Qualitative Model Approach to Computer Assisted ... - CiteSeerX

Qualitative Model Approach to Computer Assisted Reasoning in Physiology Blaz Zupan1, John A. Halter2, Marko Bohanec1

Abstract. Developing practical tools to aid in understand-

ing physiological systems is a formidable undertaking. This paper presents a method that uses a property structure for the domain being investigated. Furthermore, it employs realistic models to present examples of the behavior of the system. From these examples the principles that relate the properties are inferred through the use of machine learning. To allow prediction of property values in quantitative domain, interval logic and fuzzy logic based methods for qualitative model interpretation are proposed. The principles are expressed as qualitative rules that derive the values of the properties. The structured approach and the qualitative representation of principles provide a simpli ed means to reason about the roles of properties and meaning of principles of the physiological systems being investigated.

1 Introduction Recent advances in computational technology allow the use of complex realistic models to assist in the discovery of principles and properties within dierent physiological systems. Such models map a set of input properties (input parameters of a model) to a set of output properties (description of model behavior). However, as these models grow in sophistication, it becomes more dicult to manipulate the growing number of model parameters and equations eciently in order to reason about the roles of the properties and hypothesize about the principles of the modeled system. Moreover, the simulation of realistic models usually requires substantial computational resources. The reasoning about principles which is then based only on model simulation can be a potentially time-consuming and tiresome process. The principles of the modeled system determine how the input properties are mapped to output properties. In order to understand the process that is being modeled, it should be feasible to derive why such mapping occurs and to be able to derive and understand the principles that guide the mapping. The realistic model itself is usually based on several known mechanisms that can be viewed as atomic principles. The reason that such models were built is not only to reproduce the behavior of the realistic system, but moreover to discover how Jozef Stefan Institute, Dep. of Intelligent Systems, Ljubljana, Slovenia e-mail: [email protected], [email protected] 2 Baylor College of Medicine, Div. of Restorative Neurology and Human Neurobiology, Houston, Texas, USA e-mail: [email protected] 1

these mechanisms interact, how they are in uenced by a selection of input properties, and what the relationship is between properties. For example, a realistic computational neuronal model [8, 7] may describe the properties of sodium channels, but sole inspection of the model equations does not reveal how changes in the sodium channel activation voltage alters the sodium ion concentration dynamics, and moreover, how these dynamics in uence the conduction of the action potential. This example presents a need for properties intermediate to input and output properties and a need for the relational structure that would outline the possible dependencies among the properties. By intermediate properties we are referring to those that are not explicitly present within the model, but are considered to be useful when reasoning about model behavior. One might assume the existence of intermediate properties from medical science or can solely hypothesize it. The realistic model is precise but complex and non-transparent. It requires substantial time and computational resources for simulation. The main idea of this paper is to couple a realistic model with a corresponding qualitative model that is substantially less complex. The qualitative model's main feature is transparency, which is achieved through its hierarchical structure and qualitative rules that encode the principles which interrelate the properties. Although the qualitative model may be used for approximation of the realistic model in quantitative terms, it is much less precise. The qualitative model can be constructed manually using the existing knowledge of physiology of the system being modeled. It can also be derived from examples of behavior of the realistic model by employing machine learning techniques. The paper is organized as follows. The next section presents several frameworks that use a structured approach to reasoning and data analysis. The method proposed in this paper uses crisp and interval logic for model interpretation and is outlined in Sec. 3. Sec. 4 gives an alternative method for model interpretation that uses fuzzy logic. An example of using these methods to derive the principles for the model from the domain of computational neuroscience is given in Sec. 5. Sec. 6 concludes the paper.

