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[代码资源] 数学建模优化类算法分类_必须知道的用于疾病建模的新聚类算法

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    2023-7-31 10:17
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    数学中国浅夏
    发表于 2021-10-20 19:42 |显示全部楼层
    |招呼Ta 关注Ta

    数学建模优化类算法分类

    This article explains a new method for clustering disease data by both subtype and stage called SuStaIn (Subtype & Stage Inference). It explains the concept, summarises the maths, and provides a link to the python code.

    本文介绍了一种通过以下方法对疾病数据进行聚类的新方法 子类型和阶段都称为SuStaIn(子类型和阶段推断)。 它解释了概念,总结了数学,并提供了python代码的链接。


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    Classic clustering algorithms like K-Means and Gaussian Mixture Model (GMM) are great for modelling data when we want to find cross-sectional subtypes (aka clusters). This kind of subtyping is used a lot in medicine. A well-known general example is that of subtyping diabetes into “Type I” and “Type II” using a single blood sugar measurement. This can help doctors decide whether to prescribe insulin injections or lifestyle changes.

    当我们要查找横截面子类型(又称为聚类)时,像K-Means和高斯混合模型(GMM)这样的经典聚类算法非常适合对数据建模。 这种子类型在医学​​中被大量使用。 众所周知的一般示例是使用单次血糖测量将糖尿病分为“ I型”和“ II型” 。 这可以帮助医生决定是否开胰岛素注射或改变生活方式。

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    . V" l) l+ d# \/ y3 y& d( zFigure 1. Subtypes of a disease by phenotype using cross-sectional biodata; for example, a single measurement of blood sugar or a medical image. An example would be diabetes which is subtyped into “Type I” and “Type II”. Image created by author. 图1 。 使用横断面生物数据按表型划分的疾病亚型; 例如,一次血糖测量或医学图像测量。 一个例子是糖尿病,其被分为“ I型”和“ II型”。 图片由作者创建。

    Grouping a disease by stage is also very useful in medicine, this time for modelling disease progression. For example, a model for grouping cancer into stages 1–4 was developed using longitudinal data (multiple measurements from the same person over time). The model itself was developed using longitudinal data but once developed, allowed doctors to determine which stage a patient is at using only a single cross sectional measurement (i.e., tumour size in millimetres). Knowing the stage of cancer may help doctors decide whether radiotherapy or chemotherapy is needed.

    在医学中,将疾病按阶段分组也是非常有用的,这一次可以模拟疾病的进展。 例如,使用纵向数据(一段时间内同一个人的多次测量)开发了将癌症分为1至4期的模型。 该模型本身是使用纵向数据开发的,但是一旦开发,医生就可以使用单个横截面测量(即,以毫米为单位的肿瘤大小)来确定患者处于哪个阶段。 了解癌症的阶段可能有助于医生确定是否需要放疗或化疗。


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    Figure 2. Grouping disease by stage, using longitudinal data which measures progression over time. An example would be cancer which is usually grouped into stages 1–4. Image created by Author. 图2.使用纵向数据对疾病进行分期,纵向数据可衡量随时间的进展。 一个例子是癌症,通常分为1-4期。 图片由作者创建。

    The downside of these kinds of staging models is that they assume all patients come from the same type of the disease. I.e., they account for disease progression, but there is no account of disease subtypes.

    这些分期模型的缺点是,它们假定所有患者都来自同一类型的疾病。 即,它们解释了疾病的进展,但没有解释疾病的亚型

    Conversely, the “Cross-sectional subtypes” mentioned previously explain subtypes but not progression. I.e.,they assume all patients are at the same stage.

    相反,前面提到的“横断面亚型”解释了亚型,但没有解释进展。 即他们假设所有患者都处于同一阶段。

    So what if we want to do both, i.e. find subtypes of a disease based on how it progresses over time, and create that model using only cross-sectional data?

    那么,如果我们想同时做这两种事情,即根据疾病随时间的进展找到疾病的亚型,并仅使用横截面数据来创建该模型,该怎么办?

    引入Z分数SuStaIn(子类型和阶段推断)算法 (Introducing the Z-Score SuStaIn (Subtype & Stage Inference) Algorithm)

    The Z-Score SuStaIn is an unsupervised machine-learning technique that identifies population subgroups (clusters) with distinct patterns of disease progression based on biomarkers. This is shown abstractly in figure 3 below.

