An Overview of the Application of Mathematical and Physical Models in the Study of the Properties and Compatibility Rules of Traditional Chinese Medicines

Abstract: Based on the theories of traditional Chinese medicine and mathematics, the construction and application of sound mathematical–physical models represent a new paradigm for research on the properties of Chinese medicinal herbs and the compatibility of herbal formulas, and constitute an important step in the inheritance and innovation of TCM. In recent years, methods such as network theory, data mining, and path modeling have been widely employed in studies of the properties of Chinese medicines and their formulaic combinations. By synthesizing and summarizing recent research on the properties of Chinese medicines and herbal formula compatibility that relies on mathematical–physical models, this review aims to provide a modern scientific framework for elucidating the profound theories underlying these concepts, to deepen the understanding of their scientific essence, and to offer research directions for the internationalization and modernization of TCM.

Employing modern scientific language—such as quantitative thinking and mathematical modeling—to represent and elucidate the scientific essence of TCM theory is one of the most effective approaches for its inheritance and innovation. ^{[ 5 ］} Mathematical and physical models are now widely applied in research areas such as the properties of traditional Chinese medicines, compatibility mechanisms, and dosage. ^{[ 6-7 ］} This paper synthesizes and summarizes recent research on the theories of Chinese medicinal properties and formula compatibility based on mathematical and physical models, with the aim of elucidating the profound theories of Chinese medicinal properties and formula compatibility in modern scientific terms, deepening the understanding of their scientific essence, and providing research directions for the internationalization and modernization of traditional Chinese medicine.

1 　Application of Mathematical and Physical Models in the Theoretical Research on the Properties of Traditional Chinese Medicines

1.1 　Four characteristics

In traditional Chinese medical theory, the “four natures” refer to a drug’s cold, hot, warm, and cool properties, which are defined in relation to the “cold” or “hot” nature of the disease being treated. ^{[ 8-9 ］} In clinical practice of Traditional Chinese Medicine, the “cold” or “heat” nature of a condition is often used as the basis for prescribing medications, as stated in the Shennong’s Classic of Materia Medica: “Medicines possess four natures—cold, hot, warm, and cool—and cold conditions are treated with hot medicines, while heat conditions are treated with cold medicines.” ^{[9 ］} The Treatise on the Spleen and Stomach also explicitly states: “Within a single substance, both flavor and qi are present; within a single herb, all its inherent properties are fully embodied—its primary therapeutic action arises from this very principle.” Therefore, when pathological changes occur in the body, TCM differential diagnosis must first determine whether the pattern is one of cold or heat in order to arrive at an accurate diagnosis. ^{[ 10 ］} 。

He Fuyuan et al. ^{[ 9 ］} Based on the First Law of Thermodynamics and Hess’s Law, a mathematical model of “state functions” was established and experimentally validated. Experimental rats were administered Coptis chinensis, Evodia rutaecarpa, and Glycyrrhiza uralensis; the enthalpy of the organism before drug administration and the enthalpy after drug elimination were recorded, and the difference between these values was calculated to determine the cold–hot nature of the drugs: a positive difference indicates a cold property, whereas a negative difference indicates a hot property. The results demonstrated that Coptis chinensis and Evodia rutaecarpa are classified as cold and hot medicines, respectively. These findings suggest that the “four natures” of traditional Chinese medicines can be reflected in the difference between the change in metabolic enthalpy following drug administration and the corresponding change in the organism’s enthalpy.

Yang Yantao et al. ^{[ 11 ］} A mathematical model of the “four natures” of traditional Chinese medicines under pathological conditions was established based on the first law of thermodynamics. The HR3000 digital calorimeter was used to measure the combustion enthalpies of metabolic products in the body before and after administration of modified Xiaochaihu Tang and a cold-pattern mouse model. The cold–hot nature of the drugs was then determined according to these combustion enthalpies: if the difference between the combined formulation of Xiaochaihu Tang and a neutral drug is greater than zero, the drug is classified as cold; otherwise, it is classified as hot. The results showed that the Qin–Chai group and the Qin–Chai–Gancao group are cold-natured drugs, while the Xiaochaihu Tang group, the Shen–Jiang–Xia–Zao group, the Shen–Jiang–Zao–Xia–Gancao group, and the Qin–Chai–Shen–Jiang–Zao–Xia group are hot-natured drugs, and the Gancao group is neutral. These findings are consistent with the clinical application principles of the “theory of the four natures” in traditional Chinese medicine. ^{[ 11 ］} The research results also indicate that the “heat of combustion” model can be used to characterize the “four natures” attributes of traditional Chinese medicines.

