Chapter 1: Modeling in Bioengineering

第一章：生物工程建模

Book: Vascular Biomechanics by T. Christian Gasser (2022) 书籍：《血管生物力学》，T. Christian Gasser 著 (2022)

1.1 Introduction | 引言

Bioengineering modeling plays a prominent role in the study of biological systems and processes. Computational modeling has fundamentally changed industry and health care. While some biological properties can be directly measured, in vivo properties require models. Prospective questions such as "What would be the outcome from a certain clinical intervention in an individual patient?" can only be thoroughly studied through model-based simulations.

生物工程建模在生物系统和过程的研究中扮演着重要角色。计算建模已经从根本上改变了工业和医疗保健领域。虽然某些生物特性可以直接测量，但体内特性需要通过模型来研究。前瞻性问题，例如"某个临床干预对个体患者会产生什么结果？"，只能通过基于模型的模拟来深入研究。

1.1.1 Bottom-Up Approach | 自下而上方法

A bottom-up approach assembles pieces to give rise to more complex systems. Engineering problem solving often follows such an approach, where knowledge blocks represent basic physical principles (e.g., Newton's laws). Inductive reasoning integrates knowledge blocks leading to top-level systems. Given the vasculature being a complex hierarchical structure, bottom-up analysis requires integration of a large number of unit blocks. Quantitative predictions are challenging and often require phenomenological "scaling" parameters.

自下而上方法将各个部分组装成更复杂的系统。工程问题解决通常遵循这种方法，知识块代表基本物理原理（如牛顿定律）。归纳推理整合知识块，形成顶层系统。鉴于血管是复杂的层级结构，自下而上分析需要整合大量单元块。定量预测具有挑战性，通常需要引入唯象"缩放"参数来校准。

1.1.2 Top-Down Approach | 自上而下方法

A top-down approach breaks down a system into simpler sub-structures to gain better insight. Clinical science often uses this approach. Principles are less enforced, and a clinical method is regarded suitable as long as it can treat the patient. Clinical research questions are frequently posted retrospectively. Too much complexity hinders deductive reasoning, and pre-clinical approaches investigate less complex systems such as animal models or wet laboratory experiments.

自上而下方法将系统分解为更简单的子结构以获得更好的理解。临床科学经常使用这种方法。原理的约束较少，只要能治疗患者，临床方法就被认为是合适的。临床研究问题通常以回顾性方式提出。过多的复杂性阻碍演绎推理，临床前方法研究如动物模型或湿实验室实验等较简单的系统。

1.1.3 Opportunities and Challenges | 机遇与挑战

工程挑战：传统工程方法通常不直接适用，需要特定进一步开发。血管组织适应机械和生化环境的固有特性仍是具有挑战性的建模任务。血管器官的整体研究和生物过程的理解可能需要耦合结构、流体、化学和电气场的生物工程模型。

数据获取挑战与不确定性：生物系统实验测试具有挑战性。生物数据具有不确定性，样本内和样本间输入参数（如载荷条件和本构特性）的变异性削弱了生物工程模型的可预测性和效益。

验证挑战：生物工程模型的临床接受需要证明其临床和经济效益的可靠前瞻性验证。此类验证耗时且常受到伦理和其他限制的挑战。

产品开发挑战：工程解决方案可能市场有限或初始商业部署成本过高。可能难以集成到临床工作流程中，甚至可能威胁现有医疗公司和临床医生的常规工作。

1.2 Model Design | 模型设计

A model represents the real object or process to some degree of completeness. As A. Einstein states: "Everything should be made as simple as possible, but no simpler."

