3.3.    Heterogeneity for MMID4, MSID4 and GENTEX

3.3.1.   General

Organs are known to be heterogeneous in many of their physical properties including regional blood flow. In these models, flow heterogeneity is modeled by a set of parallel flow paths as shown in Fig. 1.1.1. Each of the up to 20 pathways has a different blood flow. The procedure used to specify the blood flows is discussed in the next section.

In addition to heterogeneity of regional blood flow, MMID4 and MSID4 also permits heterogeneity in membrane conductances and consumptions ( Section 3.3.3). If heterogeneous conductances and/or consumptions are used, the values for each path are proportional to the flow for that path. Thus, these heterogeneities can only be used in conjunction with heterogeneity of flow.

The parameters and windows related to heterogeneity are shown in Fig. 3.3.8. The number of paths, found in the model layout window, is the primary controller for heterogeneity. As the name implies, it sets the number of flow pathways used by the model. If it is set to 1, no heterogeneity is used regardless of the values of the other heterogeneity parameters.  

3.3.2.    Flow Heterogeneity

Introduction

Blood flow heterogeneity affects the shape of outflow and residue curves measured in indicator dilution experiments. Comparing the outflow curve from a multipath (heterogeneous flow) model to that from a single path (homogeneous flow) model, differences include

1. shortened appearance time (caused by the presence of paths with flows greater than the mean),
2. decreased peak height and increased dispersion (caused by distributing the total flow among paths with different flows), and
3. an elevated tail of the curve (caused by paths with flows lower than the mean).

Figure 3.3.9.    Probability density function (PDF) of local regional flows. Top: Continuous representation. Bottom: Histogram representation.

Regional blood flow in an organ can be described by a probability density function (PDF) of relative flows. (The relative flow is the regional flow divided by the mean flow to the organ.) A sample regional flow PDF, is shown in Fig. 3.3.9. The relative flow, f_i, is the regional flow divided by the mean flow for the whole organ, and the fractional mass, w(f_i), is

where Sm_j is the total weight of all the organ regions with relative flows in the i^th flow class, W is the total weight of the organ, and Df_i is the width of the i^th class. The PDF has a mean of 1.0, and an area of 1.0. As shown in Fig. 3.3.9, the PDF may be represented as a histogram or as a continuous function. The former has the advantage of showing the widths of the flow classes, while the latter allows more than one PDF to be displayed on the same axes. In this example, all the flow classes have equal widths, but this need not be the case.

Terminology used for flow heterogeneity

In the model, each flow class (maximum of 20) is modeled by a separate flow path. The relative flow for the path, f_i, is referred to by f (N), and the fractional mass, w( f_i), is referred to as w(N), where N is the path number.

Note that the area of the PDF is unity. Thus
. Since Df_i need not be constant for all pathways, it may be difficult to interpret a set of w_i's by visual inspection. To overcome this difficulty, the weighting factors for the paths in the modelare specified in terms of the w_iDf_i for that path, referred to as wd(N). Thus wd(N) is the fraction of the organ represented by the flow for the N^th pathway. For a properly normalized PDF, the sum of the wd(N)'s is 1.0.

As described below, the user may specify the PDF relative flows and weighting factors in the parameter array. These are referred to as "Fin" and "WDin" respectively and are accessed by clicking on the User specified PDF button. The user may also specify a set of relative flows to be used for the paths; these are referred to as "Fout" and their window is opened by pressing the User-specified flows button. The relative flows and weights for the paths that are actually used by the model are referred to simply as fi and wdfi and are located in the Flow heterogeneity results window. The fractional mass values, wi, for the paths are also calculated and are located in the same results window.

Calculation of flow heterogeneity parameters

The procedure used in the model for calculating the heterogeneity parameters from the user specified values is diagrammed in Fig. 3.3.10. The goals are to allow several choices for selecting a PDF that represents the distribution of flows in the tissue being modeled, allow flexibility in selecting the flows used for the pathways, and select weights for these flows that, in so far as possible, faithfully represent the selected PDF. In Phase 1, a PDF that characterizes the organ being modeled is specified. Relative flows that best represent the data being analyzed and weights for each path are selected/calculated in Phase 2.

Three choices are available in Phase 1a:

1. the PDF for a particular experiment can be loaded from the reference data,
2. a PDF can be taken from the parameter array, or
3. a mathematical distribution can be used.

Two mathematical distributions are available: a lagged normal density curve (LNDC) or a random walk (RANWOK). The LNDC is used for distributions with moderate skewness and RANWOK is used when the skewness exceeds three times the relative dispersion. In Phase 1b, the high and low ends of the PDF are clipped to eliminate paths with very small probabilities (default wdi < 0.1%). The user may specify additional clipping.