The “four natures” of traditional Chinese medicine are closely related to its content of trace elements and inorganic elements. ^{[ 8 , 12 ］} . Wu Wenli et al. ^{[ 8 ］} Four classes of Fisher discriminant equations for the “four natures” of traditional Chinese medicines were established based on 42 elements in 105 TCM formulations, and these equations were applied to explore the relationships between the “four natures” and elemental contents. This study not only quantitatively analyzed the cold–hot–warm–cool “four natures” of TCM but also examined the associations between these natures and the concentrations of chemical elements. The findings clearly demonstrate that trace elements are a key determinant of the cold–hot–warm–cool properties of TCM, with their levels reflecting, to a certain extent, the therapeutic characteristics of the herbs. These results provide valuable guidance for further research into the material basis of TCM.

Liu Jin et al. ^{[ 12 ］} Using seven inorganic elements—K, Ca, Mg, Mn, Fe, Cu, and Zn—as indicators, support vector machines combined with leave-one-out cross-validation were employed to classify the properties of 193 traditional Chinese medicines. The authors first investigated the effects of these seven elements on both neutral and non-neutral herbs; subsequently, they examined the influence of these elements on warming–heating herbs versus cooling–cold herbs. The study revealed that the concentrations of K, Ca, Mg, Mn, and Fe exert a substantial impact on distinguishing between neutral and non-neutral herbs, while the levels of Ca and Fe are particularly influential in identifying warming–heating herbs. These findings suggest that the content of inorganic elements is moderately correlated with the “four natures” classification of traditional Chinese medicines, and that this correlation can be leveraged to determine a herb’s cold–hot–warm–cool nature based on the specific concentrations of certain inorganic elements.

1.2 　Five Flavors

In the course of disease prevention and treatment, it has been observed that the “taste” of many medicinal substances is closely related to their therapeutic effects. Consequently, the “taste” is used to describe one or more functional characteristics of a drug, and this attribute is applied to guide rational clinical use of medications. ^{[ 13 ］} The “five tastes” of traditional Chinese medicine refer to pungent, sour, bitter, sweet, and salty; for most TCM herbs, these five tastes are determined through sensory evaluation by tasting. The conventional assessment method relies on experienced pharmacists who judge the tastes by tasting and smelling. However, this approach is highly subjective. Therefore, there is a need for objective, quantifiable methods to scientifically and quantitatively characterize the “five tastes” of TCM. ^{[ 14 ］} 。

Zhang Pei et al. ^{[ 15 ］} Based on modern pharmacological and clinical data for 197 traditional Chinese medicinal decoction pieces, a Bayesian network model of the “five tastes” in TCM was constructed. Using network topology diagrams and conditional probability tables, the pharmacological activities of bitter, pungent, and sweet tastes were predicted, elucidating the relationships between these tastes and relevant pharmacological indices. This approach enables the determination of the “five-taste” attributes of TCM herbs based on pharmacological and therapeutic efficacy indicators. The results demonstrate that the Bayesian network model not only effectively predicts the “five-taste” attributes of TCM herbs but also provides an intuitive representation of the complex relationships between the “five tastes” and their therapeutic effects.

Liu Yangqian ^{[ 14 ］} Holographic chemical fingerprint profiles were obtained for 40 traditional Chinese medicinal decoction pieces and 33 reference medicinal materials. The Euclidean distance–based similarity index between each decoction piece and its corresponding reference material was calculated, and this index was used to quantitatively analyze the “five flavors” of the herbs: the closer the Euclidean distance, the greater the similarity in flavor. Furthermore, multiple regression analysis was employed to establish quantitative indicators for the principal “five flavors” of traditional Chinese medicines, and a五行 (five-element) diagram depicting these “five flavors” was constructed. A predictive model capable of simultaneously providing qualitative and quantitative characterization of the “five flavors” was also developed. These findings offer new insights for further elucidating the chemical basis underlying the “five flavors” of traditional Chinese medicines and for refining the theoretical framework surrounding this concept.

