---
title: "Cheat sheet — Markdown, Quarto and equations"
subtitle: "IATDMCT Congress · Introduction to Markdown"
author: "Markus Hovd"
date: today
format:
  pdf:
    toc: true
    toc-depth: 2
    number-sections: true
    geometry:
      - margin=2cm
engine: markdown
---

Keep this next to your keyboard. Everything here is written with the syntax it describes,
so you can copy the source straight out of `cheatsheet.qmd`.

# Markdown in one table

| Purpose | Type this | Looks like |
|--------------------|--------------------------------|---------------------------|
| Heading | `## Section` | Section heading |
| Bold / italic | `**bold**` / `*italic*` | **bold** / *italic* |
| Strikethrough | `~~retracted~~` | ~~retracted~~ |
| Inline code | `` `CL = D/AUC` `` | `CL = D/AUC` |
| Bulleted list | `- item` | • item |
| Numbered list | `1. item` | 1. item |
| Task list | `- [x] done` | ☑ done |
| Link | `[text](https://url)` | [text](https://quarto.org) |
| Image | `![Caption](img.png)` | An image with caption |
| Blockquote | `> quoted` | inset quote |
| Footnote | `Text.[^a]` | superscript marker |
| Line break | two trailing spaces | new line, same paragraph |
| Divider | `---` | horizontal rule |
| Escape | `\*` `\_` `\#` `\$` `\%` | literal characters |

# Quarto extras

| Purpose | Syntax |
|-----------------------------------|--------------------------------------------------|
| YAML header | `---` … `---` at the very top of the file |
| Callout | `::: {.callout-note}` … `:::` (also `tip`, `warning`, `important`, `caution`) |
| Columns | `:::: {.columns}` + `::: {.column width="50%"}` |
| Table caption + label | `: Caption {#tbl-name}` |
| Figure label | `![Caption](img.png){#fig-name}` |
| Equation label | `$$ … $$ {#eq-name}` |
| Cross-reference | `@sec-name`, `@fig-name`, `@tbl-name`, `@eq-name` |
| Citation | `[@key]`, bibliography in the YAML |
| Code chunk | ```` ```{r} ```` … ```` ``` ````, options via `#| label:` |
| Chunk without running | ```` ```r ```` (no braces) |
| Raw HTML fallback | `<span style="color:#b30000">text</span>` |

# Equations — the essentials

| Type this | You get | Type this | You get |
|-----------------------|----------------|------------------------|-----------------|
| `$x$` | inline $x$ | `$$x$$` | display equation |
| `C_{max}` | $C_{max}$ | `C^2`, `x_i^2` | $C^2$, $x_i^2$ |
| `\frac{a}{b}` | $\frac{a}{b}$ | `\dfrac{a}{b}` | $\dfrac{a}{b}$ (taller) |
| `\sqrt{x}` | $\sqrt{x}$ | `\sqrt[3]{x}` | $\sqrt[3]{x}$ |
| `\ln`, `\exp`, `\max` | $\ln,\ \exp,\ \max$ | `\log_{10}` | $\log_{10}$ |
| `\sum_{i=1}^{n}` | $\sum_{i=1}^{n}$ | `\prod_{i=1}^{n}` | $\prod_{i=1}^{n}$ |
| `\int_{0}^{\infty}` | $\int_{0}^{\infty}$ | `\lim_{x \to 0}` | $\lim_{x \to 0}$ |
| `\alpha \beta \gamma` | $\alpha\ \beta\ \gamma$ | `\mu \sigma \lambda_z` | $\mu\ \sigma\ \lambda_z$ |
| `\Delta \partial \nabla` | $\Delta\ \partial\ \nabla$ | `\infty \pm \approx` | $\infty\ \pm\ \approx$ |
| `\times \cdot` | $\times,\ \cdot$ | `\geq \leq \neq` | $\geq\ \leq\ \neq$ |
| `\left( \right)` | $\left( \right)$ | `\left\lvert \right\rvert` | $\left\lvert \right\rvert$ |
| `\mathrm{mg/L}` | $\mathrm{mg/L}$ | `\text{steady state}` | $\text{steady state}$ |
| `\ `, `\,`, `\quad` | space, thin space, wide space | `\hat{\theta}` | $\hat{\theta}$ |

Multi-line building blocks:

````markdown
$$
\begin{aligned}
a &= b + c \\
  &= d
\end{aligned}
\qquad
\begin{cases}
x & t \le T \\
y & t > T
\end{cases}
\qquad
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
````

$$\begin{aligned}
a &= b + c \\
  &= d
\end{aligned}
\qquad
\begin{cases}
x & t \le T \\
y & t > T
\end{cases}
\qquad
\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

Spacing, which is what makes an equation readable:

| Type this | You get | What it does |
|----------------------------|------------------------------|---------------------------|
| `a\,b` | $a\,b$ | thin space |
| `a\ b` | $a\ b$ | normal space (units) |
| `a \quad b` | $a \quad b$ | wide space |
| `\ln\!\left(x\right)` | $\ln\!\left(x\right)$ | pull together with `\!` |
| `\operatorname{CL}` | $\operatorname{CL}$ | upright multi-letter label |

# Equations you will type most often

Copy the source column straight into a document. `$$ … $$` on one line is display math;
`$ … $` is inline math.

