Hyperkalaemia Management: An Overview

• The provided text includes guidelines for managing hyperkalaemia in adults, fundamental chemical and physical principles, algorithms and thermodynamics.

Management of Hyperkalaemia in Adults

• Hyperkalaemia is defined as a serum potassium level exceeding 5.5 mmol/L.

• Suspect hyperkalaemia in patients with:

  • Acute or chronic kidney disease
  • Those taking medications that raise potassium levels such as ACE inhibitors, ARBs, spironolactone, eplerenone, trimethoprim, tacrolimus, and ciclosporin
  • Showing ECG changes (peaked T waves, widened QRS, prolonged PR interval)
  • Exhibiting muscle weakness or paralysis
  • Experiencing cardiac arrhythmias.

• Hyperkalaemia investigation steps include:

  • Repeat potassium measurement
  • Check renal function (urea, creatinine, eGFR)
  • Electrolytes (sodium, calcium, magnesium)
  • Glucose, full blood count (FBC) for thrombocytosis
  • ECG for cardiac effects
  • Arterial blood gas (ABG) for acidosis
  • Review medications.

• Emergency hyperkalaemia treatment (ECG changes or K > 6.5 mmol/L):

  • Protect the heart with 10 mL of 10% calcium chloride or 30 mL of 10% calcium gluconate IV over 5-10 minutes, with ECG monitoring but caution with digoxin
  • Drive potassium into cells using 10 units of rapid-acting insulin IV and 25g glucose IV (adjust glucose as needed); nebulized salbutamol 5mg can serve as a useful adjunct.
  • Remove potassium via calcium resonium 30g orally or 50g rectally; consider haemodialysis.

• Ongoing hyperkalaemia management:

  • Review and discontinue medications raising potassium
  • Monitor daily potassium levels
  • Treat underlying causes like AKI or CKD
  • Consider loop diuretics and potassium-binding resins for chronic cases.

• Cautions regarding hyperkalaemia:

  • Rapid correction may cause hypokalaemia
  • Monitor potassium closely
  • Note slow onset of calcium resonium
  • Consider alternative resins if needed.

• Medical professionals should manage hyperkalaemia and consult with senior clinicians, nephrology for AKI/CKD, and cardiology for ECG changes/arrhythmias.

The Properties of Gases: Chemical Principles

• Gases exert pressure, defined as force per area.

• Pressure measurements:

  • SI unit is the Pascal (Pa), 1 Pa = 1 N/m².
  • 1 atm = 760 mm Hg = 760 torr = 101,325 Pa = 14.7 psi.

• Boyle's Law: pressure and volume are inversely proportional at constant temperature and amount of gas: $P_1V_1 = P_2V_2$.

• Charles's Law: volume and temperature are directly proportional at constant pressure and amount of gas: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$.

• Avogadro's Law: volume and number of moles are directly proportional at constant temperature and pressure: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$.

• Ideal Gas Law: $PV = nRT$ where $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$.

• Gas density and molar mass calculations:

  • $d = \frac{m}{V} = \frac{P \cdot Molar,Mass}{RT}$
  • $Molar,Mass = \frac{dRT}{P}$.

• Dalton's Law of Partial Pressures: the total pressure is the sum of the partial pressures of individual gases, where $P_i = X_i \cdot P_{total}$ and $X_i = \frac{n_i}{n_{total}}$.

• Kinetic Molecular Theory of Gases:

  • Gases consist of many molecules in continuous, random motion
  • Molecular volume is negligible
  • Intermolecular forces are negligible
  • Collisions are elastic
  • Average kinetic energy is proportional to absolute temperature.

• Graham's Law of Effusion: the effusion rate is inversely proportional to the square root of its molar mass: $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$.

• Real gases deviate from ideal behavior at high pressures and low temperatures.

• van der Waals equation corrects for non-ideal behavior: $(P + a\frac{n^2}{V^2})(V-nb) = nRT$.