1.3 　Channel Tropism

The theory of meridian tropism in traditional Chinese medicine is one of the core components of the pharmacological properties theory and an essential part of the theoretical system of TCM. ^{[ 16 ］} “Channel tropism” refers to the selective action of a medicinal substance on specific organs and meridians involved in pathological conditions. It is an objective principle for understanding the sites of drug action that has been distilled by generations of TCM practitioners through their long-standing clinical experience in combating disease. As the modernization and internationalization of TCM accelerate, the traditional system of channel tropism in Chinese herbal medicine still requires further refinement and expansion. ^{[ 17-19 ］} Moreover, its scientific essence urgently requires interpretation and representation using modern scientific language.

Li Fang et al. ^{[ 20 ］} A formula–meridian attribution coefficient model was established by quantifying the meridian tropism of herbal formulas and integrating the proportional contributions of each constituent herb. A total of 21 classic formulas were analyzed for their meridian attribution coefficients. Taking Mahuang Decoction as an example, the calculation revealed that it primarily targets the Lung Meridian, which is consistent with its therapeutic actions of inducing perspiration to dispel exterior pathogenic factors and dispersing the Lung to relieve wheezing. The developed model provides a methodological reference and lays a foundation for the modernization of research on the meridian tropism of traditional Chinese medicines.

Li Fang et al. ^{[ 21 ］} The analytic hierarchy process was employed to further investigate the meridian tropism intensity of herbal formulas. Based on the quantitative assessment of drug meridian tropism and relative dosages, a judgment matrix was constructed, and a weighting model for formula meridian tropism was developed. Using Sijunzi Decoction as a case study, it was found that ginseng exhibits the highest meridian tropism weight within the formula, and that Sijunzi Decoction primarily targets the Spleen meridian. These findings are consistent with the primary therapeutic indication of Sijunzi Decoction—spleen and stomach qi deficiency. The proposed model not only quantifies the meridian tropism intensity of a formula but also characterizes the position and significance of each herb within the formula’s structural composition. The results of this study are of great significance for exploring the underlying principles governing the clinical application and pharmacological actions of herbal formulas.

1.4 　Toxicity Study

In the traditional medical concept, “toxin” in a broad sense refers to the inherent bias or tendency of a medicinal substance; in a narrow sense, it denotes the adverse effects and harm that a drug exerts on the human body, namely drug toxicity and adverse reactions. ^{[ 22 ］} With the widespread use of traditional Chinese medicine, reports of adverse reactions caused by TCM have been steadily increasing both domestically and internationally, thereby drawing growing attention to the safety of TCM. ^{[ 23 ］} 。

Liu Hongjie et al. ^{[ 23 ］} Through literature retrieval, traditional Chinese medicines with and without nephrotoxicity were included in a database. A neural network model was then employed to predict and analyze the nephrotoxicity of these herbs, and correlation analyses were conducted between the predicted outcomes and intrinsic attributes such as “four natures,” “five flavors,” and “meridian tropism” to identify relevant variable factors. Based on these findings, a neural network model was constructed and validated using receiver operating characteristic curves. The results indicated that “hot nature” exerts a significantly stronger influence on the nephrotoxicity of traditional Chinese medicines than other factors, far surpassing their relative effects. The developed model not only provides a useful reference for research on the nephrotoxicity of traditional Chinese medicines but can also be applied to predict such toxicity.

Liu Hongjie et al. ^{[ 24 ］} A Logistic regression predictive model was further developed to analyze the “four natures,” “five flavors,” and “meridian tropism” of 111 nephrotoxic and 398 non-nephrotoxic traditional Chinese medicines, with validation performed using receiver operating characteristic curves. The study found that the toxicity of traditional Chinese medicines is associated with their “four natures” and “five flavors,” but not with their “meridian tropism.” Furthermore, “hot nature” and “bitter taste” were identified as risk factors for nephrotoxicity, whereas “neutral nature” and “sweet taste” were protective factors. These findings demonstrate that the established Logistic regression predictive model can effectively predict nephrotoxicity in traditional Chinese medicines, thereby laying a foundation for in-depth investigation into the mechanisms and scientific underpinnings of this phenomenon.