2 Related Work The approach presented in this paper is based on a qualitative multi-attribute decision making method DECMAK [2]. DECMAK uses decomposition of a decision problem into smaller, less complex problems. The representational structure of the

problem is a tree of nite-valued qualitative attributes. The tree hierarchically orders the attributes from inputs (tree leaves) to a single output (the root of the tree). DECMAK uses utility functions in a form of rules that de ne the aggregation of lower level attributes into the corresponding higher level attribute. DECMAK assumes that the structure and the utility functions are to be given by an expert. Both are then used to evaluate the options which are given as a set of input attributes. The overall utility of the option (output attribute) is then derived in a bottom-up fashion from the input attributes. DECMAK provides an extensive support for option evaluation, comparison and ranking. Its successor DEX [3] has been used in over 50 complex decision-making problems. A similar tree-based hierarchical approach called \signature table schema" is presented by Biermann et al. [1]. The problem representation is again a tree of attributes, this time restricting the attributes to two-valued (boolean) domains. An aggregation of lower-level to higher-level attributes is done through the use of signature tables, which are equivalent to the rule representation of DECMAK, where a single table entry represents a single rule. In contrast with DECMAK, the signature tables were not elicited from the expert but rather derived from the example inputs and outputs of a function the system had to represent. Similar representation techniques to those used in DECMAK and signature table schemas are used to represent functions of multivalued variables in [11]. Both Biermann et al. [1] and Perkowski [11] present methods that derive structures based on automatic function decomposition. Although the complexity of such decomposition process is high and good heuristics only partially known, this seems a very promising approach. DECMAK, signature table schema, and function decomposition in [11] deal with qualitative attributes only. Within machine learning, there has been substantial eort to still use qualitative (or discrete) models but to be able to predict quantitative results. First, quantitative input properties have to be converted to qualitative ones. Dougherty et al. [4] gives a survey of discretization methods. Then, a qualitative model has to be used to nd a qualitative value of output properties. Finally, these output qualitative properties are then transformed to quantitative ones. A well known approach uses a combination of model-based and instance based reasoning [13, 14]. Recently, methods that use fuzzy logic have been used for approximation and reasoning in domains with quantitative data. For example, Machado and de Rocha [10] describe a hybrid approach that combines both hierarchical structure and fuzzy reasoning. Similarly, as in DECMAK, attributes can be fuzzy variables and can be described by descriptors and corresponding degrees of belief. Dierent from DECMAK and signature table schema, the in uences of attributes are not encoded as rules but rather as neural networks. Thus, each non-input attribute requires a single neural network, where descriptors of that attribute are output nodes of the network, and descriptors of the lower-level attributes are input nodes. The weights of the connections in neural network are derived from a set of examples of input and output attributes using a punishment and reward algorithm. Common to all of the methods mentioned above is the attempt not only to approximate the decision process or some function being modeled, but to reveal its hierarchical components and their interrelation. The major dierences between

DECMAK and signature tables schema and the approach by Machado and de Rocha [10] is in the complexity and expressional power of utility functions. We might observe that the approach that uses neural networks might be more accurate in predicting the value of output attribute, while within DECMAK and signature tables it might be easier to explain why such a value was derived. This accuracy{simplicity tradeo and the fact that our interest is more in the transparency of the derivation process than in its accuracy in uenced the basic structure and the derivation methods of the framework presented which are in fact derived from the ones used in DECMAK. As for the signature table schema, we base our approach on learning the rules from examples of input and corresponding output properties. The method employed also uses rule table decomposition, and a genetic algorithm approach to either improve discretization or to nd appropriate fuzzy membership functions.

3 Method The method is based on a structure that hierarchically orders the properties and depicts their possible dependencies. The values of properties are derived from the properties they directly depend on. Here we present a crisp derivation method that uses qualitative rules, which can be either explicitly given or derived through the use of machine learning. For learning to be applied it assumes that a set of examples with input properties and corresponding output properties are obtained through simulation of the realistic model. Essential components of the method are given in Fig. 1. numerical solution of ODE or PDE -

input properties (quantitative)

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realistic computational model

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output properties (quantitative) 6

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discretization ?

input properties (qualitative)

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learning examples interval machine logic ......... learning interpretation 

qualitative model

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output properties (qualitative)

-

qualitative simulation

Quantitative realistic computational model and its corresponding qualitative model. Qualitative model may be derived from examples of behavior of the realistic model (learning) and can be used to approximate the behavior of the realistic model.

Figure 1.

