a variant of back differentiation formulae (BDF) for nonlinear stiff and sparse problems (Authors: Alan c. Hindmarsh and Andrew h. Sherman, 1987); (3) RADAU, a variant of implicit Runge-Kutta method for nonlinear stiff problems (Authors: E. Hairer and G. Wanner, 1996); and (4) an adaptive method trying to use the best-suited solver (default solver). Note that the adaptive method is safe but not necessarily the fastest.

3.4.6.   Vascular and BTEX operator errors and messages

[TO BE DEVELOPED]

3.4.7.   Reference tables for GENTEX parameters