To investigate the dermal cytotoxicity of the volatile oil components of pungent Chinese medicinal herbs, Yang Wenguo et al. ^{[ 25 ］} A mathematical model was developed using nonlinear canonical correlation analysis to integrate the pharmacological properties of traditional Chinese medicines with the dermal cytotoxicity of their volatile oils. By examining the relationships among the “four natures,” “five flavors,” and “meridian tropism” of 33 pungent herbs and their cytotoxic effects, the study revealed that the pharmacological characteristics of TCM—particularly the “meridian tropism” attribute—significantly influence the dermal cytotoxicity of the volatile oil constituents in pungent herbs. This research and its findings lay a foundation for a deeper understanding of TCM cytotoxicity from the perspective of its inherent “attributes.”

2 　Application of Mathematical and Physical Models in Research on the Compatibility of Traditional Chinese Medicine Formulas

2.1 　Single mathematical model

2.1.1 　Rough Set Theory: In the study of traditional Chinese medicine compound formulas, rough set theory treats these formulas as a form of knowledge, applying sophisticated mathematical principles to mine and learn from them, thereby handling the large volumes of nonlinear data they contain. ^{[ 26 ］} Rough set theory, when applied to the study of the relative importance of individual herbs in a compound formula, focuses on the contribution of each herb and can further identify the syndrome patterns associated with the diseases for which that herb is indicated. Consequently, rough set theory offers a new perspective for investigating the principles governing the compatibility of traditional Chinese medicinal herbs.

Liu Juan et al. ^{[ 27 ］} Data mining was conducted on traditional Chinese medicine formulas for the treatment of hepatitis B using rough set theory, and the results indicated that qi-tonifying herbs play a crucial role in these formulations. The study further revealed that Dang Gui is an important herb for treating hepatitis B and is particularly effective in addressing the syndrome pattern of qi stagnation and blood stasis. Applying rough set theory to the study of herbal formula compatibility not only enables the identification of key herbs within a formula but also facilitates the differentiation of the predominant syndrome patterns associated with the primary disease treated by that formula. However, the application of rough set theory requires the integration of prior knowledge to perform attribute reduction on the medicinal properties.

2.1.2 　Network pharmacology: Component-based research strategies in traditional Chinese medicine, grounded in network pharmacology, effectively capture the rationality of component combinations and the characteristics of drug–drug interactions among active components, including synergy, additivity, and antagonism.

Yu Jin Gao ^{[ 28 ］} Animal experiments and cell-based models have revealed that aquaporin targets may be the primary molecular basis for the contraindicated interaction between licorice and veratrum. Furthermore, the authors employed network pharmacology and molecular docking simulations to validate the critical role of aquaporin targets in the contraindication associated with the licorice–veratrum combination. Network pharmacology constructs an interaction network linking herbal medicines, their bioactive constituents, and disease-related targets, thereby elucidating the relationships between the chemical constituents of a compound formula and its therapeutic targets and enabling the prediction of key active ingredients and their corresponding targets within the formulation. In addition, molecular docking simulations can serve as a complementary validation tool, corroborating findings from pharmacological assays. Consequently, the integration of network pharmacology with traditional Chinese medicine theory offers a novel research paradigm and perspective for studying the compatibility and synergistic interactions of multi-ingredient herbal formulas. However, the conclusions derived from network pharmacological analyses still require further confirmation through clinical studies and pharmacological experiments.

2.1.3 　Neural Network Models Neural networks offer unparalleled advantages in three areas of research related to the compatibility of traditional Chinese medicine formulas: prediction of pharmacological property characteristics, identification of herb pairs, and elucidation of synergistic relationships among formula components. Specifically, neural networks have a greater number of parameters than the independent variables used in statistical methods, and they can be applied regardless of the type of variables or whether the assumptions of normality and independence are met. ^{[ 29 ］} 。

Lu Xia ^{[ 30 ］} A neural network model for predicting the pharmacological properties of traditional Chinese medicine formulas for treating colds was developed using 50 such formulas, and the model was subsequently validated with an additional 10 cold formulas. The results demonstrated that the neural network model can accurately predict the therapeutic effects of cold formulas based on the basic properties of the constituent herbs, such as their nature, flavor, and meridian tropism. For example, the model predicted that Banlangen exerts its efficacy in “clearing heat and detoxifying, as well as moistening the lungs and relieving cough,” which is consistent with the known clinical effects of Banlangen. This study integrates the pharmacological properties of TCM with neural network mathematical modeling, providing valuable guidance for research on TCM formula compatibility. Neural networks are the optimal tool for handling nonlinear data, and their independent variables offer unparalleled advantages over those of other modeling approaches. The characteristics and capabilities of neural networks align well with the “multi-component, multi-target, multi-pathway” nature of TCM formulas. However, because the algorithms underlying neural networks cannot elucidate the factors influencing the dependent variable, it is impossible to verify the weight coefficients of the input variables. ^{[ 23 ］} , it still needs to be combined with other mathematical methods and models for more comprehensive and in-depth analysis.