Qualitative model and its interpretation Qualitative input and output properties are an abstraction of corresponding quantitative ones used in the realistic model. Besides these, the qualitative model also includes intermediate properties, which may depend either on input or other

intermediate properties. The input, intermediate, and output properties and their dependencies constitute a hierarchical property structure which is directed and required to be acyclic. Within the property structure, properties are represented as nodes: input properties as leaves, intermediate as internal nodes, and output properties as root nodes of the structure. Vertices in the property structure indicate property dependence. An example of property structure is given in Fig. 2.a.

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x1

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lo hi lo hi

lo med med hi

(b)

x3

x4

x5

lo lo lo hi hi hi

lo med hi lo med hi

lo med med med med hi

(c)

Figure 2. An example property structure (a) for an output property x5 , input properties x1 , x2 , and x3 , and intermediate property x4 . The qualitative descriptors for each of the properties are flo; hig for x1 , x2 , and x3 and flo;med; hig for x4 and x5 . x5 depends on x3 and x4 and x4 depends on x1 and x2 . The dependencies (principles) are expressed through rules given for each non-leaf node of the structure (b and c).

Principles are expressed through qualitative rules. Each property has a corresponding nite set of descriptors. For example, for properties used in Fig. 2, descriptors are given in the caption of the gure. For each non-leaf node in the property structure, a set of rules is given that relates the values of properties that are immediate descendants in property structure of the node to the values of the node itself. Figs. 2.b and 2.c illustrate such rules for properties x4 and x5 , respectively. These rule sets are complete, i.e., for any combination of qualitative values of x1 and x2 a value for x4 is known. This may in general not be the case and interpolation algorithms might be used to approximate unknown rules [3]. We use a crisp qualitative interpretation of the proposed model. For this, each input property can take only one single value from its set of descriptors. Then, an output property is derived using bottom-up derivation process using given qualitative rules. For example, if x1 =lo, x2 =hi, and x3 =lo are given and a model from Fig. 2 is used, than x4 = x4 (x1 ; x2 ) is derived as med and x5 = x5 (x3 ; x4 ) is derived as med. Quantitative interpretation uses qualitative interpretation at its core, but additionally requires a derivation of qualitative input properties and quantitative output properties (see Fig. 1). The quantitative interpretation is used to approximate the behavior of the realistic model and predict its derivation of output properties.

Quantitative interpretation is based on interval logic. For each input and output property, its quantitative domain is divided into a nite set of disjunct intervals, each interval representing a distinct qualitative descriptor. Each quantitative value from a property's domain maps to one and only one interval. The exact position of quantitative value within the interval is given by an interval coecient. If q is a quantitative value for some property and it belongs to the interval (l; h), than the interval coecient k is determined as k = (q ? l )=(h ? l ). By de nition, interval coecients have values from 0 to 1. Using the corresponding intervals, the qualitative values of input properties are rst derived from their quantitative values. Next, the qualitative value of the output property is determined through crisp derivation. This also determines which interval the output value belongs to. An exact position within this interval, i.e., its quantitative value, is then determined using its interval coecient which is an average of interval coef cients of input properties. I.e., if ki are interval coecients of n input properties, then the interval coecient of output P property is ko = i ki =n. If an output property's quantitative value is determined to be within the interval (l; h), then this value is q = l + ko (h ? l). The use of interval logic as de ned above uses a qualitative model as a piecewise quantitative approximator. This, besides the simplicity, was also the main reason why interval logic was chosen to be used for quantitative interpretation. We illustrate the use of interval logic with an example. Suppose x1 ; x2 ; x3 ; x5 2 [0; 1] and the intervals are given in Fig. 3. Suppose we have given x1 = 0:25, x2 = 0:8, and x3 = 0:7. Then, x1 =lo with interval coecient k1 = (0:25 ? 0)=(0:5 ? 0) = 0:5, x2 =hi with k2 = (0:8 ? 0:5)=(1 ? 0:5) = 0:6, and x3 =hi with k3 = (0:7 ? 0:5)=(1 ? 0:5) = 0:4. For these and using the model in Fig. 2, we derive x5 = x5 (x3 ; x4 (x1 ; x2 )) = x5 (hi;med) =med. k5 is an average of k1 , k2 , and k3 , i.e., k5 = (0:5 + 0:6 + 0:4)=3 = 0:5. Finally, the quantitative value of x5 is 0:5 + k5 (0:75 ? 0:5) = 0:625. x1 0 x2 x3 0 x5 Figure 3.

lo

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1

Intervals for input properties x1 , x2 , x3 and output property x5 .

