2c:null 2d:["$","title",null,{"children":"Mathematical Model to Simulate Mechanical Ventilation Waveforms | Sami Safadi | Observable"}] 32:[] 30:{"id":"d78a53c315456b83","slug":"mechanical-ventilation-waveforms","title":"Mathematical Model to Simulate Mechanical Ventilation Waveforms","thumbnail":"d0c4c8a4a48acc34225d6e9ee7a536ea9013dd540f050280a2375e8cad343699","access_level":"public","create_time":"$D2024-06-02T15:00:55.208Z","update_time":"$D2024-06-03T03:05:31.422Z","view_time":null,"first_public_time":"$D2024-06-02T15:22:22.088Z","has_unpublished_edits":null,"fork":true,"stars":1,"starred":false,"owner":{"type":"team","id":"1d804b626bd08da7","login":"samisaf","name":"Sami Safadi","avatar_url":"https://avatars-next.observableusercontent.com/a2adb3491825954cb8c600a2e043624c461158aa29ae90abb5d3fab14240f3de","tier":"starter_2024"},"version":581,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"a139fe285b2a1eaf","login":"samisaf","name":"Sami Safadi","avatar_url":"https://avatars-next.observableusercontent.com/a2adb3491825954cb8c600a2e043624c461158aa29ae90abb5d3fab14240f3de","tier":null}],"members":[{"type":"team","id":"1d804b626bd08da7","login":"samisaf","name":"Sami Safadi","avatar_url":"https://avatars-next.observableusercontent.com/a2adb3491825954cb8c600a2e043624c461158aa29ae90abb5d3fab14240f3de","tier":"starter_2024","role":"editor"},{"type":"individual","id":"a139fe285b2a1eaf","login":"samisaf","name":"Sami Safadi","avatar_url":"https://avatars-next.observableusercontent.com/a2adb3491825954cb8c600a2e043624c461158aa29ae90abb5d3fab14240f3de","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Mathematical Model to Simulate Mechanical Ventilation Waveforms","cells":[{"id":352,"value":"# Mathematical Model to Simulate Mechanical Ventilation Waveforms\n© 2024 Sami Safadi, M.D. For a complete simulator, please visit https://ventsim.cc/\n \n## Introduction\nThe ventilator generates positive pressure which is applied to the respiratory system ${tex`P_{rs}`}. Assisted ventilation can be total, partial or absent, depending on respiratory muscles’ ability to generate additional pressure to the respiratory system ${tex`P_{rs}`}.\n\n${tex`P_{rs}`} of a ventilated patient is the sum of the pressure generated by the ventilator (${tex`P_{aw}`}) and the pressure developed by the respiratory muscles (${tex`P_{mus}`}). Note that ${tex`P_{mus}`}. is negative, as inspiratory muscles act by producing a decrease in pressure below the airways.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":375,"value":"## Inspiration\nDuring inspiration, the relationship between pressure, and flow is described by the equation of motion: \n${tex`P_{rs} = P_{aw} + (-P_{mus}) = \\dot{V}\\times R + \\frac{V}{C} + PEEP`}\n\nNote that flow is the change of volume per time, and hence the notation ${tex`\\dot{V} = Flow`}. Also, note that ${tex`PEEP`} is a constant representing the alveolar end expiratory pressure.\n\nThe term ${tex`\\dot{V} \\times R`} reads as **flow times resistance** and corresponds to the pressure dissipated across the airway and the endotracheal tube, and to overcome the frictional forces generated with gas flow.\n\nThe term ${tex`{V}/{C}`} reads as **volume divided by compliance** and corresponds to the pressure that must be applied to overcome elastic forces of the lungs. It depends on both the volume that is in the lungs in excess on resting volume (${tex`FRC`}), and on the respiratory system compliance.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":377,"value":"## Expiration\nExpiration is passive, and the flow is described using the exponential decay equation which can be written as a first order ordinary differential equation ${tex`\\dot{V} = -\\lambda V`} or ${tex`\\dot{V} = -{1\\over\\tau} V`}. A solution to the above equation is ${tex`V= V_0 e^{-\\lambda t}`}, which can also be written as ${tex`V= V_0 e^{-t\\over\\tau}`}.\n\nNote that *tau* is a property of the respiratory system and is a constant equaling the **compliance times resistance**. This can be written as ${tex`\\tau=C \\times R= -V / \\dot{V}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":578,"value":"## Simulation\nBelow is the simulation, and the code used to generate the plots. The code is concise and should be easy to understand (hopefully).\n\n\n \n \n \n\n\n\n\n\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":60,"value":"vl.hconcat(plotData('Paw', 'Palv'), plotData('Flow'), plotData('Volume')).render()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"dataframe = mode === 'ACVC' ? simulateVC(n, duration, rr, tv, flow, wave, peep, r, c, v0) : simulatePC(n, duration, rr, pi, it, peep, r, c, v0)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":441,"value":"function simulateVC(n, duration, rr, tv, flow, wave, peep, r, c, v0) {\n const dt = duration / n; \n const breathTime = 60.0 / rr;\n const ticks = [...Array(n).keys()];\n const it = wave === 'SQ'? (tv / 1000) / (flow / 60) : 2 * (tv / 1000) / (flow / 60);\n const Time = ticks.map(t => t * dt);\n const timeInCycle = Time.map(t => t % breathTime);\n const phaseInCycle = timeInCycle.map(t => t < it ? 