2.1.4 　Bayesian networks use probability distributions to represent the strength of dependencies, making them one of the most effective methods for representing, reasoning about, and modeling uncertain knowledge. ^{[ 29 ］} 。

Wang Xiaoyan ^{[ 31 ］} A Bayesian network was constructed and applied to analyze the compatibility patterns between the cold–hot properties of traditional Chinese medicines and their pharmacological characteristic markers. Representative cold- and heat-inducing herbs recognized in TCM were selected, and their chemical constituents as well as pharmacodynamic indices measured in normal rats and in rats with deficiency–heat and deficiency–cold syndromes following administration were investigated to explore the relationships among cold–hot properties, pharmacological effects, and chemical composition. The results demonstrated that cold–hot TCM not only corrects pathological cold–heat imbalances in diseased organisms but also elicits normal cold–heat responses in healthy subjects; moreover, the chemical constituents of TCM with similar pharmacological effects exhibit commonalities in their compatibility patterns. The Bayesian network leverages existing knowledge of TCM to predict the complex, implicit relationships inherent in herbal combinations. By applying this mathematical model, it is possible to uncover the clinical value of classic TCM formulas and major TCM product categories, thereby guiding clinical practice, enhancing therapeutic efficacy, and laying the foundation for the development of new TCM drugs. However, its validity still requires verification through pharmacological experiments and clinical trials.

2.1.5 “Yin-Yang Sphere–Eight Principles Three-Level Structural System” Mathematical Model by Guo Shuqiang ^{[ 32 ］} Based on a comparative analysis and deeper understanding of the traditional Chinese medicine yin-yang Taiji diagram, we have constructed the “Yin-Yang Sphere” model and integrated it with the Eight Principles (cold, heat, deficiency, excess, exterior, interior, yin, and yang) from TCM theory, thereby establishing the “Yin-Yang Sphere–Eight Principles Three-Level Structural Mathematical Model.” At the same time, we applied this newly developed model to explore the mathematical principles embedded in the herbal compound formulas recorded in the Treatise on the Spleen and Stomach. First, we built a “Medicinal Power” database based on the “medicinal potency” of each formula, where “medicinal potency” refers to the magnitude of the therapeutic effect of a particular herb on its corresponding syndrome. Next, we imported the formulas selected from the Treatise on the Spleen and Stomach into the input module of the “Yin-Yang Sphere–Eight Principles Three-Level Structural Mathematical Model,” and used spatial positioning to examine the dynamic trajectories of herbs, formulas, and syndromes within a three-dimensional space. This study employs a mathematical framework to provide a clear and precise articulation of the traditional TCM principle of syndrome differentiation and individualized treatment. ^{[ 32 ］} 。

2.2 　Composite Mathematical Model

Although the analytical results obtained from single mathematical and physical models can help identify the key herbs in a traditional Chinese medicine formula, pinpoint the critical targets underlying compatibility contraindications, and elucidate the principles of combining cold- and heat-natured herbs, they are insufficient for quantitative characterization. Therefore, the integration of different mathematical and physical models or methodologies represents one of the most effective approaches for achieving quantitative analysis, thereby facilitating a deeper and more comprehensive elucidation of the scientific underpinnings of traditional Chinese medicine formula compatibility.

2.2.1 　Orthogonal Design Combined with Modeling, Zeng Yong et al. ^{[ 33 ］} By integrating orthogonal design with path analysis, an “orthogonal design–path analysis” model was established and used to elucidate the formulation principles of Mahuang Decoction. The specific analytical procedure involved first employing orthogonal design to conduct a factor-splitting experimental design, followed by applying the path-analysis model to examine the formulation composition underlying the diaphoretic effect of Mahuang Decoction. The results demonstrated that ephedra plays a decisive role in the decoction, with cinnamon twig ranking second, which is consistent with the formulation principle of “ephedra as the chief herb, cinnamon twig as the minister, apricot seed as the assistant, and licorice as the messenger.” These findings help reveal the synergistic and integrative effects of herbal combinations in traditional Chinese medicine and lay a foundation for further exploration of TCM theory. However, due to the substantial workload involved, orthogonal design is best suited for experiments with a limited number of levels.