Derivation of qualitative model from quantitative examples

Given  input and output property names, their descriptors, names of intermediate properties, and property structure,  intervals that map every descriptor of input/output property to a range of quantitative values, and  examples of behavior of the realistic model in the form of tuples (quantitative value of input properties, corresponding quantitative value of output property), the task of learning is to derive

 the rules for every internal node of property structure, and  the descriptors of intermediate properties. The learning takes place in two steps: Step 1. Derivation of qualitative table T of tuples (qualitative value of input properties, corresponding qualitative value of output property) from quantitative learning examples. Step 2. Derivation of the rules of the property structure and the descriptors of intermediate properties from qualitative table T using decomposition algorithm. In Step 1, for every possible combination of qualitative values of input properties, the learning method determines a corresponding output property value. Each tuple of qualitative values of input properties determines the tuple of quantitative intervals. The learning examples such that their input properties' values map to such tuple of intervals are selected. Next, the selected learning examples are checked to which interval their output property maps and the most frequent mapping, i.e., the most frequent interval (qualitative value) of output property for given input intervals, is found. This is also the output property's value that corresponds to a given set of qualitative values of input properties. Step 2 of learning algorithm maps the qualitative table T to the property structure. When a property structure has no intermediate properties, the qualitative table found in Step 1 is already the rule table needed to derive the output property. If any intermediate properties exist, than the mapping of table T to the rules of property structure is done through a series of decomposition steps. Each such step takes a rule table f that maps any value of a set of properties I to the output property o, i.e., o = f (I ), and decomposes this table to two rule tables g and h, so that o = f (I ) = g(I1 ; h(I2 )), where I1 and I2 are subsets of properties in I . For rule table decomposition, we have adopted the techniques described in [1, 11]. Here we describe the decomposition algorithm through an example. Suppose that Step 1 founds a qualitative table for x1 , x2 , x3 , and x5 to be: x1

x2

x3

x5

lo lo lo lo hi hi hi hi

lo lo hi hi lo lo hi hi

lo hi lo hi lo hi lo hi

lo med med med med med med hi

and we need to map it into rules for the structure in Fig. 2.a. This is done through the decomposition of function x5 = x5 (x1 ; x2 ; x3 ) to a function x5 = x5 (x4 (x1 ; x2 ); x3 ). First, a partition table is derived from a table given above: (x1 ; x2 ) x3

lo hi

(lo,lo) lo med lo

(lo,hi) med med med

(hi,lo) med med med

(hi,hi) med hi hi

label Each column in the partition table expresses a behavior of combination (x1 ; x2 ) with respect to x5 . Identical columns

represent same behavior. We can thus label the columns so that identical columns are labeled with the same label. These labels are actually descriptors of x4 we have found, and from these we can derive rules for x4 = x4 (x1 ; x2 ) and x5 = x5 (x3 ; x4 ). In our case, the partition table above maps to rules for x4 and x5 that are exactly those given in Fig. 2.

Derivation of intervals

The above learning algorithm assumes to be given intervals that map every qualitative value of input/output properties to the range of quantitative values. Intervals might be directly derived from learning examples (for a survey of discretization methods see [4]) or can be given explicitly. The selection of intervals in uences the error the model makes when predicting output properties from their corresponding input properties. To minimize this error, we propose to couple the learning algorithm with the genetic algorithm. Here, the chromosomes encode the intervals' bounds, and the tness of each chromosome (each set of intervals) is determined by using the learning algorithm and then estimating the estimation error the qualitative model would make. As proposed in [16] we de ne this error as: norm =

e

X

i





MAX(oi ; oi ) ? 1 MIN(oi; oi) 0



0

where oi denotes a quantitative value of output property o for -th learning example as derived using realistic computational model, and oi a value estimated by the use of qualitative model. The method uses a standard genetic algorithm with the mutation and crossover operators as provided in PGApack genetic algorithm library [9]. i

