Definition 1: Relative dosage. Standardized herb dosage that compares the actual dosage with its normal dosage range. Relative dosage is a standardized value ranging (0, 1]. Normally the larger the actual dosage of an herb, the larger the relative dosage prescribed related to patients’ symptoms and clinician’s medication experience. Considering that the negative exponential function features non-linear, continuous, monotone decreasing and asymptotic, the calculation is defined as

(1) Y(x)=1−e−λxβ,

where x and Y(x) are the actual dosage and the standardized dosage, respectively. Y(x) indicates that the efficacy of an herb is enhanced with the increase of herb dosage to 1 as the standardized maximum value. λ is chosen to define a range that Y(x) increases rapidly with x, and the increase becomes less significant when x goes further beyond the particular range. The parameters are chosen as follows, β = 2 and

(2) λ=ln2/[(a+b)/2]2,

where [a, b] is the routine dosage range of an herb [15]. According to the original dosage, the unit Liang is converted to 15.625 g [16], and other units are scaled with literature research and measurements [17].

Definition 2: Relative interaction intensity. The relative dosage of a particular herb to the sum of relative dosage of all herbs in a prescription. Relative interaction intensity describes the relative dosage of an herb in a prescription in proportion to all the other relative dosages, and its range of value is (0, 1]. For a prescription with n number of herbs, and the relative dosage of an herb is denoted as x_j, so the relative interaction intensity of i^th herb is

(3) xi/∑j=1nxj,i∈(1,n)

Based on the calculation of herb dosage in each formula according to Def. 1, taking Cassia Twig Decoction as an example, according to the original record it is composed of cassia twig 46.875 g, debark peony root 46.875 g, fresh ginger 46.875 g, Chinese date 12 g, and licorice 31.25 g. Based on research of dosage in Treatise on Exogenous Febrile and Miscellaneous Diseases, the routine dosage ranges of cassia twig and debark peony root are [3.9 g, 78.125 g], [3.9 g, 93.75 g], respectively, and their relative dosage in this prescription are 0.5956 and 0.4721, respectively, and their relative action intensities are 0.2541 and 0.2014, respectively. As the relative interaction intensity of an herb depends on the dosage of itself and of other herbs used in combination, the relative interaction intensity varies in different prescriptions even if the herb dosage itself stays the same. For example, although the dosage of cassia twig in Xiaoqinglong decoction is the same as that in cassia twig decoction, its relative action intensity in Xiaoqinglong decoction is 0.1276.

Definition 3: Intra-layer Network. Intra-layer networks include herb-efficacy, prescription-herb, prescription-function networks. PH network demonstrates a set of herbs and their associations based on the composition of prescriptions under analysis. Prescription-herb network is defined as

(4) NPH = <Y,Ry>whereY={y1,y2,…,yn} and Ry={ri,j}∀i,j∈n,i≠j,

where Y is a set of nodes representing herbs appearing in all prescriptions, and R_y for herb-to-herb relationships. r_i,j = <n₁, φ_m> is in the form of a tuple representing the relationship between herb y_i and herb y_j where n₁ is the total number of co-existence of y_i and y_j in all prescriptions under analysis, and φ_m includes a set of prescriptions where they co-exist and the corresponding original and relative dosage of herb y_i and herb y_j.

Prescription function network demonstrates associations between functions according to their co-occurrences in prescriptions. It is represented as

(5) NPH = <G,Rg>,where G={g1,g2,…,gn} and Rg={gi,j}∀i,j∈n,i≠j,

where G is a set of functions, and R_g is a set of relationships between them where g_i,j includes the number of co-occurrences and a set of prescriptions where they co-exist. Likewise, herbal efficacy network shows associations between efficacies defined as N_HE = <E, R_x> where E is a set of efficacies, and R_x represents their relationships between them including co-occurrences from herbs in the scope of the analysis.

Definition 4: Interlayer Network. PH-HE interlayer network represents the relationships between and prescription-herb and herb-efficacy network. PH-HE interlayer network is represented as

(6) NPH−HE =<Y,E,Ry−x>,

where Y and E are a set of herbs and a set of herbal efficacies respectively, and matrix R_y−x represents the relationships between Y and E. If the herb y_i from Y has efficacy x_i from E, the corresponding element in R_y−x is set as 1 denoting there is a weightless edge between them.

PF-PH interlayer network presents relationships of key herbs in prescriptions and functions of the prescriptions where they are in. It is represented as

(7) NPF−PH =<Y,G,Ry−g>,

where Y and G are a set of herbs and a set of prescription functions, respectively. Herbs in each prescription are sorted according to their relative interaction intensities, and main herbs accounting for more than 60% cumulative relative interaction intensities are linked to establish associations with the functions of corresponding prescription. R_y−g stores these herb-function connections in the matrix, each element in the form of a tuple as R_y−g = <n, φ_m> where n is the total number of co-occurrences of particular herb y and a particular function g taking account of all prescriptions, and φ_m is the set of prescriptions where they co-exist.