2.2.2 　Uniform Design Combined with Modeling: Uniform design is an experimental design method that integrates number-theoretic techniques with multivariate statistical methods. It features a uniform distribution of experimental points and requires a relatively small number of trials, and has been applied to the study of formulation compatibility in traditional Chinese medicine compound prescriptions. ^{[ 34 ］} Regression analysis is the primary method for analyzing uniform design data. ^{[ 35 ］} 。

Path analysis is an ideal tool for studying the pharmacodynamic network systems of traditional Chinese medicine. ^{[ 36 ］} . Qiu and his delegation ^{[ 37 ］} Taking Mahuang Tang as the research subject, this study first employed uniform design to formulate various herbal combinations; subsequently, it investigated the sweat-inducing effects of these different combinations in SD rats and their relaxant effects on guinea pig tracheal smooth muscle, as well as the correlations among the pharmacological effects of the various combinations; finally, a path analysis model was applied to examine the interrelationships among the individual herbs in the formula and between each herb and the corresponding pharmacodynamic outcomes. The results confirmed the pivotal role of Ephedra as the chief herb in Mahuang Tang and demonstrated the feasibility of the path analysis approach in the study of traditional Chinese medicine formulas.

2.2.3 　Central composite design, compared with full-factorial experiments, requires fewer experimental runs while still providing as much information as possible. Central composite design is often used in combination with other methods; for example, when combined with response surface methodology, it can effectively enhance experimental precision and serves as an excellent analytical tool. ^{[ 38 ］} . Sun Fang ^{[ 39 ］} and Wu Lin ^{[ 40 ］} A “star-point design–response surface optimization” model was established and applied to optimize the combinatorial formulation of individual ginkgo leaf flavonoid compounds. Using this approach, cell-based experiments were conducted to investigate the antitumor and antioxidant activities of flavonoid monomer combinations under a “multi-factor, multi-level” experimental design. Subsequently, models were constructed using “three-dimensional response surfaces” and “two-dimensional contour plots,” followed by validation through additional cell experiments to identify the optimal formulation. The results demonstrate that this method exhibits strong predictive performance and high accuracy, thereby providing a valuable framework for research on traditional Chinese medicine compatibility.

2.2.4 　Network Model—Fast Newman Algorithm, Hu Fang et al. ^{[ 41 ］} This study employs a network model combined with the FastNewman algorithm to identify the core drugs for treating obesity. First, a network model of anti-obesity medications is constructed based on a review of the literature and systematic data compilation. Subsequently, the FastNewman algorithm is applied to partition the data, revealing that the core drugs for obesity treatment include Poria, Atractylodes, Hawthorn, and Alisma, among others. The findings clearly elucidate the patterns of drug–drug combinations, thereby providing both methodological approaches and theoretical foundations for interpreting the principles underlying herbal medicine in Traditional Chinese Medicine.

2.2.5 　Modeling Using the Minimum-Angle Partial Least Squares (PLS) Algorithm Optimized by Particle Swarm Optimization – Li Xiaoke ^{[ 42 ］} A method combining PLS with a particle swarm multi-objective optimization algorithm was designed and applied to investigate traditional Chinese medicine compound formulas for the treatment of coronary heart disease and angina pectoris. The study found that the Huoxue Xuanbi formula is the core compound for treating these conditions. Based on a comprehensive summary and systematization of formulation techniques and herb-selection strategies, this research integrates traditional clinical experience in TCM with modern mathematical methods to identify and develop novel TCM compound formulas characterized by rational drug composition and optimal compatibility patterns.