0

Utilization of qualitative model

Some of the dierent ways for utilization of qualitative models as presented in this paper are:

 The discovery of principles: the principles that relate the

properties can be revealed by the inspection of qualitative rules. The rules can either be viewed textually or graphically, or can be transformed to a decision tree. All of these and other rule inspection methods have been introduced in DEX and DECMAK [3, 2].  Structure exploration: once the rules for speci c structure are known, this structure can be changed to a dierent one by removal of existing and/or by introduction of new intermediate properties. For example, structure from Fig. 2 can be changed from x5 = x5 (x3 ; x4 (x1 ; x2 )) to x5 = x5 (x6 (x3 ; x1 ); x2 ) and then the new relationships (principles) relating x3 and x1 to x6 and x6 and x2 to x5 might be investigated.  Testing and discovery of the constraints that may exist between input and output parameters (hypothesis testing). The constraints that we are testing for are simple monotonicity and local minima/maxima constraints of types de ned in [5]. The structured approach reduces the complexity of constraint testing. For example, to test for the speci c constraint between an input property xi and an output property xo , it is only necessary to inspect the rules that apply for properties that are on the path from xi to xo .

 Explanation of the derivation of output properties for a

given set of input properties through the use of intermediate properties and the rules that were used to derive them.  The estimation of in uence of the parameters to a certain output parameter using the known methods that derive informativity of parameters from rules.  Analysis of the dierences of two (or more) input parameter sets that resulted in dierent output sets.  Quantitative prediction: given a set of input properties' values, these are mapped to qualitative descriptions and then propagated through the property structure up to the output properties. Such predictions can be orders of magnitude faster than the ones that require simulation.

Implementation The system described in this section was implemented in a C programming language under HP and SGI UNIX workstations. The system uses a PGApack genetic algorithm library [9] and employs Geomview 3D visualization tool [12] for graphical display of rules. Graphic interface was implemented using XForms library [15].

5 Example

This section illustrates the use of the proposed method and its capability to derive the principles that de ne the relationships among properties. In this example, a realistic nerve ber model is used. The sophisticated distributed parameter model was developed at Baylor College of Medicine in Houston and is used to predict the functional implications of neuronal structural and biophysical properties [6]. The model is given in the form of a system of multiple cross-coupled parabolic partial dierential equations that are solved by an implicit numerical integration method. The problem addressed in this paper is to observe the in uences of sodium ionic channel properties (voltage activation av and inactivation iv) and the in uence of sodium permeability (naperm) to the peak sodium current (ina) in a node segment of the neuron. Fig. 4 shows the property structure used.

4 A Fuzzy Approach to Model Interpretation Fuzzy logic can be used instead of crisp interpretation of the qualitative model and interval logic for estimation of quantitative values. For fuzzy interpretation of qualitative model, a degree of belief for each property's descriptor is given (input properties) or computed (intermediate and output properties). Derivation of intermediate and output properties then uses known fuzzy reasoning. The method has been described in [2], and here we give only an example. Suppose x2 =lo/0.5,hi/0.5 (degree of belief for x2 to be lo is 0.5 and to be hi is 0.5), x1 =hi/1 (implying the degree for x1 to be lo is 0) and x3 =lo/0.4, hi/0.6. Then x4 =med/0.5,hi/0.5 (using third and fourth rule in rule table for x4 ) and x5 =med/0.72, hi/0.28 (using second and third, and fth and sixth rule in rule table for x5 ). MIN is used as a fuzzy conjunctive operator. For fuzzy quantitative interpretation of the model, the discretization and interval logic interpretation are replaced with the fuzzi cation and defuzzi cation processes. Both these require membership functions to be given for each distinct qualitative value of input and output properties. Given a set of quantitative input properties, fuzzi cation nds their fuzzy qualitative representation. Next, fuzzy qualitative values of output properties are derived from fuzzy qualitative values of input properties through use of fuzzy qualitative interpretation as described above. Finally, defuzzi cation is used to map fuzzy qualitative output property to its quantitative representation. The fuzzy logic approach to qualitative model interpretation is described in more detail by Zupan et al. [16]. There, we also outline a learning method based on a genetic algorithm that, given quantitative learning example, learns rules and membership functions for a given property structure.