Prescription-Herb-Function Network (PHFN) is a multidimensional heterogeneous network integrating herb-efficacy, prescription-herb, prescription-function networks and their interlayer networks.

In order to realize elastic and scalable massive prescription analysis, the prescription data storage and analysis system adopts a distributed database and computing framework, which can be combined with a variety of analysis tools to realize interactive or batch data analysis, machine learning model construction, and graph analysis. As the association between nodes is sparse, in order to optimize the storage efficiency and ensure that the data can be queried interactively, the Cassandra database is adopted as shown in Fig. 1.

112 prescriptions in Treatise on Exogenous Febrile and Miscellaneous Diseases are collected to illustrate a multidimensional heterogeneous PHFN network, and part of data in networks are shown in Tables 6 to 7. The following clinical terminologies of TCM in this paper are referred to Chinese-English Dictionary of State Standard Clinical Terminologies of Traditional Chinese Medicine [18]. To initialize association-relation in networks, in preprocessing a total of 89 kinds of medicinal materials were obtained, including cinnamon twig, peony, roasted licorice, ginger, jujube, pueraria, magnolia officinalis, almond, processed aconite, etc. 146 functions of prescriptions are obtained after standardization, including relieving muscles, dispelling wind, harmonizing Ying and health, engender fluid, relaxing meridians, relieving exterior, dispersing lung, relieving asthma, supporting yang, etc. Cassia twig and peony root is a common combination, and their relative dosage in different prescriptions is shown in Table 6. They are used in comparable dosage in Guizhi (cassia twig) Decoction to promote sweating and relieve muscles, but varies when used for warming and tonifying deficiency in No. 35 prescription Xiaojianzhong Tang. It can be inferred that the purpose of treatment has a great influence on the usage and dosage of prescriptions.

Table 7 shows that the function of promoting sweating to relieve muscles co-occurs with dispelling wind four times in Nos. 0, 1, 4 and 5 prescriptions, in names are Guizhi (cassia twig) Decoction, Guizhi (cassia twig) plus Gegen (pueraria lobata) decoction, Guizhi (cassia twig) decoction without Shaoyao (peony root), Guizhi (cassia twig) and Fuzi (processed aconite root) Decoction without Shaoyao (peony root) respectively. Cassia Twig Decoction (Guizhi Tang) is a classical prescription with functions of harmonizing defensive and nutritive levels of the body, and expelling pathogens from the exterior. It is composed of cassia twig (Guizhi), debark peony root (Baishao), fresh ginger (Shengjiang), Chinese date (Dazao), and licorice (Gancao). Cassia twig is warm and acrid treating the main symptoms, expelling pathogenic wind-cold from muscles and skin. Debark peony root is slightly cold, sour, and sweet, which strengthens the interior, both of them in combination serve to harmonize the body. Licorice harmonizes the actions of the other herbs. Research shows that different proportion of herb pairs produce different material basis [15].

Table 8 shows the associations-relations of main herbs and functions of prescriptions. The main herbs are collected by their cumulative relative interaction intensities of more than a threshold after sorting them in descending order. The threshold is determined by experiment to ensure that the usage of extracted main herbs cover main effects of prescriptions even if they are not matched by herbs with highest value. In the case of Xiaochaihu (Bupleurum) Decoction, the foremost herb bupleurum with relative interaction intensity 0.189 is ranked after pinellia rhizome 0.208 and scutellaria root 0.197. According to the overall analysis of the values of relative interaction intensities, we found that a threshold in the range between 0.6 and 0.8 is appropriate. For example, the relative interaction intensities of cassia twig, licorice and peony in Guizhi (cassia twig) Decoction were 0.263, 0.255 and 0.171, respectively, and that of ephedra and cassia twig in prescription No. 7 Mahuang (Ephedra) Decoction were 0.506 and 0.289, respectively. In order to ensure the accuracy of correlation between herbs and prescription effects without much data noise, the threshold is set at 0.6. In Table 8, the effects of promoting sweating and muscle relieving appear 5 times in Nos. 0, 1, 2, 4, 5 prescriptions. Table 9 lists part of herb efficacy associations on the cases of ephedra, gypsum and rhubarb.

Tables 6 to 8 show that the scale of intra and inter layer networks are related to the number of nodes in different categories, and the number of herbs, efficacies and functions for a particular prescription is limited compared to the total number of nodes in the category. For the PF network, the values of the relative action intensities related to dosage and the number of herbs in prescriptions determine the scale of the herb nodes to be associated with functions. The average number of herbs in all prescriptions is 4.83, and that contributes to prescription functions is 2.6. These herbs are connected to 2 to 4 function nodes, both much smaller than the scale of corresponding intra-layer networks which results in sparse correlations. Similarly, the associations of inter-layer networks are also sparse as the nodes are only connected to limited nodes having similar or medical-related functions.