2.2.6 　A Combined Subjective–Objective Weighting Method—Uniform Design—Least Absolute Shrinkage and Selection Operator (LASSO) Model of Group Effect Relationships, by Yang Ming et al. ^{[ 43 ］} A combined subjective–objective weighting approach was adopted, integrating the Analytic Hierarchy Process (AHP) with the Criteria Importance Through Inter-Criteria Correlation (CRITIC) method for determining indicator weights based on inter-criterion correlations. Experiments were designed using uniform design, and a group-effect relationship model was constructed via nonlinear LASSO. This model was then applied to identify the optimal combination of five herbs in lipid-lowering granules, with the computationally derived optimal formulation subsequently validated experimentally. The results demonstrated that the LASSO-based relational model exhibits strong predictive performance and is well suited for analyzing pharmacological experimental data in traditional Chinese medicine characterized by nonlinearity, high dimensionality, discreteness, and small sample sizes. This study provides valuable research insights and methodologies for optimizing the formulation of complex traditional Chinese medicinal prescriptions.

3 　Conclusion

Based on mathematical theory and the theoretical framework of traditional Chinese medicine, the construction and application of well-founded mathematical models represent one of the most effective approaches to the modernization of TCM research. In the context of the formulation and compatibility of complex TCM prescriptions, such applications primarily include predicting pharmacological properties, characterizing the contribution of each herb to a specific therapeutic effect of the entire formula, clarifying the hierarchical role of each herb within the formulation, identifying core herbs, determining optimal herbal ratios, deriving the optimal formula and its corresponding optimal therapeutic efficacy, and elucidating the quantitative relationships among constituent herbs and their effects. By applying mathematical methods to TCM formulas, we can not only more effectively reveal the underlying principles of herbal compatibility and express these principles in mathematical terms, but also achieve more objective and precise drug evaluation.

Existing research indicates that applying mathematical and statistical models to elucidate the scientific principles underlying the formulation and compatibility of traditional Chinese medicine (TCM) prescriptions is one of the most effective approaches and has already yielded notable results. However, several challenges remain in the application of such models. For instance, orthogonal experimental design is suitable only for experiments with a limited number of factor levels, while response surface methodology is applicable solely to continuous variables; moreover, the design of mathematical models requires the specialized knowledge and experience of experts, and without such expertise it is difficult to develop appropriate mathematical models. Therefore, researchers in TCM should continuously enhance their proficiency in mathematical methods and related knowledge.

Secondly, there remain several unresolved issues in the study of traditional Chinese medicine. For instance, through appropriate herbal combinations, medicinal formulas can enhance therapeutic efficacy, reduce toxicity, and adapt to complex clinical conditions, thereby achieving holistic, integrated regulation and treatment tailored to specific syndromes and patterns. How, then, can we quantitatively analyze the synergistic effects and detoxifying actions among individual herbs in a formula—how, in other words, can we objectively quantify and characterize such synergistic interactions? Moreover, how can we bridge the gap between the theoretical principles of TCM and modern, quantitative rules governing herbal formulation and combination, and further advance this integration? Furthermore, as the saying goes, “The secret of the Han prescription lies in the dosage.” Does the synergistic interaction among components in a TCM compound formula also exhibit a dose–response relationship? And if so, how should this dose–response relationship be characterized? In addition, no factor operates in isolation or remains static; rather, drugs exert their effects at different sites and at different stages within the human body, consistent with the TCM emphasis on the body’s overall balance. Do the rules governing the combination of TCM formulas possess dynamic and time-dependent characteristics, and if so, how can these be systematically characterized? To address these questions, it is essential to integrate clinical prescribing practices and real-time clinical monitoring, while also employing appropriate mathematical models, so as to further elucidate the underlying principles and developmental patterns of TCM.

In recent years, driven by the increasing cross-disciplinary integration and mutual influence among various fields, as well as the ongoing modernization and internationalization of traditional Chinese medicine, a critical challenge—and also a key focus—in the future research and development of traditional TCM theory is how to take TCM theoretical principles as the guiding framework while simultaneously incorporating mathematical and computational models. On this basis, it is essential to transcend conventional modes of thinking, pursue technological advancements and methodological innovation, and employ precise methodologies and unambiguous language to accurately articulate and describe the properties and compatibility rules of Chinese medicinal materials. This represents both the crux and the greatest difficulty in establishing an objective, science-based approach to studying the pharmacological properties and compatibility patterns of TCM.

References (omitted)

Source: Wei Fuxiao, Liu Huanle, Fan Yuhui, Li Shunyong, Qin Xuemei, and Liu Xiaojie. An Overview of the Application of Mathematical and Physical Models in the Study of the Properties and Compatibility Rules of Traditional Chinese Medicines [J]. Pharmaceutical Evaluation Research, 2021, 44(1):205–212.