| Quantity | Source |
|----------------------------|------------------------------------------------------------|
| First-order decay | `$C(t) = C_0 e^{-k_e t}$` |
| IV bolus, one compartment | `$$C(t) = \frac{D}{V_d}\,e^{-k_e t}$$` |
| Elimination rate constant | `$$k_e = \frac{CL}{V_d}$$` |
| Half-life | `$$t_{1/2} = \frac{\ln 2}{k_e}$$` |
| Clearance | `$$CL = \frac{D}{AUC}$$` |
| Volume of distribution | `$$V_d = \frac{D}{C_0}$$` |
| AUC after an IV bolus | `$$AUC_{0 \to \infty} = \frac{C_0}{k_e} = \frac{D}{CL}$$` |
| Trapezoidal AUC | `$$AUC_{0 \to t_n} = \sum_{i=1}^{n-1} \frac{C_i + C_{i+1}}{2}\,\Delta t_i$$` |
| AUC extrapolated to infinity | `$$AUC_{0 \to \infty} = AUC_{0 \to t_{last}} + \frac{C_{last}}{k_e}$$` |
| Mean residence time | `$$MRT = \frac{AUMC}{AUC}$$` |
| Steady-state volume | `$$V_{ss} = MRT \cdot CL$$` |
| Steady state, infusion | `$$C_{ss} = \frac{R_0}{CL}$$` |
| Approach to steady state | `$$f_{ss}(t) = 1 - e^{-k_e t}$$` |
| Loading dose | `$$D_L = \frac{C_{target}\,V_d}{S}$$` |
| Trough at steady state | `$$C_{min,ss} = \frac{F D}{V_d}\,\frac{e^{-k_e \tau}}{1 - e^{-k_e \tau}}$$` |
| Oral (Bateman) profile | `$$C(t) = \frac{F D k_a}{V_d (k_a - k_e)}\left(e^{-k_e t} - e^{-k_a t}\right)$$` |
| Time to peak, oral | `$$t_{max} = \frac{\ln(k_a / k_e)}{k_a - k_e}$$` |
| Peak after an oral dose | `$$C_{max} = \frac{F D}{V_d}\left(\frac{k_a}{k_a - k_e}\right)\left(e^{-k_e t_{max}} - e^{-k_a t_{max}}\right)$$` |
| Bioavailability | `$$F = \frac{AUC_{oral}\,CL}{D_{oral}}$$` |
| Cockcroft–Gault | `$$CL_{CR} = \frac{(140 - age)\,WT}{72\,S_{Cr}} \times 0.85\ \text{if female}$$` |
| Dose adjustment | `$$D_{adj} = D_{std} \times \frac{CL_{CR,pt}}{CL_{CR,ref}}$$` |
| Weight-based clearance | `$$CL_i = CL_{pop}\left(\frac{WT_i}{70}\right)^{0.75} e^{\eta_i}$$` |
| Proportional residual error | `$$C_{ij} = C_{pred,ij}\,(1 + \varepsilon_{ij})$$` |
| Between-subject variability | `$$CV\% = \sqrt{e^{\omega^2} - 1} \times 100\%$$` |
| Michaelis–Menten | `$$-\frac{dC}{dt} = \frac{V_{max} C}{K_m + C}$$` |
| Target attainment | `$$\frac{AUC_{24}}{MIC} \ge 400$$` |
| Bayesian posterior | `$$p(\theta \mid y) \propto p(y \mid \theta)\,p(\theta)$$` |
| MAP objective | `$$\hat{\theta} = \arg\max_{\theta}\; \sum_{j=1}^{m} \left[-\frac{\left(C_j^{obs} - C_j^{pred}(\theta)\right)^2}{2\sigma_j^{2}}\right]$$` |

Rendered, so you can check your output at a glance:

$$C(t) = \frac{D}{V_d}\,e^{-k_e t} \qquad
t_{1/2} = \frac{\ln 2}{k_e} \qquad
CL = \frac{D}{AUC}$$

$$AUC_{0 \to t_n} = \sum_{i=1}^{n-1} \frac{C_i + C_{i+1}}{2}\,\Delta t_i \qquad
C_{ss} = \frac{R_0}{CL} \qquad
-\frac{dC}{dt} = \frac{V_{max} C}{K_m + C}$$

# Punctuation that bites

- `_` and `*` are formatting characters: write `AUC\_MIC`, not `AUC_MIC`.
- `%` must be `\%` inside math if you ever export to PDF.
- `$` in prose: currency is `\$5`, and an unmatched `$` swallows the rest of the paragraph.
- Units belong in `\mathrm{}`. Multi-letter variables too: `\mathrm{AUC}`, `\mathrm{CL}`.
- Functions are commands: `\ln(2)`, `\exp(x)`, `\max_i`.
- Subscripts with more than one character need braces: `C_{max}`, `t_{1/2}`.
- Display math wants its own paragraph — blank line before and after.
- `\left` and `\right` are a pair; a missing one breaks the equation.
- Prefer `\mu`, `\beta`, `\alpha` to typed Unicode when the document may become a PDF.

# Command line

```bash
quarto preview file.qmd          # live preview while you write
quarto render file.qmd           # build the file next to its source
quarto render                    # build the whole project (uses _quarto.yml)
quarto render file.qmd --to docx # a different output format
quarto render file.qmd --to typst
quarto check                     # is the toolchain complete?
quarto install tinytex           # LaTeX for PDF output
```

Settings that matter for a talk:

```yaml
format:
  beamer:
    slide-level: 2             # `##` starts a slide; without it the deck collapses
    aspectratio: 169
    header-includes: |         # footer with page numbers
      \setbeamertemplate{footline}{%
        \hspace*{2ex}Your Name · IATDMCT Congress%
        \hfill\insertframenumber{}/\inserttotalframenumber\hspace*{2ex}%
      }
```

`{.smaller}`, `{.incremental}` and background colours are reveal.js features and are
ignored in a PDF — fit a slide by cutting words.