模型以一定完整性程度代表真实物体或过程。正如爱因斯坦所说："万事应力求简单，但不宜过于简单。"

1.2.1 Simplifications | 简化

Typical vascular biomechanics model simplifications relate to:

血管生物力学模型简化的典型方面：

Vascular Geometry | 血管几何：从微米到厘米直径的血管遍布全身。建模者必须分离长度尺度并明确建模血管几何的尺度。
Material Properties | 材料特性：血管壁表现出复杂的力学特性，受组织学、生物学和临床因素影响。
Boundary and Initial Conditions | 边界和初始条件：模拟对象仅覆盖真实问题的某个域。
Loading Conditions | 载荷条件：血管暴露于复杂的时变机械载荷。
Numerical Methods | 数值方法：需要适当的数值方法来求解控制方程。
Output Data Analysis | 输出数据分析：模型产生大量数据（如整个壁面的应力和应变）。

1.2.2 Strategies | 策略

A model may be seen as a transformation matrix that transforms vector x of input variables to vector y of output variables. Models can be ranked by their transparency:

模型可被视为将输入变量向量x转换为输出变量向量y的变换矩阵。模型可按透明度排序：

| Model Type | Description | 描述 | |------------|-------------| | White-Box Model | Breaks down response into underlying physical mechanisms; fully discloses inner working | 将响应分解为底层物理机制；完全披露内部运作 | | Black-Box Model | Uses empirical descriptions without physical basis; no physical explanation | 使用经验描述，无物理基础；无物理解释 | | Gray-Box Model | Implements physical representation with phenomenological approximations | 实现带有唯象近似的物理表示 | | Surrogate Model | Simplified model for specific tasks; minimizes time at loss of accuracy | 针对特定任务简化的模型；以精度损失换取时间最小化 |

1.3 Model Development and Testing | 模型开发与测试

1.3.1 Sensitivity Analysis | 敏感性分析

Biomechanical models are complex and depend on many uncertain input parameters. A parameter sensitivity analysis investigates how input uncertainties propagate toward model output.

生物力学模型复杂且依赖于许多不确定的输入参数。参数敏感性分析研究输入不确定性如何传播到模型输出。

Local Sensitivity | 局部敏感性： $$\Delta y = s(x_0) \cdot \Delta x$$ where s(x₀) = ∂y/∂x|x₀ denotes the sensitivity vector.

Global Sensitivity | 全局敏感性： Explores the model's output for the entire parameter space using Sobol's variance-based sensitivity analysis.

使用Sobol's方差敏感性分析探索整个参数空间的模型输出。

1.3.2 Verification | 验证

Verification aims at testing the correctness of the model's mathematical implementation. It tests for analytical errors, software coding errors, parameter input failure, and ensures appropriateness of numerical techniques.

验证旨在测试模型数学实现的正确性。测试分析错误、软件编码错误、参数输入故障，并确保数值技术的适当性。

1.3.3 Validation | 验证

Validation aims at testing that a model predicts the desired features of the real object or process. It ensures the model represents the real object up to the level specified by the IMA.

验证旨在测试模型是否预测真实物体或过程的期望特征。它确保模型在IMA指定的程度上代表真实物体。

1.4 Statistics-Based Modeling | 统计建模

Biological data is uncertain and statistics-based modeling describes bioengineering phenomena.

1.4.1 Correlation Amongst Variables | 变量相关性

Pearson's Product-Moment Correlation Coefficient | 皮尔逊积矩相关系数： $$r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2} \sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}$$

Correlation coefficient ranges from -1 to +1. Positive/negative coefficients denote positive/negative correlations.

相关系数范围从-1到+1。正/负系数表示正/负相关。

Spearman's Rank Correlation Coefficient | 斯皮尔曼等级相关系数： $$r_s(x_i, y_i) = r(rg(x_i), rg(y_i))$$

Indicates monotonic (linear or non-linear) relation by calculating Pearson's correlation on rank data. Indicates monotonic (linear or non-linear) relation by calculating Pearson's correlation on rank data.

通过计算等级数据的皮尔逊相关性来指示单调（线性或非线性）关系。

1.4.2 Regression Modeling | 回归建模

Simple Linear Regression | 简单线性回归： $$y_i = b_0 + b_1 x_i + e_i$$

Coefficients identified through least-square optimization: 系数通过最小二乘优化识别： $$\sum_{i=1}^{n} e_i^2 = \sum_{i=1}^{n}(y_i - b_0 - b_1 x_i)^2 \rightarrow MIN$$

Coefficient of Determination | 决定系数： $$R^2 = \frac{1}{n} \frac{\left[\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})\right]^2}{s_x s_y}$$

Range: 0 ≤ R² ≤ 1

Significance of Regression | 回归显著性： $$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$

1.4.3 Hypothesis Testing | 假设检验

Null Hypothesis H₀: Sample observations result purely from chance. 备择假设 Ha: Sample observations are influenced by some non-random cause.