In Phase 2a, the relative flows that will be used by the model, f_i, are selected. These flows can be specified by the user or generated by a mathematical algorithm that selects flows equally spaced in either the flow or the transit time domains, or at some point intermediate between those limits. If the flows are specified by the user, they may be scaled and shifted to bring them into the range of flows spanned by the PDF. In Phase 2b, the PDF is interpolated to obtain the weighting factors for the paths, wdi. Finally f i and wdi are normalized to ensure that mass is conserved in the model. Not shown on the diagram is the calculation of the w_i's that correspond to the wd_i's.

While considerable emphasis has been placed on faithfully representing the specified PDF by the relative flows and weights used for the paths, the procedure is not exact. Thus, it is strongly recommended that the user plot the PDF's as described below and visually compare the results. (See Plotting the regional flow PDF's.)

Specifying the heterogeneity model (Phase 1a)

Selection of the heterogeneity model is controlled by the PDF Model selection button in the Heterogeneity inputs window.

Using a PDF from the Reference data

If Use reference datais selected, the PDF will be read from the reference data. The data are loaded into the area reserved for a user-specified PDF discussed below. All restrictions and considerations related to a user-specified PDF apply. Note that any data in the user-specified PDF locations are overwritten.

Using a user-specified PDF

User-specified PDF: In keeping with established practice at the NSR labs for standardizing experimentally determined flow PDF's, a maximum of 30 values are permitted. The number of values actually used is specified in the Number of values field. The f_i's and wd_i's are shown in the User-specified PDF pop-up window, Fin and WDin, respectively.

These data can be directly edited by the user to change the shape of the PDF. The values need not be normalized to an area and mean of 1.0 as the model will normalize the PDF before using it in any further calculations. Note that the values in the parameter array will be overwritten if a reference data PDF is used or if the Copy to user PDF? switch, discussed below, is on.

Smoothing the PDF

A reference data or user-specified PDF may be smoothed prior to its usage in further calculations. Smoothing is turned on by setting the PDF smoothing? switch to Yes. If smoothing is on, a running three point smoother with weights [0.25, 0.50, 0.25] is used.

Using a mathematically generated PDF

If no PDF is available from the reference data and usage of a user-specified PDF is not desired, two mathematical functions are available, a lagged normal density function and a random walk density function. The shape of the PDF is controlled by these parameters:

Permissible values of the skewness depend on the function selected. For LNDC, skewness must be in the range 0.0~1.9. For Random walk, skewness must be greater than 3.0 * RD.

Since negative values of relative flow are not permitted, using large values of RD, especially in conjunction with a small value of skewness, will result in the PDF being significantly clipped on the low end. Examples of PDF's generated with LAGNDC and RANWOK are shown in Fig. 3.3.13.

Error conditions

If an error is detected in the parameters that control the heterogeneity model, a warning message is issued and a lagged normal distribution is used.

Clipping the PDF tails (Phase 1b)

To avoid useless calculations, flow paths with low weights are removed by clipping the tails of the PDF. By default, standard clipping is done at the points at which wd_i is less than 0.001. The user may specify custom clipping that is controlled by these parameters:

The clipping limits, minimum fi and maximum fi, must be less than 0.5 and greater than 1.5 respectively. If either is outside its range, it is reset to the limit and used in the subsequent calculations. The altered value is stored in the parameter array and a warning message is printed.

Additional clipping should be used with caution. Since the results will be renormalized in Phase 2b, excessive clipping can result in a final distribution that has a significantly altered shape and statistics than that originally specified.

Specifying the relative flows for the paths (Phase 2a)

The specifications of the f_i's in Phase 1 are used to determine the shape of the PDF. In Phase 2a, the relative flows that are actually used for the paths are selected, as determined by the pathway flows Selection method.

If user-specified flows are selected, the values to be used are entered in the User-specified flows pop-up window. Up to 20 values may be specified, but the number actually used is determined by the number of paths selected in the model layout window. The values must be greater than 0.0 and in ascending order. If an error is detected, a warning message is printed and weighted spaced flows are used as described below. If some of the user specified flows fall outside the clipping limits specified above, then the set of user-specified flows are scaled and/or shifted so that they fall within the clipping limits. This preserves the relative spacing of the flows over the clipped flow range.

