1.   GENTEX Overview

1.1.   What Is GENTEX?

GENTEX is a generalized nonlinear blood-tissue exchange model that can be used to examine the behavior of four types of tracers: one that remains in the red blood cells (RBC's), one that remains in the plasma, one that can leave the plasma but remains extracellular, and up to 5 fully permeant tracers that enters cells. For permeant tracers, their mother substances are also accounted for in the model. The model can be used to study the physiology of the exchange process or as a tool to help analyze actual experimental data. The exchange and metabolic processes, including linear, nonlinear and time-varying processes, may be examined by viewing the instantaneous outflow concentration, the instantaneous extraction, or the amount of tracer remaining in the exchange system (i.e., the residue function).

The model provides ways to examine tracer delay and dispersion between the injection site and the target organ and the effects of flow heterogeneity on exchange within the organ. Fig. 1.1.1 shows the general layout of the model. Each of the multiple tracers flows through the common arterial vessels, the parallel pathways each containing an arteriole, an capillary-tissue unit, and an venule, and then through the common venous vessels.

In formal notation, the outflow concentration-time curve seen from an exchange unit in response to an impulse input is denoted as h (t). We use h (t) as a general symbol to denote the normalized (unity area) outflow dilution curve even when the input has forms other than an impulse. A subscript is added to designate specific species. Thus, h_D(t) denotes the "permeating" or "diffusible" species; h_E(t) signifies the extracellular tracer; h_rbc(t) is used to represent the RBC tracer; and h_R(t) is used to represent the intravascular or plasma "reference" tracer.

1.2.   Vascular Components

1.2.1.   Nonexchaning vessels

The non-exchanging blood operators

The nonexchanging vessels in GENTEX are modeled by blood operators. Each nonexchanging vessel is described by its volume and relative dispersion (RD). Nonexchanging here means no exchange between the blood and the tissue. However, for diffusible nontracer substances and tracers, the exchange and mectabolic processes are allowed in the RBC's and plasma, and the parameters are the same that are described in the capillary-tissue unit. If a particular operator is not desired in a given simulation, it can be effectively removed by setting its volume to zero. Although the large vessels are normally used to describe vascular delay and dispersion, they can also describe pure delay (no dispersion) by setting RD to zero.

Arterial vessels

Three types of arterial vessels are shown in Fig. 1.1.1. The first is the input tubing. This models a cannula or catheter through which tracers and/or nontracers are injected. While not strictly an arterial vessel, it is upstream of the organ and can be described in the same way as the large vessels.

The second type of arterial vessel, downstream from the input tubing, is the artery. Substances flowing out of the artery is distributed to the various flow paths where it enters the third type of arterial vessel, the arterioles, and then the capillary-tissue units. The distribution of flow among the pathways is controlled by the flow heterogeneity parameters described in Section 3.3.2.

Venous vessels

Fig. 1.1.1 shows that the venous vessels in GENTEX are similar to the arterial vessels. Each exchange unit drains into a venule. All venules drain into the common vein, then into the output tubing.

1.2.2.   Capillary-tissue units

The blood--tissue exchange unit (BTEX) operator

The basic tissue unit in which blood-tissue exchange takes place, shown in Fig. 1.2.2, consists of five regions: the RBC's, the plasma, the endothelium, the interstitium, and the parenchymal cells. Note (1) that the discrete RBC's are modeled as a continuous region; (2) that there are two flowing regions: RBC's and plasma that can have different velocities; and (3) that the interstitium (ISF) is split into two parts. The first, ISF₁, lies between the endothelial and parenchymal cells. The second, ISF₂, is connected to ISF₁ and the parenchymal cell. This is considered a five region model despite the split of the ISF into two parts.

This unit is a covection-exchange-diffusion-reaction model that is described by a set of partial differential equations (PDE) of species concentrations with respect to time and axial position along the capillary length. For each species in each region, there is one PDE describing convection, axial diffusion, transmenbrane transport, and chemical reactions. If the reaction involves multiple species, it is reflected in the PDE's for all the involved species.

Numerically, the model uses multiple segments along the capillary length for axial resolution. The segmentation is same for all regions. The temporal resolution is matched with the spatial resolution based on the flow velocity. In brief, the fluid sliding downstream by exactly one segment during one numeric time step (Lagrangian fluid sliding algorithm). With this sliding algorithm, a time-splitting scheme is implemented to separate the convection and other processes numerically. Other processes, except for the axial diffusion, are solved as ordinary differential equations (ODE's) for one time step after the convection or fluid sliding.

Shown in Fig. 1.2.3 is an illustration of how the algorithm works in the situation of one flowing region (plasma in this case) interacts with two extravascular regions. The exchange unit is divided into a series of axially distributed segments. At the beginning of each time step, the fluid in the capillary is advanced one segment, i.e., the sliding phase. The exchange and other processes are then calculated for the current time step, the exchanging phase. The sliding and exchanging phases will repeat for the subsequent time steps. The rationale for this approach has been discussed by Bassingthwaighte (1974 ) and Bassingthwaighte et al. (1989 ). The model provides for up to 60 axial segments. Note that GENTEX implements a modified Lagrangian sliding element algorithm because it has two flowing fronts that can have different velocities, i.e., RBC's move faster than the plasma. The plasma velocity is used in calculating the numeric time and spatial step. For the RBC region, the faster moving region, the fluid elements slide in accord with the RBC to plasma velocity ratio. There is instant mixing in all RBC segments if a partial slide occurs.