Álgebra & Poisson distribution

• Félix del Puerto García and Àngel García Fraile are the authors of the Álgebra book.
• The Poisson distribution describes the probability of events occurring in a fixed interval
• The probability mass function is given by: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
• The mean E(X) and Variance var(X) are both lambda.
• Formula can be used if events are randomly and Independently distributed

Algorithmic Game Theory

• Algorithmic Game theory is a mathematical frame work which analyses stategic interactions and the outcome of the participants choices
• Its composed of concepts to model and and predict behaviours, specifically
  • Players: The individuals or entities involved in the game.
  • Strategies: The possible actions or plans of action available to each player.
  • Payoffs: The outcomes or rewards received by each player based on the combination of strategies chosen by all players.
  • Equilibrium: A stable state in the game where no player has an incentive to unilaterally change their strategy.
• Selfish Routing:
  • New roads can create new Nash equilibria (that are less efficient than the original equilibrium)

Implicit Function Theorem

• Implicit Function Theorems states that there are exist some functions for all x belonging to U and g(x_0) = y_0
• Contraction Mapping Principle: can be used to prove the Picard-Lindelöf Theorem, which says that the initial value problem has a unique solution if f is Lipschitz in x.
• $\qquad \begin{cases} \dot{x}(t) = f(t, x(t)) \ x(t_0) = x_0 \end{cases}$
• Sard's Theorem states:
  • Critical points that are part of $f(C)$ have a measure of zero

Trabalho e Energia, Física

• Trabalho: medida da energia transferida quando uma força causa o deslocamento de um objeto
• Formula is given as: $W = F \cdot d \cdot cos(\theta)$
• Energia Cinética: energia presente em um objeto que demonstra movimento, formula given as $K = \frac{1}{2} m v^2$
• Teorema Trabalho-Energia: O trabalho total realizado sobre um objeto é igual à variação da sua energia cinética with the formula $W_{total} = \Delta K = K_f - K_i

Introduction to CFD

• Computational Fluid Dynamics numerically solves equations predicting fluid flow, heat and mass transfer, and chemical reactions
• CFD is used
  • in design and improve product, understand and analyse a new flow and reduce cost
• Governing Equations:
  • Continuity Equation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{V}) = 0$
  • Momentum Equation (Navier-Stokes): $\frac{\partial (\rho \mathbf{V})}{\partial t} + \nabla \cdot (\rho \mathbf{V}\mathbf{V}) = -\nabla p + \nabla \cdot \tau + \rho \mathbf{g}$
  • Energy Equation: $\frac{\partial (\rho h)}{\partial t} + \nabla \cdot (\rho h\mathbf{V}) = \nabla \cdot (k\nabla T) + S_h$

Algorithmes gloutons

• Algorithms are used to resolve optimization issues
• Use choices which seem best, and make a sequence of locally optimal decisions
• Some appications of the algorithm are e.g.
  • Problème du rendu de monnaie i.e. the problem of the currency rendering
  • Problème du sac à dos fractionnaire i.e. the problem of having a fractional back pack
  • Arbre couvrant minimal i.e. the minimal spanning tree (Kruskal and Prim's algorith)
  • Plus court chemin i.e. the shorted path algorithm (algorithme de Dijkstra)
  • Ordonanncement de tâches i.e. scheduling tasks

The Laws of Thermodynamics

• The zeroth law is thermodynamic systems in equilibrium are in equilibrium with each other

• The first law, an increase in internal energy is equal to the heat added to the system and minus any work done. $\Delta U = Q - W$

• The seconde law, the entropies of interacting system increases. $\Delta S \geq 0$

• The third law, zero at absolute zero

• Heat Transer Mechanics: -Conduction, heat transfer with a material without any bulk motion using Fourier's Law $\frac{dQ}{dt} = -kA\frac{dT}{dx}$

  • Convection, heat transfer by flud moving, using the formula, $Q = hA(T_s - T_f)$
  • Radiation, heat trasfer by measuring the electromagnet radiation, using the formula $Q = \epsilon \sigma A T^4$