    Z-Score SuStaIn是一种无监督的机器学习技术,可基于生物标记物识别具有不同疾病进展模式的人群亚组(集群)。 如下图3所示。

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    Figure 3. The circles represent groups of biomarkers. The colours (green, orange, blue) represent different subtypes i.e., different progression patterns. Note: the subtypes produced in SuStaIn are not based on clinically defined phenotypic disease subtypes (such as Type I and Type II diabetes), but rather on how they each progress 图3 。 圆圈代表生物标志物的组。 颜色(绿色,橙色,蓝色)代表不同的亚型,即不同的渐进模式。 注意:SuStaIn中产生的亚型不是基于临床定义的表型疾病亚型(例如I型和II型糖尿病),而是基于它们 over time. Image created by author. 随着时间的进展。 图片由作者创建。

    Biomarker: “Any substance, structure or process that can be measured in the body or its products and can influence or predict the incidence of outcome or disease” (WHO, 2011). This includes everything from blood sugar measurements and heart rate to sweat levels and electrical signals in the brain.

    生物标志物:“可以在体内或其产品中测量并可以影响或预测结果或疾病发生率的任何物质,结构或过程”(世卫组织,2011年)。 这包括从血糖测量和心率到出汗水平和大脑中的电信号的所有内容。

    To illustrate this on a real example, we can look at how 3 subtypes of Alzheimers were found based on how each progresses in terms of neural biomarkers (see image below). The biomarkers (features) here are atrophy (shrinkage) levels across various brain regions, measured by MRI scans. The colours are explained in the figure caption.

    为了在一个真实的例子上说明这一点,我们可以根据每种神经生物学标记的进展情况,看看如何找到3种亚型的阿尔茨海默氏症(见下图)。 这里的生物标记(特征)是大脑各个区域的萎缩(收缩)水平,通过MRI扫描测量。 颜色在图形标题中说明。

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    & ^, q# u4 k9 d' N# k9 c5 p' bFigure 4. This image shows 图4 。 此图像根据每种类型的进展情况显示了 3 3种 subtypes 亚型 of Alzheimers disease based on how each type progresses (measured using MRI brain images). The colour coding represents biomarker degeneration where white indicates no pathology, r 的阿尔茨海默氏病(使用MRI脑图像进行测量)。 彩色编码表示生物标志物变性,其中白色表示无病理, ed shows moderate pathology, p Red表示中度病理,p ink shows medium and 表示中等, blue shows maximum pathology. Image adapted from Young et al (2018). 蓝色表示最大病理。 图片摘自Young等人(2018)。

    As we can see, 3 distinct patterns of disease progression have emerged, based on different patterns of neurological decay. For example, as highlighted in Figure 5 below, subtype 2 starts to show shrinkage in the occipital lobe (the small, far left corner on the brain image) at stage 9, yet subtype 3 doesn’t show atrophy in that area at all, even by stage 25. These are very distinct types of disease progression which may require different medical interventions.

    我们可以看到,根据神经衰弱的不同模式,出现了3种不同的疾病进展模式。 例如,如下面的图5中突出显示的,亚型2在第9阶段开始显示枕叶萎缩(大脑图像的最左端的小角),而亚型3根本没有显示出该区域的萎缩,甚至到了第25阶段。这些都是非常不同的疾病进展类型,可能需要不同的医疗干预措施。

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    Figure 5. Subtype 2 shows atrophy in the occipital lobe at stage 9 but subtype 3 doesn’t show atrophy in this area even by stage 25. Image adapted from Young et al (2018). 图5 。 亚型2在第9阶段显示枕叶萎缩,但亚型3甚至在第25阶段在该区域也没有萎缩。图片摘自Young等人(2018)。但是如何使用横截面数据来模拟疾病随时间的发展呢? (But how can disease progression over time be modelled using cross-sectional data?)

    SuStaIn is a clustering algorithm that works in tandem with a disease progression model, in this case the Z-Score model (hence the name Z-Score SuStaIn). Z-Score SuStaIn models disease progression using z-scores of the biomarker data with which it measures the extent to which each biomarker deviates from normality, i.e. how the biomarker differs from that of healthy individuals (the control group). This can be seen in Figure 4 above where the different colours represent the number of standard deviations each biomarker (in this case brain shrinkage level) is from the control mean. Therefore the temporal stages (and the subtypes) are essentially hidden variables that we draw out from the data. This is how we measure progression using just cross-sectional data.