To compare with other herb dosage standards, we compare our results with a rule that standardizes each herb dosage by d_i/(d_max+d_min ), where di is the actual dosage of herb d_i in a prescription, d_max is its maximum usual dosage, and d_min is the minimum usual dosage [19]. The relative interaction intensities are then calculated and sorted based on this standardized herb dosage, and main herbs are then extracted with the same threshold of 0.6. The comparison is that in 20 prescriptions the main herbs by this method have one more herb compared to our methods. The main herbs screened are very similar under the same screening conditions of the cumulative sum of relative action intensity, but the herbs selected by our model are stricter.

PHFN model demonstrates rich information of prescription dataset. Herb combinations are more various than function or efficacy combinations. As networks are sparse with most of prescription functions or herbal efficacies taken only once or twice, for illustration herb combinations for more than three times in prescriptions are shown in Fig. 2. In a variety of herbs as shown in Fig. 2, most frequent herb combinations are between cassia twig, licorice root, fresh ginger, peony root and jujube. They are also more frequent as main herbs in connection with prescription functions. Licorice is a common herb believed to be used in various kinds of syndromes, often with fresh ginger, Chinese date, and ginseng. The combination of cassia twig and licorice is a famous herb pair with warm nature to enhance the function of warming heart and activating blood circulation [20]. Based on herbs in Fig. 2, their associations in PF-PH Interlayer network are shown in Fig. 3.

The significant associations in Fig. 3 are to name a few, the node of releasing the exterior with licorice root, jujube, cassia twig and ephedra, the node of reviving yang with aconite root, the node of harmonizing the stomach with licorice root and pinellia rhizome, the node of sweating and relieving muscles with cassia twig, etc. This provides reference on the selection of herbs according to the efficacy combination for prescriptions. The fact that a prescription is prescribed with a selection of herbs with specific dosage in composition gives more information than the combination itself, so it is important to distinguish the dosage and the effect to understand their overall function and compatibility. However, most dosage prescribed are based on experience. We introduce relative dosage to standardize dosage in uniform range of (0,1], and also based on the value of relative dosage we introduced relative interaction intensity to distinguish the importance of herbs. Further analysis shows that the dosage of herbs is one of the important factors affecting the curative effects of herbs, while other factors including the medicinal properties and efficacies also play major roles for the overall function.

In our network, the TCM information components, such as prescription functions, herbs, and efficacies are represented as nodes, and their co-occurrences or contributions are represented as weighted edges. The associations can be analyzed with graph theory to extract their attribute information. It can be inferred that the sum of weights of all edges in PH network is equal to the sum of the total number of possible combinations of any two herbs appearing in the prescriptions. Furthermore, a function node from a prescription is at least connected with an herb node to form PH interlayer network. Though 112 classical prescriptions in Treatise on Exogenous Febrile and Miscellaneous Diseases are not huge in data volume, it is credible to visualize the associations at different layers. The network if containing more prescriptions data will partly serve as a database of CM classic medication with valuable associations in the metadata. It not only integrates the formula knowledge (e.g., herbs and dosage, herbal efficacy) to visualize the overall functions by networks, but also can help acquire the core herbs, therapeutic effects, and relationships from the large-scale medical data.

Most of the recent network modeling are based on network pharmacology to integrate drug-disease and traditional knowledge of herbal medicines to identify the rationale of herb combinations. Cheng et al. [21] introduced network distance between targets of a drug pair, including closest, separation, shortest, kernel and center to analyze the proximity measure of drug-drug relationships on a network-based modeling of drug combinations. Wang et al. [22] constructed a protein–protein interaction network for a given herb pair by retrieving the associated ingredients and protein targets, and determined ingredient distances with cross-level protein network-based distances based on measurements introduced by Cheng. Jafari et al. [23] constructed a bipartite network based on natural products and chemical ingredients for community detection. Cheng et al. considered only targets and their proximity to drug pairs, but not from the perspective of CM. Wang and Jafari considered the measurements of distances of CM natural products based on targets proximity, but not from the perspective of CM theory and its diagnosis and treatment approaches. Li et al. [24] introduced a distance-based mutual information model to determine herb interactions based on their frequencies and distance, but dosage information not retained. Compared to existing models, our model is in the theory of traditional Chinese medicine with integration of frequencies, combinations and curative effects of all herbs based on a full prescription level combined with dosage information among the herb combinations.

The study on selection of multiple herbs are important for understanding the enhanced and harmonized therapeutic effect. The network is an initial step to integrate herbs, prescriptions, and functions with potential extensions including domain knowledge, such as herb properties (e.g., herb nature and flavor, channel tropism), symptoms and diagnosis to establish a whole treatment framework. Huge volumes of literature and records of the theoretical concepts and practical skills provide data support for analysis of the relationships in prescription composition. Future research will include other network measures, such as correlation and feature analysis, clustering and community detection in weighted networks. We will also focus on the feature extraction and modeling to analyze a treatment network in depth integrating diagnostic information to analyze relationships between treatment and differentiation.