RBC tracer exchange units

The red blood cell (RBC) reference tracer and plasma tracer) is confined in the RBC's. The parameters affecting the vascular tracer are shown in Fig. 1.2.5. The apparent volume of distribution within the RBC region (V´_rbc) can be different from the real RBC volume (V_rbc). The tracer can undergo axial diffusion (D_rbc).

Vascular tracer exchange units

Although there are two intravascular tracers (RBC tracer and plasma tracer), the vascular tracer in GENTEX is referred to the one that is confined in the plasma. The parameters affecting the vascular tracer are shown in Fig. 1.2.5. The apparent volume of distribution within the plasma region (V´_p ) can be different from the real plasma volume (V_p). The tracer can undergo axial diffusion (D_p) and may also be bound to receptors on the endothelial surface. Such binding, if reversible, may be simulated by allowing movement into the interstitium similar to the movement of the extracellular tracer. Hence, PS_g and V´_isf are also included as parameters for the intravascular tracer.

Extracellular tracer exchange units

The extracellular reference tracer is confined in plasma and interstitial region. The parameters affecting the extracellular tracer concentrations are shown in Fig. 1.2.6. The extracellular tracer has an apparent volume of distribution in plasma (V´_p ) and interstitial region (V´_isf). It can undergo axial diffusion in plasma (D_p) and interstitial regions (D_isf). The interstitial region is splitted into two parts: V´_isf1 and V´_isf2. The fraction of V´_isf2 of the total interstitial region is fV_isf2, so V´_isf2 = V_isf fV_isf2 and V´_isf1 = V´_isf - V´_isf2. The extracellular tracer can move among the three regions, and its movement depends on two PS products.

Permeant species exchange units

The parameters affecting the concentration of the permeant species in each region are shown in Fig. 1.2.7. 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's are the contants for converting the anatomical volumes to volumes of distribution. For example, if the chemical species distribute only in the water space, W_rbc and W_p are the water fractions of the RBC's and the plasma, respectively.

In each region, the tracer has the potential to undergo axial diffusion (D). The species may be metabolized or undergo chemical reactions that result in its being cleared from the region. In GENTEX, users can click on the G or B buttos to set up chemical reactions. G represents the intraregion consumption or "gulosity" in which three types of consumptions can be set up: linear reactions, Michaelis-Menten kinetic type reactions, and enzyme binding & reactions. B represents binding in which quasi-equilibrium or slow on-and-off bindings can be set up. Movement of the species between regions can be linear transport that is governed by constant permeability-surface area product (PS), or nonlinear transport facilitated by carrier proteins. Users can set up desired transport types and parameters by clicking on the PS buttons. In addtion to the transport between adjcent regions, a molecue of a permeant species may move between the capillary and the interstitium through the clefts between the endothelial cells. In this case, movement depends on the constant PS product (PS_g).

1.3.   Additional Features

1.3.1.   Input function generation

General

GENTEX permits the user to generate an input function in a number of ways. These are discussed briefly below. For a given simulation run, the user selects one of the available methods. Generation of the input function is discussed in detail in Section 3.2 of this manual and is the subject of exercises in Section 2.2.2.

Function generator

GENTEX gives the user access to a generalized function generator. It will generate a variety of functions including several types of pulses and pulse trains and a number of density functions.

Curves from reference data

If the user specifies that reference data is to be loaded by XSIM, curves from these data can be used as the input function for one or more of the tracers and nontracers.

Deconvolution

When there are reference data that contain an outflow curve for an intravascular or excellular tracer, the user can direct GENTEX to deconvolve those data with the model transfer function and use the result as the input function.

1.3.2.   Heterogeneity

General

When the user configures GENTEX to use more than one flow path, the flow paths may have different flows, membrane conductances, and tracer consumptions. These heterogeneities are used to simulate tracer transport and exchange in a whole organ. Heterogeneity is discussed in Section 3.3 of this manual.

Flow

In a multipath configuration of GENTEX, the flow for each path must be specified. These flows can be specified by using a probability density function, by a curve in the reference data, or by the user entering the flow for each path.

1.3.3.   Parameter tools

Parameter Evaluation

In certain instances, it may be useful to have several parameters have the same numeric value (e.g., the capillary permeability for the three tracers). Parameter expression evaluation can link the value of "slave" parameters to that of a "master" parameter. Changing the value of the master automatically changes the values of the slaves. Note: a parameter, including scalars, may not be slaved onto another parameter that is already slaved to a parameter or constant.

Two types of parameter tools are provided in GENTEX: The eval field for each parameter, and a set of general scalars. The former allows a parameter to be set to equal a function of one or more other parameters.

Parameter scalars

While GENTEX has a wide range of input parameters, situations arise in which a user wants to use a parameter that is not provided by the program. General scalars are provided for this purpose. These scalars allow the user to create a new result parameter that is a linear combination of other parameters. The set of scalars are simply a set of parameters that are not tied to the model, but can be set like any other parameter, either by entering a value directly or by slaving the scalar to another parameter in the Eval field.

1.4.   Summary