    SuStaIn是与疾病进展模型(在本例中为Z-Score模型(因此称为Z-Score SuStaIn))协同工作的聚类算法。 Z分数SuStaIn使用生物标记数据的z分数来模拟疾病进展,以此来衡量每种生物标记偏离正常程度的程度,即生物标记与健康个体的差异程度 (对照组)。 可以在上面的图4中看到,其中不同的颜色代表每个生物标记物与对照平均值的标准差(在这种情况下,为大脑收缩水平)。 因此,时间阶段(和子类型)本质上是我们从数据中提取的隐藏变量。 这就是我们仅使用横截面数据来衡量进展的方式。

    The Z-Score SuStaIn algorithm produces an output similar to that in Figure 4 in two alternating steps. First it selects the sequence of events with the maximum likelihood given the (z-scored) data. Secondly it subtypes individuals and estimates progression patterns in each subgroup. The algorithm then alternates between subtyping individuals with similar progression patterns and re-estimating the progression patterns in each subgroup until it converges, producing several subtypes with different progression patterns (as shown in Figures 3 and 4).

    Z-Score SuStaIn算法在两个交替的步骤中产生与图4类似的输出。 首先,在给定(z评分)数据的情况下,选择具有最大可能性的事件序列。 其次,它对个体进行亚型化,并估计每个亚组中的进展模式。 然后,该算法在具有相似进度模式的子类型化个体之间交替,然后重新估计每个子组中的进度模式,直到收敛为止,从而产生具有不同进度模式的多个子类型(如图3和4所示)。

    The key advantages of this approach are:

    这种方法的主要优点是:

    • It can allow classification of patients based on both disease subtype and disease stage. If developed into an application, this could essentially allow clinicians or researchers to take an MRI scan of a patient, run it through the trained model, and be told which subtype and stage the patient is at in their given disease.

      它可以允许根据疾病亚型和疾病阶段对患者进行分类。 如果开发成应用程序,这实际上可以使临床医生或研究人员对患者进行MRI扫描,通过训练有素的模型运行它,并告知患者所患疾病的亚型和阶段。

    • Disease progression can be measured using only cross-sectional data which is much (much!) easier to obtain than longitudinal data, especially in medicine. This means it can be used to understand a wider range of diseases.

      只能使用横截面数据来衡量疾病的进展,该横截面数据比纵向数据要容易得多(尤其是在医学中)。 这意味着它可以用来了解更广泛的疾病。

    • It gives us a much more fine-grained model of disease progression than existing models. The model that underpins the Z-Score Model, called the EBM (Event Based Model; explained in “The Maths” section), provides the key information doctors need; i.e. whether a person is ill or not. This is shown when biomarker measurements transition to abnormal levels. However, by using z-scores, the SuStaIn model shows how people transition through multiple intermittent stages of normality to abnormality, giving us a more fine grained insight into progression. This fine grained insight allows us to also develop more accurate progression-based subtypes from the data which could lead to more targeted drug development. For example, instead of developing one blanket drug for all types of a disease (which may come with maximum side effects) drugs can be developed specific to each subtype, and effects can measured in line with the progression pattern of that subtype. This could potentially minimise unnecessary side effects as well as being altogether more effective in treating or alleviating symptoms.

      与现有模型相比,它为我们提供了更为精细的疾病进展模型。 Z-Score模型的基础模型称为EBM(基于事件的模型;在“数学”部分中进行了解释),提供了医生所需的关键信息。 即一个人是否生病。 当生物标志物的测量值转变为异常水平时,将显示出来。 但是,通过使用z得分,SuStaIn模型可以显示人们如何从正常的多个间歇阶段过渡到异常,从而使我们对进展有了更细粒度的了解。 这种细粒度的见解使我们还可以从数据中开发出更准确的基于进展的亚型,从而可能 至 更有针对性的药物开发。 例如,可以针对每种亚型开发专门针对每种亚型的药物,而不是针对所有类型的疾病开发一种总括性药物,并且可以根据该亚型的进展模式来评估其作用。 这可能潜在地使不必要的副作用最小化,并且在治疗或缓解症状方面更加有效。

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    The main drawback is that Z-score SuStaIn only works on data with normally distributed variability in the control population. If the data isn’t Gaussian, results will be poor. (In this case it would be better if the recently developed nonparametric Kernel Density Estimation Mixture Model (KDEMM) by Firth et al, (2018) was used instead of the Z-Score Model).

    主要缺点是Z分数SuStaIn仅适用于 对照人群中具有正态分布变异的数据。 如果数据不是高斯,则结果将很差。 (在这种情况下,最好使用Firth等人(2018)最近开发的非参数内核密度估计混合模型(KDEMM)代替Z分数模型)。

    数据 (The Data)

    The input data for Z-Score SuStaIn could be any kind of biomarker, measured by for example, blood tests, MRI scans, cardiograms etc.