If mathematically generated f_i's are selected, the user can choose from four different methods of spacing the f_i's:

1. equally spaced in the flow domain, Equal F Spacing( Fig. 3.3.14a, b),
2. equally spaced in the transit time domain, Equal t spacing( Fig. 3.3.14b, c),
3. intermediate between (1) and (2), Scaled ( Fig. 3.3.14d, e), or
4. weighted flows, Weighted ( Fig. 3.3.14f, g).

Equal flow domain spacing results in many paths with short transit times and, thus, few paths with long transit times. This may lead to deformation of the tails of outflow curves due to undersampling of long transit times. Equal flow domain spacing almost always preserves the statistics of the PDF selected. Equal transit time spacing inadequately preserves the PDF statistics, but may give a better representation of the tails of the curves. Using a scale factor between 0 and 1 gives f_i's that are linearly interpolated between the equal flow and transit time spacing. Fig. 3.3.14d, e show the results when scale factor is set to 0.6. Selecting the weighted method selects flows that are weighted to be equally spaced in transit time at low flows and equally spaced in flow at high flows as seen in Fig. 3.3.14f, g. This choice is recommended as it preserves the statistics of the flow PDF and gives the more flows at the ends of the PDF that effect the upslope and tail of the curves.

Generating the weights for the flow paths (Phase 2b)

The weights for the pathways are obtained by interpolating the PDF generated in Phase 1 at the values of relative flow generated in Phase 2a. After the interpolation, the results are renormalized to insure conservation.

Flow heterogeneity results

The normalized PDF resulting from Phase 2b is stored in the Flow heterogeneity results window, Fig. 3.3.15. Relative flows, f_i, path weights, w_i, and fractional masses, wd( f_i), are displayed. The flow limits and the statistics of the PDF are shown in the Flow histogram section.

Copying the heterogeneity results

The relative flows and weights used for paths, fi and wdfi in the results window, can be copied to the User-specified PDF window, Fin and WDin. This action is controlled by the Copy to user PDF? switch in the Heterogeneity inputs window. When it is set to Yes, the heterogeneity results are copied to the user-specified PDF section. No action is taken when the switch is set to No.

If the results are copied, the number of user-specified values is set to the current number of paths. These data can then be edited and, if the pathway flows selection method is set to equal F spacing, used as the input PDF for subsequent model runs.

When the copy is complete, the copy switch is turned off. This avoids continuous copying and renormalization of the PDF that can result in undesirable shifts in the relative flows.

Plotting the regional flow PDF's

The input PDF and the PDF used by the pathways is available for plotting. Relevant parameters are:

Parameters PDF_hist, PDF_cont, HST_hist and HST_cont are updated at each step of the simulation and can be plotted in the usual manner for XSIM outputs by entering the parameter name in a plot area Y parameter field. An example of plotting HST_hist and PDF_cont is shown in Fig. 3.3.16. It is recommended that the continuous version of the input PDF and the histogram version of the model PDF be plotted so that they can be easily distinguished.

The x-axis scaling is controlled by Plot maximum. The x-axis minimum is always 0.0. If Plot maximum is set to 0 or less, the default value of 3 is used for the x-axis maximum. (Note: The x-axis labeling is still controlled by the scaling of the independent variable.)

3.3.3.    Heterogeneous membrane conductances and consumption

Introduction

Recent studies have indicated, though not conclusively proven, that regional membrane conductances, permeability-surface area (PS) products, are related to regional flow. One mechanism for this is increased surface area as additional capillaries are recruited. Similarly, it may be expected that there is a relationship between the regional consumption, or utilization, of a substrate and its delivery to the region (i.e., some optimal balance between flow and consumption).

These effects are modeled in the model by including mathematical relationships between flow and PS or G in each path. The function used is ,

where P is either PS or G, P_mean is the average value of the parameter specified in the parameter array, a is a_rct or a_opt, f_i is the relative flow for the pathway, and P_i is the actual value of the parameter used for that path. Note that a = 0 makes the parameter constant for all path and a = 1 makes the parameter directly proportional to flow.

These relationships are controlled by the parameters:

Recruitment heterogeneity

Setting a_rct produces heterogeneity in PS_g for all tracers and in the endothelial cell permeabilities, PS_ecl and PS_eca, for the permeant tracer.

Optimality heterogeneity

Setting a_opt only affects the parameters for the permeant tracer. It produces heterogeneity in PS_ecl, PS_eca, PS_pc1, PS_pc2, G_ec, and G_pc. Note that a_rct and a_opt both affect PS_ecl and PS_eca for the permeant tracer. If a_opt is nonzero, it overrides the control specified by a_rct.

The Recruitment window is accessed from the Parameters pull-down menu.

3.3.4.   Heterogeneity errors and messages

[TO BE DEVELOPED]