'INSP' : 'EXP');\n const [Volume, Flow, Paw, Palv] = [Array(n), Array(n), Array(n), Array(n)]\n Volume[0] = v0;\n for (const t of ticks) {\n if (phaseInCycle[t] === 'INSP') {\n Flow[t] = wave === 'SQ' ? flow / 60 : (flow / 60) * (1 - timeInCycle[t] / it);\n Volume[t + 1] = Volume[t] + Flow[t] * dt;\n } else {\n Flow[t] = -Volume[t] / (r * c);\n Volume[t + 1] = Volume[t] + Flow[t] * dt;\n }\n Palv[t] = Volume[t] / c + peep;\n Paw[t] = Flow[t] * r + Palv[t];\n }\n return ticks.map(t => ({Time: Time[t], Paw: Paw[t], Palv: Palv[t], Flow: Flow[t] * 60, Volume: Volume[t] * 1000}));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":490,"value":"function simulatePC(n, duration, rr, pi, it, peep, r, c, v0) {\n const dt = duration / n;\n const breathTime = 60.0 / rr;\n const ticks = [...Array(n).keys()];\n const Time = ticks.map(t => t * dt);\n const phaseInCycle = Time.map(t => (t % breathTime < it ? 'INSP' : 'EXP'));\n const Paw = phaseInCycle.map(i => (i === 'INSP' ? pi + peep : peep));\n const [Volume, Flow] = [Array(n), Array(n)]\n Volume[0] = v0; \n for (const t of ticks){\n Flow[t] = (Paw[t] - Volume[t] / c - peep) / r;\n Volume[t + 1] = Volume[t] + Flow[t] * dt; \n } \n const Palv = Volume.map(v => v / c + peep);\n return ticks.map(t => ({Time: Time[t], Paw: Paw[t], Palv: Palv[t], Flow: Flow[t] * 60, Volume: Volume[t] * 1000}));\n\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"function linePlot(x, y, style=\"solid\"){\n const lineMark = vl.markLine({strokeDash: style==\"dashed\" ? [5, 5] : null});\n return lineMark.encode(vl.x().fieldQ(x),vl.y().fieldQ(y),vl.tooltip([x, y]));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"function plotData(y, z=null) {\n const yLine = linePlot('Time', y);\n const zLine = linePlot('Time', z, \"dashed\");\n return z?vl.layer(yLine, zLine).data(dataframe):vl.layer(yLine).data(dataframe);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"import {vl} from '@vega/vega-lite-api'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":451,"value":"viewof mode = Inputs.radio([\"ACVC\", \"ACPC\"], {label: \"Ventilator Mode:\", value: \"ACVC\"})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":288,"value":"viewof duration = Inputs.number({label: \"Duration (seconds)\", value: 10, min: 0, max: 100, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":295,"value":"viewof n = Inputs.number({label: \"N (points)\", value: 1000, min: 0, max: 10000, step: 10})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":"viewof r = Inputs.number({label: \"Resistance (cmH2O/L/sec)\", value: 10, min: 0, max: 100, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":334,"value":"viewof c = Inputs.number({label: \"Compliance (L/cmH2O)\", value: 0.05, min: 0, max: 1, step: 0.01})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":337,"value":"viewof v0 = Inputs.number({label: \"FRC (L)\", value: 0, min: 0, max: 100, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":309,"value":"viewof rr = Inputs.number({label: \"Respiratory Rate (1/min)\", value: 10, min: 0, max: 100, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":313,"value":"viewof tv = Inputs.number({label: \"Tidal Volume (mL)\", value: 500, min: 0, max: 1000, step: 50})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":320,"value":"viewof peep = Inputs.number({label: \"PEEP (cmH2O)\", value: 5, min: 0, max: 20, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":317,"value":"viewof flow = Inputs.number({label: \"Flow (L/min)\", value: 30, min: 0, max: 100, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":323,"value":"viewof wave = Inputs.radio([\"SQ\", \"DEC\"], {label: \"Flow Pattern:\", value: \"SQ\"})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":478,"value":"viewof pi = Inputs.number({label: \"Pi (cmH2O)\", value: 10, min: 0, max: 50, step: 1})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":480,"value":"viewof it = Inputs.number({label: \"Inspiratory Time (sec)\", value: 1.0, min: .05, max: 5, step: 0.5})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1"}} 31:"$30:body"

Simulation Mode & Duration	Lung Parameters	Ventilator Settings
\n ${viewof mode}${viewof duration}${viewof n} \n	\n ${viewof r}${viewof c}${viewof v0} \n	\n ${viewof rr}${viewof peep} \n ${viewof tv}${viewof flow}${viewof wave} \n ${viewof pi}${viewof it} \n