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Property structure for the peak sodium current (output property) as a function of sodium permeability, inactivation and inactivation voltage (input properties).

Figure 4.

The problem is now to discover how iv and av in uence nach, and how naperm and nach in uence ina. To derive these principles, 100 examples of the form (naperm, iv, av) were

presented to the system. Input properties were expressed as the osets to corresponding parameters of normal mammalian myelinated nerve ber (for the values and units see [7]). For each example they were selected arbitrary from ranges [0.5,1.5] for naperm, [-10,10] for av, and [-20,20] for iv. The experimental results show that there is high non-linearity because of two dierent states of the neuron: ring and non- ring. High negative values of ina appear when the neuron red and the values close to 0 are in the cases when the neuron did not re. We have rst used interval logic interpretation and derived corresponding intervals with the use of genetic algorithm. The intervals derived from the data are shown in Fig. 5. The rules for nach and ina found were:

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av lo med hi lo med hi lo med hi

0.5 -10

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ina

0 lo 549

Figure 5.

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av lo med hi lo med hi lo med hi

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Notice that the system discovered and used four descriptors for naperm, which we have named xlo, lo, med, and hi. The approximation error depended slightly on the setting of parameters for genetic algorithm and was usually around 20 to 30. Examples with ina close to 0 contributed the most to this error. We have also used fuzzy logic interpretation and learned membership functions and rules for this example. In this case, we had to de ne the descriptors for nach in advance. We have used nach2 flo,med,hig. The approximation error depended on the parameters of genetic algorithm and was usually around 10. The derivation process and results are described in more detail in [16]. Here we give just the rules learned: iv lo lo lo med med med hi hi hi

match with a known physiological eect of activation and inactivation voltage on the peak sodium current [6].

ina lo med med lo med hi med med hi

Although the results of the two dierent types of learning were dierent, the basic principles learned were similar. In both cases ina was found to monotonically increase with nach, and, importantly and to no surprise, nach increases with iv and decreases with av (see Fig. 6). Because of the monotonical dependence of ina from nach, ina increases with iv and decreases with av. This discovered principles qualitatively

Rules for derivation of nach from iv and av obtained by learning with fuzzy interpretation of the qualitative model.

Figure 6.

6 Summary

We have presented a method that is used to assist in discovery of the principles of physiological systems. The method uses a structure of properties. Additional information used for automatic derivation of principles is supplied in the form of the examples that map input properties to output and are, in the case of realistic models, derived through simulation. We have proposed two methods that can use a qualitative model for prediction of quantitative values. We expect that due to higher complexity, fuzzy interpretation might be more precise than the one with interval logic, but will also require more time for learning. Interval logic interpretation is less complex, however, we illustrated that both methods can potentially discover the same principles. In both fuzzy and interval logic case the approximation errors were rather high. These can be circumvented by increasing the number of descriptors. This would decrease approximation error but would also make the model less transparent due to the high number of rules learned. But, most importantly, the experiments with higher number of descriptors resulted in the discovery of the same principles found in less complex models. Namely, in all cases, the principles expressed the same in uence of activation and inactivation voltages to peak sodium current. Although the outlined approach is targeted for computer assisted reasoning with a general class of realistic models, the idea for its development and the rst tests were done on a speci c model. This realistic nerve ber model was developed at Baylor College of Medicine in Houston and is used to predict the functional implications of neuronal structural and

biophysical properties. The idea we are pursuing is to embed both realistic model and methods presented in this paper in a framework that would be eciently used in the exploration of properties and principles in neurobiology as well as in other physiological systems.

ACKNOWLEDGEMENTS

Support for this eort was generously provided by the Ministry of Science and Technology of the Republic of Slovenia (BZ and MB), the W.M. Keck Center for Computational Biology, the Whitaker Foundation (JAH), and Developmental Support from Information Technology Program at Baylor College of Medicine (JAH and BZ).

REFERENCES

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