Type I Error: Incorrectly rejecting H₀ (false positive)
Type II Error: Failing to reject H₀ (false negative)

1.4.4 Mean Difference Test | 均值差异检验

One-Sample t-test | 单样本t检验： $$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$

Two-Sample t-test (Welch t-test) | 两样本t检验（韦尔奇t检验）： $$t = \frac{\bar{x} - \bar{y}}{s} ; s = \sqrt{\frac{s_x^2}{n_x} + \frac{s_y^2}{n_y}}$$

1.5 Artificial Intelligence | 人工智能

AI (Machine Learning) creates machines to perform tasks without explicit instructions. AI algorithms mimic cognitive functions such as learning and problem solving.

1.5.1 Learning and Prediction | 学习与预测

Supervised Learning | 监督学习：从包含输入和期望输出的数据中获取知识
Unsupervised Learning | 无监督学习：从仅包含输入的数据中获取知识

1.5.2 Artificial Neural Network (ANN) | 人工神经网络

ANNs are models inspired by brain structure. Nodes represent neurons, connected by edges representing synapses. Each node processes received information and signals connected nodes. Typically aggregated into layers: input layer → hidden layers → output layer. Uses backpropagation algorithm for training.

人工神经网络是由大脑结构启发的模型。节点代表神经元，由代表突触的边连接。每个节点处理接收到的信息并向连接的节点发信号。通常聚合成层：输入层→隐藏层→输出层。使用反向传播算法进行训练。

Logistic Function | 逻辑函数： $$y = [1 + \exp(-z)]^{-1}$$

1.5.3 Bayesian Network | 贝叶斯网络

A BN is a probabilistic model representing variables and their conditional dependencies through a Directed Acyclic Graph. Uses Conditional Probability Tables to express conditional dependence.

贝叶斯网络是通过有向无环图表示变量及其条件依赖的概率模型。使用条件概率表表达条件依赖。

1.5.4 Decision Tree | 决策树

A large number of observations may be used to "grow" a decision tree through recursive splitting toward homogenizing the response variable in emerging sub-cohorts. Uses cost function: 大量观测值可用于通过递归分割来"生长"决策树。使用的成本函数：

\[\sum_{i \in A}(k_i - k^A)^2 + \sum_{i \in B}(k_i - k^B)^2 \rightarrow MIN\]

Splitting ceases and pruning is applied to avoid overfitting and reduce complexity. 分割停止并应用剪枝以避免过拟合和降低复杂性。

1.6 Case Study: Biomechanical Rupture Risk Assessment | 案例研究：生物力学破裂风险评估

Abdominal Aortic Aneurysm (AAA) | 腹主动脉瘤

AAA is a serious condition causing many deaths, especially in men over 65. Progressive treatment (surgical or endovascular AAA repair) cannot be offered to all patients. AAA repair is recommended if rupture risk is deemed to exceed intervention risk.

腹主动脉瘤是一种严重疾病，尤其在65岁以上男性中导致许多死亡。进行性治疗（手术或血管内AAA修复）不能提供给所有患者。如果认为破裂风险超过干预风险，则推荐AAA修复。

Current AAA Risk Assessment Limitations | 当前AAA风险评估的局限性

Current practice uses aneurysm's largest transverse diameter and its change over time. Repair recommended if diameter exceeds 55 mm or grows faster than 10 mm/year. AAAs smaller than 55 mm can and do rupture, while many larger than 55 mm remain quiescent. The poor sensitivity of diameter-based criteria leads to sub-optimal cost-effectiveness.

当前实践使用动脉瘤的最大横径及其随时间的变化。如果直径超过55毫米或增长速度快于10毫米/年，则推荐修复。直径小于55毫米的AAA可能且确实会破裂，而许多大于55毫米的动脉瘤保持静止。基于直径标准的敏感性差导致次优的成本效益。

Biomechanical Rupture Risk Assessment (BRRA) | 生物力学破裂风险评估

Failure Hypothesis | 失效假说： Raising tension in vessel wall to supra-physiological levels leads to micro-scale damage that cannot heal, accumulating weak links until macro-defects form and rupture occurs.