    Z-Score SuStaIn的输入数据可以是任何种类的生物标记,例如可以通过血液检查,MRI扫描,心电图等进行测量。


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    - C' ~! k3 _" f6 e  `Figure 6. Image created by author. 图6.作者创建的图像。

    After standard data cleaning, 3 things need done to the data before it can be put into the algorithm:

    经过标准的数据清理之后,需要对数据做三件事才能将其放入算法中:

    • Adjust for the effects of covariates. In brain volume measurements for example, a covariate could be age as this variable affects brain volume independent of disease (our brains tend to naturally shrink a little as we get older). The effects of non-disease related factors like this should be stripped out so we can measure only the effect of the disease on the biomarker measurements. (Code for doing this via regression is in the GitHub repo at the end of the article).

      调整协变量的影响。 例如,在脑容量测量中,协变量可以是年龄,因为该变量影响脑容量而与疾病无关(随着年龄的增长,我们的大脑往往会自然缩小一点)。 像这样与非疾病相关的因素的影响应该被剔除,因此我们只能在生物标志物测量中测量疾病的影响。 (通过回归完成此操作的代码在文章结尾的GitHub存储库中)。

    • Z-score the data. This entire algorithm is based on measuring disease progression based on how biomarkers deviate from the mean. Therefore we need to input the data in the form of z-scores; i.e. the number of standard deviations a data point is from the mean (in this case, the control mean).

      Z评分数据。 整个算法基于生物标志物如何偏离均值来衡量疾病的进展。 因此,我们需要以z分数的形式输入数据; 也就是说,数据点与平均值(在这种情况下为对照平均值)之间的标准偏差数。

    • Make sure all features increase with disease. If any features decrease with disease multiply the data in these features by -1. For example, the biomarker brain fluid will likely increase as the biomarker frontal lobe volume decreases (this is because when parts of the brain shrink, the body fills the empty space with fluid). If there is a mix of increasing and decreasing features, the algorithm will get confused and produce inaccurate results.

      确保所有功能随疾病而增加。 如果任何特征疾病而减少,则将这些特征中的数据乘以-1。 例如,随着生物标志物额叶体积的减少,生物标志物的脑液可能会增加(这是因为当大脑的某些部分收缩时,身体会用液体填充空的空间)。 如果混合使用增加和减少的功能,则该算法将感到困惑并产生不准确的结果。


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    The python code for all these steps and the analysis itself is in the SuStaIn Jupyter Tutorial, also linked at the end of this article.

    所有这些步骤以及分析本身的python代码都在SuStaIn Jupyter Tutorial中,该链接也在本文结尾处。

    数学 (The Maths)

    An extra section for those interested in the details of SuStaIn.

    对于那些对SuStaIn详细信息感兴趣的人,还有一个额外的部分。

    SuStaIn is a likelihood based subtyping model which works in tandem with a disease progression model —in this case the Z-score model.

    SuStaIn是基于可能性的子类型模型,可与疾病进展模型(在本例中为Z评分模型)协同工作。

    The Z-score model aims to predict the order of events that maximises the likelihood of the data, where an event is the transition of a biomarker from one stage to the next (based on z-score).

    Z评分模型旨在预测事件顺序,以最大化数据的可能性,其中事件是生物标记从一个阶段到下一个阶段的过渡(基于z评分)。

    The Z-score model is built upon the Event Based Model (Fonteijn et al, 2014), therefore to understand the Z-score model, it’s easier to first understand the EBM. The EBM measures disease progression in a binary manner by measuring the point at which biomarkers transition from normal to abnormal.

    Z评分模型基于事件模型(Fonteijn et al,2014),因此要了解Z评分模型,先了解EBM会更容易。 EBM通过测量生物标志物从正常状态转变为异常状态的点,以二进制方式测量疾病的进展。


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    5 Q8 a- y% R; s- rFigure 7. The Event Based Model (EBM). Image taken from Young et al (2018). 图7.基于事件的模型(EBM)。 图片取自Young et al(2018)。

    The equation above shows the Event Based Model (EBM) where:

    上面的等式显示了基于事件的模型(EBM) ,其中:

    • X is the data set of raw biomarker measurements (not z-scores)

      X是原始生物标志物测量值的数据集(不是z得分)

    • S is the sequence in which biomarkers transition from normal to abnormal.

      S是生物标记从正常转变为异常的序列。

    • p(X|S) is the most likely order of events that maximises the likelihood of the data.

      p(X | S)是最可能的事件顺序,它使数据的可能性最大化。

    • p(k) is the prior likelihood of being at stage k at which the events E1,…Ek have occurred (a uniform prior is used).

      p(k)是发生事件E1,... Ek的阶段k的先验可能性(使用统一先验)。

    • p(x|E) and p(x|¬E) are the probability the event has occurred (the biomarker has transitioned to abnormal) and not occurred given a data point x within X. These are modelled as normal distributions.

      p(x | E)和p(x |¬E)是事件发生的可能性(生物标记已转变为异常),并且在给定X内的数据点x的情况下未发生。将它们建模为正态分布。


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    When only a small number of biomarkers is used, all possible orderings can be enumerated and the order of events, Ŝ, is the sequence which maximises P(X|S) as in Figure 7, above.

    当仅使用生物标志物的数量少,所有可能的排序可以枚举和事件,S的顺序,是最大化P(X | S)的序列如在图7中,上方。

    However, because the number of possible sequences grows factorially as the number of biomarkers increases, Markov Chain Monte Carlo (MCMC) sampling of p(S|X) is used to find Ŝ when the number of sequences to sample is too large to enumerate. (This process is explained on page 4 of this article).

    但是,由于可能的序列数会随着生物标记数的增加而析因地增长,因此, Markov Chain Monte Carlo(MCMC) P(S | X)的采样被用来求S时样品序列的数目过大而无法一一列举。 (此过程在本文的第4页上进行了说明)。

    Once we have Ŝ, it can be used to estimate a disease stage for individuals given their biomarker measurements. The disease stage, k, is defined as, the stage (i.e. the number of events that have occurred so far) that has the highest probability given the data and our sequence. This is calculated by maximising the equation in Figure 7 above, as shown in Figure 8, below:

    一旦有了Ŝ,就可以用其生物标记测量来估计个体的疾病阶段。 疾病阶段k被定义为在给定数据和我们的序列的情况下具有最高概率的阶段(即,迄今为止已发生的事件数)。 这是通过最大化上面图7中的公式来计算的,如下图8所示:

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    4 {6 r' C: |) i2 PFigure 8. Optimising in relation to disease stage, k. (Image taken from Firth et al, 2018). 图8 。 关于疾病阶段的优化,k。 (图片取自Firth等人,2018)。

    The Z-score Model then builds on the EBM model as shown below:

    然后, Z评分模型基于EBM模型构建,如下所示:


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    . ]) y" i! _* ]3 g4 S' pFigure 9. The Z-score SuStaIn model 图9. Z分数SuStaIn模型

    There are two key innovations in the Z-score model (compared to EBM). The first is that X now equals the z-scores of the data rather than the raw data. The second is that S now equals the sequence in which biomarkers move through multiple stages rather than just from normal to abnormal. So we are now able to measure progression, not by a single transition from normal to abnormal, but by transitions between multiple stages of disease progression. This gives us a much more fine grained model of how a disease develops.

    Z评分模型(与EBM相比)有两项关键创新。 首先是X现在等于 数据的z得分,而不是原始数据。 第二个是S现在等于生物标志物经历多个阶段而不是从正常到异常的顺序。 因此,我们现在能够测量进展,而不是通过从正常到异常的单个转变,而是通过疾病发展的多个阶段之间的转变来测量。 这为我们提供了关于疾病如何发展的更为精细的模型。

    SuStaIn is then fitted by simultaneously optimising subtype membership, subtype trajectory and the posterior distributions of both (see Figure 10 below). It is fitted hierarchically, whereby the number of subtypes that can be supported by the data is selected via model selection criteria CVIC (cross-validation information criterion) which is obtained from ten-fold cross-validation.

    然后,通过同时优化子类型成员资格,子类型轨迹和两者的后验分布来拟合SuStaIn (请参见下面的图10)。 它是分层装配的,从而可以通过从十倍交叉验证中获得的模型选择标准CVIC(交叉验证信息标准)来选择数据可以支持的子类型的数量。

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    9 Z; Y# q0 D% L) g) G& }Figure 10: The SuStaIn model 图10: SuStaIn模型

    Where:

    哪里:

    • C is the number of clusters (subtypes)

      C是簇数(子类型)

    • f is the proportion of subjects assigned to a particular cluster (subtype)

      f是分配给特定类别(子类型)的主题所占的比例

    • M is the overall SuStaIn model

      M是整体SuStaIn模型

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    A full explanation of these mathematical models can be seen in the SuStaIn intro paper (Young et al, 2018).

    这些数学模型的完整说明可以在SuStaIn简介论文中看到(Young等,2018)。

    结论 (In conclusion)

    If you are looking to subtype disease or model disease progression using biomarkers, it could be very worthwhile including SuStaIn in your model evaluation.

    如果您正在寻找使用生物标志物的亚型疾病或模型疾病进展的模型,那么将SuStaIn纳入模型评估中可能是非常值得的。

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