将血管壁张力提高到超生理水平导致微尺度损伤，损伤无法愈合，积累弱连接直到形成宏缺陷并发生破裂。

Risk Index | 风险指数： $$\xi = \sigma_M / Y$$ where σM is von Mises stress and Y is wall strength. Maximum value called Peak Wall Rupture Index (PWRI).

其中σM是冯·米塞斯应力，Y是壁强度。最大值称为峰值壁破裂指数（PWRI）。

Key Modeling Assumptions | 关键建模假设

Organ-Level Model | 器官级模型：使用厘米尺度的连续体方法，非线性FEM求解
Vascular Tissue Model | 血管组织模型：使用各向同性两参数Yeoh模型

Clinical Validation Results | 临床验证结果

Operator variability: 2.7% for PWRI predictions
RRED was 14.0 mm larger in ruptured vs non-ruptured cases (p < 0.001)
Quasi-prospective studies showed BRRA could discriminate between ruptured and stable cases
Female AAA ruptures at smaller diameters; BRRA considers this wall weakening effect

1.7 Summary and Conclusion | 总结与结论

Biomechanical modeling is key in life science exploration. A model represents reality up to the degree of complexity determined by the IMA. The IMA guides model development and testing. The modeler's goal should always be keeping a model as simple as possible.

生物力学建模是生命科学探索的关键。模型代表现实的复杂程度由IMA决定。IMA指导模型开发和测试。建模者的目标应始终是保持模型尽可能简单。

Key Insights | 关键洞察：

Model Complexity | 模型复杂性：系统误差随复杂性增加而减少，但不确定性和随机误差增加。最佳复杂性是系统误差和随机误差之间的权衡。
Input Data Quality | 输入数据质量：输入信息总是存在不确定性，改进输入数据质量的措施（如标准化数据获取协议）通常比增加模型复杂性更成功。
Validation is Critical | 验证至关重要：血管生物力学建模尚未获得广泛的临床接受。需要与IMA相关的严格验证。
Black-Box vs White-Box | 黑盒与白盒：黑盒建模方法已获得显著的临床兴趣，但统计相关性不应与因果关系混淆。白盒建模方法考虑血管系统的物理学，允许更深入的理解。

"Every model is wrong, but only a few are useful." — George E. P. Box "每个模型都是错的，但只有少数是有用的。" — 乔治·E·P·鲍克斯

Formula Table | 公式表

Eq.	Formula	公式
(1.1)	Δy = s(x₀) · Δx	局部敏感性
(1.2)	R = (pᵢ - pₒ)/q = 128lη/(πd⁴)	Hagen–Poiseuille定律
(1.8)	r = Σ(xᵢ - x̄)(yᵢ - ȳ) / √[Σ(xᵢ - x̄)²Σ(yᵢ - ȳ)²]	Pearson相关系数
(1.9)	rₛ(xᵢ, yᵢ) = r(rg(xᵢ), rg(yᵢ))	Spearman等级相关系数
(1.10)	yᵢ = b₀ + b₁xᵢ + eᵢ	简单线性回归
(1.11)	R² = (1/n)[Σ(xᵢ - x̄)(yᵢ - ȳ)]²/(sₓsᵧ)	决定系数
(1.13)	t = r√(n-2)/√(1-r²)	回归显著性
(1.17)	H₀: p(f) = 1/6; Hₐ: p(f) ≠ 1/6	假设检验（骰子示例）
(1.18)	t = (x̄ - μ₀)/(s/√n)	单样本t检验
(1.19)	t = (x̄ - ȳ)/s; s = √(sₓ²/nₓ + sᵧ²/nᵧ)	两样本t检验
(1.22)	y = [1 + exp(-z)]⁻¹	逻辑函数（ANN）
(1.24)	Σ(kᵢ - kᴬ)² + Σ(kᵢ - kᴮ)² → MIN	决策树成本函数

Notes compiled from chapter_01.txt of Vascular Biomechanics by T. Christian Gasser (2022) 笔记整理自T. Christian Gasser著《血管生物力学》(2022)第一章